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# Economic Decision Analy ISYE 6230

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This 0 page Class Notes was uploaded by Maryse Thiel on Monday November 2, 2015. The Class Notes belongs to ISYE 6230 at Georgia Institute of Technology - Main Campus taught by Staff in Fall. Since its upload, it has received 9 views. For similar materials see /class/234208/isye-6230-georgia-institute-of-technology-main-campus in Industrial Engineering at Georgia Institute of Technology - Main Campus.

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Date Created: 11/02/15

S Administrivia Teaching assistant Martin Smith gtg593dprismgatechedu Of ce hours Tuesday 330430 Wednesday 200300 Location ISYE 403 I Recap Last cass Games in normal form NBC vs ABC old vs new technology Prisoner39s ma Dominant and dominated actions best response Equilibrium in dominated actions Nash Equilibrium Pure vs mixed strategies This class Games with continuous action sets Comparison ofduopoly games with monopoly 5 Example A scarce manufacturing resource is required by two departments A and B YJZO quantity of the resource used by departmentj Payoff to departmentj from using one unit of the resource zoomars2 Department j s maximization problem Max Y200YY72 From FOC 200YY722YYYVJ 2003YJZ4YJYVJY720 Example cont 200 3y2 myB YB2 o 1 200 3Y5 4mB YA o 2 2112 ZYAZ 2Y5 o gt YAYB From 1 200 3y2 4YA2 YA o gt 8YA2 200 gt YA YB 5 gt Payoff for department j 500 A solution with higher payoff YA YB 4 gt Payoff for department j 544 5 The Tragedy of the Commons Immigrant villages in New England in the 17th century privately owned hornesteads and gardens unityown pastures called oomrnons where all of the villagers39 livstock could graze Incentive to avoid overuse of their private lands so they would remain productive in the future Result The commons were overgrazed and degenerated tn the point that they were no longer able to support the villagers39 cattle The failure of private incentives to provide adequate maintenance 0 pu lic rsources is known to economists as quotthe tragedy of the commonsII SI Examples of the Tragedy of the Commons Congestion on urban highways Overpopulation Poution The depletion of sh stocks in international waters SI Duopoly models Two competing rms selling a homogeneous good The marghacostof producing each unit of the good c an 1 2 The market price P is determined by inverse market demand PabQ ifagtbQ P0 otherwise Both rms seek to maximize pro ts Cournot Firms set quantities simultaneously Bertrand Firm set prics simultaneously Stackelberg Firms set quantitis rm 1 followed by rm 2 Cournot Competition The market price P is determined by inverse market d mand PabQ ifagtbQ P0 otherwise Each rm decides on the quantity to sell market share q1 and q2 Q q1q2total market demand Both rms seek to maximize pro ts 5 Cournot Competition Best response of Firm 1 Suppose rm 2 produces q2 Firm 1 s pro ts if it produces q1 are 1 Pclq1 abq1 q2q1 clql Residual revenue Cost How to choose q1 to maximize n1 First note that 111 is concave dZIIIdql2 2b lt0 First order conditions FOC d lldq a 2bq1 qu 61 Residual marginal revenue Marginal cost 0 q1ac12b q22 R1012 5 Cournot Competition Best response of Firm 2 Suppose rm 1 produces q1 Firm 2 s pro ts if it produces q2 are 2 P62q2 abq1 q2q2 6qu Residual revenue Cost First order conditions d 79131112 a 39 2bq2T bql T 12 RMR MC 0 gt q2aCz2b l112 R2011 Example Cournot Competition P 130C1q2 so a130 b1 c1 c2 c 10 Suppose Firm 2 thinks that Firm 1 will set q140 Residual demand of Firm 2 P QOq2 Residual revenue of Firm 2 RR90q2q2 Residual marginal revenue RMR902q2 Setting RMRMC10 902q210 a q240 Cournot Competition Graphical solution p Cournot Equilibrium I C1a39512b cI22 q2ac22b q12 Solving together for q1 and q2 qcla2c1C23b q52a2c2c13b Market demand and price QC qC1 qC2 26 61 c23b P a bQC ac1c23 g Example Cournot Competition P 130q1q2 so a130 b1 c1 c c 10 The rms39 best response functions cl1abq2C2b 130q210260q22 q2abq1c2b 130q11026oq12 Solving for q1 and q2 q q240 Q80 P50 Firms profits HI 12 501o4o 1600 Cournot Competition Graphical solution q1 R1q2 60q22 q2 R2q1 60q12 2q1 60q12 Cournot equilibrium q2 60q22 I Cournot Equilibrium with N rms maxq 7W1 qaabq 52 414 704 1 First order conditions 172lyqxinqucx 0 Vi1 N 1 7 Substitute 0217 17qu ibQic 0 Vi1N Sum over N NaibQibNQizkx 0 SI Cournot Equilibrium with N firms Na 20 x Q N1b N1b PC a 20 N1 N1 If each firm has the same cost 45 QC aic CiaNc C7 7 q N N1b N1 g Bertrand Equilibrium Model Firms set prices rather than quantities PabQ Customers buy from the firm with the cheapest price The market is split evenly if rms offer the same price Best response Firm 1 s pro t function P1P139 51 Q1 To ensure q1gt0 recall PabQ and QaPb To ensure nonnegative pro ts P12 c1 Firm 1 should choose clS P1 S a Similarly rm 2 should choose g Best response cont Firm i39s demand depends on the relationship between Plan P2 0 ifogtPl 1 2P1 ifRltPJ il2 P 1fP P P b J Firm 1 should choose c1 S P1 S P2 ifpossible Firm 2 should choose c2 S P2 S P1ifpossible 5 Bertrand equilibrium For both rms to sell positive quantities pro tably clS PlSPzand c2 S PZSP1 Suppose c c1 c2 P c q1 q2 ac2b Suppose c1lt c2 P1 cZe P2 2 c2 q1 a c2eb q2 0 Example P 130C1q2 a130b1 c1 c2 c 10 P10 q1 q2 aP2b 60 Q120 Firms profits HI 12 0 SI Quantitysetting monopolist Single rm monopolist selling a single good The marg nd castof producing each unit of the goo c The rm decides on the quantity to sell Q market demand The market price P is determined by inverse market demand PabQ ifagtbQ P0 otherwise The rm seeks to maximize pro ts g Quantitysetting monopolist The rm39s pro ts if it produces Q are n PcQ abQJQ Q Revenue Cost How to choose Q to maximize 1r First note that n is concave dz dQZ 2b lt0 First order conditions FOC d 1tdQ a 2bQ c Marginal revenue Marginal cost gt c 2b P ac2 PrincipalAgent Problem x W UWj 5ai 6 9 Principal Agent 9 Pquot 6 W1 WX1 UW1 e2 1 W2 WX2 UW2 C32 4 9 Wm WXm UWn ean Outcomes Actions l E l Principalagent problem If the agent doesn t accept the contract his payoff is his reservation utility u If the agent accepts the contract he chooses between n possible actions a1 an These actions produce m possible outcomes X1 Xm There is a stochastic relationship between actions and outcomes called technology When the action is ai the principal observes outcome X with probability Pij If the principal observes outcome X she pays the agent wj The agent s payoff is Uweai where Uw is the utility of wage w to the agent and eai is the cost of action ai to the agent U is increasing differentiable and concave Assuming the principal is riskneutral her payoff is Xj Wj What if the agent s actions can be 3 observed The principal can design a contract where the wages are conditioned on the actions ie wai 1T1 max Px w E m1nw 11 I I 1 Participation constraint U wi 601 2 Q Incentive constraint Uw ea 2 Uwk eak Vk ii To induce the agent to choose action at set w such that Uw Q 601 and set all other wk sufficiently low Unobservable actions Principal s problem Step 1 Given an action a how to set the wages such that the agent chooses ai and the principal s payoff is maximized Participation constraint BjUwj ea 2 Q Incentive constraint 21 PjUwj ea 2 21 ijUwj eak v k 72139 Principal s objective minimize the expected cost of inducing the agent to choose ai m m max 2le xj wj min F1 Pijwj Cal Principal s problem Step 1 Cai is the minimal cost to the principal of inducing the agent to take action a Cai is convex 9 the original maximization objective is concave Wellbehaved mathematical program with a concave objective function maximization and linear constraints Principal s problem Step 1 Fora given action ai maxZ1Bjxj wj ZLPJUmp emgzg u 21PIUwj eaiZZIUwj eak Vkii A LltwzgtZIPjltxj wjgtltZ111Ultwjgt eltagt Qgt szltZjLIszltwjgt eltaigt iPgUltwjgteltakgtgt 01121 pooqnexn 41 39 d quot d W WM W W I I dens LU3COJC sedpuud 5 M e d IJWZrMHeeo Zn Me I El Mae 2 0 d Wjil Me 21 Mae f M9 Ti 1 078 121 me JJWZUMHQ d7739 CI fme 2 Me e 0 d Wne Mae 121 W fmnfd va mg21 3ij va fmanjZrM foo213 0 um I dens LU3COJC sedpuud Principal s problem Step 2 I From step 1 maXZj1PUxj wj E j1Pijxj min j1Pijwj Rai Cai What is the action ai that maximizes RaiCai Two actions and two outcomes Suppose the agent has two possible actions a and b and there are two possible outcomes x1 and x2 Suppose that action b is preferred by the principal For action b 2 max ZJ1PIJJXJ wj PJlUw1P2Uw2 eb 2Q u PmUw1 szUW2 6b 2 Pa1Uw1 Pa2Uw2 ea it Two actions and two outcomes Incentive compatibility constraint lUw1PbZUwZ 91 2Pa1Uw1 PaZU wz ea When is the agent indifferent between a and b 1Uw1PbZUwZ eb Pa1Uw1PaZUwZ ea ea Jwleb ea P P P a2 b2 a2 Uw Pal 31 UWeb 2 P P 1 P 132 oz 132 since Pa1PaZ Pb1 332 1 g Two actions and two outcomes 91 ea Uw Uw Z 1 sz PaZ Action b is preferred to action a UWi UW2 UWi g Two actions and two outcomes Paiticipation constraints Action b PblUw1szUw2 8b 2 g Action 61 PalUw1PazUw2 ea 2 g Action b Uw2 2 iUw1eb Jrg Pm Pm Two actions and two outcomes ebea Uwz Uw1 PM Paz UWi UW2 compatibility constraint Uw2 iUWI ebg PM PM UWi g Two actions and two outcomes Principal s objective function For action b maXB71x1 w1B72x2 w2 Let fbe the inveise of U fUw wj maXB71x1 fUw1Byzx2 fUw2 Two actions and two outcomes Uw2 UWi UW2 Participation constraint Principal s isopro t cuxves PM 1 ltUltwlgt szltxz7fltUltwZ c 39 Um Two actions and two outcomes maXPbix1 fUw1 szxz fUw2 Slope of the objective function P afUwl P 3fUW23UW2 b1 8Uw1 b2 8Uw1 8Uw1 P afUwl aUw2 aUwl alfmwl 8Uw1 P 8fUw2 szf39Uw2 b2 8Uw1 39 Two actions and two outcomes If Uw1 Uw2 45 line 3UW2 Pblf39UW1 i 3UW1 szf39UW2 PM Same as the slope of the participation constraint The principal39s isopro t curve is tangent to the agent39s participation constraint along the 450 line g Two actions and two outcomes Uw2 UWi UW2 Participation constraint IncenmIe I compatibility constraint Ultwgt Two actions and two outcomes P P V y 1 i a likelihood ratio U w Pb Pb If the incentive constraint is not binding i0 U39wj1l andwj w Substituting into the incentive constraint the agent prefers action 17 over action a Zn BJJUM 91 2 Zn Panw ea gt ea 2 91 Two actions and two outcomes 1 135 Pa u 1 1 J likelihood ratlo U W ij bj If the incentive constraint is binding xi gt 0 and U wj depends on the likelihood ratio LR The optimal incentive scheme is a linear function of LR Sensitivity analysis Changes in the agent s costs What is the impact of the agent s costs on the outcome Lw nu Pb1xi w1szx2 w2 PMUW1 szUWz eID EH PMUWHHBJzUW z 6b Pa1UW1 PazUWz e0 aLw1 y 1 aLw1 2 l1 Beb 36a 1 Recap Last Thursday Extensive form ofa game Classroom exercise nd equilibrium Twostage Prisoner39s Dilemma Repmted games Finiter repeated games Infinitely repeated games Today Continue with In nitely repeated games Prisoner39s dilemma trigger strategies 5 Definitions In an in nitely repeated game Goo5 a player s slrategzspeci es the player s actions in each stage for each possible history of play through the previous stages In the in nitely repeated game Goo5 each subgame beginning at stage t1 is is identical to the original game Goo5 g Infinitely Repeated Prisoner s Dilemma Prisoner 2 H C oooperate D defect g C 4 0 3 i D 5 o 1 1 Strategy Play C in the rst stage In the tm stage if the outcome of all t1 preceding stages has been CC then play C otherwise play D g Trigger strategies for Prisoner s Dilemma Assuming player 1 adopts the trigger strategy what is the best response of player 2 Player 2 best response in stage t1 If the outcome in stage t is DD Play D forever If the outcomes of stages 1t are CC Play D gt receive 5 in this stage switch to DD forever a er a 5 51 511 B315 51 B Play C gt receive 4 in this stage and face the exact same game same choices in stage t2 g Trigger strategies for Prisoner s Dilemma Let V be the payoff of player 2 from making the optimal choice in the subgame starting in stage t1 given that the outcomes in the previous stages have been CC Play c gt v4 5v gt v 41 a Play D gt V 5 51 a Play c if41 a 2 5 51 a gt if a 2 14 g Trigger strategies for Prisoner s Dilemma Two types of subgames i Subgames where the outcomes of all previous stages have been CC The trigger strategies are Nash equilibrium for this class of subgames as well as for the original game ii Subgames where the outcome of at least one earlier stage differs from CC Player s strategies are to repeat DD forever which is also a Nash equilibrium for the original game Observation El FeaSIbe payo sin the stage game Even if the stage game G has a unique Nash The payoffs 71 72 nquot are fea5be in the stage equilibrium there may be subgameperfect game G if they are a convex combination of the outcomes of the infinitely repeated game in which purestrategy payoffs of G no stage s outcome is a Nash equilibrium of G l Example I Feasible payoffs in the Prisoner s Dilemma Prisoner 2 t C cooperate D defect Player 2 payorfs g c 4 4 0 5 O 9 LL D 5 0 1 1 What are the purestrategy payoffs 44055011 50 Player 1 payoffs il Friedman 5 Theorem il Feasible payoffs in the Prisoner s Dilemma Let G be a finite static game of complete Player 2 payoffs Information Let e1 e2 equot denote the payoffs Xquot 05 WWW WW 50 Playefrf 1 payo S from a Nash equilibrium of G and let x1 x2 denote any other feasible payoffs from G If xl39gt ej for every playerj and if 5 is sufficiently close to 1 then there exists a subgameperfect Nash equilibrium of the infinitely repeated game Goo5 that achieves x1 x2 xquot as the average payoff 11 Proof of Friedman s Theorem Let ael aez aen be the Nash equilibrium of G that yields the equilibrium payoffs e1 e2 6 Le am be the collection of actions that yields the equilibrium payoffs x1 x2 x Trigger strategy for player i Play agtq in the rst stage In the tm stage if the outcome 0 a l t1 preceding stages has been axl a am then play ax otherwise play as Show that the trigger strategies induce a NE Show that the equilibrium is subgame perfect 5 Proof of Friedman s Theorem cont If player i deviates in stage t by choosing ad Payoff in stage t d Payoffin future stages 5e 52e Be 1 5 Total discounted payoff V d 5 e 1 5 If playeri plays agtq in stage t Receive a payoff x in this stage face the same ame in the next stage V x 5 V gt V x 1 5 Playing gtd is optimal iff gtlt 1 52 d 5 e 1 5 gt 5 2 d gtdd e Proof of Friedman s Theorem cont Suppose all other players other than playeri use the trigger strategy Best response of playeri in stage t Ifthe outcome of the previous stage differs from axl a am Play a2 forever Ifthe outcoms of all previous stages are axl axz lam mm araaaagt7d 41614 d M aaaaa gt7 aa 7e g Proof of Friedman s Theorem cont It is Nash equilibrium for all players to play the trigger strategy if and only if 5 2 max d gt0d e Subgame perfectness Ifthe outcome of the previous stage differs from Ifthe outcoms of all previous stages are 3X1 3x21 I 3m Play the trigger strategy Repeated Cournot Game Cournot stage ame wo competing firms selling a homogeneous good The magmaCostof producing each unit of the good c The market price P is determined by inverse market demand PaQ ifagtQ P0 othenNise Each firm decides on the quantity to sell market share 5 and Q2 Q q1q2 total market demand Both firms seek to maximize pro ls Unique NE ofthe stage game qCac3 Q 2ac3 Monopoly quantity qMacZ g Repeated Cournot Game cont The stage game is repeated in nitely many times The rms have discount factor 5 Trigger strategy Produce half the monopoly quantity qMZ in the rst stage In the tm stage produce qMZ ifboth rms have produced qMZ in all previous stages otherwise produce qc Show that the trigger strategy induces a subgame perfect NE SupplyChain Coordination A typical supply chain Manufacturing Distribution Channelsretailers Endusers Materials products services information money Supply chain activities Design Manufacturing Procurement Planning and forecasting Order ful llment Distribution supply chain DSC l Equilibrium solution for the decentralized Supplier s wholesale price w wac2 Retailer s quantity q qac4b Market price P P3ac4 Supplier s profit HS Hsac28b Retailer s profit HR HRac216b Total SC profits H H 3ac216b Comparison of the decentralized DSC and centralized supply chains CSC 51 Decentralized Centralized Supply Supply Chain Chain CSC DSC Supplier s wholesale price w wac2 Retailer s quantity q qac4b q ac2b Market price P P3ac4 Pac2 Supplier s profit HS Hsac28b Retailer s profit HR HR acZ16b Total SC profits H H 3ac216b Hac24b PrincipalAgent Problem x W UWj 5ai 6 9 Principal Agent 9 Pquot 6 W1 WX1 UW1 e2 1 W2 WX2 UW2 C32 4 9 Wm WXm UWn ean Outcomes Actions l E l Principalagent problem If the agent doesn t accept the contract his payoff is his reservation utility u If the agent accepts the contract he chooses between n possible actions a1 an These actions produce m possible outcomes X1 Xm There is a stochastic relationship between actions and outcomes called technology When the action is ai the principal observes outcome X with probability Pij If the principal observes outcome X she pays the agent wj The agent s payoff is Uweai where Uw is the utility of wage w to the agent and eai is the cost of action ai to the agent U is increasing differentiable and concave Assuming the principal is riskneutral her payoff is Xj Wj What if the agent s actions can be 3 observed The principal can design a contract where the wages are conditioned on the actions ie wai 1T1 max Px w E m1nw 11 I I 1 Participation constraint U wi 601 2 Q Incentive constraint Uw ea 2 Uwk eak Vk ii To induce the agent to choose action at set w such that Uw Q 601 and set all other wk sufficiently low Unobservable actions Principal s problem Step 1 Given an action a how to set the wages such that the agent chooses ai and the principal s payoff is maximized Participation constraint BjUwj ea 2 Q Incentive constraint 21 PjUwj ea 2 21 ijUwj eak v k 72139 Principal s objective minimize the expected cost of inducing the agent to choose ai m m max 2le xj wj min F1 Pijwj Cal Principal s problem Step 1 Cai is the minimal cost to the principal of inducing the agent to take action a Cai is convex 9 the original maximization objective is concave Wellbehaved mathematical program with a concave objective function maximization and linear constraints Principal s problem Step 1 Fora given action ai maxZ1Bjxj wj ZLPJUmp emgzg u 21PIUwj eaiZZIUwj eak Vkii A LltwzgtZIPjltxj wjgtltZ111Ultwjgt eltagt Qgt szltZjLIszltwjgt eltaigt iPgUltwjgteltakgtgt 01121 pooqnexn 41 39 d quot d W WM W W I I dens LU3COJC sedpuud 5 M e d IJWZrMHeeo Zn Me I El Mae 2 0 d Wjil Me 21 Mae f M9 Ti 1 078 121 me JJWZUMHQ d7739 CI fme 2 Me e 0 d Wne Mae 121 W fmnfd va mg21 3ij va fmanjZrM foo213 0 um I dens LU3COJC sedpuud Principal s problem Step 2 I From step 1 maXZj1PUxj wj E j1Pijxj min j1Pijwj Rai Cai What is the action ai that maximizes RaiCai Two actions and two outcomes Suppose the agent has two possible actions a and b and there are two possible outcomes x1 and x2 Suppose that action b is preferred by the principal For action b 2 max ZJ1PIJJXJ wj PJlUw1P2Uw2 eb 2Q u PmUw1 szUW2 6b 2 Pa1Uw1 Pa2Uw2 ea it Two actions and two outcomes Incentive compatibility constraint lUw1PbZUwZ 91 2Pa1Uw1 PaZU wz ea When is the agent indifferent between a and b 1Uw1PbZUwZ eb Pa1Uw1PaZUwZ ea ea Jwleb ea P P P a2 b2 a2 Uw Pal 31 UWeb 2 P P 1 P 132 oz 132 since Pa1PaZ Pb1 332 1 g Two actions and two outcomes 91 ea Uw Uw Z 1 sz PaZ Action b is preferred to action a UWi UW2 UWi g Two actions and two outcomes Paiticipation constraints Action b PblUw1szUw2 8b 2 g Action 61 PalUw1PazUw2 ea 2 g Action b Uw2 2 iUw1eb Jrg Pm Pm Two actions and two outcomes ebea Uwz Uw1 PM Paz UWi UW2 compatibility constraint Uw2 iUWI ebg PM PM UWi g Two actions and two outcomes Principal s objective function For action b maXB71x1 w1B72x2 w2 Let fbe the inveise of U fUw wj maXB71x1 fUw1Byzx2 fUw2 Two actions and two outcomes Uw2 UWi UW2 Participation constraint Principal s isopro t cuxves PM 1 ltUltwlgt szltxz7fltUltwZ c 39 Um Two actions and two outcomes maXPbix1 fUw1 szxz fUw2 Slope of the objective function P afUwl P 3fUW23UW2 b1 8Uw1 b2 8Uw1 8Uw1 P afUwl aUw2 aUwl alfmwl 8Uw1 P 8fUw2 szf39Uw2 b2 8Uw1 39 Two actions and two outcomes If Uw1 Uw2 45 line 3UW2 Pblf39UW1 i 3UW1 szf39UW2 PM Same as the slope of the participation constraint The principal39s isopro t curve is tangent to the agent39s participation constraint along the 450 line g Two actions and two outcomes Uw2 UWi UW2 Participation constraint IncenmIe I compatibility constraint Ultwgt Two actions and two outcomes P P V y 1 i a likelihood ratio U w Pb Pb If the incentive constraint is not binding i0 U39wj1l andwj w Substituting into the incentive constraint the agent prefers action 17 over action a Zn BJJUM 91 2 Zn Panw ea gt ea 2 91 Two actions and two outcomes 1 135 Pa u 1 1 J likelihood ratlo U W ij bj If the incentive constraint is binding xi gt 0 and U wj depends on the likelihood ratio LR The optimal incentive scheme is a linear function of LR Sensitivity analysis Changes in the agent s costs What is the impact of the agent s costs on the outcome Lw nu Pb1xi w1szx2 w2 PMUW1 szUWz eID EH PMUWHHBJzUW z 6b Pa1UW1 PazUWz e0 aLw1 y 1 aLw1 2 l1 Beb 36a Recap Last class January 27 2004 Proof of Friedman s theorem Repeated Cournot game Wage setting Today January 29 2004 Repeated game example Wage setting Extensive form of a game Information sets Example Wage setting Stage game One firm one worker The firm offers the worker a wage w The worker accepts or rejects the rm s offer Reject the worker becomes selfemployed at wage w0 Accept Work disutility e or Shirk disutility 0 u If the worker works supplies effort Output is highy If the worker shirks Output is high with probability p and low0 with probability 1 p The firm does not observe the worker s effort decision The output of the worker is observed by both parties Example Wage setting cont Payoffs FirmWorker Work Supply effort High output ywwe Shirk High output yw w Low output w w What is the subgameperfect equilibrium in this stage game For any w 2 wo worker accepts employment and shirks Firm offers w0 or any other wltw0 Example Wage setting cont Strategies Firm Offer ww in the first stage In stage t offer ww if the history of play is highwage high output all previous offers have been w all previous offers have been accepted and all previous outputs have been high otherwise offer w0 Worker If wgtw0 accept the firm s offer and supply effort if the history of play including the current offer is high wage highoutput shirk otherwise If wltw0 choose self empoyment Example Wage setting cont Suppose firm offers w 2 w0 Worker accepts Work Supply effort Vewe 8 Ve gt Ve we15 Shirk vs w 6pvs lpwO16 2 Vs 15W 51P Wo1 5p1 5 Worker should supply effort if Ve2VS gt w 2 w0 e e1551p If p0 we1 5 2 w w051 5 Example Wage setting cont When is it the best response for the firm to offer w From worker s best response w 2 w0 e e1861p 1 y2w gt y 2 w0 e e1861p 2 The strategies induce a NE if 1 and 2 hold Is this a SPNE Example Wage setting cont What are the subgames Subgames beginning after a highwage highoutput history Subgames beginning after all other histories Extensive form of a game The set of players The order of moves The players payoffs as a function of the moves that were made The set of actions available to the players when they move Each player s information when he makes his move The probability distributions over any exogenous events Nature Example 1 1 21 00 41 32 Player 1 moves first After observing player 1 s action player 2 moves Player 1 action set UD Player 2 action setLR Player 1 strategies UD Player 2 strategies LL LR RL RR Normal form representation of extensiveform games Player2 H LL LR M RR 1 U 21 21 00 00 E D 11 32 11 32 Player 2 s strategies correspond to a contingent plan made in advance g Example 2 1 L 21 00 41 32 Player 1 moves first player 2 moves next Player 2 does not know player 1 s action when he chooses his action 21 11 00 32 Player 2 moves first player 1 moves next Player 1 does not know player 2 s action when he chooses his action Example Player 1 chooses an action from the feasible set L R Player 2 observes player 1 s action and then chooses an action from the feasible set L R Player 3 observes whether or not the history of actions is RR and then chooses an action from the feasible set L R 5 Example cont Information set An information set for a player is a collection of decision nodes satisfying The player has the move at every node in the information set When the play of the game reaches a node in the information set the player with the move does not know which node in the information set has or has not been reached 5 Example cont Player 2 has two information sets both singletons Player 3 has two information sets one of them is singleton Subgame in an extensive form game A subgamein an extensive form game begins at a decision node nthat is a singleton information set but is not the game s first decision node includes all the decision and terminal nodes following n in the game tree but no nodes that do not follow n and does not cut any information sets ie if a decision node 7 follows n in the game tree then all other nodes in the information set containing 7 must also follow n and so must be included in the subgame QUANTITY FLEXIBILITY 1116 Supplier sell at w per unit Demand is distributed with cdf and pdf Supplier s unit cost is 05 retailer s unit cost is CR and c 05 CR Unit selling price is p Centralized supply chain CSC 1 pSq cq p q 7 fog 7 cq 82 5 p7pFltqgt7cplt17Fltqgtgt7c0 cplt17Fltqgtgt 1 Quantity Flexibility Refund the retailer w CR for units unsold up to min 1 6g 5q expected number of units sold by the retailer DZq sellq Dltqgt sellD 5917 m q qxfxdx lt2 PNDZq Let us compute the term fog d dx xfx 0q dx q Fq 0q Fxdx 0q xfxdx s 2mm q M 7 qFltxgtdx lt3 From 2 and 3 What is the retailer s expected compensation D 7 0 17 5q 1 D lt 1 7 6q 5 leftover inventory gt 6g 5 compensated for 6g 1 7 6q lt D lt q 5 leftover inventory lt 6g 5 compensated for q 7 D D gt q 5 no inventory left Expected of units for which the retailer gets compensation 1R F1 WW q q 7 Jcfpdp 176 Since limtq 7 acfrdr q176qf1dr 7176qfd and 1 W00 q Fq 7 F17 am 176 we have 1R 7 Fltlt17 wag W 7 qFltlt17 6m 7 6 xfmdx 2 Let s compute 1176 q q 175 xfxdx xfxdx7 xfxdx 176M 0 0 q 175 7 gm 7 Fltxgtdx 7 17 6gtqFltlt1 7 am 7 Hamlet 7 gm 7 lt1 7 6gtqFltlt17 am 7 q Fltxgtdx 5 175 Hence7 inserting 117th from Equation 5 into IR we get q IR 7 F17 5Q6q qFq 7 qF17 59 7 qFQ17 5qF17 5M H FWW 5 IR176qFxdx 6 Ret ailer s pro t function HRq p Sq 7 w CRq w CR1qi6 Fxdx 8115 7p 82 29 7 w cRgt w CRgtltFltqgt 7 Fltlt17 6gtqgtlt17 6 p17Fq7wCRl17FQF1759175H 0 p1 7 FW 7 w CRl17 FOJ F17 5917 5l For 1 to be optimal for the retailer it needs to satisfy the retailers FOC7 ie7 5 w plt17 W 0 17 Fq M17 6q17 6 R A 3 Can we claim that if the supplier chooses w in this way the retailer will choose 1 This is the case only the retailer s pro t function is concave fgg eijwaVy4wxp o aiww W f15q175 fg wqumwwwcmOi 075Mlt0 q V v gt0 gt0 wcRDp ie pricR as lon as g wcR20 1e 11127013 What do you think happens to retailer s pro t as 6 T 7 As 6 T the retailer gets protection for more units but at a higher cost Supplier s pro t Us w Cslq w l CR q Fxdx 15q p i m 60 w1i mF 02 P1 F 1 CR From equation 1 we have pl 7 c ie if 6 0 we have u 05 H5 0 and HR H Compliance If the retailer orders q he will certainly not accept more than q units from the supplier However what if the supplier provides less than q units This might happen due to shortages or other problems at the supplier or the supplier might simply nd it more pro table to deliver a different quantity given the contractual terms and her pro t maximization objec tive ln voluntary compliance the supplier delivers an amount not to exceed the retailer s order quantity to maximize her pro ts ln forced compliance the supplier delivers exactly 4 11 Recap Last class February 6 2003 Battle of the sexes under incomplete information Firstprice sealedbid aucb39ons Equilibrium recap Onecard poker game Today February 13 2002 Onecard poker game results and analysis Onecard poker game analysis PrincipalAgent problem g ISYE6230 onecard poker strategies Evenone bet whenever they had an ace Most commons strategy A bet K passquot The second most common strategy quotAlways betquot Third most common strategy Start with quotA bet K passquot and bet more aggressively late Alternate between betting and passing ifthe card is K Always bet Calculate the probability that the opponent has an A and bet depending on that probability ISYE 6230 onecard poker game Total number of games played 224 Average 1027 Number of different outcomes 55 131 BP 4o PB 3s PP 1o Example Onecard poker Player 2 Player 1 Example Onecard poker cont The payoffs depend on players actions Ed on card combinations Need to compute expected payoffs for each outcome Example Onecard poker cont Payoffs for outcome BB under different card combinations Expected payoff for player 0250025ab0 Similarly expected payoff for player 2 is 0 Expected payoffs for 33 00 1 25ab02500 Onecard poker in normal form Player 2 The unique NE is BB Principalagent examples Restaurant owner waiter Software company salesman Auto manufacturer customer leasing a car Insurance company insured S PrincipalAgent Problem Example The principal offers wage w If the agent accepts the offer Agent can put highquot e25 or low e0 effort Agent39s utility Uwewe Agent39s resenation level of utility 81 Principal39s payoff 270 if the agent works hard 70 if the agent doesn39t work hard Firstbest contract The agent won39t accept the job unless the wage exceeds his resenation utility wzsl Employing this agent is worthwhile to the principal only if the agent works hard otherwise the principal only gets 70 For the agent to work hard his utility from working hard should exceed his resenation utility Uwe 2 81 w25281gtw2106 Firstbest contract offer 106 9 to the agent and trust that he will work hard Moral hazard Firstbest contract Offer the agent 106 What is the problem with this contract Moral hazardquot the agent takes a decision or action that affects his or her utility as well as the principal39s the principal only ob erves the outcome as an imperfect signal of the action taken and the agent does not necessarily choose the action in the interest of the principal Alternative Offer a contract where the wage depends on the effort level Contract conditioned on effort level Offer two wage rates wH ifthe agent exerts high effort wL iflhe agent exerts low effort How to choose wH so that accepting the offer and working hard is desirable for the agent wH 25 2 81 participation constraint individual rationality constraint wH 25 2 wL incentive constraint Contract conditioned on outcome Suppose the agent is a salesman representing the principal to a client Three possible outcomes based on the effort level of the salesman The client places no order 0 The client places a small order 100 nt places a large order 400 Probabilities for different outcomes under each effort level No order Smallorder Large order Expected 0 100 400 order size ngh effort l 01 l 03 l 6 270 ILow effort I 06 03 01 70 Contract conditioned on outcome A contract where wage depends on the obsenable outcome No order gt agent pays the principal 164 Small order gt agent pays the principal 64 Large order gt principal pays agent 236 Contract conditioned on outcome Principal39s expected revenue if the agent works hard 270 Expected pro t 164 How much does the principal s revenue differ from the expected revenue under each outcome No order gt 0270 270 270106 164 Small order gt 100270 170 170106 64 Large order gt 400270 130 130106 236 Principal39s profit No order gt 164 Small order gt 164 Large order gt 164 Contract conditioned on outcome Agent39s choices and expected payoffs under each choice assuming the agent is riskneutral Reject the contract and get reservation utility 81 Accept the contract and don39t work hard 01235035405154 0 94 Acce t the contract and work hard 0523503540115425 81 The principal always gets 164 Contract with positive wages Suppose the agent only aocepls positive wages What are the wa s w w5 and wL corrsponding to no order small order and large order outoorrles that maximize the principal39s payo Participation constraint W501W 25281 Inoentive constraint 06 WL 03 W5 01 WEl 25 2 06 wEl 03 w5 01 wL Nonnegativity constraint w W5 wL 2 0 Principal39s o 39ective Maximize 06400 wL03100 w5010w Equivalently Minimize 06 wL 03 w5 01 wEl Recap Games in normal form NBS vs ABC Prisoner s dilemma Tragedy of the commons Dominant or dominated actions outcomes Best response Nash equilibrium pure vs mixed strategy Duopoly models Cournot Stackelberg Bertrand comparison with monopoly Multistage games with observed actions Stackelberg Strategic investment Extensive form of a game information sets Subgame perfect Nash equilibrium Repated games finitely or infinitely Prisoner s dilemma Cournot competition Trigger strategies Friedman s theorem Example Oneca rd poker A player is dealt an ace or king with equal probability A SI Example Onecard poker Two players One deck of cards half aces half kings Pay a to play Each player is dealt a card face down After seeing hisher card each player simultaneously Action B bets b or Action P passes Payoffs BP or PB gt betting player gets the pot 88 or PP gt higher card gets the pot in case of a tie the pot is split Record the results Player 1 Player 2 Payoffs Game 1 AceBet KingBet ab ab Game 2 KingPass KingBet a a Game 3 KingPass KingPass 0 0 etc Player 2 Player 1 O Options f 3 Example Onecard poker P 39 Example Onecard poker cont The payoffs depend on players actions w on card combinations Need to compute expected payoffs for each outcome 11 Recap Last Tuesday February 4 2003 Subgames in extensive form games Games of incomplete information Coumot competition under incomplete information Battle of the sexes under incomplete information Today February 6 2003 Battle of the sexes under incomplete information Firstprice sealedbid aucb39ons Equilibrium recap Onecard poker game Example Battle of the sexes Bo b Football Fo l Alice 0390 wire tA is privately known to Alice and tB is privately nown to o tA and tB are independent draws from a uniform distribution on 0x 5 Example Battle of the sexes Bob Strategies Alice Play ballet iftA exceeds a critical value cA otherwise play football Bob Play football iftB exceeds a critical value cB otherwise play ballet Example Battle of the sexes Alice s expected payoffs Alice plays Ballet cBx2tA Alice plays Football 1 cBx Play ballet if cBx2tAgt 1 cBx gt tAgtxcB3 cA Bob s expected payoffs Bob plays Ballet 1cAx Bob plays Football cAx2 t3 Play football if cAx2tBgt 1 cAx gt tBgtxcA3 cB Example Battle of the sexes tA gt xcB3 cA tB gt xcA3 cB cA cB c c23cx0 c 3 94x2 What is the probability that Alice plays ballet 1 cAx 1 394x2x What is the probability that Bob plays football 1 cBx 1 394x2x In the limit an these probabilities approach 23 g FirstPrice SealedBid Auction Two bidders one good Bidder i39s valuation for the good is v is known only by bid er i Valuations are independently and uniformly distributed on 01 Each bidder i submits a nonnegative bid b The higher bidder wins and pays his bid Other bidder pays and receivs nothing In case ofa tie the winner is determined by a coin ip Bidder i39s payoff ifwins and pays p is v p Bidders are riskneutral All of this information is common knowledge SI FirstPrice SealedBid Auction Acb39on spaces 39 A1 01 Type spaces 39 T1 2 01 Belies p1t2t1 1219 P2t1 t2 P201 Player i39s expected payoff function v 7b if 5 gt51 blvb2VhV2 Vim2 if b 51 0 if 5 gt51 FirstPrice SealedBid Auction Strategy for player i b v Strategies b1v1 b2v2 are a Bayesian Nash equilibrium if for each v in 01 b v solvs max v b Probb gt bv v b Probb bv2 Consider a linear equilibrium b v a c v i12 Assuming playerj adopts the strategy bv aJcv player i39s best response max v b Probb gt bv v b Probb gt acv SI FirstPrice SealedBid Auction Assuming playerj adopts the strategy bv aJcv player i39s best response max v b Probb gt aJcv st by S minacJ v Probb gt acv Probvltb ac b acJ max v b b acJ st b S minacJ v From FOC b v ifv Sa by v aZ otherwise q FirstPrice SealedBid Auction Player i s best response b v if wSa b wa2 otherwise I Can Between 0 and 1 Greater than or equal to 1 by a chJ 2 1 Less than or equal to Zero b v v aJ2 We have also b ac v aJ212q gt a aJZ c 12 g FirstPrice SealedBid Auction Player i s best response a350 ac v aJ212v gt a aJZ c 12 Playerj s best response aSO aCa 212VJ gt a aZ c12 We have a aJ 0 and c c12 and b v vZ i12 q Equilibrium recap Static games of complete information Nash equilibrium Dynamic games of complete information Subgame perfect Nash equi i rium Static games of incomplete information Bayesian games Bayesian Nash equilibrium Dynamic games of incomplete information Perfect Bayesian equilibrium 11 Recap Last Tuesday February 4 2003 Subgames in extensive form games Games of incomplete information Coumot competition under incomplete information Battle of the sexes under incomplete information Today February 6 2003 Battle of the sexes under incomplete information Firstprice sealedbid aucb39ons Equilibrium recap Onecard poker game Example Battle of the sexes Bo b Football Fo l Alice 0390 wire tA is privately known to Alice and tB is privately nown to o tA and tB are independent draws from a uniform distribution on 0x 5 Example Battle of the sexes Bob Strategies Alice Play ballet iftA exceeds a critical value cA otherwise play football Bob Play football iftB exceeds a critical value cB otherwise play ballet Example Battle of the sexes Alice s expected payoffs Alice plays Ballet cBx2tA Alice plays Football 1 cBx Play ballet if cBx2tAgt 1 cBx gt tAgtxcB3 cA Bob s expected payoffs Bob plays Ballet 1cAx Bob plays Football cAx2 t3 Play football if cAx2tBgt 1 cAx gt tBgtxcA3 cB Example Battle of the sexes tA gt xcB3 cA tB gt xcA3 cB cA cB c c23cx0 c 3 94x2 What is the probability that Alice plays ballet 1 cAx 1 394x2x What is the probability that Bob plays football 1 cBx 1 394x2x In the limit an these probabilities approach 23 g FirstPrice SealedBid Auction Two bidders one good Bidder i39s valuation for the good is v is known only by bid er i Valuations are independently and uniformly distributed on 01 Each bidder i submits a nonnegative bid b The higher bidder wins and pays his bid Other bidder pays and receivs nothing In case ofa tie the winner is determined by a coin ip Bidder i39s payoff ifwins and pays p is v p Bidders are riskneutral All of this information is common knowledge SI FirstPrice SealedBid Auction Acb39on spaces 39 A1 01 Type spaces 39 T1 2 01 Belies p1t2t1 1219 P2t1 t2 P201 Player i39s expected payoff function v 7b if 5 gt51 blvb2VhV2 Vim2 if b 51 0 if 5 gt51 FirstPrice SealedBid Auction Strategy for player i b v Strategies b1v1 b2v2 are a Bayesian Nash equilibrium if for each v in 01 b v solvs max v b Probb gt bv v b Probb bv2 Consider a linear equilibrium b v a c v i12 Assuming playerj adopts the strategy bv aJcv player i39s best response max v b Probb gt bv v b Probb gt acv SI FirstPrice SealedBid Auction Assuming playerj adopts the strategy bv aJcv player i39s best response max v b Probb gt aJcv st by S minacJ v Probb gt acv Probvltb ac b acJ max v b b acJ st b S minacJ v From FOC b v ifv Sa by v aZ otherwise q FirstPrice SealedBid Auction Player i s best response b v if wSa b wa2 otherwise I Can Between 0 and 1 Greater than or equal to 1 by a chJ 2 1 Less than or equal to Zero b v v aJ2 We have also b ac v aJ212q gt a aJZ c 12 g FirstPrice SealedBid Auction Player i s best response a350 ac v aJ212v gt a aJZ c 12 Playerj s best response aSO aCa 212VJ gt a aZ c12 We have a aJ 0 and c c12 and b v vZ i12 q Equilibrium recap Static games of complete information Nash equilibrium Dynamic games of complete information Subgame perfect Nash equi i rium Static games of incomplete information Bayesian games Bayesian Nash equilibrium Dynamic games of incomplete information Perfect Bayesian equilibrium PrincipalAgent Problem Principalagent examples Restaurant owner waiter Software company salesman Auto manufacturer customer leasing a car Insurance company insured Example The principal offers wage w If the agent accepts the offer Agent can put high e25 or low e0 effort Agent s utility Uwewe Agent s reservation level of utility 81 Principal s payoff 270 if the agent works hard 70 if the agent doesn t work hard Firstbest contract The agent won t accept the job unless the wage exceeds his reservation utility I Employing this agent is worthwhile to the principal only if the agent works hard otherwise the principal only gets 70 For the agent to work hard his utility from working hard should exceed his reservation utility Uwe 2 81 w 25281 aw2106 Firstbest contract offer 106 2 to the agent and trust that he will work hard s Moral hazard Firstbest contract Offer the agent 106s What is the problem with this contract Moral hazard the agent takes a decision or action that affects his or her utility as well as the principal s the principal only observes the outcome as an imperfect signal of the action taken and the agent does not necessarily choose the action in the interest of the principal Alternative Offer a contract where the wage depends on the effort level Contract conditioned on effort level Offer two wage rates wH if the agent exerts high effort wL if the agent exerts low effort How to choose wH so that accepting the offer and working hard is desirable for the agent wH 25 2 81 participation constraint individual rationality constraint wH 25 2 wL incentive constraint to a client The client places no order 0 The client places a small order 100 The client places a large order 400 g Contract conditioned on outcome Suppose the agent is a salesman representing the principal Three possible outcomes based on the effort level of the salesman Probabilities for different outcomes under each effort level No order Small order Large order Expected 0 100 400 order size High effort 01 03 06 270 Low effort 06 03 01 70 39 Contract conditioned on outcome A contract where wage depends on the observable outcome No order gt agent pays the principal 164 Small order gt agent pays the principal 64 Large order gt principal pays agent 236 SI Contract conditioned on outcome Principal s expected revenue if the agent works hard 270 Expected profit 164 How much does the principal s revenue differ from the expected revenue under each outcome No order gt 0270 270 270106 164 Small order gt 100270 170 170106 64 Large order gt 400270 130 130106 236 Principal s pro t No order gt 164 Small order gt 164 Large order gt 164 Si Contract conditioned on outcome Agent s choices and expected payoffs under each choice assuming the agent is riskneutral Reject the contract and get reservation utility 81 Accept the contract and don t work hard O1236O364O6164 0 94 Accept the contract and work hard O6236O3640116425 81 The principal always gets 164 SI Contract with positive wages Suppose the agent only accepts positive wages What are the wages wo ws and wL corresponding to no order small order and large order outcomes that maximize the principal s payoff Participation constraint 06wL03w501w 25281 Incentive constraint 06wL03w501w 252 06 w0 03 w5 01 wL Nonnegativity constraint wO ws wL 2 0 Principal s objective Maximize 06400 wL03100 w5010w Equivalently Minimize 06 wL 03 w5 01 w0 Risk aversion What if the agent is riskaverse A person who prefers to get the expected value of a gamble for sure instead of taking the risky gamble is risk averse Eg getting 25 for sure vs getting 0 with probability 075 and 100 with probability 025 The agent and the principal may have different beliefs about the probabilities of different outcomes under different effort levels Example Risk averse agent Agent s reservation utility 10 Agent s possible actions if accepts the contract work hard e2 don t work hard e0 Two possible outcomes L and H Principal offers wages wL and wH based on the outcome Example Risk averse agent Probabilities of H and L outcomes Agent does not work hard H with probability 04 L with probability 06 Agent works hard Principal s belief H with probability 08 L with probability 02 Agent s belief H with probability 07 u L with probability 03 5 PrincipalAgent Problem Dates Covered February 19 28 2002 Suppose a restaurant owner the principal hires a waiter the agent to run the restaurant in his or her absence If the waiter does not work hard or 7 shirks fewer customers will come and the owner will lose potential revenue If the waiter does work hard more customers will come and the owner will absorb greater revenue than otherwise However this additional revenue is as a direct result of the waiter s effort who depending on the terms of employment could potentially reap no bene ts from the increased effort Thus the restauranteur will want to design a contract and offer it to the agent in the hopes that it will provide incentive to the waiter to work hard such that both parties will mutually bene t The preceding is a speci c example of the principalagent problem which can generically be described in the following manner The principal designs the terms of the contract and offers it to the agent The agent decides whether or not to accept the contract Should the agent accept the contract he or she decides on the level of effort that will be exerted The rm s revenue is observed and the principal pays the agent based on the terms agreed upon in the contract Example 1 Let s rst look at the problem from the agent s perspective We will de ne e the level of effort exerted by the waiter as 7 0 if shirhs 6 T 2 if workshard In addition we de ne the agent s utility as U 7 w 7 e if devotese 7 10 if rejectscontract where w is the wage offered in the contract Thus the agent will either 7 take it or leave it From the principal s perspective we de ne Re as the revenue for the restaurant as a function of the waiter s effort H ife2 R5 L ife0 The potential pro t for the principal is 7T Re 7 w Of course the principal will design the contract with the objective of maximizing 7139 We want to de ne wH as the wage rate the principal promises to pay when the effort is high and wL as the wage rate that the agent will receive if the effort is low We will nish this example during the next lecture End of 2 19 Lecture Start of 2 21 Lecture 46 Now we will nish the example we started during the previous lecture The principal will design a contract offer two wage rates one wH when the agent exerts high effort and another wL when the agent exerts low effort First the principal will design the participation contraint wH 7 2 2 10 to ensure that the agent participates agrees to accept the contract This constraint is designed such that the difference between the high wage rate and maximal effort is greater than or equal to any other alternative that the waiter has In short even if the agent has to exert maximal effort it would be in his or her best interest to accept the principal s contract offer In addition to wanting the agent to accept the contract the principal also wants the agent to agree to work hard Recall our assumption from the previous lecture that pro ts generated when the agent works hard 7TH are greater than pro ts generated when the agent 7 shirks 711 since customers are more likely to be repeat visitors when there is high customer service Thus the principal will also construct an incentive constraint wH 7 2 2 wL which gives the agent the incentive to work hard This constraint is designed such that the difference between the high wage rate and the maximal effort is greater than or equal to the low wage rate In short the high wage rate is set such that the agent is better off by accepting it and working hard as opposed to accepting the low wage rate and shirking The desired combined effect of these two contraints is to produce a contract such that the agent will both accept the principal s offer and agree to work hard In this example the participation and incentive constraints generate a high wage rate wH of 12 and a low wage rate wL of 10 The corresponding pro t functions are 7THHUIHH712 77LL7wLL710 where H and L are the revenues associated with the agent working hard and 7 shirking respectively The preceding principal agent problem also illustrates a concept called moral hazard A moral hazard occurs when the agent takes a decision or action that affects his or her utility as well as the principal s the principal only observes the outcome an imperfect signal of the action taken and the agent given a spontaneous choice does not necessarily choose Pareto optimality Since the action is unobservable the principal cannot force the agent to choose an action that is Pareto optimal He or she can only in uence the choice of an action by conditioning the agent s utility to the outcome which is the only observable variable In our example the principal does not observe whether or not the agent is working hard or 7 shirking action The only thing that the principal observes is the revenue generated as a result of the agent s effort Thus it is possible that the agent will accept the high wage agree to work hard but 7 shirk The principal will not know this until he or she observes the revenue gures and calculates pro t the outcome Hence there is a 7 moral hazard 47 associated with these sorts of contracts Moral hazards essentially occur in all economic relationships Some potential complications to the principal agent problem include asymettric informa tion risk differences and revenue uncertainty We will now look at an example that has the latter complication Example 2 Suppose we have the same parameters as in Example 1 only the revenues H and L are uncertain 7 as indicated by the following 7 states of nature Ra 7 H with probability 08 7 L with probability 02 Rm 7 H with probability 04 T L with probability 06 Here the agent affects the probability of each realization of Re depending on the level of effort chosen As a result the utility function of the agent must be modi ed and is now 10 if rejects contract U Ew 7 e if devotes e The wage is now an expected value Ew because of the uncertainties associated with the revenue function Based on the revenue functions 7 08wH 02wL when 6 2 w 7 04wH 06wL when 6 0 E Thus the new participation constraint is 08wH 02wL 7 2 2 10 and the new incentive constraint is 08wH 02wL 7 2 2 04wH 06wL Rewriting the participation constraint in terms of wL yields wL 60 7 4wH and rewriting the incentive constraint in terms of wL yields Solving this as a system of equations yields wL 8 and then using back substitution we obtain wH 13 The uncertainty results in the agent incurring more risk which in turn generates a higher wH value Example 1 wH 12 For homework model this as an extensive form game and solve SPE 48 5 1 Risk Aversion Suppose there are two consumers 239 and j Consumerz39 is more risk averse than consumer j when cansumerj prefers a xed sum 0f money over a lattery then consumer 239 also prefers the xed amaunt This is explained using the following example Example 3 Suppose that the principal and agent think about risk differently One way to model this is the use of 7 subjective probability that measures the probabilities that each player assigns to the realization of the states of nature Suppose the principal has the same revenue functions as above RPQ 7 H with probability 08 7 L with probability 02 and RP0 7 H with probability 04 7 L with probability 06 The agent however has a different belief about how much revenue will be generated as a result of his or her effort The agent believes that the principal s revenue functions will be as follows RAQ 7 H with probability 07 7 L with probability 03 RA0 RP0 In this case the agent is more risk averse since he or she is more skeptical about the realization of the 7 high state of nature In fact it is typically assumed that the principal is 7 risk neutral ie less risk averse and the agent is risk averse Thus in this case the participation constraint will be 07M 03wL 7 2 2 10 a wH and the incentive constraint will be 07M 03wL 7 2 2 04M 06wL a wH 23 wL The objective of the principal is to minimize expected wage if H 7 L is suf ciently large mianWL E0w 08wH 02wL which yields the optimum values wH 14 and wL The principal s expected wage will exceed the agent s reservation utility plus effort E0w 08wH 02wL 1266 gt 10 12 12 Plotting all of this yields the following graphic Feasible Region no 60 22 125 7 39 50 W W 3 0 95 92 b4 9 C121 0 Figure 21 Risk Averse Principal Agent Example The following list of items is a general summary of a principal agent model 1 The agent chooses between H possible actions 11 an 2 These actions produce m possible outcomes x1 vm The outcome is a signal that gives the principal information on the action that the agent chooses 3 There is a stochastic relationship between actions and outcomes called a 7 technol ogy such that when the agent chooses action 11 the principal observes outcomes W with probability pij 4 1f the principal observes outcome mj he or she will pay the agent a wage wj and keep the remaining CW 7 wj 5 Assuming the agent s utility Uw 7 a is separable U is increasing and concave 6 Assuming the principal is risk neutral his or her utility is In 7 w First we look at the agent s problem When the principal offers a contract wj the agent chooses his or her best action by solving maxi71an1 PijUWj ail If the agent chooses 11 this gives n 7 1 incentive compatibility constraints ICk 2 1PijUwj Ii 2 Zglpijo j ak Vk 739 239 Suppose Q is the utility for the agent if he or she does not accept the contract Then the participation constraint known in this case as an individual rationality constraint HR is 2 PijUWj Ii 2 2 Now we look at the principal s problem The principal wishes to choose a contract wl wm that maximizes the expected utility while taking into account the dependence of the contract on the agent s decisions Thus the objective function is man1wmi 2 Pijo wj7 subject to Ck and IRW from the agent s problem End of 2 21 Lecture Start of 2 26 Lecture If we x ai the Lagrangian of this maximization problem is Lw7 7 M 2 H39jo wj Zkgm AMZJTZl PijUWj M 2 1 ijUWj MN12 1 PijUWj az39 Q 3 Taking the rst order condition with respect to wj we get 5 Pz39j ZZ1kiAkPijUiwj ijU UIj MPz39jU Wj E 0 7 P a may M 221k7 i Ml P which makes wj as a function ofj depend on the ratio 1 In equilibrium at least one of the k values must be positive otherwise we could neglect the incentive constraints and the moral hazard problem would vanish Thus if the agent has only two possible actions as in the previous examples our expression simpli es to P WMAl p ijl Note that in statistics an expression of the form i Zj is known as a likelihood ratiowhz39ch 7 measures the likelihaad 0f abserm39ng 2 given that the agent chaases a to the likelihaad 0f abserm39ng 2 given that the agent chooses 17 Thus the higher the value of the ratio the more likely the agent will choose a 7 and vice versa Recall that we said Uw is increasing and concave This indicates that the ratio U 1m is 7 increasing and that wj will tend to be higher as the ratio increases This is illustrated by the following graphic UW Uw Figure 22 As likelihood ratio deceases left hand side increases in wj Since this is a monotone increasing function of IQ the optimal incentive scheme is a linear function of the likelihood ratio However if the number of actions n gt 2 the argument will not necessarily hold since wage depends on a weighted average of likelihood ratios It should be noted however that the intuition will hold in most cases In order to make the argument more concrete we need to assume that the cumulative distribution function cdf is convex on actions 11 an That is for 239 lt j lt k and 6 01 13 ai 1 7 Mak This condition gives le S Pil 1 7 APkl Vl 1m An example of how to construct an incentive scheme for two actions and two outcomes is as follows w1w wg wsx2 7x1s S 1 Here the agent receives a base wage w and a bonus proportional to the increase in surplus when he or she suceeded in satisfying the principal s demands Using the Lagrangian statistics mentioned earlier we can determine how the principal s optimizaed values change with respect to the agent s effort in the two action two outcome case The Lagrangian rst order conditions with respect to the agent s costs effort yield 6L 6111 7 5 A M7 based on the Envelope Theorem which was outlined extensively in lSyE 6229 This helps answer the question 7 Which is better the 7 carrot or the 7 stick The 7 carrot provides incentive to the agent to perform the action the principal wants by decreasing the cost effort of the chosen action a2 The 7 stick punishes the agent for not taking the action desired by the principal by increasing the cost of the alternative action a1 by the same magnitude As evidenced by the rst order conditions the 7 carrot relaxes two constraints IC and IR while the stick only relaxes one 1C Hence a small decrease in the cost of the chosen action always increases the principal s utiity by a larger amount than an increase of the same magnitude in cost of the alternative action In other words the carmt is always better than the stz39ck Suppose now we have a change in the probability distribution dPaj The effect on the principal s utility of such a change is dL 7A dPaJUwj Thus when the C constraint is binding A gt 0 the interests of the principal and agent are diametrically opposed with respect to changes in the probability distribution In other words any action that helps the agent unambiguously hurts the principal 52 Multiple Agents We will conclude our discussion on the Principal Agent problem with a look at the case of multiple agents We will do so by looking at the following example Example 4 Consider an order picking operation in a warehouse with N pickers The value V to the warehouse depends on the effort levels of the N pickers V 5 i39 To simplify the analysis assume that Ztil wi V7 where wi is the wage for picker 239 Furthermore assume all pickers have the same utility Uiwi7eii1N 53 ISYE 6230 Economic Decision Analysis II 1 Spring 2004 What is this course about The interactions of multiple players decision makers and the resulting dynamics in a market environment Suppliers manufacturers retailers consumers etc tee s Interactions Customers Business partners Company XYZ 39 Other entities in the market What is this course about The interactions of multiple players decision makers and the resulting dynamics in a market environment Suppliers manufacturers retailers consumers etc Strategic behavior Each player in the market acts on selfinterest eg tries to maximize its own pro t In choosing an action a player considers the potential responsesreactions of other players 5 Main tool for analysis Game Theory Intel vs AMD Advanced Micro Devices AMD has slashed prices of its desktop and mobile Athlon processors just days after a similar move by rival Intel quotWe39re going to do what it takes to stay competitivequot on prices said an AMD representative AMD39s move is also designed to clear out inventory and make way for faster new desktop and mobile processors AMD39s aggressive pricechopping means the company doesn39t want to give up market share gains even at the cost of losses on the bottom line analysts said ZDNet News May 30 2002 Intel vs AMD cont During the first quarter of 2002 AMD increased processor shipments from the fourth quarter of 2001 topping 8 million but processor revenue declined by 3 sequentially In effect the company sold more chips for less money than in the fourth quarter Burger King vs McDonald s Burger King Corp will put its agship Whopper hamburger on sale for 99 cents beginning Friday The move is likely to intensify and prolong the burger price wars that have been roiling the US fastfood industry in recent months Burger King officials had said earlier that while they were reluctant to discount the Whopper they had little choice given a 1 menu at archrival McDonald39s Corp that included a Whopperlike hamburger called the Big 39N Tasty Chicago 5un 77mes January 3 2003 Tesco vs Asda Tesco announced plans to slash 80 million from prices of more than 1000 products with some prices falling by more than 30 The cuts came as rival Asda also said it was slashing selected prices The cuts echo memories of the supermarket price wars played out in 1999 as stores fought to capture more customers and increased market share News of the price cuts raise fears that Sainsbury39s and Safeway would lose market share this year to Tesco Sunday Telegraph January 5 2003 US Auto manufacturing Through June 7 2002 US vehicle output was up 77 from a year ago Ward s Automotive Reports GM39s secondquarter production is up 12 from a year ago The company says thirdquarter output will increase 06 Ford39s secondquarter production is up 4 and the company expects the third quarter to be up 16 DaimlerChrysIer39s Chrysler Group but said production through May 2002 is up 32 Analysts warn that automakers may end up spending more on rebate deals in late summer to clear out too many cars being built now USA Today June 14 2002 Oil production OPEC decided to slash its crude oil production by 15 million barrels a day 6 The issue came to a head this autumn as the weakening world economy together with the uncertainty caused by the Sept 11 attacks on the United States dragged down prices some 30 percent The cut is expected to lift OPEC39s benchmark price to 22 a barrel the group39s minimum target price Rival independent producers such as Russia and Norway promised reciprocal cuts that were smaller than expected OPEC which has already reduced its official production by 35 million barrels a day this year is weary of doing so only to see producers outside the group increase their market share as a result CBS News December 28 2001 Players In a supply chain Suppliers Horizontal interactions Manufacturing Warehousing Distribution Customers I Blockbuster and movie studios Q Order qty Rental fee Studio Blockbuster Demand supplier retailer W Wholesale price Before 1998 W65 Peak demand usually lasts lt10 weeks breakeven 22 rentals Low inventory stockouts 20 of customers were unable to rent the movie they wanted After 1998 W8 Blockbuster gives 3045 of the revenues back to the studio Stockouts dropped Blockbuster increased its overall market share from 25 to 31 and its cash flow by 61 Winwin situation for all parties involved 5 Extensive form of a game The set of players The order of moves The players payoffs as a function of the moves that were made The set of actions available to the players when they move Each player s information when he makes his move The probability distributions over any exogenous events Nature Example 1 1 21 00 41 32 Player 1 moves first After observing player 1 s action player 2 moves Player 1 action set UD Player 2 action setLR Player 1 strategies UD Player 2 strategies LL LR RL RR Normal form representation of extensiveform games Player2 H LL LR M RR 1 U 21 21 00 00 E D 11 32 11 32 Player 2 s strategies correspond to a contingent plan made in advance Example 2 1 2 L 21 00 Ll 32 21 11 00 32 Player 1 moves first player 2 Player 2 moves first player 1 moves next Player 2 does not moves next Player 1 does not know player 1 s action when know player 2 s action when he chooses his action he chooses his action Classroom exercise 1 30 02 Multistage Prisoner s Dilemma Prisoner2 C cooperate D defect g C 4 4 0 o 9 0 D 5 0 1 1 The stage game will be repeated M times The payoffs of the players are the sum of their payoffs from all stages Games of Incomplete Information Bayesian Games In a game of complete information the players payoff functions are common knowledge In a game of incomplete information at least one player is uncertain about another player s payoff function Cournot competition with incomplete information The market price P is determined by inverse market demand Pa Q if agtQ P0 othenNise Each firm decides on the quantity to sell market share q1 and q2 Q q1q2 total market demand Both firms seek to maximize profits The marginal cost of producing each unit of the good c1 and c2 c1 is common knowledge however c2 is known only by firm 2 Firm 1 believes that c2 is high cH with probability p and Iow cL with probability 1p gt Firm 1 s belief about firm 2 s cost is common knowledge v V Cournot Competition with Incomplete Information Best response of Firm 2 Suppose rm 1 produces q1 Firm 2 s pro ts if it produces q2 are 732 P39C2CI2 a39CI1 QZJQ2 C2q2 Residual revenue Cost First order conditions d nZdq2 a 2q2 q1 c2 RMR MC 0 gt Clza39C239 CID2 If Firm 2 s type is high qH2acH q12 If Firm 2 s type is low qL2acL q12 Cournot Competition with incomplete information Best response of Firm 1 Firm 1 s expected profits if it produces q1 are 1n PC1q1 piaq1 qH2q1 139pa39CI1 qL2CI139 C1C1 First order conditions FOC d nldq1pa qu qHz 1p a qu qL2 C1 O gt q1 pa C1 qH22 1pa clqL22 Cournot with complete information q1a cl q22 Recap Last class January 8 2004 Course goals and objectives Examples of strategic business interactions Introduction to game theory What is it A collection of tools for predicting outcomes for a group of interacting agents where an action of a single agent directly effects the payoffs of the other participating agents Elements of a game Normal form game Assumptions risk neutrality rationality common knowledge Classroom game Today January 13 2004 Dominant and dominated actions Best response Nash equilibrium 5 Example NBC vs ABC NBC News Sitcom U News EDA 45 20 4amp 10 2 Sitcom 50 50 45 FiUo Regardless of ABC s action NBC chooses sitcom Sitcom is a dominant action for NBC Regardless of NBC s action ABC chooses news News is a dominant action for ABC Si Definition Dominant and dominated actions 0 A particular action 5quot e A is a dominant action for playerz39 if no matter what all other players are playing playing 5quot maximizes zquot s payoff EYE 2 iaquota 39 for all ai 6 Ai o A particular action 5quot e A is a weakly dominated action for playerz39 if no matter what all other players are playing there eXists another action 51quot 6 Ai such that i i aii S iampi aii Example NBC vs ABC NBC News Sitcom U News ioo 45 20 4amp 10 2 Sitcom 50 50 45 FiUo Sitcom is a dominant action for NBC News is a dominant action for ABC NewsSitcom is an equ brum 7 damnant actions Si Definition Equilibrium in dominant actions 1 2 N 0Anoutcomeaa a a isan equilibrium in dominant actions if 5quot 6 Ai is a dominant action for playeri Example Old vs new technology Firm 2 New Current quot39 New 0 0 a a E LT Current a a 0 0 Regardless of Firm 1 s action Firm 2 chooses new technology Dominant action for firm 2 is to choose new technology Example Old vs new technology Firm 2 New Current quot39 New 0 0 a a E LT Current a a 0 0 Regardless of Firm 2 s action Firm 1 chooses new technology Dominant action for rm 2 is to choose new technology Example Old vs new technology Firm 2 New Current v l New 0 0 a a E Ll Current a a 0 0 New technology is a dominant action for both players NewNew is an equiIbnum in dominant actions Classroom game Player 2 H x Y X 8 7 1006 E Y 7 6 65 Is there a dominant action for player 2 For player 1 What is the equilibrium 5 Definition Best response 0 In an N player game the best response function of playerz39 is such that R161 2 argmax iaia i In other words given the actions 614 of players 1 2 il i1 n R161 is the best re action from playerz39 Example Y 00 12 Player 2 X Y 1 o o X N Player 1 Player 1 plays X gt Player 2 s best response R2XX Player 1 plays Y gt Player 2 s best response R2YY Player 2 plays X gt Player 1 s best response R1XX Player 2 plays Y gt Player 1 s best response R1YY What is the dominant action for player 1 Player 2 6 5 Example cont Player 2 X Y 4 00 gtlt LN Player 1 T Y oo 12 There is no equilibrium in dominant actions What is the likely outcome of this game Nash Equilibrium 0 An outcome a 51151251Nis a Nash Equilibrium NE if no player would nd it bene cial to deviate from a strategy provided that all other players do not deviate from their strategies played at the Nash extreme 7ri5tquot5fquot27ri aidquot for all at 6 At Finding Nash Equilibrium Calculate the best response functions of every player Check which outcomes lie if any on the best response function of every player Example Player 2 X Al Player 1 00 The outcomes XX and YY are Nash equilibrium 5 Observation What is the relationship between an equilibrium in dominant actions and a NE An equilibrium in dominant actions is a NE but the converse is not necessarily true There will be only one equilibrium in dominant actions while there can be multiple NEs Example Player 2 Player 1 There are neither equilibrium in dominant actions nor pure strategy Nash equilibrium Example Mixed strategy equilibrium Player 2 X Y v l X 20 02 I gt 2 LL Y o 2 2 o Player 2 Play X with probability 05 and Y with probability 05 Player 1 expected payoff if plays X 0520501 Player 1 expected payoff if plays Y 0500521 Player 1 is indifferent between playing X and Y Player 1 Play X with probability 05 and Y with probability 05 Player 2 is indifferent between playing X and Y Neither player has an incentive to deviate from the strategy of randomly selecting between X and Y Example Mixed strategies Player 2 H Y 0 L 1 4 1 1 1 o m E R I I I Are there any dominant or dominated pure strategies for player 2 Player 2 mixed strategy Play X with probability 37 Z with probability 47 Player 1 s expected payoff is 1 whether he plays L or R Expected payoff of player 2 Player 1 plays L 374470127 Player 1 plays R 370473127 The mixed strategy dominates the pure strategy Y 10 Existence of mixed strategy equilibrium Theorem Every finite normalform game has a mixed strategy equilibrium inding the mixed strategy equilibrium 5F Bob Ballet Football 8 Ballet 3 1 0 0 2 Football 0 0 1 g Alice q1q Bob p1p Alice s expected payoff If Alice chooses ballet 2p If Alice chooses football 1 p 2p1 p gt p13 gt Bob chooses football with probability 23 Bob s expected payoff If Bob chooses ballet q If Bob chooses football21 q q21 q gt q23 gt Alice chooses ballet with probability 23 Two pure strategy and one mixed strategy equilibria Comparison of outcomes If there are multiple equilibria are some of them preferable for all players Definition Pareto dominance o The outcome 0 Pareto dominates the outcome at if for every playeri 72quot a 2 72quot a and there exists at least one player j where gt a 0 An outcome a is called Pareto efficient if there does not eX1st any outcome wh1ch Pareto dom1nates outcome at Example Prisoner s dilemma Prisoner 2 t C cooperate D defect g C 1 1 3 0 o 2 n D 0 3 2 2 1 1 Paretodominates 22 The rest are Pareto noncomparable Even though 1 1 Paretodominates 22 22 is the unique Nash equilibrium g All the examples we looked at so far had discrete action sets Next Examples of games with continuous action sets Lu Example A scarce manufacturing resource is required by two departments A and B Yj20 quantity of the resource used by department j Payoff to departmentj from using one unit of the resource ZOOYAYB2 Department j s maximization problem Max Yj200YjYJ2 From FOC 200YjYj22YjYjYj 2003Yj24YjYjYj20 Example cont 200 3YA2 myB YBZ o 1 200 3YB2 myB YAZ o 2 212 ZYA2 ZYB2 0 gt YAYB From 1 200 3YA2 4YA2 YA2 0 gt 8YA2 200 gt YA YB 5 gt Payoff for department j 500 A solution with higher payoff YA YB 4 gt Payoff for department j 544 Twostep procurement Retailer orders Retailer orders quantity additional quantity if i necissary T PERIOD 1 T PERIOD Z T New information Retailer sells may arrive the product in the market 1 Model Definition Single supplier S Unit cost 030 Wholesale price in periods 1 and 2 W1 and w2 Single retailer R n Unit cost cR0 Linear inverse demand function Order quantity in periods 1 and 2 PQabQ q l and q2 Both the retailer and the supplier are profit maximizers All the information is common knowledge 5 PrincipalAgent Problem Dates Covered February 19 28 2002 Suppose a restaurant owner the principal hires a waiter the agent to run the restaurant in his or her absence If the waiter does not work hard or 7 shirks fewer customers will come and the owner will lose potential revenue If the waiter does work hard more customers will come and the owner will absorb greater revenue than otherwise However this additional revenue is as a direct result of the waiter s effort who depending on the terms of employment could potentially reap no bene ts from the increased effort Thus the restauranteur will want to design a contract and offer it to the agent in the hopes that it will provide incentive to the waiter to work hard such that both parties will mutually bene t The preceding is a speci c example of the principalagent problem which can generically be described in the following manner The principal designs the terms of the contract and offers it to the agent The agent decides whether or not to accept the contract Should the agent accept the contract he or she decides on the level of effort that will be exerted The rm s revenue is observed and the principal pays the agent based on the terms agreed upon in the contract Example 1 Let s rst look at the problem from the agent s perspective We will de ne e the level of effort exerted by the waiter as 7 0 if shirhs 6 T 2 if workshard In addition we de ne the agent s utility as U 7 w 7 e if devotese 7 10 if rejectscontract where w is the wage offered in the contract Thus the agent will either 7 take it or leave it From the principal s perspective we de ne Re as the revenue for the restaurant as a function of the waiter s effort H ife2 R5 L ife0 The potential pro t for the principal is 7T Re 7 w Of course the principal will design the contract with the objective of maximizing 7139 We want to de ne wH as the wage rate the principal promises to pay when the effort is high and wL as the wage rate that the agent will receive if the effort is low We will nish this example during the next lecture End of 2 19 Lecture Start of 2 21 Lecture 46 Now we will nish the example we started during the previous lecture The principal will design a contract offer two wage rates one wH when the agent exerts high effort and another wL when the agent exerts low effort First the principal will design the participation contraint wH 7 2 2 10 to ensure that the agent participates agrees to accept the contract This constraint is designed such that the difference between the high wage rate and maximal effort is greater than or equal to any other alternative that the waiter has In short even if the agent has to exert maximal effort it would be in his or her best interest to accept the principal s contract offer In addition to wanting the agent to accept the contract the principal also wants the agent to agree to work hard Recall our assumption from the previous lecture that pro ts generated when the agent works hard 7TH are greater than pro ts generated when the agent 7 shirks 711 since customers are more likely to be repeat visitors when there is high customer service Thus the principal will also construct an incentive constraint wH 7 2 2 wL which gives the agent the incentive to work hard This constraint is designed such that the difference between the high wage rate and the maximal effort is greater than or equal to the low wage rate In short the high wage rate is set such that the agent is better off by accepting it and working hard as opposed to accepting the low wage rate and shirking The desired combined effect of these two contraints is to produce a contract such that the agent will both accept the principal s offer and agree to work hard In this example the participation and incentive constraints generate a high wage rate wH of 12 and a low wage rate wL of 10 The corresponding pro t functions are 7THHUIHH712 77LL7wLL710 where H and L are the revenues associated with the agent working hard and 7 shirking respectively The preceding principal agent problem also illustrates a concept called moral hazard A moral hazard occurs when the agent takes a decision or action that affects his or her utility as well as the principal s the principal only observes the outcome an imperfect signal of the action taken and the agent given a spontaneous choice does not necessarily choose Pareto optimality Since the action is unobservable the principal cannot force the agent to choose an action that is Pareto optimal He or she can only in uence the choice of an action by conditioning the agent s utility to the outcome which is the only observable variable In our example the principal does not observe whether or not the agent is working hard or 7 shirking action The only thing that the principal observes is the revenue generated as a result of the agent s effort Thus it is possible that the agent will accept the high wage agree to work hard but 7 shirk The principal will not know this until he or she observes the revenue gures and calculates pro t the outcome Hence there is a 7 moral hazard 47 associated with these sorts of contracts Moral hazards essentially occur in all economic relationships Some potential complications to the principal agent problem include asymettric informa tion risk differences and revenue uncertainty We will now look at an example that has the latter complication Example 2 Suppose we have the same parameters as in Example 1 only the revenues H and L are uncertain 7 as indicated by the following 7 states of nature Ra 7 H with probability 08 7 L with probability 02 Rm 7 H with probability 04 T L with probability 06 Here the agent affects the probability of each realization of Re depending on the level of effort chosen As a result the utility function of the agent must be modi ed and is now 10 if rejects contract U Ew 7 e if devotes e The wage is now an expected value Ew because of the uncertainties associated with the revenue function Based on the revenue functions 7 08wH 02wL when 6 2 w 7 04wH 06wL when 6 0 E Thus the new participation constraint is 08wH 02wL 7 2 2 10 and the new incentive constraint is 08wH 02wL 7 2 2 04wH 06wL Rewriting the participation constraint in terms of wL yields wL 60 7 4wH and rewriting the incentive constraint in terms of wL yields Solving this as a system of equations yields wL 8 and then using back substitution we obtain wH 13 The uncertainty results in the agent incurring more risk which in turn generates a higher wH value Example 1 wH 12 For homework model this as an extensive form game and solve SPE 48 5 1 Risk Aversion Suppose there are two consumers 239 and j Consumerz39 is more risk averse than consumer j when cansumerj prefers a xed sum 0f money over a lattery then consumer 239 also prefers the xed amaunt This is explained using the following example Example 3 Suppose that the principal and agent think about risk differently One way to model this is the use of 7 subjective probability that measures the probabilities that each player assigns to the realization of the states of nature Suppose the principal has the same revenue functions as above RPQ 7 H with probability 08 7 L with probability 02 and RP0 7 H with probability 04 7 L with probability 06 The agent however has a different belief about how much revenue will be generated as a result of his or her effort The agent believes that the principal s revenue functions will be as follows RAQ 7 H with probability 07 7 L with probability 03 RA0 RP0 In this case the agent is more risk averse since he or she is more skeptical about the realization of the 7 high state of nature In fact it is typically assumed that the principal is 7 risk neutral ie less risk averse and the agent is risk averse Thus in this case the participation constraint will be 07M 03wL 7 2 2 10 a wH and the incentive constraint will be 07M 03wL 7 2 2 04M 06wL a wH 23 wL The objective of the principal is to minimize expected wage if H 7 L is suf ciently large mianWL E0w 08wH 02wL which yields the optimum values wH 14 and wL The principal s expected wage will exceed the agent s reservation utility plus effort E0w 08wH 02wL 1266 gt 10 12 12 Plotting all of this yields the following graphic Feasible Region no 60 22 125 7 39 50 W W 3 0 95 92 b4 9 C121 0 Figure 21 Risk Averse Principal Agent Example The following list of items is a general summary of a principal agent model 1 The agent chooses between H possible actions 11 an 2 These actions produce m possible outcomes x1 vm The outcome is a signal that gives the principal information on the action that the agent chooses 3 There is a stochastic relationship between actions and outcomes called a 7 technol ogy such that when the agent chooses action 11 the principal observes outcomes W with probability pij 4 1f the principal observes outcome mj he or she will pay the agent a wage wj and keep the remaining CW 7 wj 5 Assuming the agent s utility Uw 7 a is separable U is increasing and concave 6 Assuming the principal is risk neutral his or her utility is In 7 w First we look at the agent s problem When the principal offers a contract wj the agent chooses his or her best action by solving maxi71an1 PijUWj ail If the agent chooses 11 this gives n 7 1 incentive compatibility constraints ICk 2 1PijUwj Ii 2 Zglpijo j ak Vk 739 239 Suppose Q is the utility for the agent if he or she does not accept the contract Then the participation constraint known in this case as an individual rationality constraint HR is 2 PijUWj Ii 2 2 Now we look at the principal s problem The principal wishes to choose a contract wl wm that maximizes the expected utility while taking into account the dependence of the contract on the agent s decisions Thus the objective function is man1wmi 2 Pijo wj7 subject to Ck and IRW from the agent s problem End of 2 21 Lecture Start of 2 26 Lecture If we x ai the Lagrangian of this maximization problem is Lw7 7 M 2 H39jo wj Zkgm AMZJTZl PijUWj M 2 1 ijUWj MN12 1 PijUWj az39 Q 3 Taking the rst order condition with respect to wj we get 5 Pz39j ZZ1kiAkPijUiwj ijU UIj MPz39jU Wj E 0 7 P a may M 221k7 i Ml P which makes wj as a function ofj depend on the ratio 1 In equilibrium at least one of the k values must be positive otherwise we could neglect the incentive constraints and the moral hazard problem would vanish Thus if the agent has only two possible actions as in the previous examples our expression simpli es to P WMAl p ijl Note that in statistics an expression of the form i Zj is known as a likelihood ratiowhz39ch 7 measures the likelihaad 0f abserm39ng 2 given that the agent chaases a to the likelihaad 0f abserm39ng 2 given that the agent chooses 17 Thus the higher the value of the ratio the more likely the agent will choose a 7 and vice versa Recall that we said Uw is increasing and concave This indicates that the ratio U 1m is 7 increasing and that wj will tend to be higher as the ratio increases This is illustrated by the following graphic UW Uw Figure 22 As likelihood ratio deceases left hand side increases in wj Since this is a monotone increasing function of IQ the optimal incentive scheme is a linear function of the likelihood ratio However if the number of actions n gt 2 the argument will not necessarily hold since wage depends on a weighted average of likelihood ratios It should be noted however that the intuition will hold in most cases In order to make the argument more concrete we need to assume that the cumulative distribution function cdf is convex on actions 11 an That is for 239 lt j lt k and 6 01 13 ai 1 7 Mak This condition gives le S Pil 1 7 APkl Vl 1m An example of how to construct an incentive scheme for two actions and two outcomes is as follows w1w wg wsx2 7x1s S 1 Here the agent receives a base wage w and a bonus proportional to the increase in surplus when he or she suceeded in satisfying the principal s demands Using the Lagrangian statistics mentioned earlier we can determine how the principal s optimizaed values change with respect to the agent s effort in the two action two outcome case The Lagrangian rst order conditions with respect to the agent s costs effort yield 6L 6111 7 5 A M7 based on the Envelope Theorem which was outlined extensively in lSyE 6229 This helps answer the question 7 Which is better the 7 carrot or the 7 stick The 7 carrot provides incentive to the agent to perform the action the principal wants by decreasing the cost effort of the chosen action a2 The 7 stick punishes the agent for not taking the action desired by the principal by increasing the cost of the alternative action a1 by the same magnitude As evidenced by the rst order conditions the 7 carrot relaxes two constraints IC and IR while the stick only relaxes one 1C Hence a small decrease in the cost of the chosen action always increases the principal s utiity by a larger amount than an increase of the same magnitude in cost of the alternative action In other words the carmt is always better than the stz39ck Suppose now we have a change in the probability distribution dPaj The effect on the principal s utility of such a change is dL 7A dPaJUwj Thus when the C constraint is binding A gt 0 the interests of the principal and agent are diametrically opposed with respect to changes in the probability distribution In other words any action that helps the agent unambiguously hurts the principal 52 Multiple Agents We will conclude our discussion on the Principal Agent problem with a look at the case of multiple agents We will do so by looking at the following example Example 4 Consider an order picking operation in a warehouse with N pickers The value V to the warehouse depends on the effort levels of the N pickers V 5 i39 To simplify the analysis assume that Ztil wi V7 where wi is the wage for picker 239 Furthermore assume all pickers have the same utility Uiwi7eii1N 53 Games of Incomplete Information Bayesian Games In a game of complete information the players payoff functions are common knowledge In a game of incomplete information at least one player is uncertain about another player s payoff function Cournot competition with incomplete information The market price P is determined by inverse market demand Pa Q if agtQ P0 othenNise Each firm decides on the quantity to sell market share q1 and q2 Q q1q2 total market demand Both firms seek to maximize profits The marginal cost of producing each unit of the good c1 and c2 c1 is common knowledge however c2 is known only by firm 2 Firm 1 believes that c2 is high cH with probability p and Iow cL with probability 1p gt Firm 1 s belief about firm 2 s cost is common knowledge v V Cournot Competition with Incomplete Information Best response of Firm 2 Suppose rm 1 produces q1 Firm 2 s pro ts if it produces q2 are 732 P39C2CI2 a39CI1 QZJQ2 C2q2 Residual revenue Cost First order conditions d nZdq2 a 2q2 q1 c2 RMR MC 0 gt Clza39C239 CID2 If Firm 2 s type is high qH2acH q12 If Firm 2 s type is low qL2acL q12 Cournot Competition with incomplete information Best response of Firm 1 Firm 1 s expected profits if it produces q1 are 1n PC1q1 piaq1 qH2q1 139pa39CI1 qL2CI139 C1C1 First order conditions FOC d nldq1pa qu qHz 1p a qu qL2 C1 O gt q1 pa C1 qH22 1pa clqL22 Cournot with complete information q1a cl q22 S Administrivia Homework 2 posted on the course page Exam 1 February 11 Closed book closed notes You are allowed to use One page of nets singlesided Basic calculators Due next Thursday Jan 6 at the beginning ofclass 5 Repeated Cournot game Trigger strategy Produce halfthe monopoly quantity qMZ in stage In e tm stage produce qMZ if both have produced qMZ in all previous stags otherwise produce qc the rst rms Playing the trigger strategy is SPNE iff 5 2 917 What if 5 lt 917 Repeated Cournot Game cont Best response of rm i If the last stage outcome is other than q qquot Play qC forever Ifall previous stags39 outcomes Deviate 5 1 Play q are q q vi rt 5 vi gt vi rt 1 5 Playing the trigger strategy is NE iff m1 a 211D nca 1 5 Substitute and solve for qquot qquot 955ac395 Recall qCac3 qM acZ Recap Last class 12803 In nitely repeated Prisoner39s Dilemma Feasible payoffs Friedman39s theorem In nitely repeated Cournot game Trigger strategies Today13003 In nitely repeated Cournot game Example wagesetting Extensive form representation g Repeated Cournot Game cont Trigger strategy Produce q in the rst stage In the tm sta e produce qquot if both rms have produced qquot in all previous stages otherwise produce qc Profit of one rm If both produce qquot a2qlt q 75 If both produce qC ac29 75 If rm j produces qand rm i deviates max 339 Cir CI K39C Cli Cli 339 CI K39C 2 79 a qc24 Example Wage setting Stage game One rm one worker The rm offers the worker a wage w The worker accept or reject the frm39s offer Reject the worker becomes selfemployed at wage wEl Accept Work disutility e or Shirk disutility o rker works ilpplies effort Output is high y If the worker sh39rks Output is high with probability p and low0 with probability 17p The rm does not obsene the worker39s effort decision The output of the worker is obsened by both parties Example Wage setting cont Payoffs FirmWorker Work Supply effort High output ywwe Shirk High output yw w Low output w w What is the subgameperfect equilibrium in this stage game For any w 2 we worker accepts employment and s Irks Firm offers w0 or any other wltw0 Example Wage setting cont Strategies Firm Offer ww in the rst stage In stage t offer ww if the history of play is highWage Ivy7 outputall previous offers have been w all previous have been accepted and all previous outpuls otherwise offer w0 Worker I the history of play including the current offer is high wage highoutput shirk otherwise If wltwm choose self employment Example Wage setting cont Suppose rm offers w 2 wO Worker accepts Work Supply effort we 5 V6 gtVe we1B Shirk vs w 6pvs 1p wan6 vs 16w 61pwO1sp1s Worker should supply effort if vezvs gt w 2 wO e e1BB1p If p0 we1 B 2 w wOBl B Example Wage setting cont When is it the best response for the rm to offer w7 From worker39s best response w 2 wO e e1BB1p 1 yzwquot gt y 2 wO e e1BB1p 2 The strategies induce a NE if 1 and 2 hold Is this a SPNE 5 Example Wage setting cont What are the subgames Subgames beginning a er a highwage highoutput history Subgames beginning after all other histories 9 Extensive form of a game The set of players The order of moves The players payoffs as a function of the moves that were ma e The set of actions available to the players when they move Each player s information when he makes his move The probability distributions over any exogenous events Nature Example 1 1 21 00 711 32 Player 1 action set UD Player 2 action setLR Player 1 strategis UD Player 2 strategies LL LR RL RR P ayer 1 moves first After observing player 139s action player 2 moves Normal form representation of extensi orm games Player 2 Player 2 s strategies correspond to a contingent plan made in advance Example 2 1 21 010 711 32 21 11 00 32 Player 1 moves first player 2 Player 2 moves first player 1 moves next Player 1 does not know player 239s action when he chooses his action he chooses his action Example Player 1 chooses an action from the feasible set LR Player 2 obsenes player 1 s action and then chooses an action from the feasible set L R Player 3 obsenes whether or not the histow of actions is RR39 and then chooses an action from the feasible set LquotR Example cont Information set An information set for a player is a collection of decision nodes satisfying The player has the move at every node in the set information When the play of the game reachs a node in the information set the player with the move does not know which node in the informaan set has or has not been reached El Recap Last class January 20 2004 Duopoly models Multistage games with observed actions Subgame perfect equilibrium Extensive form of a game Two stage prisoner s dilemma Today January 22 2004 Finitely repeated games Infinitely repeated games Prisoner s dilemma Friedman s Theorem Repeated Cournot game Repeated games 0 Let G A1Aquot 72391 7239 denote a static game of completeinformation in which players 1 n simulateneously choose their actions a1 aquot from action spaces A1Aquot and receive payoffs 7r1a1aquot 7139 a1 aquotWe call G the stage game of the repeated game 0 Given a stage game G let GT denote the nitely repeated game in which G is played T times with the outcomes of all preceding plays observed before the next play begins The payoffs for GT are the discounted sum of the payoffs from the T stage games 5 Repeated games Result If the stage game G has a unique Nash equilibrium then for any nite T the repeated game GT has a unique subgameperfect outcome the Nash equilibrium of G is played in every stage Example Player 2 M L R H L 11 30 00 E M 05 44 00 R 00 00 33 The stage game is played twice The firststage outcome is observed before the second stage begins El Example Player 2 M L R H L 11 30 00 E M 05 44 00 R 00 00 33 Partial strategy for stage 2 Play R in stage 2 if stage 1 outcome is MM otherwise play L in stage 2 Example Player 2 L M R H L 12 11 E M 16 11 11 R 11 11 44 Modi ed stage 1 game Subgame perfect equilibria LLLL MMRR RRLL El Observation Let G be a static game of complete information with multiple Nash equilibria There may be subgameperfect outcomes of the repeated game GT in which for any tlt T the outcome in stage t is not a Nash equilibrium of G Definitions In the nitely repeated game GI39 a player s strategyspecifies the player s actions in each stage for each possible history of play through the previous stages In the nitely repeated game GT a subgame beginning at stage t1 is the repeated game in which G is played Tt times denoted by Grt 5 Example Player 2 M L R H L 11 30 00 E M 05 44 00 R 00 00 33 All possible outcomes histories at the end of stage 1 LL LM LR ML MM MR RL RM RR M L L L L R L L L L Play M in the first stage Play L in the second stage unless the first stage outcome is MM Si Infinitely Repeated Prisoner s Dilemma PrisonerZ C cooperate D defect g C 4 4 0 5 o 9 0 D 5 0 1 1 The game is repeated infinitely For each t the outcomes of the previous t1 stage games are observed Payoffs 5 Discounted payoffs Let 8 be the value today of a dollar to be received one stage later Eg 511r where r is the interest rate per stage Given the discount factor 8 the present value of the in nite sequence of payoffs 11 n2 n3 IS n1 81t2 821t3 Zhbw Smut Discounted payoffs Suppose after each stage is played the game continues to the next stage with probability 1p and stops with probability p Expected present value of next stage s payoff lpn1r Expected present value of the payoff two stages later lp2n1r2 Let 8 1p1r n1 8n2 82n3 reflects the time value of money and the possibility that the game will end 5 Average payoffs l V n1 81t2 821t3 Zhbw Smut If we received an average payoff of 1t in every stage then V n 8n 82n n1 8 82 nl 8 nl 8 Zhbw Smut 1 1 8 Zhbw Smut Example Payoffs 4 4 4 4 4 Average payoff 4 Net present value 4 1 8 Infinitely repeated games 0 Given a stage game G let Gltgtltgt 6 denote the in nitely repeated game in which G is repeated forever and the players share discount factor 6 For each I the outcomes of the t l preceding plays of the stage game are observed before the t stage begins Each player s payoff in Gltgtltgt 6 is the present value of the player s payoffs from the infinite sequence of stage games SI Infinitely Repeated Prisoner s Dilemma Prisoner 2 t C cooperate D defect g C 4 4 0 5 o 2 0 D 5 0 1 1 Strategy Play C in the rst stage In the tth stage if the outcome of all t1 preceding stages has been CC then play C otherwise play D Definitions In an infinitely repeated game Goltgt8 a player s strategyspecifies the player s actions in each stage for each possible history of play through the previous stages In the infinitely repeated game Goltgt8 each subgame beginning at stage t1 is is identical to the original game Goltgt8 Trigger strategies for Prisoner s Dilemma Assuming player 1 adopts the trigger strategy what is the best response of player 2 Player 2 best response in stage t1 If the outcome in stage t is DD Play D forever If the outcomes of stages 1t are CC Play D gt receive 5 in this stage switch to DD forever after gt 5 51 521 5315 81 5 Play C gt receive 4 in this stage and face the exact same game same choices in stage t2 Si Trigger strategies for Prisoner s Dilemma Let V be the payoff of player 2 from making the optimal choice in the subgame starting in stage t 1 given that the outcomes in the previous stages have been CC Play C gt V4 8V gt V 41 8 Play D gt V 5 81 8 Play c if 41 5 2 5 51 5 gt if 5 2 14 SI Trigger strategies for Prisoner s Dilemma Two types of subgames i Subgames where the outcomes of all previous stages have been CC The trigger strategies are Nash equilibrium for this class of subgames as well as for the original game ii Subgames where the outcome of at least one earlier stage differs from CC Player s strategies are to repeat DD forever which is also a Nash equilibrium for the original game Observation Even if the stage game G has a unique Nash equilibrium there may be subgameperfect outcomes of the in nitely repeated game in which no stage s outcome is a Nash equilibrium of G 10 Feasible payo s in the stage game The payoffs n1 n2 1t are fea5be in the stage game G if they are a convex combination of the purestrategy payoffs of G Example PrisonerZ C cooperate D defect g C 4 4 0 5 o 9 0 D 5 0 1 1 What are the purestrategy payoffs 39 44 11 g Feasible payoffs in the Prisoner s Dilemma Player 2 payoffs 50 Player 1 payoffs Recap Last class January 22 2004 Finitely repeated games In nitely repeated games Prisoner39s dilemma Friedman39s Theorem Today January 27 2004 Proof of Friedman39s theorem Repeated Oournot game Wage setting g Friedman s Theorem Let G be a finite static game of complete information Let e1 e2 equot denote the payoffs from a Nash equilibrium of G and let x1 x2 xquot denote any other feasible payoffs from G If xJ39gt ej for every playerj and if 5 is sufficiently close to 1 then there exists a subgameperfect Nash equilibrium of the infinitely repeated game Goltgt5 that achieves x1 x2 xquot as the average payoff l Feasible payoffs in the Prisoner s Dilemma Player 2 payoffs 05 lll iiii 44 W60 Elaavoersl Proof of Friedman s Theorem Let ael aez aen be the Nash equilibrium of G that yields the equilibrium payoffs e1 e2 equot Let axl axz ax be the collection of actions that yields the equilibrium payoffs x1 x2 xquot Trigger strategy for player i Play axi in the first stage In the tth stage if the outcome of all t1 preceding stages has been axl axz am then play axi otherwise play aei Show that the trigger strategies induce a NE Show that the equilibrium is subgame perfect Proof of Friedman s Theorem cont Suppose all players other than player i use the trigger strategy Best response of player i in stage t If the outcome of the previous stage differs from axll agtlt2I I axn Play aei forever If the outcomes of all previous stages are axl axz I axn i I maX 39 anaxHaiax7i1am d LEE1 i i i d 2 7139 awavaaxiaxvi1am gt 7139 aelam e l A Proof of Friedman s Theorem cont If player i deviates in stage t by choosing adi Payoff in stage t di Payoff in future stages 5ei 52ei Sei 1 5 Total discounted payoff Vi di 5 ei 1 5 If player i plays axi in stage t Receive a payoff xi in this stage face the same game in the next stage Vi xi 8 Vi gt Vi xi 1 5 Playing xi is optimal if and only if Xi 1 82 di 8 ei 1 8 gt 8 2 di Xidi ei Proof of Friedman s Theorem cont It is Nash equilibrium for all players to play the trigger strategy if and only if 8 2 maxi di xidi ei Subgame perfectness If the outcome of the previous stage differs from axll agtlt2I I axn Play aei forever If the outcomes of all previous stages are axl axz I axn Play the trigger strategy 5 Repeated Cournot Game Cournot stage game Two competing firms selling a homogeneous good The marginal cost of producing each unit of the good c The market price P is determined by inverse market demand PaQ if agtQ P0 othenNise Each firm decides on the quantity to sell market share q1 and q2 Q q1q2 total market demand Both firms seek to maximize profits Unique NE of the stage game qCac3 Q 2ac3 Monopoly quantity qMac2 Repeated Cournot Game cont The stage game is repeated in nitely many times The rms have discount factor 8 Trigger strategy Produce half the monopoly quantity qMZ in the first stage In the tth stage produce qMZ if both rms have produced qMZ in all previous stages otherwise produce qC Show that the trigger strategy induces a subgame perfect NE 16 Repeated Cournot Game cont Profit of one firm If both produce qMZ a c28 rcMZ If both produce qc a c29 rcc Best response of firm i If the last stage outcome is other than qMZ qMZ Play qc forever If all previous stages outcomes are qMZ qMZ Deviate max a qi qMZ c qi gt qi 3a c8 n0 9a c264 n0 nCSl 5 Play qMZ rcMZ 5W gtVi nM215 Repeated Cournot Game cont Pro t of one rm If both produce qMZ ac28 rcMZ If both produce qC ac29 TEC Best response of firm i If the last stage outcome is other than qMZ qMZ Play qC forever If all previous stages outcomes are qMZ qMZ Deviate Vi 15D TEC5 1 5 Play qMZ Vi rcM 21 5 Playing the trigger strategy is NE iff nM 21 5 2 70 THE 1 8 gt 8 2 917 Repeated Cournot game Trigger strategy Produce half the monopoly quantity qMZ in the first stage In the tth stage produce qMZ if both rms have produced qMZ in all previous stages otherwise produce qC Playing the trigger strategy is SPNE iff 8 2 917 What if 8 lt 917 Repeated Cournot Game cont Trigger strategy Produce qquot in the rst stage In the tth stage produce qquot if both rms have produced qquot in all previous stages otherwise produce qC Pro t of one rm If both produce qquot a2qc q nquot If both produce qC ac29 nC If rmj produces qquot and rm i deviates max a qr qc q gt qi a qc2 rcD a qc24 5 Repeated Cournot Game cont Best response of firm i If the last stage outcome is other than q q Play qC forever If all previous stages outcomes are q q Deviate Vi 15D TEC5 1 5 Play q Vi Tr 5Vi gt Vi r1 5 Playing the trigger strategy is NE iff 1t1 8 2 nD ms 1 8 Substitute and solve for q q 955aC395 Recall qCac3 qMac2 Example Wage setting Stage game One firm one worker The firm offers the worker a wage w The worker accepts or rejects the rm s offer Reject the worker becomes selfemployed at wage w0 Accept Work disutility e or Shirk disutility 0 u If the worker works supplies effort Output is highy If the worker shirks Output is high with probability p and low0 with probability 1 p The firm does not observe the worker s effort decision The output of the worker is observed by both parties Example Wage setting cont Payoffs FirmWorker Work Supply effort High output ywwe Shirk High output yw w Low output w w What is the subgameperfect equilibrium in this stage game For any w 2 wo worker accepts employment and shirks Firm offers w0 or any other wltw0 Example Wage setting cont Strategies Firm Offer ww in the first stage In stage t offer ww if the history of play is highwage high output all previous offers have been w all previous offers have been accepted and all previous outputs have been high otherwise offer w0 Worker If wgtw0 accept the firm s offer and supply effort if the history of play including the current offer is high wage highoutput shirk otherwise If wltw0 choose self empoyment Example Wage setting cont Suppose firm offers w 2 w0 Worker accepts Work Supply effort Vewe 8 Ve gt Ve we15 Shirk vs w 6pvs lpwO16 2 Vs 15W 51P Wo1 5p1 5 Worker should supply effort if Ve2VS gt w 2 w0 e e1551p If p0 we1 5 2 w w051 5 Example Wage setting cont When is it the best response for the firm to offer w From worker s best response w 2 w0 e e1861p 1 y2w gt y 2 w0 e e1861p 2 The strategies induce a NE if 1 and 2 hold Is this a SPNE Recap Last Thursday February 13 2003 Results and analysis of the onecard poker game Principalagent problem Moral hazard Contracis with wags conditioned on outcoms Today February 18 2003 Principalagent problem general model PrincipalAgent Problem is a w Uw e ea a 6 Pquot e 3232 223 w2wx2 2 2 Wmwxm WWquot 6 Outcomes Actions Principalagent problem If the agent doesn39t accept the contract his payoff is his rservation utility Q If the agent accepts the contract he chooses between n 39 a possible actions a1 n These actions produce m possible outcoms xm There is a stochastic relationship between actions and outcomes called technologyquot When the action is a the principal observes outcome xJ with probability P If the principal observes outcome x she pays the agent w The agent39s payoff is uwea where U is the utility of wage w o the agent and ea is the cost of action a to the u is increasing differentiable and concave Assuming the principal is riskneutral her payoff is xJ wJ g Principal s problem Step 1 Given an action a how to set the wages such that the agent chooses a and the principal39s payoff is maximized Pa1ticipation constraint 212Uw17 ea 2 Q Incentive constraint 21 13105 7 ea 2 21 PkJUwJ a eak Vk if Principal39s objective minimize the expected cost of inducing the agent to choose 0 maxZJZRJOcJ 7w E minZJZPw Cq U J Principal s problem Step 1 Ca is the minimal cost to the principal of inducing the agent to take action a Ca is convex 9 the original maximization objective is concave Wellbehaved mathematical program with a concave objective function maximization and linear constrain s Principal s problem Step 2 From step 1 m m m maxzsiRro r in E 11ij mmzisi wr Ra Ca What is the action a that maximizes RaCa What if the agent s actions can be SI observed Principal s problem Step 1 The principal can design a contract where the For agiVe aetio 0 1 wages are conditioned on the actions ie wa maxilla iwj max 1112quoter E 11an ZIUwjieazg y P ti 39 ti tr 39 tU 7 2U m m in Pa oncons am WI 901 2112Uwj7eaZZHBVUWJ7eakVkiim A Incentive constraint Ultw7elta2Uwoieltak Wm m m aw 2 2113 x a w 42 1UW a M 7w To induce the agent to choose action ax set wx such that 21742 R U010 9a 2 RC U010 1 ak Uw Q ea and set all other wk suf cientlylow 1 11 5 Principal s problem Step 1 g Principal s problem Step 1 w JIFZLRAXWZIUW2aQ m 1 1 Pk 227ij EJUwj7eaiz PUwj2ak UWJ 7 liyjj is F g 3Lw u77 3UW 3UW7 3WW 7 7 13 BWJ WT 5 BWJ 23 1 BWJ P19 aW J 0 F lkehhoodratio a 3Uw 712 3W amp57ij0 m i 7 7 ii 1 3W y l Rj ioaamw iing kp R1 3w J Two actions and two outcomes Two actions and two outcomes Suppose the agent has two possible actions a and Ince Ve compatibility COHStTai ti b and there are two possible outcomes x1 and x2 airwasp gzyw2eb 2 Suppose that action b is preferred by the principal lUwlRIZUw2iea For action I 2 When is the agent indifferent between a and b maxz Pb x 7w F1 1 f B71Uwl ButW2 eb PailW1 llW2 ea PMUwlI32Uw27eb2g u p 7 eb7ea 7 w eb7ea l Um 1 7 39 W1 i PbiUWiszUW2Teb2 Bszgzz Bszpz BszRzz RnUwlPa2Uw2iea 1 Since Rn 1312 31 372 1 Two actions and two outcomes eb ea Uw2 B 2 Pa 2 Action b is preferred to UWi UWz action a UWi Two actions and two outcomes Participation constraints Action b PblUw1 PbZUw2 8b 2 Q Action 61 PalUw1 PHZUw2 8a 2 Q Action b Uw2 2 iUw1eb Jrg Pm P 172 Two actions and two outcomes eb ea U w1 39 Pig Pu UWi UW2 Incentive compatibility constraint P 81 U UW2 T M UW1 I352 I352 UWl Two actions and two outcomes Principal s objective function For action b maXB71x1 w1B72x2 w2 Let fbe the inveise of U fUw wj maXB71x1 fUw1B72x2 fUw2 Two actions and two outcomes UWz 5 W Uw UW2 Particip ation constraint UWi Two actions and two outcomes maXPbIOG fUW1szx2 fUWz Slope of the objective function afUW1 P afUW2 aUWz BUM BUM BUM P BfUM BUM BUM Pblf39UM 3Uw1 P 3fUwZ szf Uw2 BUM T1351 0 Example Low effort 210 In Low output 10000 In Medium output 20000 High effort 211 In High output 40000 Cost of effort to the agent 10000a Agent s reservation utility 90 Agent s expected payoff Ewage cost of effort Principal s expected payoff Eoutput wage Example cont Under low effort Agent s cost0 9 Offer wg0 to the agent Principal s expected payoff 05100000520000 015000 Under high effort Principal s expected revenue O520000054000030000 Principal s expected payoff Erevenue wage For participation Ewage 2 gcost of effort10000 For the principal to prefer high effort over low Ewage lt 15000 5 Example Bonus payment Principal s proposed contract If the output is low or medium w0 If the output is high w24000 Participation constraint for high effort 0500524000100002000gt0 Incentive constraint 0500524000100002000gt0500500 Principal s payoff 18000 Agent s payoff 2000 What is the best w that maximizes the principal s payoff 050O5w1000020 9 w20000 Principal s payoff 20000 Agent s payoff 0 Example Revenue sharing Principal s proposed contract If the output is less than x18000 w0 If the output R 2 x wRx Participation constraint 052000018000054000018000 100002000gt0 Incentive constraint 052000018000054000018000 100002000gt 0500520000180001000 Principal s payoff 18000 Agent s payoff 2000 What is the best x that maximizes the principal s payoff Example Revenue sharing cont xgt20000 Participation constraint is not satisfied 0500540000X 10000 1000005Xlt0 if Xgt20000 10000ltxSZ0000 Participation 0520000 x0540000 x 10000 20000 x 2 0 Incentive 0520000 x0540000 x 10000 2 0500520000X x320000 9 To maximize principal s payoff x20000 Principal s payoff 20000 Agent s payoff 0 X S 10000 Participation 0520000 x0540000 x 10000 20000 x 2 0 Incentive 0520000 x0540000 x 10000 2 0510000 x0520000 x To maximize principal s payoff x10000 Principal s payoff 10000 Agent s payoff 10000 Optimal X 20000 Piecerate system Principal s proposed contract wocR if the output is R Participation constraint for high effort 05a2000005x400001000030000x10000 2 O 9 0213 Incentive constraint 05a2000005x400001000030000x10000 2 05x1000005x20000 9 0223 To maximize the principal s payoff x23 Principal s payoff 10000 Agent s payoff 10000 Is it optimal for the principal to induce high effort under this contract Managerial compensation under Cournot competition Cournot competition The market price P is determined by inverse market demand Pa Q if agtbQ P0 otherwise The marginal costof producing each unit of the good is c Each firm decides on the quantity to sell market share q1 and q2 Q q1q2total market demand Both firms seek to maximize pro ts Revenue of rmiR pqi a Qq Pro t of f1rmi 72 2R cq The owner of f1rmi offers the following compensation to her manager MI u0472 l 0Ri Managerial compensation uncler Cournot competition cont 1 The owner of firm 139 needs to choose ui and 05 to maximize firm profits net of managerial compensation ow max 7 M 2 For given q j and 04 the manager of rm i chooses ql to maximizeMl ul047ri 1 04 Ri M atriaR cqilt1 04gtRiiiiRi o4cqii M iaq qq 0cq m il2 2 FOC a Zqi qj0400 gt qr Managerial compensation under Cournot competition cont In equilibrium aac 2ac qia1az i12 Qa a a0lzc 2010a010 2020 201 040 0120 1 2 I 3 3 3 Pa1azw 3 Managerial compensation uncler Cournot competition cont Stage 1 Owners choose u and a How to choose ui To simplifyassume ul Mi 0 Owner 5 objective Owner 77quot pa1a2cqi ac04 01 3 ac0j 204 3 3 60 01 005 From FOC 04 1 4c PrincipalAgent Problem Principalagent examples Restaurant owner waiter Software company salesman Auto manufacturer customer leasing a car Insurance company insured Example The principal offers wage w If the agent accepts the offer Agent can put high e25 or low e0 effort Agent s utility Uwewe Agent s reservation level of utility 81 Principal s payoff 270 if the agent works hard 70 if the agent doesn t work hard Firstbest contract The agent won t accept the job unless the wage exceeds his reservation utility I Employing this agent is worthwhile to the principal only if the agent works hard otherwise the principal only gets 70 For the agent to work hard his utility from working hard should exceed his reservation utility Uwe 2 81 w 25281 aw2106 Firstbest contract offer 106 2 to the agent and trust that he will work hard s Moral hazard Firstbest contract Offer the agent 106s What is the problem with this contract Moral hazard the agent takes a decision or action that affects his or her utility as well as the principal s the principal only observes the outcome as an imperfect signal of the action taken and the agent does not necessarily choose the action in the interest of the principal Alternative Offer a contract where the wage depends on the effort level Contract conditioned on effort level Offer two wage rates wH if the agent exerts high effort wL if the agent exerts low effort How to choose wH so that accepting the offer and working hard is desirable for the agent wH 25 2 81 participation constraint individual rationality constraint wH 25 2 wL incentive constraint to a client The client places no order 0 The client places a small order 100 The client places a large order 400 g Contract conditioned on outcome Suppose the agent is a salesman representing the principal Three possible outcomes based on the effort level of the salesman Probabilities for different outcomes under each effort level No order Small order Large order Expected 0 100 400 order size High effort 01 03 06 270 Low effort 06 03 01 70 39 Contract conditioned on outcome A contract where wage depends on the observable outcome No order gt agent pays the principal 164 Small order gt agent pays the principal 64 Large order gt principal pays agent 236 SI Contract conditioned on outcome Principal s expected revenue if the agent works hard 270 Expected profit 164 How much does the principal s revenue differ from the expected revenue under each outcome No order gt 0270 270 270106 164 Small order gt 100270 170 170106 64 Large order gt 400270 130 130106 236 Principal s pro t No order gt 164 Small order gt 164 Large order gt 164 Si Contract conditioned on outcome Agent s choices and expected payoffs under each choice assuming the agent is riskneutral Reject the contract and get reservation utility 81 Accept the contract and don t work hard O1236O364O6164 0 94 Accept the contract and work hard O6236O3640116425 81 The principal always gets 164 SI Contract with positive wages Suppose the agent only accepts positive wages What are the wages wo ws and wL corresponding to no order small order and large order outcomes that maximize the principal s payoff Participation constraint 06wL03w501w 25281 Incentive constraint 06wL03w501w 252 06 w0 03 w5 01 wL Nonnegativity constraint wO ws wL 2 0 Principal s objective Maximize 06400 wL03100 w5010w Equivalently Minimize 06 wL 03 w5 01 w0 Risk aversion What if the agent is riskaverse A person who prefers to get the expected value of a gamble for sure instead of taking the risky gamble is risk averse Eg getting 25 for sure vs getting 0 with probability 075 and 100 with probability 025 The agent and the principal may have different beliefs about the probabilities of different outcomes under different effort levels Example Risk averse agent Agent s reservation utility 10 Agent s possible actions if accepts the contract work hard e2 don t work hard e0 Two possible outcomes L and H Principal offers wages wL and wH based on the outcome Example Risk averse agent Probabilities of H and L outcomes Agent does not work hard H with probability 04 L with probability 06 Agent works hard Principal s belief H with probability 08 L with probability 02 Agent s belief H with probability 07 u L with probability 03 A Introduction to Game Theory collection of tools for predicting outcomes for a group of interacting agents where an action of a single agent directly effects the payoffs of the other participating agents I What is a game Many types of games board games card games video games field games eg football etc A zerosum game is one in which the players39 interests are in direct conflict eg in football one team wins and the other loses A game is nonzero sum if players interests are not always in direct conflict so that there are opportunities for both to gain We focus on games where There are 2 or more payers There is some choice of action where strategy matters The game has one or more outcomes eg someone wins someone loses The outcome depends on the strategies chosen by all players there is strategic intera din7 What does this rule out Games of pure chance eg lotteries slot machines Strategies don39t atter Games without strategic interaction between players eg Solitaire 5quot Elements of a game The QaKers how many players are there does naturechance play a role A complete description of what the players can do the set of all possible actions A description of the Qagoff consequences for each player for every possible combination of actions chosen by all players playing the game 5quot Example NBC vs ABC NBC News Sitcom News 55 45 52 48 e Sitcom 50 50 45 55 Players NBC and ABC Set of all possible actions sitcom news Payoffs market shares for each outcome I Normal form game Notation o A set of N players 2 1 N 0 Each playerz39 e N has an action set A afaax such that of 6 Al is a particular action by 139 oEach playerihas a payoff 7r a where a a1a2aN is the game outcome 1 2 3971 39l N a a a39 a a denotestheactior1sof i oa all players except i in the outcome I Assumptions Payoffs are known and fixed Players are risk neutral ie maximize expected payoffs Example a risk neutral person is indifferent between 25 for certain or a 25 chance of earning 100 and a 75 chance of earning 0 All players behave rationally They understand and seek to maximize their own payoffs The rules of the game are common knowledge Each player knows the set of players strategies and payoffs from all possible combinations of strategies call this information X Common knowedge means that each player knows that all players know X that all players know that all players know X that all players know that all players know that all players know X and so on ad Ih nl um Classroom game Player 2 H X Y 1 X 8 7 1006 E Y 7 6 6 5 Player1 X9 Y21 PlayerZ X 18 Y 12 Outcomes XX6 XY4 YX 11 YY9 Classroom game Player 2 H X Y 3 X 7 5 6 E Y 7 6 6 Player 1 X18 Y12 PlayerZ X21 Y 9 Outcomes XX14 XY3 YX9 YY4 5quot Example NBC vs ABC NBC News Sitcom News EDA 45 20 we Sitcom 50 50 45 F o Regardless of ABC s action NBC chooses sitcom Sitcom is a dominant action for NBC Regardless of NBC s action ABC chooses news News is a dominant action for ABC Definition Dominant and dominated actions 0 A particular action 5quot 6 Ai is a dominant action for playerz39 if no matter What all other players are playing playing 5quot maximizes 139 s payoff EYE 2 iaia i for all ai e A o A particular action 539 6 All is a weakly dominated action for playerz39 if no matter what all other players are playing there eXists another action 51quot e A such that i ia i E i ia i 5quot Example NBC vs ABC NBC News Sitcom News EDA 45 20 we Sitcom 50 50 45 F o Sitcom is a dominant action for NBC News is a dominant action for ABC NewsSitcom is an eqUYbrum 7 damnant actions Definition Equilibrium in dominant actions oAn outcomea 571572L7Nis an equilibrium in dominant actions if if 6 Ai is a dominant action for playerl39 5quot Example Old vs new technology Firm 2 New Current v l New 0 0 a a E LT Current a a 0 0 Regardless of Firm 1 s action Firm 2 chooses new technology Dominant action for firm 2 is to choose new technology 5quot Example Old vs new technology Firm 2 New Current quot39 New 0 0 a a E LT Current a a 0 0 Regardless of Firm 2 s action Firm 1 chooses new technology Dominant action for rm 2 is to choose new technology 5quot Example Old vs new technology Firm 2 New Current v l New 0 0 a a E i Current a a 0 0 New technology is a dominant action for both players NewNew is an equ brum in dominant actans 5quot Classroom game Player 2 H x Y X 8 7 1006 E Y 7 6 65 Is there a dominant action for player 2 For player What is the equilibrium 5quot Definition Best response 0 In an N player game the best response function of playerz39 is such that R161quot 2 argmax iaia i In other words given the actions a i of players 1 2 il i1 n R161quot is the best re action from playerz39 Player 2 5quot Example X Y X i1 oo Player 1 Y 00 12 Player 1 plays X gt Player 2 s best response R2XX Player 1 plays Y gt Player 2 s best response R2YY Player 2 plays X gt Player 1 s best response R1XX Player 2 plays Y gt Player 1 s best response R1YY What is the dominant action for player 1 Player 2 5quot Example cont Player 2 X Y H 03 X i lt 00 5 l l n Y 0 0 gt 1 2 There is no equilibrium in dominant actions What is the likely outcome of this game aquot Nash Equilibrium 0 An outcome a 51151251Nis a Nash Equilibrium NE if no player would nd it bene cial to deviate from a strategy provided that all other players do not deviate from their strategies played at the Nash extreme 7ri5ti5t i27riai5fi for all at 6 At 5quot Finding Nash Equilibrium Calculate the best response functions of every player Check which outcomes lie if any on the best response function of every player 5quot Example Player 2 X Y X i 1 o o Player 1 Y 00 12 The outcomes XX and YY are Nash equilibrium 5quot Observation What is the relationship between an equilibrium in dominant actions and a NE An equilibrium in dominant actions is a NE but the converse is not necessarily true There will be only one equilibrium in dominant actions while there can be multiple NEs 5quot Example Player 2 Player 1 There are neither equilibrium in dominant actions nor pure strategy Nash equilibrium I Example Mixed strategy equilibrium Player 2 X Y v l X 20 02 I gt 2 LL Y o 2 2 o Player 2 Play X with probability 05 and Y with probability 05 Player 1 expected payoff if plays X 0520501 Player 1 expected payoff if plays Y 0500521 Player 1 is indifferent between playing X and Y Player 1 Play X with probability 05 and Y with probability 05 Player 2 is indifferent between playing X and Y Neither player has an incentive to deviate from the strategy of randomly selecting between X and Y I Example Mixed strategies Player 2 H X g L 14 11 10 E R 10 11 13 Are there any dominant or dominated pure strategies for player 2 Player 2 mixed strategy PlayX with probability 37 Z with probability 47 Player 1 s expected payoff is 1 whether he plays L or R Expected payoff of player 2 Player 1 plays L 374470127 Player 1 plays R 370473127 The mixed strategy dominates the pure strategy Y 5quot Existence of mixed strategy equilibrium Theorem Every finite normalform game has a mixed strategy equilibrium 5quot Finding the mixed strategy equilibrium Bob Ballet Football 93 Ballet 00 lt Football 0 o l g Alice q1q Bob p1p Alice s expected payoff If Alice chooses ballet 2p If Alice chooses footba1 p 2p1 p gt p13 gt Bob chooses football with probability 23 Bob s expected payoff If Bob chooses ballet q If Bob chooses footba21 q q21 q gt q23 gt Alice chooses ballet with probability 23 Two pure strategy and one mixed strategy equilibria 5quot Comparison of outcomes If there are multiple equilibria are some of them preferable for all players 5quot Definition Pareto dominance o The outcome 0 Pareto dominates the outcome at if for every playeri 72quot c 2 72quot a and there exists at least one player j where 7239 a gt 7239 a 0 An outcome at is called Pareto ef cient if there does not eX1st any outcome wh1ch Pareto dom1nates outcome at Sustainability Issues and Some Research Questions Beril Toktay April 20 2004 Waste tracked in US Municipal solid waste Waste discarded by households hotelsmotels and commercial institutional and industrial sources Consists of everyday items such as product packaging grass clippings furniture clothing bottles food scraps newspapers appliances paint and batteries In 2000 the US generated approximately 232 million tons of MSW Hazardous waste 7 waste that is ignitable corrosive reactive or toxic 40 million tons of RCRA hazardous waste by top 2000 businesses in 2000 Radioactive waste 7 In 2000 approximately 600000 cubic meters htipepagovindicatorsroehtmlroeLdeahtm Other types of waste Extraction waste Industrial nonhazardous waste Household hazardous waste Agricultural waste Construction and demolition waste Medical waste Oil and gas waste Sludge Energy inefficiencies htlpepagovindicatorsroehtmlroeLdeahtm Focus on Energy Good News Between 1973 and 1986 the US economy grew by 36 percent with no increase in energy use IfAmericans had not become more energy efficient annual energy bills would have been 150 billion higher Focus on Energy Bad News In 1994 the United States imported more than 50 percent of its petroleum America still wastes upwards of 300 billion a year worth of energy more than the entire military budget far more than the federal budget deficit and enough to increase personal wealth by more than 1000 per American per year Amory B and L Hunterrowns CUMATE Making Sense and Making Money Focus on Energy Bad News The US economy is not even 10 energy efficient as the laws of physics allow Just the energy thrown off as waste heat by US power stations equals the total energy use of Japan Energy production and use account for nearly 80 percent of air pollution more than 88 percent of heat trapping greenhouse gas emissions and more environmental damage than any other human activity A12 A RT m TT TT m andvm kn MarJunel999 Resource Inefficiencies Only 1 ofthe energy consumed by today s cars is actually used to move the driver only 1520 of the power generated by burning gasoline reaches the wheels the rest is lost in the engine and drive train and 95 of the resulting propulsion moves the car not the dnver Only about 1 of all the materials 220 billion tons mobilized to serve America is actually made into products and is still in use 6 months after sale A12 A RT in TT TT m andvm kn MarJunel999 What Can You Do If over the next 15 years everyone were to buy only those energyef cient products marked in stores with EPA39s distinctive ENERGY STAR label we could shrink our energy bills by a total ofabout 100 billion over the next 15 years and dramatically cut greenhouse gas emissions Producing aluminum from recycled aluminum consumes 90 percent less energy than producing it from raw materials and generates 95 percent less air pollution hug magy navy milawmmessLoolsLools 10 himl What Can You Do Each ton of glass produced from raw materials generates about 385 pounds of mining waste using 50 percent recycled glass reduces this waste by almost 80 percent One gallon of used motor oil when recycled yields the same amount of re ned lubricating oil25 quarts as 42 gallons of crude oil If America re ned the billion gallons of motor oil they use every year we would save 13 million barrels of oil every day which represents halfthe daily output ofthe Alaska Pipeline Recycle your used motor oil hug 818ch navy milawarmessLoolstools 10 hlml What Can You Do I It is estimated that 50 percent to 80 percent of the tires rolling on US roads are underinflated Driving with tires that are underinflated increases quotrolling resistancequot wasting up to 5 percent of a car39s fuel We could save up to 2 billion gallons of gasoline annually simply by properly inflating our tires hug magy navy milawavmessLoolsLools 10 hlml IncreasingiEfficiency Adding lowemissivity lowE coatings to all windows in the United States would save the equivalent of 500000 barrels of oil per day onethird the amount of oil we import from the Persian Gulf Boosting the fuel efficiency of cars in the United States by a mere 15 milespergallon would save more oil than is estimated to lie under the Arctic National Wildlife Refuge The EPA Advocates Source Reduction Source Reduction refers to any change in the design manufacture purchase or use of materials or products including packaging to reduce their amount or toxicity before they become municipal solid waste Source reduction also refers to the reuse of products or materials Industry and the Environment Typical framing Inherent and fixed tradeoff ecology versus the economy Social benefits from strict environmental standards versus Industry s private Costs that lead to costs for prevention higher prices and and cleanup reduced competitiveness Industry and the Environment Properly designed environmental standards Innovations that lower the total cost of a product or improve its value Resource productivity 1 Increased competitiveness Parallel to TOM Initial reaction Higher quality means more inspection and rework or inevitable defects from the production line 9 quot TQM approach Defects are a sign of inefficient product and process designquot Redesign the process build quality into the process source reduction instead of inspection and redesign identification and processing of waste Examples 1991 law on benzene emissions Opposed by coal tar distillers Only way was to add costly gas blankets One firm found a way of removing benzene from tar in the first processing step Saved 33 million Green and Competitive Ending the Stalemate Porter and va1 der Line Harvard Business Review Sep 7 Oct 1995 Examples 7 Reducing solvent emissions by 90 3M avoided solvents by using waterbased solutions Shortened time to market Competitive advantage due to moving early Green and Compe tive Ending me Slalemate Porter and V81 der Line Harvard Business Review Sep 7 Oct 1995 Examples Electronics companies to eliminate CFCs Raytheon found alternate cleaning agent to be used in closedloop system Improved product quality Reduced operating costs Green and Compe tive Ending me Slalemate Porter and van der Line Harvard Business Review Sep 7 Oct 1995 Innovationfriendly Regulation Focus on outcomes not technologies Enact strict rather than lax legislation Regulate close to the enduser and encourage upstream solutions Use phasein periods Use market incentives Develop regulations in sync or slightly ahead of them Green and Compe tive Ending me Stalemate Porter and va1 der Line Harvard Business Review Sep 7 Oct 1995 Examples WEEE Directive of the European Union Stipulates producer responsibility for recovery Is expected to encourage innovations for ease of recovery reduction of hazardous materials information availability etc All firms selling in Europe are affected Challenges in Designing ClosedLoop Supply Chains Inventory Management for Q Remanufacturable Products Issue In the procurement of new components for remanufacturable products delay in procurement products in field unobservable return flows unpredictable Tokbay Wein and Zenios Management Science 47 9 2001 12687 1281 11 IMRT Research Question Develop an ordering policy for new component procurement How much is it worth tracking products in the field Tokbay Wein and Zenios Management Science 47 9 2001 126871281 IMRT Approach Statistical estimation methods for censored data Closedqueuing network modeling Developed several policies for adaptive estimation and control Simulationbased comparison using disguised data from Kodak Tokbay Wein and Zenios Management Science 47 9 2001 12687 1281 12 Market Segmentation and Remanufacturing Technology Selection 5 Issues Retreaded tires are valued less than new tires gtmarket segmentation Not all used tires are retreadable TireTread CESng gttechnologY se39eCtiO RetreadingReplace Reuse Debo Toktay and Van Wassenhove INSEAD Working Paper MSRTS Research Questions What are the key profitability drivers in a product portfolio consisting of a new and a remanufactured product How do they interact What pricing strategy and production technology choice best fit the target market What is the impact of a change in remanufacturing cost and other parameters Debo Toktay and Van Wassenhove INSEAD Working Paper 13 MSRTS Approach Develop multiperiod optimization model with Heterogeneous customers Lower valuation of remanufactured product Production and remanufacturing cost as a function of remanufacturability level Volume dependence over time Prices and remanufacturability level as decision variables Characterize optimal solution and do sensitivity analysis DeboTokbay and Van Wassenhove INSEAD Working Paper MSRTS One Result It may be optimal to price new product at a low margin or even at a loss Why Creates opportunity for future remanufactured product sales where most of the profit is made Implication Single profit center may dominate two separate profit centers Debo Tokbay and Van Wassenhove INSEAD Working Paper 14 Internal Transfer Pricing Computer industry Separate remarketing organization Sales org buys from manufacturing at SC and sells returns to remarketing at same SC Telecommunications Separate remarketing organization Sales org buys from manufacturing at SC and sells returns to remarketing at 0 transfer price Internal Transfer Pricing Copierprinting solutions Oce Manufacturing sells to OpCo s at standard cost There is an Asset Recovery department Asset Recovery posts internal transfer price to OpCo for purchasing used products Managing Asset Recovery at Oc NV A Business View on Cosed Loop Suppy Chains REVLOG Group 15 ITP Research Question Under a decentralized structure what is the appropriate cost allocation mechanism that aligns division incentives with those of the firm ITP Approach Develop twodivision decentralized model D1 manufactures and sells first D2 remanufactures and sells later Stackelberg Game Solve centralized benchmark quantify level of inefficiency from Stackelberg equilibrium Develop cost allocation mechanism to achieve centralized benchmark profit 16 Model Company Structure Decisions 5 sell new c M D1 gt new product I returns Price Pn C sell remanufactured remanufactured product price p salvage Model Company Structure Decisions 5 sell new c M D1 gt new product I returns Price Pn 0 sell remanufactured remanufactured product price p salvage 17 g Model Company Structure Decisions C M C D1 5e new gt new product returns price p fc 32 sell remanufactured remanufactured product price pr l salvage emand model Consumer has lower willingnesstopay for a remanufactured product P7 1 6 l perceived depreciation dr1 18 ModeIDecisions and cash flows Dispose 1 Produce and y sell new prod Returned 511 Oia0n q amprggsable Remanufacture Salvage quot and sell Sr din S qsn 0 D1 D2 r me Period 1 Period 2 v FirstBest Solution maxSer SnPn3n Csn Srpr5r Crsr qsn 3r 3r S qsn Proposition 8 lt 618 unconstr if or v 2 1 q1 c Bvql 6 5 15 constr ow 19 s Decentralized Scenario oc 1 maxsn snpn8n CSn 8rpr 8 r CTS F 8 5quotquot l 5r S qgn 3n l 7 Proposition If v gt 0 then n lt s and 7 S 3 Proportional Cost Allocation ocC D1 maxsn snpnsn ozcsn gt 3n DZImaXsr1 Oian l Srpr5r 0T3 l 95 57 37 3 qgn gt874 20 ptimal Cost Allocation Proposition An oc1oc cost allocation scheme achieves the firstbest solution with 0 5716 if unconstrained C c qCrl5q1 C1 q2 16 1f constra1ned K g Interpretation of 08 C Vs Firm s marginal cost c 39 gt D1 s marginal cost Proposition 04 D1 maxsn snpnsn ozcsn maxsn RNsn Cgvsfklsn FOC REVCS 03763 O gt D1 chooses Sn 21 Interpretation of oc I l Uneonstrained case 04 FOC Riv8n C 5W O Y V marginal lifecycle marginal revenue cost to rm When next to D1 unit is salvaged gtllt 06q0r1 5q 1 2 Constramed case or C1 q216 RI Sn C qcr Q6q1 O N 1 q21 5 marginal lifecycle marginal revenue cost to rm When next to D1 unit is remanufaetured g Conclusions Cost allocation should be such that the marginal revenue from making one more new product the marginal cost to the firm of producing that product When salvage value is negligible and remanufacturing unconstrained decentralization is not an issue oc1 When cost is allocated to a second department it must be allocated so it is a sunk cost not a variable cost 22

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