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# Game Theory ECON 309

UMass

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This 11 page Class Notes was uploaded by Mr. Kay Bergstrom on Friday October 30, 2015. The Class Notes belongs to ECON 309 at University of Massachusetts taught by Woojin Lee in Fall. Since its upload, it has received 10 views. For similar materials see /class/232320/econ-309-university-of-massachusetts in Economcs at University of Massachusetts.

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Date Created: 10/30/15

Games under incomplete information Bayesian games The analysis of games of complete information assumes that the complete description of the game is common knowledge In those games all players are supposed to know in particular the exact payoffs that their opponents can obtain In determining a Nash equilibrium each player s information about her opponents payoffs is given Now suppose that agents are incompletely informed about their opponents payoffs and sometimes about their own payoffs too which depend on a random vector 6 In this game not only the action combination a but also 9 determines a player s payoff Hence for every player i e the payoff function is given by uia6 Harsanyi transformation Harsanyi1967 suggested a method of transforming games of incomplete information into games of imperfect information for which best responses and equilibrium behavior are well defined The basic idea is simple A player with incomplete information about some other player s payoff will be treated as if she were uncertain of the type of player she will face If one assumes that there is an artificial player called Nature that chooses a particular type of all players according to some joint probability distribution and if each player cannot observe the move of Nature but is partially informed about the types then the players face the environment of a game with imperfect information Hence the incompleteness of information about payoffs is transformed into uncertainty about the move of nature Strategies versus Actions or Moves An action or move is a choice a player can make A strategy is a detailed set of plans for playing the game that specifies the move for every contingency A strategy maps out a plan of action under all eventualities In games under incomplete information a strategy is a profile of actions contingent on types Example Entry Deterrence I under Incomplete Information Incumbent Expand Don t Expand Entrant Enter 1 0c 1 1 Stay Out 0B 0 3 Entrant doesn t know the precise values of 0c and B but knows that there are two possibilities Scenario 1 prob 23 Scenario 2 prob 13 a2 4 oc 1 0 Incumbent is a low cost firm type 6L Incumbent is a high cost firm type 6H Entrant believes that there are two different games Scenario 1 prob 23 Scenario 2 prob 13 Expand Don t Expand Expand Don t Expand Enter 1 1 1 1 Enter 152 I Stay Out 0 0 03 Stay Out 0 4 0 3 Incumbent has a dominant strategy Incumbent has a dominant strategy Don t Expand Expand Therefore Entrant s expected payoff from Stay out is greater than its expected payoff from Enter 2 1 1 2 1 Enter 1 1 Stay out 30 0 0 Hence the pure strategy BNE of the game is Stay 0utEXpand OLlDon tEXpand 0H Separating eguilibrium Alternative Way Harsanyi transformation Incumbent has 4 strategies EX 6L EX 6H EX 6L Don t 9H Don t 6L EX 9H Don t 6L Don t9H Entrant has 2 strategies Enter Stay out The pure strategy BNE is a NE of this expanded game EX EX EX Don39t Don39t EX Don39t Don39t E 1 2 1 21 1 1 1 11 3390 0 40 0 4 3 0 3 0 0 3 3 Ti Ti Ti All dominated by Ex Don t Remark We computed the pure strategy Bayesian Nash equilibrium We can compute the mixed strategy BNE if we want Example Entry Deterrence ll under Incomplete Information Incumbent Expand Don t Expand Entrant Enter 1 h l 2 Stay Out 0 k 0 3 Entrant doesn t know the precise values of h and k but knows that there are two possibilities Scenario 1 prob 23 Scenario 2 prob 13 h15k35 h0k2 Incumbent is a low cost firm type 0L Incumbent is a high cost firm type 0H Hence Entrant believes that there are two different games Scenario 1 prob 23 Scenario 2 prob 13 Expand Don t Expand Expand Don t Expand Enter 1 0 1 2 Enter 115 12 Stay out 0 2 03 Stay Out 035 03 Incumbent has a dominant strategy Incumbent does not have a dominant Don39t Expand strategy Incumbent best responses are Expand if E stays out and Don t Expand if E enters Now Entrant s BR for Don t0LDon t0H is Enter since 1gtO Also Entrant s BR for EX6LDon t6H is Stay Out since 2 1 1 0gt 131 3 Conclusion There are two pure strategy BNE One pooling the other sepa rati ng Enter Don t 9L Don t 6H s 0 Ex 0L Don t 0H Alternative wa using Harsan i transformation EX7 EX EX7 Don39t Don39t7 EX Don39t7 Don39t E 17 157 0 7057 2 427 0 17 27 2 SO 07 357 2 07 3573 07 37 2 07 373 Hence there are two pure strategy BNE Enter Don t 0L Don39t 6H a nd S O EX 9L Don t 9H Example Cournot Game under Incomplete Info There are two shrimpers in New Haven catching homogeneous shrimps The inverse demand function is given by Pa Q Shrimper 2 knows his own cost function and shrimper 1 5 but shrimper 1 does not know shrimper 2 s cost function although she knows her own Thus information is asymmetric More specifically assume that C1Cq1 and C2yqzwhereyeCLCH CHgtCL Suppose PjCH6 and PjCL1 6 Shrimper 2 s BR is obtained by maximizing 7r2oc q1 q2 yq2 Thus Note that different types have different BR s Shrimper 1 does not know shrimper 2 s type hence maximizes its expected profit E75160 q1q2HCq116aq1qCq1 0 q1Eq2Cq1gt where Elqzl9q2H 1 9612L Thus shrimper 1 s best response is za C Eq2Za C0q519q 3 2 2 91 By solving 1 2 3 simultaneously we have qf9 a 2C Ey a 2C0CH1 0CL 3 3 2C C 1 0 aTlcchl a ZCL C g 3 6 q q0L CH CL BNE q qng q6L separating eguilibrium Comparison with the Cournot NE under Complete Info If y is known for sure then oc 2C lQ1 3 oc 2C lq2 3 Hence q 6Hgtq IFCH amp 126L lt q IFOL 1 The Battle of the Bismarck Sea In 1943 Rear Admiral Kimura has been ordered to transport Japanese troops across the Bismarck Sea to New Guinea and Admiral Kenny wants to bomb the troop transports Kimura must choose between a shorter northern route or a longer southern route to New Guinea and Kenny must decide where to send his planes to look for the Japanese If Kenny sends his planes to the wrong route he can recall them but the number of days of Kenny a Does Kenny have a strictly or weakly dominated strategy No b What about Kimura Kimura has a weakly dominated strategy South c Can you solve the game using the notion of strictly or weakly dominant equilibrium Yes Kenny may delete the column of South and then choose North d Can you solve the game using the notion of Nash equilibrium de ned in terms of best responses I highlighted best replies of each player There is one cell that is a mutual best reply Thus North North is a NE e Can you solve the game using the notion of Nash equilibrium de ned in terms of nodeviation North North is a NE because there is no player who would like to deviate from this North South is not a NE because Kenny would like to deviate South North is not a NE because Kenny has an incentive to deviate South South is not a NE because Kimura has an incentive to deviate 2 The Battle of the Sexes You and your france wish to go out together Two concerts are available one of music by Bach and one of music by Stravinsky You prefer Bach and your france prefers Stravinsky If you and your france go to different concerts each of you is equally unhappy listening to the music of either composer The payoff structure of the game is You a Does anyone have a dominated strategy No one has a dominated strategy b Can you solve the game using the notion of dominant equilibrium No c Can you solve the game using the notion of Nash equilibrium Yes There are two NE Bach Bach and Stravinsky Stravinsky 3 Hawk Dove Two animals ofthe same species compete for a resource eg food whose value is v gt 0 Each animal can be either aggressive hawkish or passive dovish If both animals are hawkish they ght until one is seriously injured the winner obtains the resource without sustaining any injury whereas the loser suffers a loss of c lt v The two animals are equally likely to win so each one s expected payoffis v c Ifboth animals are dovish then each obtains the resource with probability without a ght Finally if one animal is hawkish while the other is dovish then the hawk obtains the resource without a ght One animal Dove v v b Solve the game using the notion of Nash equilibrium There are two pure strategy Nash equilibria Hawk Hawk and Dove Dove 4 l The policy pair 05 05 is the unique Nash equilibrium and the two parties have 50 chances of winning at the equilibrium It is easy to see that this is a Nash equilibrum Case 1 Suppose party 1 deviates leftward from 05 to 05s where sgt0 given that party 2 keeps 05 Then all voters with policy preferences from 058 2 to 1 vote for party 2 whereas all voters with policy preferences from 0 to 0582 vote for party 1 Party 1 loses for sure by the deviation Case 2 Suppose party 1 deviates rightward from 05 to 05s sgt0 given that party 2 keeps 05 Then all voters with policy preferences from 05s 2 to 1 vote for party 1 whereas all voters with policy preferences from 0 to 05s2 vote for party 2 Party 1 loses for sure by the deviation In like manner one can show that party 2 does not bene t by deviating unilaterally from 05 05 Since no party bene ts from any deviation this is a Nash equilbrium 2 There is no pure strategy NE if there are three Downsian parties Consider any strategy pro le in which each party locates at a separate point Such a strategy pro le is not an equilibrium since the two parties nearest the ends would edge in to squeeze the middle pa1ty s vote share If a strategy pro le has any two parties at the same point the third party would be able to acquire a share of at least 05s by moving next to them and if the third player39s share is that large one of the doubledup parties would deviate by jumping to its other side and capturing its entire vote share Finally a strategy pro le in which all three parties are at the same point is not also a NE Why

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