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# Class Note for ECON 702 at UMass(1)

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Date Created: 02/06/15

III Games under Incomplete Information 1 Introduction The analysis of games under complete information assumes that the complete de scription 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 0 01701 In this game not only the action combination s but also 0 determines a player s payoff Hence for every player 139 E the payoff function is given by u st9 I where s E S X SI In many economic applications we usually deal with special cas 11 es in which player i s payoff is independent from 94 In principle however player i s payoff may depend not only on her own type 9i but also on the types of the other players 94 Since a player can no longer predict for sure what would be best responses of the other players she cannot determine what constitutes optimal behavior for herself 2 Harsanyi transformation Harsanyi 1967 suggested a method of transforming games of incomplete informa tion into games of imperfect information for which best responses and equilibrium be havior are well defined Note 1 Imperfect information Players are uncertain about the other player s action 2 Incomplete information Players are uncertain about payoffs of the other players His 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 arti cial 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 incom pleteness of information about payoffs is transformed into uncertainty about the move of nature Harsanyi transformation A game with incomplete information can be transformed into a game with imperfect information by introducing Nature that assigns types 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 speci es the move for every contingency A strategy maps out a plan of actions under all eventualities 3 Bayes Theorem If some individuals do not know the payoff functions of other players how do indi viduals choose their strategies then A natural step towards an answer to this question is to place some belief about the type of the other players Even though a player may not know the type of the other players she can derive an updated probability assessment of the type combination of all players based on the initial joint probability distribution over types and on the information she receives A typical updating rule used in the literature is the Bayes rule Suppose PA is a prior belief on A and PAlB is the posterior ie updated belief on A after observing B Then the relationship between PA and PAlB is the follow P B l A P A O B P B P B ing PA B PA This is true because PA l B and PA B PM PBA PB 39 Also we note that PB P A O B P A0 O B holds as an identity See the P B l A Thus the updated belief is proportional to the initial belief and the scale factor is figure below Thus PA l B PA can be rewritten as 1309 l APA PM PAmBPA0 m3 PB APA PB APAPB A0PA0 This is the socalled Bayes rule Example Is Stuart infected There is a debate over universal screening for infection by HIV HIV screening consists of a blood test for the presence of antibodies to the HIV virus Like nearly all medical tests however this test is not foolproof there is a chance of false negative and false positive results The test generates a false negative result if the lab detects no antibodies to the HIV vi rus when the person is indeed infected The test generates a false positive result if the test detects the presence of antibodies to the HIV virus when the person is not infected Assume that the false positive error rate is 0005 and a false negative error rate is 0 If 0001 of the adult population in the US is infected with the HIV virus what is the probability that a randomly chosen person is not infected with the HIV virus given that the test result is positive Suppose we use the following notations E the event that a person is not infected with AIDS F the event that a person gets positive test results Then we need to calculate PE l Note that the false positive error is PFlE0005 and the false negative error is PFC EC 0 Also we know that PE9999 Thus PF EPE PE l WW 0005gtlt9999 333 0005 X 9999 100 X 001 Thus the probability that even though the test result is positive a randomly chosen per son is not infected with the HIV virus is very high An intuition behind this obtained by P F E rewriting PE I F c where PF O E is uninfected indi PFmEPFmE viduals who get positive results and PF EC is infected individuals who get posi tive results Infected pop who get positive resulm F EC Uninfected pop who get positive resulm F I E Infected pop who do not get positive resulm FC EC Example Entry Deterrence I under incomplete information Incumbent Expand Don t Expand Entrant Enter 1 0L 1 1 Stay Out 0 B 0 3 Incumbent knows his own type but Entrant doesn t know the precise values of 0L and Entrant thinks that there are two possibilities Scenario 1 prob 23 oz 2 5 4 Incumbent is a low cost firm type 19L Suppose the joint distribution chosen by Nature is given by 136 9H andPt9 19L Scenario 2 prob 13 oz 715 0 Incumbent is a high cost rm type 19H Then Entrant updates its belief by using Bayes rule thus 1 P6H6and P6L6 In this example applying Bayes Theorem is trivial Entrant believes that there are two different ames Scenario 1 prob 23 Scenario 2 rob 13 Expand Don t Expand Expand Don t Expand Enter 12 11 Enter 1 1 1 1 Stay Out 0 4 0 3 Stay Out 0 0 0 3 Type39L incumbent has a dominant 30mm TypeH incumbent has a dominant action Expand Don t Expand Entrant s expected payoff from Stay out is greater than its expected payoff from Enter 2 1 1 2 1 Enter 1 1 Sta out 0 0 0 33 3 y 33 Hence the pure strategy BNE of the game is Stay out Expand 19L Don39t Expand 19H Separating eguilibrium An alternative way Incumbent has four possible strategies EX 0L EX9H EX 0L Don39t 0H Don t 0L EXt9H Don t 0L Don39 tt9H Entrant has two strategies Enter Stay out The pure strategy BNE is a NE of this expanded game EX EX EX Don t Don t EX Don t Don t Enter 1 2 1 13 21 13 11 1 11 Stay out 0 40 0 43 0 30 0 33 Remark We computed the pure strategy Bayesian Nash equilibrium We can compute the mixed strategy BNE if we want Example Entry Deterrence II under incomplete information Incumbent Expand Don t Expand Entrant Enter 1 h 1 2 Stay Out 0 k 0 3 Entrant doesn t know the precise values of h and k but knows that there are two possi bilities Scenario 1 prob 23 Scenario 2 rob 13 h15k35 h0k2 Incumbent is a low cost rm type 19L Incumbent is a high cost firm type 19H Hence Entrant believes that there are two different games Scenario 1 prob 23 Scenario 2 rob 13 Expand Don t Expand Expand Don t Expand Enter 1 15 12 Enter 10 12 Stay Out 0 35 0 3 Stay Out 0 2 0 3 Type39L incumbent does Qt haVe a domi39 TypeH incumbent has a dominant strate nant strategy Its best responses Don t Ex gy Don t Expand pand if E enters and Expand if E stays out Now Entrant s BR for Don t 19L Don39t 19H is Enter since 1gt0 But Don t Expand is TypeL incumbent s BR to Entrant s Enter Thus Enter D0n tt9LDon39tt9H is a BNE This is a pooling equilibrium Entrant s BR for Ex 19L D0n39tt9H is Stay Out since 11 lt 0 But TypeL incumbent s BR to Entrant s Stay Out is Expand Thus S 0 Ex 19L Don39t 19H is also a BNE This is a separating equilibrium An alternative way Ex Ex Ex Don t Don t Ex Don t Don t Enter 1 15 0 13 152 13 20 1 22 Stay out 0 352 0 353 0 32 0 33 Example 3 So far we considered games in which only one player does not know the other player s type We now consider a game in which both players do not know the oth er player s type The analysis is somewhat cumbersome but we can do it Suppose player 1 does not know whether player 2 is a yestype or a notype Likewise player 2 does not know whether player 1 is a yestype or a notype Correspondingly there are four possible states Assume that joint probabilities assigned by Nature are given Pmw 1 6 W Hw and P01 7102 71 H2m From these probabilities we can deduce conditional probabilities as follows Player 1 s estimate of 02 s probability Player 1 s estimate of 01 s probability P0y0y 13 1 P0y70y 2 HMQWA4 L l i HwwA4 L l P01y 23 2 P02y 3 P0 0 P0 0 P92n101y 1 1 72 13 l P91n192y 1 72 101 P91y 23 2 P92y 3 P0 0 P0 0 1 n72 1 y72 P01n 13 2 P02n 3 P0 0 P0 0 P92n191n 1 72 16l P91n192n 1 72 Ll P01n 13 2 P02n 3 There are four possible games Player 2 is a yestype Player 2 is a noty e Player 1 is a yestype B S B S B 11 m0 B 10 m2 S 00 12 S 01 10 P02y101y and P02n101y and 2 2 Hww HQ Mm Player 1 is a notype B S B S B Q1 10 B mo 12 S 10 02 S 11 00 P02 y101n and P02n101y and 1 1 P01 n102yg P01 n102ng The expected payoffs of player 1 are Yestype of player 1 Notype of player 1 The expected payoffs of player 2 are Yes Combining all these tables yields eof la er2 B B B S S B S S B B 20 10 11 12 11 00 02 02 B S 21 2313 1052343 1052313 00 2343 s B 00 1323 0511323 0514323 124323 s S 01 01 0505 00 0505 21 1020 Thus there are two pure strategy BNEs B B BS and SB SS In the above two examples both the action spaces and the type spaces are discrete The next example considers a case in which the type spaces are discrete but the actions spac CS are continuous Example Cournot Game under incomplete information There are two shrimpers in New Haven catching homogeneous shrimps The inverse demand function is given by P 0 Q Shrimper 2 knows his own cost function and shrimper l s but shrimper 1 does not know shrimper 2 s cost function although she knows her own Thus information is asymmetric More specifically assume that C1 Cq1 and C2 yq2 whereyECLCH CH gtCL Suppose PCyCHt9 and PCyCLl t9 Note If shrimper 1 updates his belief on shrimper 2 s type using Bayer rule then P C yC t9 P7CHlC PC HT6 and PC yCL 16 P7CLlC PW Hence the revision of prior beliefs is trivial in this case Shrimper 2 s BR is obtained by maximizing 712 a ql qz yq2 Thus q2aqi70r 2 2 L91L aqlCL 2 Note that different types have different BR s Shrimper 1 does not know shrimper 2 s type hence maximizes its expected profit E771 6a q2HCIQ116aQ1 qu Cq1 aq1 ECa where Eq2 9g l t9q2L Thus shrimper l s best response is 10 a c Eltqzgta c eqflt1 6gtq 2 2 3 Q1 By solving 1 2 3 simultaneously we have a ZC Eya 2C6CH1 6CL 6 q1 3 3 Ul ZC C 1 9 q26H HTCHCL a ZC C 9 q26LECHCL BNE qr qt9H q 6L segarating eguilibrium Comparison with the Cournot NE under Complete Info If y is known for sure then a 2C y 1 f a 2yC Hence q t9Hgtq 7cH amp q9Lltq ML QZ BR of shIiInDerl L q x H BR oftVDe L QZ x x x x x Q1 BR of the H 11 Example Bertrand game with incomplete information There are two pricesetting firms Assume that profit functions are given by TF1 1317132 1 and W2p17p2b where a and b are random variables Firm 1 does not know the parameter of rm 2 b E bH7bL and rm 2 does not know the parameter of firm 1 a E 0H7 11L Suppose the joint probability that a particular pair of parameter values is selected by Nature is given by Pa 1H7b bH 05 Pa 1H7b bL 0125 Pa upb bH 0125 Pa upb bL 025 From the joint probabilities and using the Bayes rule we can compute the condi tional probabilities as follows Firm 1 s estimate of b s probability Firm 2 s estimate of a s probability P 1 71 05 4 4 PbH 15H pol ME PaHbH PaHbL 0625 5 5 1 pbL aHwl P0L15Hg PaH7bH PaH7bL 0625 5 1 Pa 0 7 PaLbH 7 0125 l 13 1 g L L 7 PaLbH PaLbL 7 0375 7 3 2 P b pltb a 7 P L L 7 025 7 Z W L 3 L L PaLbH PaLbL 0375 3 Each firm s strategy is a pair of typecontingent actions p1aH7p1aL and P2bH7P2bL We now derive each type s expected payoff Consider type 1H of firm 1 Its expected profit at price p1 is 1p17p2byaHPbH 1 1P17P2bL 1HPbL 10H 12 Type 1H of firm 1 chooses 13111H by maximizing the above expected payoff while holding p2bH7p2bL constant this will give type 1H s best response to 2241297224129 Similarly the expected payoff of type 1L of firm 1 at price p1 is mltppp2ltbygtmLgtPltbH an nlltppp2ltbLgtmLgtPltbL m and type 1L offirm l chooses p1aL by maximizing the above expected payoff while holding p2bH7p2bL constant this de nes type 1L s best response to 13459719450 In like manner we can deduce the expected payoffs of the two types of firm 2 at price p2 as follows W2p1aH7p2bHPaH W 750310107232 bHPaL ME for type bH and 2p1aH7p2bLPaH 10 750310107232 bLPaL W for tpr bL Type bH optimally chooses p2 bH while holding p1 11H7p1aL constant and type bL chooses p2 bL while holding p1aH7p1aL constant This procedure defines type bD s best response to p1aH7p1aL for i H7L The BayesNash equilibrium is defined as 131 11H7p1 11L7p2bH7p2bL such that 131012 is type 12 s best response to p2bH7p2bL and 1321 is type b2 s best response to p1 0H7p1aL We will study this problem in the homework 13 So far We considered games in which the type spaces are discrete The next three examples consider cases in Which the type spaces are continuous We start from an example in which the action spaces are discrete Example Contribution to the public good V Megaw has ems was 69922 stvalumeo madam z Mme 17 m fawn gage P156 33 479 db WW7 Ts 5 M MW citemas a mew ab 1 75a tea 5 4 0 5 ew FWva 1 FMLQEC Tax and a home ices 9 rs Arm quotWM mm m m cpl 9 F m 96 a ae ltllt A F of ab P W x TS 529 SJ 6 ag W4 9 31 5 3a same Samp D 33 3 I v 3 0 0 0 I Ha S 19 1 9 Iea New Fwe 9hr quots A Ha m 349 KW 357313 ol I quotWC5 Ma SAQJ eJLen 95 Pg 3 wa9j loPyobquot 5o 30 1266460 it will wase 3 i F6 9 gt bnwmej 1 9 f xlt Iwa83 l H u A 14 Lot 6 Pnbejl The Se fquot F6 9lte 0 Wm M 6 wag CantaMe 23 K5 Fm 3 MM Way 4 mxww 01 1 5M5 quotW4 08am I wb39 I Sm PnLCs ewq Pne ltltef my me ovm 91mg ef IF6 03 man ef l Wef gt m 0M we QM efef e we e I FCl Fte v 15 Now We consider two examples in which the actions spaces are continuous Example Cournot game with continuous types r Coum39 Game amp wus k Two Sims 1 hex W 39B x waIayse alawmi 5 P39 a wgfpggl MC 3 CA TS d rmm 39me 4L4 wilamzl S2 40m otrsmm smokm F l rs BEw Tyye SFche Mexs 0 myth m 3934 no 5 33 cm a BRA 39 go 2ii Bwa cr eweaieo l pang T55 fg ao gt3 a 538B d 75 L Gnu38MB 385 ZMoM C GADZBZB 35 2 60 91W0 020 3Bgt5B n 813 60 5 gg 601 3731035 6 139quot35 ii KI 0 Hm 5R2 35 mpg Hem lu ENE Ts 3m o C 50 Example Firstprice Sealed bid Auction Er wsf fymz Smeao 574 7AucH9n777 7 Cuwa 92 lmkam bnugms 3577a LmzeaLxa gotEJ 77 gaoyJ wa 7 512 vale pf a 3m f m sawEel AJ ua1m7 EM brdofezv squs a 19744459 149 khaki3n 09c h 1449 14 had biJJPu7ako7 77r g75 HA0 fl i 39 e biJ 7 7kg b Is wTJIEYIV39YIW J57 af xa Fn mmd7o 7 1771 4 be in valuairm7 A 67 LJIes 7 La 7 ad wa 92111 wTM77fmba6ia 4nd Magma Tyre S 46 Gzlllg ac df39 7 7 9m Cv w7rs End Alaf 77 7 m fnLMSa 1az39Nmsa MGM A fme 947a 797 an 41 Te 07 be lasing 2 swig s ALfs 3 IR7MJMW lasiTgaL Zi 7 77mg c 7 biuHEL 3 Fym 75713 br39 bull s 7 jump 5 7 1 1 77 1 1577162 b rmaltbv lvnj H343 M141 mems 7 O o l hovwise 7 Sfeagafm img 7 4714917 m7m7mm chore ea 7 7 7 ma V maize 7M7 In Aeei39 57d is Manamaulna gemM92 7 7me7C739u375 7 179 177 7 77 7 Nous 1 ex Ed l77gt ampa 424703 7719 7 77 7 E Md 39 747 M ILU 7dvv77 7 7 177 IT Bg l v U k 77uL7gt 7 7 gt777 Risamg H114 G19 E Wai axw dighiluwkim G W39 77 7 77 7 0m 7176 ouij 7 77 17 r wsvm Ju L Ve V m n 35 Psm bx m 21s 36 956155 5 is h 33 m 8 a nit i wYeQwa ka fnxiewr mel mw ww a E S 8 m m Scots 4w EN 3 mm chpwh n hm r a WWWEE Ragga agua mv 3L 1 k h 1 yY W ml ma 1 8m Ssgm a 436 mwa i i crim l i 39 u 34an Avg w w 6 Egg 3 n warhelmww SPF a 9 31 1 H Mi V Ni v 1 m Wt ww of Famean Es J 9 gm 1 1 r in a 47 m mh xm V m3 93V Tr mw10 x mivm ew NMTE a a 311 quot WE WVWEig m A m m manyv gr stWW nJm LH Fw ma a r in E sz FoaP Lon vigrw an LPL s nbd mm a ql a1 Hv tm Iv Wai lemWNwinmw M f immu w H V Lam M Cy A agsvnmu V 9150 w am Vinny Y La rv Hm 4 Formal representation of games under incomplete information To formalize the games of incomplete information suppose i is a set of possible types for player i and let 8 l X 71 Nature chooses one 9 919I from 8 Each player is informed about the joint distribution of 9 called the prior belief on 8 the density of which we denote by u 8 gt R Here we allow different u s across individuals although in most cases u s are assumed to be identical for all indi viduals Nature provides to each player with some partial information about her choice using a message correspondence or a signal correspondence h I gt Tl where Tl C 8 is a set of messages or signals individual i receives 1119 is called the mes sage set of player i and it contains player i s information about the state of nature For instance if player i receives a signal 1 6 T1 according to a signal correspondence h he deduces that the state 9 nature chose is in the set h1 II E 9 e 8 I 1 6 1119 If T1 8 and h 9 9 then i has full information On the other hand if h 9 8 then i has no information except his prior Indeed h defines player i s information partition H1 We assume that each player assigns positive probability to every member of hf1t1 Example 1 In typical Bayesian games T1 l and h1991 That is play ers messages are independent and each player learns only about his own types 2 In correlated games T1 8 Hence players receive some correlated message x if 9 x To g1ve a concrete example suppose G x y Z If 1719 and yz 1f9yorz xay if9x0ry 1129 the player 1 1s fully 1nformed when 9 x 1s chosen but 2 1f 9 Z partially informed when y or Z is chosen Likewise player 2 is fully informed when Z is chosen but partially informed when x or y is chosen In this example player 1 s infor mation partition is xy 2 and player 2 s information partition is xyz 19 Now a pure strategy of individual i in a game of incomplete information could be defined a decision function S1 mapping Hz to SI In other words we could define it as a mapping that maps elements of H1 ie the possible signals ireceives h into SI But note that if 939 6 1119 then the strategy function S1 prescribes the same action in state 939 and 9 Hence instead of defining strategy in this way it is more con venient to define pure strategies as maps from 8 to S with the additional property that 9139 E h 91 3 S1 91 S1 9139 The formal term for this property is that strategies are adapted to the information structure How do individuals choose their strategies then Even though a player may not know the type of the other players she can derive an updated probability assessment of the type combination of all players based on the initial joint probability distribution over types and on the information she receives Let f1 9 l h be player i s updated belief of a particular type combination 9 given that player i has receive a message 1 E h 9 If it is based on the Bayesian updat M 9 ing then it should be f19lhl9 M9d9 if e e hf1t1 and Jleeevxme f 9 l h 0 otherwise Now each player s expected utility is given by V 3137 39at 131379f 9 l h96le 959V Eh 9 De nition 1 A dominant strategy equilibrium in the interim sense is a list of decision functions 31 9s9 such that for all i E I all h E H and all si 84 u s91s719f 9l h19d9 Z u sls719f 9l h 9d9 LQEOlt h 9 sewer 9 holds for all si ESi 2 A correlated Nash equilibrium is a list of decision functions 31 9s 9 such that for all i E and all h1e H1 20 uls91s991971f19l h19d9 2f u sls991971f19 h16d9 Leanna 9 lt95 ltih 9 holds for all si ESi Now let s specify to typical Bayesian games In these games Tl 81 and hl991Hence M 9 fl9lhxefl9lezW ewea 6 I I 1 1 I and the expected payoff is v S1 371 91 L u slsl91971f 9 l 91al97 There are two concepts of equilibrium as in games of complete imperfect informa tion De nition 1 A dominant strategy equilibrium in the interim sense is a list of decision functions 31quot91 s91 such that for all i 6 all 91 6 T1 and all si 684 e u ltsiltegtseemm Bade 2 L u ltsseegtf lteWe holds for all si ESi 2 A Bayes Nash equilibrium is a list of decision functions 31quot91 s91 such that for all i E and all 91 ET a u sltegts 999f 9 l e we 2 Q m m ltegtee f e l e we holds for all si ESi The above definition is given in the interim sense We can also define these con cepts in the eXpost sense De nition 1 A dominant strategy equilibrium in the eXpost sense is a list of decision functions 31 9s9 such that for all i E I and all 9 E 8 s 9 argmaxu 313759 for all si 84 holds 5 21 2 A Bayes Nash equilibrium in the eXpost sense or a robust Bayes Nash equili brium is a list of decision functions s9s9such that for all i6 and all 9 e 8 s 9 arg maxu s s 99 holds Obviously an eXpost equilibrium is an interim equilibrium but not conversely 22 5 Reinterpretation of mixed strategy Nash equilibria in games of complete in formation A mixed strategy NE in a game of complete information can be interpreted as a pure strategy BNE in a closely related game With a little bit of incomplete information That is mixed strategy equilibria of complete information game can be interpreted as the limiw of pure strategy equilibria of slightly perturbed games of incomplete information M16 0F th Eggs 0J1 K K V Foo msfswavieag NE39 COWOVM gk g m Wch Shakad NF E 75 FZ3 Bugle 0 4i Sexes lmazmake je Iat a 6 FT 151 L o o 0V 0 o I yen F214 Mag 6 N 1amp2ij 455 chH 493055 Dew l g R939Rxl SF60 OFWJ L ML s er F30 0 HA s grlz Ora Fmb 9F W99 gt ma a Fm 3fb2390rma laf r 9c gt PVDIV5TQP Or 2 3c 39Falc wall gon e F799 1 0 rmb395OFEalC1Bfgtllynbf9 9 gt g Mwsgafow 0 xVlefSJiQc 3 e MQ wg z 9 i rm C5wa 3 5 23 v N019 ail IMMSE Ofem may Mb 9 Fig5i We Hech J J LLi 9 2 man 7 39 6 9g gtEQE7 2 7 Q We Fm Cl Z7Co Pemow ha 3 OF 9 ad 3990 e 7 904 b F Ts JisAwEkdeoo LB game 0 x 7 E 3 3 9235zo ea 9 M z HenIa Tw5 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