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# Note for ECON 309 at UMass

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Date Created: 02/06/15
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 Harsanvi 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 EXD0n39t Don39t EX Don39t Don39t E 1 21 21 1 1 1 11 3390 0 40 0 4 3 0 3 0 0 3 3 TT TT TT 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 0gt 11 Conclusion There are two pure strategy BNE One pooling the other sepa rating Enter Don t 9L Don t 9H 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 Don39 t 0L Don39 t 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 C2 yqz Whereye CLCH CH gtCL 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 a 2C Ey a 2C0CH1 0CL 3 a ZCH C 3 6 a ZC C 0 qf9 3 q0H CH CL q 0L BNE q qng q6L separating eguilibrium Comparison with the Cournot NE under Complete Info If y is known for sure then a 2Cj 7lql3 oc 2C lq2 3 Hence q 6Hgtq IFCH amp 126L lt q IFOL

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