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GT ISYE 6230 - 6230 Dynamic Games Incomplete

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11Equilibrium recap¡ Static games of complete informationl Nash equilibrium¡ Dynamic games of complete informationl Subgame-perfect Nash equilibrium¡ Static games of incomplete information (Bayesian games)l Bayesian Nash equilibrium¡ Dynamic games of incomplete informationl Perfect Bayesian equilibrium2Dynamic Games of Incomplete Information¡ Each player i has a type ti, where p(ti) is the probability that player i’s type is ti¡ At the beginning of the game, each player learns his type, but is given no information about his opponents’ types¡ (At each stage of the game) players simultaneously choose their actions¡ A strategy maps the set of possible types (and histories) into actions23Example 11LML`R`L`R`22(2,1)(0,0)(0,2)(0,1)R(1,3)1,31,3R0,10,2M0,02,1LR'L'Player 2Player 14Perfect Bayesian Equilibrium¡ Requirement 1: l At each information set, the player with the move must have a belief about which node in the information set has been reached by the play of the game¡ Requirement 2: l Given their beliefs, the players’ strategies must be sequentially rational. That is, at each information set, the action taken by the player with the move (and the player’s subsequent strategy) must be optimal given the player’s belief at that information set and the other players’ subsequent strategies (where a “subsequent strategy” is a complete plan of action covering every contingency that might arise after the given information set has been reached)35Example1LML`R`L`R`2[1-p](2,1)(0,0)(0,2)(0,1)R(1,3)[p]6Perfect Bayesian Equilibrium¡ Definition: l For a given equilibrium in a given extensive-form game, an information set is on the equilibrium path if it will be reached with positive probability if the game is played according to the equilibrium strategies and is off the equilibrium path if it is certain not to be reached if the game is played according to the equilibrium strategies (where “equilibrium” can mean Nash, subgame-perfect, Bayesian, or perfect Bayesian equilibrium)¡ Requirement 3: l At information sets on the equilibrium path, beliefs are determined by Bayes’ rule and the players’ equilibrium strategiesl Bayes’ Rule: P(A|B)=P(A,B)/P(B)47Perfect Bayesian Equilibrium¡ Requirement 4:l At information sets off the equilibrium path, beliefs are determined by Bayes’ rule and the players’ equilibrium strategies where possible¡ Definition: A Perfect Bayesian equilibrium consists of strategies and beliefs satisfying Requirements 1 through 4.8Example 22LML`R`L`R`3[1-p](1,2,1)(3,3,3)(0,1,2)(0,1,1)A(2,0,0)[p]1D1,11,2R3,32,1LR'L'Player 3Player 259Example¡ Unique subgame perfect NE of entire game is (D,L,R’)l With p = 1, satisfies R1 - R3l Also satisfies R4 since no information set is off the equilibrium path¡ Consider strategies (A,L,L’) with belief p=0l Satisfies R1 – R3 ¡ player 3 has a belief and thus acts optimally¡ players 1 & 2 act optimally given other player’s strategies¡ R1 – R3 don’t specify 3’s belief because 3’s information set is not reachedl Belief p = 0 is inconsistent with player 2’s strategy Ll R4 forces player 3’s belief to agree with player 2’s strategy, so this strategy does not meet requirements10Example 32LML`R`L`R`3[1-p]A[p]1DA’611Equilibrium recap¡ Static games of complete informationl Nash equilibrium¡ Dynamic games of complete informationl Subgame-perfect Nash equilibrium¡ Static games of incomplete information (Bayesian games)l Bayesian Nash equilibrium¡ Dynamic games of incomplete informationl Perfect Bayesian equilibrium12Recap on Information Asymmetries¡ Moral Hazard: l Agent has private information on actionsl Private information doesn’t better the agent, but the principal can be worse off¡ Adverse Selection: l Agent has private information on type or costsl Private information distorts contracts since the agent attempts to take advantage of it¡ Would you ever want to reveal private information?713Signaling¡ An agent may be willing to reveal private information if he obtains greater utility¡ Will not want to reveal information if the signal is too costly14Signaling Games¡ Nature draws a type tifor the Sender from a set of feasible types T={t1,…,tI} according to a probability distribution p(ti), where p(ti)>0 for every i and p(t1)+ …+p(t1)=1.¡ The Sender observes tiand then chooses a message mjfrom a set of feasible messages M={m1,…,mJ)¡ The Receiver observes mj(but not ti) and then chooses an action akfrom a set of feasible actions A={a1,…,aK}.¡ Payoffs are given by Us(ti,mj,ak) and UR(ti,mj,ak)815Signaling ExampleNLemon [0.5]High Quality [0.5]QSNo QSQSNo QS11buynot buybuynot buy2buynot buy2buynot buy2[p][q][1-q][1-p]16Signaling Game Pure Strategies¡ Sender:l Strategy 1: Play m1if nature draws either typel Strategy 2: Play m1if nature draws t1and play m2if nature draws t2l Strategy 3: Play m2if nature draws t1and play m1if nature draws t2l Strategy 4: Play m2if nature draws either type¡ Receiverl Strategy 1: Play a1no matter what message the Sender choosesl Strategy 2: Play a1if the Sender chooses m1and play a2if the sender chooses m2l Strategy 3: Play a2if the Sender chooses m1and play a1if the sender chooses m2l Strategy 4: Play a2no matter what message the Sender chooses917Signaling Requirements¡ Requirement 1: After observing any message mjfrom M, the Receiver must have a belief about which types could have sent mj. Denote this belief by the probability distribution µ(ti|mj), where µ(ti|mj) ¸ 0 for each tiin T, and ∑ti 2 Tµ(ti|mj)=1.18Signaling Requirements¡ Requirement 2R: For each mjin M, the Receiver’s action a*(mj) must maximize the Receiver’s expected utility, given the belief µ(ti|mj) about which types could have sent mj. That is, a*(mj) solves maxak 2 A∑ti 2 Tµ(ti|mj)UR(ti,mj,ak). ¡ Requirement 2S: For each tiin T, the Sender’s message m*(ti) must maximize the Sender’s utility, given the Receiver’s strategy a*(mj). That is, m*(ti) solvesmaxmj 2 MUS(ti,mj,a*(mj)).1019Signaling Requirements¡ Let Tjdenote the set of types that send the message mj¡ Requirement 3: For each mjin M, if there exists tiin T such that m*(ti)=mj, then the Receiver’s belief at the information set corresponding to mjmust follow from Bayes’rule and the Sender’s strategy: µ(ti|mj)=p(ti)/∑ti 2 Tjp(ti).¡ Definitiion: A pure-strategy perfect Bayesian equilibrium in a signaling game is a pair of strategies m*(ti) and a*(mj) and a belief µ(ti|mj) satisfying


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GT ISYE 6230 - 6230 Dynamic Games Incomplete

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