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## Extensive Game with Imperfect Information

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**Extensive Game with Imperfect Information**Part I: Strategy and Nash equilibrium**Adding new features to extensive games:**• A player does not know actions taken earlier • non-observable actions taken by other players • The player has imperfect recall--e.g. absent minded driver • The “type” of a player is unknown to others (nature’s choice is non-observable to other players)**1**R L M 2 2 2,2 R L R L 0,0 0,2 1,1 3,1 Player 1’s actions are non-observable to Player 2**1,1**0,-1 -2,-1 0,-3 hire hire No hire No hire 2 2 MBA MBA 1/2 1/2 c 1 1 high low No MBA No MBA 2 2 hire No hire hire No hire 0,0 -2,2 0,0 1,2 Nature’s choice is unknown to third party**Extensive game with imperfect information and chances**• Definition: An extensive game <N,H,P,fc,(Ti),(ui)> consists of • a set of players N • a set of sequences H • a function (the player function P) that assigns either a player or "chance" to every non-terminal history • A function fc that associates with every history h for which P(h)=c a probability distribution fc(.|h) on A(h), where each such probability distribution is independent of every other such distribution. • For each player i, Ti is an information partition and Ii (an element of Ti) is an information set of player i. • For each i, a utility function ui.**Strategies**• DEFINITION: A (pure) strategy of player i in an extensive game is a function that assigns to each of i's information sets Ii an action in A(Ii) (the set of actions available to player i at the information set Ii). • DEFINITION: A mixed strategy of player i in an extensive game is a probability distribution over the set of player i’s pure strategies.**Behavioral strategy**• DEFINITION. A behavioral strategy of player i in an extensive game is a function that assigns to each of i's information sets Ii a probability distribution over the actions in A(Ii), with the property that each probability distribution is independent of every other distribution.**Mixed strategy and Behavioral strategy: an example**β1(φ)(L)=1; β1(φ)(R)=0; β1({(L,A),(L,B)})(l)=1/2; β1({(L,A),(L,B)})(r)=1/2**Mixed strategy choosing LL with probability ½ and RR with**½. The outcome is the probability distribution (1/2,0,0,1/2) over the terminal histories. This outcome cannot be achieved by any behavioral strategy. non-equivalence between behavioral and mixed strategy amid imperfect recall**Equivalence between behavioral and mixed strategy amid**perfect recall • Proposition. For any mixed strategy of a player in a finite extensive game with perfect recall there is an outcome-equivalent behavioral strategy.**Nash equilibrium**• DEFINITION: The Nash equilibriumin mixed strategies is a profile σ* of mixed strategies so that for each player i, ui(O(σ*-i, σ*i))≥ ui(O(σ*-i, σi)) for every σi of player i. • A Nash equilibrium in behavioral strategies is defined analogously.**1**R L M 2 2 2,2 R L R L 0,2 0,2 1,1 3,1 A motivating example Strategic game**(L,R) is a Nash equilibrium**According to the profile, 2’s information set being reached is a zero probability event. Hence, no restriction to 2’s belief about which history he is in. 2’s choosing R is optimal if he assigns probability of at least ½ to M; L is optimal if he assigns probability of at least ½ to L. Bayes’ rule does not help to determine the belief 1 R L M 2 2 2,2 R L R L 0,2 0,2 1,1 3,1 The importance of off-equilibrium path beliefs**belief**• From now on, we will restrict our attention to games with perfect recall. • Thus a sensible equilibrium concept should consist of two components: strategy profile and belief system. • For extensive games with imperfect information, when a player has the turn to move in a non-singleton information set, his optimal action depends on the belief he has about which history he is actually in. • DEFINITION. A belief system μ in an extensive game is a function that assigns to each information set a probability distribution over the histories in that information set. • DEFINITION. An assessment in an extensive game is a pair (β,μ) consisting of a profile of behavioral strategies and a belief system.**Sequential rationality and consistency**• It is reasonable to require that • Sequential rationality. Each player's strategy is optimal whenever she has to move, given her belief and the other players' strategies. • Consistency of beliefs with strategies (CBS). Each player's belief is consistent with the strategy profile, i.e., Bayes’ rule should be used as long as it is applicable.**Perfect Bayesian equilibrium**• Definition: An assessment (β,μ) is a perfect Bayesian equilibrium (PBE) (a.k.a. weak sequential equilibrium) if it satisfies both sequential rationality and CBS. • Hence, no restrictions at all on beliefs at zero-probability information set • In EGPI, the strategy profile in any PBE is a SPE • The strategy profile in any PBE is a Nash equilibrium**Sequential equilibrium**• Definition. An assessment (β,μ) is consistent if there is a sequence ((βn,μn))n=1,… of assessments that converge to (β,μ) and has the properties that each βn is completely mixed and each μn is derived from using Bayes’ rule. • Remark: Consistency implies CBS studied earlier • Definition. An assessment is a sequential equilibrium of an extensive game if it is sequentially rational and consistent. • Sequential equilibrium implies PBE • Less easier to use than PBE (need to consider the sequence ((βn,μn))n=1,… )**The assessment (β,μ) in which β1=L, β2=R and**μ({M,R})(M)= for any (0,1) is consistent Assessment (βε,με)with the following properties βε1 = (1-ε, ε,(1-)ε) βε2 = (ε,1- ε) με({M,R})(M)= for all ε As ε→0, (βε,με)→ (β,μ) For ≥1/2, this assessment is also sequentially rational. 1 R L M 2 2 2,2 R L R L 1,1 0,2 0,2 3,1 Back to the motivating example**1**1 R C L L M 2 R 2 M 2 2 3,3 R 3,3 L R L R L R L 0,0 1,0 5,1 0,1 0,0 1,0 5,1 0,1 Two similar games Game 1 has a sequential equilibrium in which both 1 and 2 play L Game 2 does not support such an equilibrium Game 1 Game 2**Structural consistency**• Definition. The belief system in an extensive game is structurally consistent if for each information set I there is a strategic profile with the properties that I is reached with positive probability under and is derived from using Bayes’ rule. • Remark: Note that different strategy profiles may be needed to justify the beliefs at different information sets. • Remark: There is no straightforward relationship between consistency and structural consistency. (β,μ) being consistent is neither sufficient nor necessary for μ to be structurally consistent.**Signaling games**• A signaling game is an extensive game with the following simple form. • Two players, a “sender’ and a “receiver.” • The sender knows the value of an uncertain parameter and then chooses an action m (message) • The receiver observes the message (but not the value of ) and takes an action a. • Each player’s payoff depends upon the value of , the message m, and the action a taken by the receiver.**Signaling games**• Two types • Signals are (directly) costly • Signals are directly not costly – cheap talk game**Spence’s education game**• Players: worker (1) and firm (2) • 1 has two types: high ability H with probability p Hand low ability L with probability p L . • The two types of worker choose education level e H and e L (messages). • The firm also choose a wage w equal to the expectation of the ability • The worker’s payoff is w – e/**Pooling equilibrium**• e H = e L = e* L pH (H - L) • w* = pHH + pLL • Belief: he who chooses a different e is thought with probability one as a low type • Then no type will find it beneficial to deviate. • Hence, a continuum of perfect Bayesian equilibria**Separating equilibrium**• e L = 0 • H (H - L) ≥ e H ≥L (H - L) • w H = H and w L = L • Belief: he who chooses a different e is thought with probability one as a low type • Again, a continuum of perfect Bayesian equilibria • Remark: all these (pooling and separating) perfect Bayesian equilibria are sequential equilibria as well.**The signal is costly**Single crossing condition holds (i.e., signal is more costly for the low-type than for the high-type) When does signaling work?**Refinement of sequential equilibrium**• There are too many sequential equilibria in the education game. Are some more appealing than others? • Cho-Kreps intuitive criterion • A refinement of sequential equilibrium—not every sequential equilibrium satisfies this criterion**Two sequential equilibria with outcomes: (R,R) and (L,L),**respectively (L,L) is supported by belief that, in case 2’s information set is reached, with high probability 1 chose M. If 2’s information set is reached, 2 may think “since M is strictly dominated by L, it is not rational for 1 to choose M and hence 1 must have chosen R.” 1 R L M 2 2 2,2 R L R L 5,1 0,0 0,0 1,3 An example where a sequential equilibrium is unreasonable**1,1**1,0 3,0 0,1 F N F N 2 Q Q 0.9 0.1 c 1 1 strong weak B B 2 N F N F 1,1 1,0 0,0 3,1 Beer or Quiche**If player 1 is weak she should realize that the choice for B**is worse for her than following the equilibrium, whatever the response of player 2. If player 1 is strong and if player 2 correctly concludes from player 1 choosing B that she is strong and hence chooses N, then player 1 is indeed better than she is in the equilibrium. Hence player 2’s belief is unreasonable and the equilibrium is not appealing under scrutiny. 1,1 1,0 3,0 0,1 F N F N 2 Q Q 0.9 0.1 c 1 1 strong weak B B 2 N F N F 1,1 1,0 0,0 3,1 Why the second equilibrium is not reasonable?**Spence’s education game**• All the pooling equilibria are eliminated by the Cho-Kreps intuitive criterion. • Let e satisfy w* – e*/ L > H – e/ L and w* – e*/ H > H – e/ L (such a value of e clearly exists.) • If a high type work deviates and chooses e and is correctly viewed as a good type, then she is better off than under the pooling equilibrium • If a low type work deviates and successfully convinces the firm that she is a high type, still she is worse off than under the pooling equilibrium. • Hence, according to the intuitive criterion, the firm’s belief upon such a deviation should construe that the deviator is a high type rather than a low type. • The pooling equilibrium break down!**Spence’s education game**• Only one separating equilibrium survives the Cho-Kreps Intuitive criterion, namely: e L = 0 and e H =L (H - L) • Why a separating equilibrium is killed where e L = 0 and e H >L (H - L)? • A high type worker after choosing an e slightly smaller will benefit from it if she is correctly construed as a high type. • A low type worker cannot benefit from it however. • Hence, this separating equilibrium does not survive Cho-Kreps intuitive criterion.