Extensive Form Abstract Economies and Generalized Perfect Recall

Transcription

1 Extensive Form Abstract Economies and Generalized Perfect Recall Nicholas Butler Princeton University July 30, 2015 Nicholas Butler (Princeton) EFAE and Generalized Perfect Recall July 30, / 1

2 Motivation Generalize the notion of an extensive form game to accommodate important non-standard models Games played by automata Games with rational inattention Multi-self models of decision making Provide a better understanding of perfect recall Relationship to generalized convexity Existence of equilibrium in behavioral strategies Generalizations to settings without an information partition Nicholas Butler (Princeton) EFAE and Generalized Perfect Recall July 30, / 1

3 Overview Extensive Form Games vs. Extensive Form Abstract Economies Perfect Recall and Convexity Generalized Perfect Recall Equilibrium in Extensive Form Abstract Economies Tightness of Generalized Perfect Recall Partitional Convexity and Rational Inattention Incredible Self Restraint and Generalized Perfect Equilibrium Nicholas Butler (Princeton) EFAE and Generalized Perfect Recall July 30, / 1

4 Extensive Form Games An Extensive Form Game is a tuple G xi, A, T, ρ, π 0, u, Hy I is a set of players A is a set of actions T is a tree whose elements are called nodes. ρ maps nodes to the player who acts at that node π 0 is the distribution of moves of nature u is a utility index H is an information partition Strategies in extensive form games are mappings from information sets to local strategies or actions. Nicholas Butler (Princeton) EFAE and Generalized Perfect Recall July 30, / 1

5 Extensive Form Abstract Economies An Extensive Form Abstract Economy (EFAE) is a tuple G xi, A, T, ρ, π 0, u, Qy where I, A, T, ρ, π 0, u are the same as in an extensive form game Q tq i u ipi is a collection of feasibility correspondences A feasibility correspondence maps the behavioral strategies of other players into a set of feasible strategies. Q i : B i ÝÑ 2 B i Strategies in an EFAE are mappings from nodes to local strategies or actions. Nicholas Butler (Princeton) EFAE and Generalized Perfect Recall July 30, / 1

6 Interpretation of Q Q describes constraints on players strategies resulting from Physical constraints Information acquisition (rational inattention) Finite automata Nonstandard consistency conditions Information partitions Budget constraints Every extensive form game can be expressed as an EFAE for an appropriately defined Q. Nicholas Butler (Princeton) EFAE and Generalized Perfect Recall July 30, / 1

7 Standard Strategies A standard pure strategy implements the same action at nodes in the same information set. Si s ph iq ts i P S i w 1 P hpwq ñ s i pwq s i pw 1 qu The support of a standard mixed strategy is a subset of Si sph iq. Σ s i ph iq tσ i P Σ i s i R Si s ph iq ñ σ i ps i q 0u A standard behavioral strategy implements the same local strategy at every node in the same information set. Bi s ph iq tb i P B i w 1 P hpwq ñ b i p wq b i p w 1 qu Set Q i Bi sph iq to implement an extensive form game as an EFAE Nicholas Butler (Princeton) EFAE and Generalized Perfect Recall July 30, / 1

8 Perfect Recall and Convexity 1 L R 1 1 L R L R σ 1 pl, Lq 1 2 σ 1pL, Rq 1 4 σ 1pR, Lq 1 6 σ 1pR, Rq 1 12 σ 1 1 pl, Lq 1 4 σ1 1 pl, Rq 1 12 σ1 1 pr, Lq 1 2 σ1 1 pr, Rq 1 6 Nicholas Butler (Princeton) EFAE and Generalized Perfect Recall July 30, / 1

9 Perfect Recall and Convexity The information set imposes a constraint on the behavioral strategies of the decision maker. b 1 pl, Lq σ 1 pl, Lq σ 1 pl, Lq ` σ 1 pl, Rq 2 3 σ 1 pr, Lq σ 1 pr, Lq ` σ 1 pr, Rq b 1pL Rq b1 1 σ pl, Lq 1 pl, Lq σ 1 pl, Lq ` σ 1 pl, Rq 3 4 σ1 1 pr, Lq σ1 1 pr, Lq ` σ1 1 pr, Rq b1 1 pl Rq However, the constraint is not satisfied for nontrivial convex combinations of the two mixed strategies. Let σ σ 1 ` 1 2 σ1 1 σ 2 pl, Lq σ 2 pl, Lq ` σ 2 pl, Rq 9 13 ă 8 11 σ 2 pr, Lq σ 2 pr, Lq ` σ 2 pr, Rq Nicholas Butler (Princeton) EFAE and Generalized Perfect Recall July 30, / 1

10 Notation B i - behavioral strategies of player i b i pa wq - probability of choosing action a at node w P i pwq - predecessors of w P T where player i acts apw 1, wq - the action taken at w 1 on the way to w Ppw xq - the probability of reaching w P T under strategy profile x Z - the set of terminal nodes of T Definition: x i x 1 i if and only if Ppz x i, x i q Ppz x 1 i, x iq for all z P Z and all strategy profiles of others x i Nicholas Butler (Princeton) EFAE and Generalized Perfect Recall July 30, / 1

11 Behavioral Convexity The node dependent weight modifier is P i pw b i q ź w 1 PP i pwq b i papw 1, wq w 1 q The node dependent mixing coefficient of player i is Λ i pb i, bi 1 λp, λ, wq The behavioral mixing function of player i is i pw b i q λp i pw b i q ` p1 λqp i pw b 1 i q B i pb i, b 1 i, λqpa wq Λ ipb i, b 1 i, λ, wqb ipa wq ` p1 Λ i pb i, b 1 i, λ, wqqb1 i pa wq Nicholas Butler (Princeton) EFAE and Generalized Perfect Recall July 30, / 1

12 Behavioral Convexity Definition: Behavioral Convexity A set V i Ă B i is behavioral convex if v i, v 1 i P V i implies B i pv i, v 1 i, λq P V i for every λ P r0, 1s Definition: Feasible Mixed Strategy A mixed strategy σ i is feasible if there exists a b i P Q i such that σ i b i Proposition 4: (Equivalence of Convexity Notions) The set of feasible mixed strategies is convex if and only if the set of behavioral strategies is behavioral convex Nicholas Butler (Princeton) EFAE and Generalized Perfect Recall July 30, / 1

13 Perfect Recall Definition: Perfect Recall H satisfies perfect recall if for every h P H (PR1) If w 1 P hpwq then w is not a predecessor of w 1 and conversely (PR2) If w 1 P hpwq and w is a predecessor of w then there exists w 1 P hpw 1 q such that w 1 is a predecessor of w 1 and ap w, wq ap w 1, w 1 q Kuhn (1953) showed that perfect recall is necessary and sufficient for mixed and behavioral strategies to be equivalent Nicholas Butler (Princeton) EFAE and Generalized Perfect Recall July 30, / 1

14 Behavioral Convexity and Perfect Recall Theorem 1: (Equivalence of Convexity and Perfect Recall) If G is irreducible then G satisfies perfect recall if and only if the set of behavioral strategies is behavioral convex. A tree is called irreducible if it contains no trivial nodes Perfect recall is necessary and sufficient for the set of feasible mixed strategies to be convex Nicholas Butler (Princeton) EFAE and Generalized Perfect Recall July 30, / 1

15 Theorem 1: Proof Sketch Perfect recall implies behavioral convexity Kuhn s Theorem implies mixed and behavioral strategies are equivalent Show that this implies the feasible mixed strategies are convex Apply Proposition 4 Behavioral convexity implies perfect recall Contrapositive: imperfect recall implies not behavioral convex Construct a non-convexity for any game with imperfect recall Strategy: consider separately absentmindedness, imperfect recall of actions, and imperfect recall of past knowledge Nicholas Butler (Princeton) EFAE and Generalized Perfect Recall July 30, / 1

16 Generalized Perfect Recall Perfect Recall ensures existence of an equilibrium in extensive form games Restricts information partitions Not applicable to EFAE Behavioral Convexity is equivalent to perfect recall in extensive form games Geometric restriction Applicable to EFAE Definition: Generalized Perfect Recall An extensive form abstract economy has generalized perfect recall if Q i is behavioral convex valued for every i P I Nicholas Butler (Princeton) EFAE and Generalized Perfect Recall July 30, / 1

17 Continuity Let X Ă R n and Y Ă R m, C : Y Ñ 2 X be a compact valued correspondence, and f : X ˆ Y Ñ R. Definition: Upper Hemicontinuous The correspondence C is upper hemicontinuous if for any y n Ñ y with x n P Cpy n q there exists a convergent subsequence of tx n u with limit in Cpyq. Definition: f -Lower Hemicontinuous The correspondence C is f -lower hemicontinuous if for any y n Ñ y and x P Cpyq, there exists ty nk u a subsequence of ty n u, and x nk P Cpy nk q such that f px nk, y nk q Ñ f px, yq. Nicholas Butler (Princeton) EFAE and Generalized Perfect Recall July 30, / 1

18 Maximum Theorem Definition: f -Continuous A correspondence is f -continuous if it is both upper hemicontinuous and f -lower hemicontinuous Lemma 1: (Modified Maximum Theorem) Let f : X ˆ Y Ñ R, be jointly continuous and C : Y Ñ 2 X be a compact valued correspondence. Let f pyq max xpcpyq f px, yq and C pyq arg max xpcpyq f px, yq. If C is f -continuous then f is continuous, and C is non-empty, compact valued, and upper hemicontinuous. The proof is similar to that of the standard maximum theorem Nicholas Butler (Princeton) EFAE and Generalized Perfect Recall July 30, / 1

19 Canonical EFAE Definition: Canonical Feasibility Correspondence A feasibility correspondence Q i is canonical if (Q1) Q i is compact and nonempty valued (Q2) Q i is U i -continuous (Q3) If b j bj 1 for all j i then Q i pb i q Q i pb 1 i q (Q1) + (Q2) ensures that the best response correspondence is upper hemicontinuous (Q3) is a relatively weak condition imposing equivalence classes on Q identical to those of B imposed by. Nicholas Butler (Princeton) EFAE and Generalized Perfect Recall July 30, / 1

20 Equilibrium in EFAE Definition: Equilibrium in Behavioral Strategies An equilibrium in behavioral strategies is a b P B such P I b i P arg max b i PQ i pb i q U i p b i, b i q Theorem 2: (Existence of Equilibrium in Behavioral Strategies) If G is finite, canonical, and satisfies generalized perfect recall then it has an equilibrium in behavioral strategies Nicholas Butler (Princeton) EFAE and Generalized Perfect Recall July 30, / 1

21 Theorem 2: Proof Sketch Transform then Optimize Transform into a normal form abstract economy Show that the hypotheses of the maximum theorem hold for mixed strategy best response correspondences Generalized perfect recall and Proposition 4 imply mixed strategy best response is convex Apply fixed point theorem and check feasibility of fixed point Optimize then Transform Apply maximum theorem to behavioral strategy best response correspondences Transform behavioral best response into mixed best response Show that the mixed best response satisfies hypotheses of fixed point theorem, and check feasibility of fixed point Nicholas Butler (Princeton) EFAE and Generalized Perfect Recall July 30, / 1

22 Tightness of Generalized Perfect Recall P1 L R P2 l r l r P2 l r l r l r l r P1 L R L R L R L R L R L R L R L R 0, 1 1, 0 1,0 1,0 1,0 0,1 1,0 1,0 1,0 1,0 0,1 1,0 1,0 1,0 1,0 0,1 Nicholas Butler (Princeton) EFAE and Generalized Perfect Recall July 30, / 1

23 Tightness of Generalized Perfect Recall Player 1 faces the restrictions b 1 pl Lllq b 1 pl Llrq b 1 pl Lrlq b 1 pl Lrrq b 1 pl Rllq b 1 pl Rlrq b 1 pl Rrlq b 1 pl Rrrq Player 2 faces the restrictions b 2 pl Llq b 2 pl Rlq b 2 pl Lrq b 2 pl Rrq b 2 pl Lq b 2 pl Rq ď ɛ For a given ɛ call this EFAE G ɛ For ɛ ą 0, player 2 does not have generalized perfect recall Nicholas Butler (Princeton) EFAE and Generalized Perfect Recall July 30, / 1

24 Tightness of Generalized Perfect Recall Proposition 5: (Tightness of Generalized Perfect Recall) If ɛ ą 0 then G ɛ does not have an equilibrium in behavioral strategies Fixing the strategy of player 2, player 1 can guarantee winning at least 75% of the time Fixing the strategy of player 1, player 2 can guarantee winning strictly more than 25% of the time Player 2 self signals with his first action. G 0 has a unique equilibrium in behavioral strategies where both players mix uniformly at every node Nicholas Butler (Princeton) EFAE and Generalized Perfect Recall July 30, / 1

25 Partitional Convexity M t1,..., mu P tp 1,..., P K u is a partition of M Z Ă R m If i, j P I are in the same partition element of P write i» j. Let z, z 1, z 2 P Z. Definition: Partitional Convexity The set Z is P-partitional convex if z, z 1 P Z and zi 2 λ i z i ` p1 λ i qzi 1 where λ P r0, 1s m such that i» j implies λ i λ j then z 2 P Z. Lemma 2: (Characterizing Partitional Convexity) The set Z is P partitional convex if and only if Z ś K k 1 Z k where each Z k Ă R P k is convex Nicholas Butler (Princeton) EFAE and Generalized Perfect Recall July 30, / 1

26 More Notation A partitional decomposition of a feasibility correspondence Q i is a pair p Q i, H i q such that Q i p q Q i p q X Bi s ph iq The pure strategies of player i that do not preclude w are denoted by, R i pwq ts i P S i w 1 P P i pwq ñ s i pw 1 q apw 1, wqu The partition of nodes induced by R i and H i is R i tw Ă T i w, w 1 P W ô R i pwq X Si s ph iq R i pw 1 q X Si s ph iqu Nicholas Butler (Princeton) EFAE and Generalized Perfect Recall July 30, / 1

27 R i -Partitional Convexity Proposition 6: (Partitional Convexity and Perfect Recall) If G is irreducible, an information partition H i satisfies perfect recall if and only if H i is finer than R i Proposition 7: (Partitional Convexity and GPR) If H i satisfies perfect recall and Q i is R i -partitional convex valued then Q i satisfies generalized perfect recall Nicholas Butler (Princeton) EFAE and Generalized Perfect Recall July 30, / 1

28 Constraint Partitions Let W i twi 1,..., W K i i u be a partition of T i called a constraint partition with elements called constraint sets. The local behavioral strategies of player i at constraint set Wi k ź ΔpA i pwqq B k i wpw k i A local feasibility correspondence is a Q k i : B i ÝÑ B k i. Any feasibility correspondence can be written as follows for some partition pair pw i, H i q is Q i Q i X B s i ph iq źk i k 1 Q k i X B s i ph iq Nicholas Butler (Princeton) EFAE and Generalized Perfect Recall July 30, / 1

29 Standard Partitions Definition: Standard Partition Pairs The partition pair pw i, H i q is standard if (S1) H i is finer than W i (S2) W i is finer than R i Proposition 8: (Standard Partitions and GPR) Any feasibility correspondence composed of convex valued local feasibility correspondences with a standard partition pair satisfies generalized perfect recall. Nicholas Butler (Princeton) EFAE and Generalized Perfect Recall July 30, / 1

30 Mutual Information The mutual information between nodes in a constraint set Wi k actions taken at that constraint set ApWi k q is IpW k i, ApW k i qq ÿ wpw k i ÿ apapwq and the b i pa wqpi k b pwqlogp i pa wq ř wpwi k b i pa wqpi k pwq q Where pi k is the conditional probability of arriving at w given strategy profile b, given what player i knows. Nicholas Butler (Princeton) EFAE and Generalized Perfect Recall July 30, / 1

31 Rational Inattention The set of nodes that can be reached under b i is Π i pb i q A capacity c ik P R` describes the amount of attention player i has to allocate at his kth constraint set. Definition: Rational Inattention Constraint A rational inattention constraint is a Q i satisfying $ & Qi k tbi k P Bi k IpWi k, ApWi k qq ď c ik u pb i q % B k i if W k i otherwise. Proposition 9: (Equilibrium with Rational Inattention) X Π i pb i q H If pw i, H i q is standard, then rational inattention constraints satisfy the hypotheses of Theorem 2. Nicholas Butler (Princeton) EFAE and Generalized Perfect Recall July 30, / 1

32 Incredible Self Restraint 1 L M R L R L R L R -4,-4 0,4 2,-1-4,-5 2,-1-4,-1 b 2 pl Mq b 2 pl Rq ď 1{2 An equilibrium of this EFAE is b 1 pl ɛq 1 b 2 pr Lq 1 b 2 pr Mq 1{2 b 2 pr Rq 1 Nicholas Butler (Princeton) EFAE and Generalized Perfect Recall July 30, / 1

33 Generalized Perfect Equilibrium The behavioral strategies which tremble with probability at least ɛ B i pɛq tb i P B i b i pa wq ě ɛu An ɛ-modified feasibility correspondence is a Q i p, ɛq Q i p q X B i pɛq G ɛ is the EFAE produced by replacing Q i p q with Q i p, ɛq Definition: Generalized Perfect Equilibrium A strategy profile b P B is a generalized perfect equilibrium if b is an equilibrium of G, and there exists ɛ n Ñ 0 and b n Ñ b such that b n is an equilibrium of G ɛn for all n. Nicholas Butler (Princeton) EFAE and Generalized Perfect Recall July 30, / 1

34 Existence of GPE Definition: Admits Trembling A feasibility correspondence Q i admits trembling if there exists δ ą 0 such that for ɛ ď δ the modified feasibility correspondence Q i p, ɛq is nonempty valued. Theorem: (Existence of GPE) If G is finite, canonical, admits trembling, and satisfies generalized perfect recall then it has a generalized perfect equilibrium. The equilibrium in the previous example is not a GPE since along any sequence player 2 must choose b 2 pr Mq 0 b 2 pr Rq ď 1{2 Nicholas Butler (Princeton) EFAE and Generalized Perfect Recall July 30, / 1

35 Conclusion In extensive form games perfect recall is equivalent to behavioral convexity Extensive form abstract economies generalize extensive form games by allowing for endogenous constraints with general form Normal Form Game Ñ Abstract Economy Extensive Form Game Ñ EFAE Equilibria exist in EFAE under mild technical conditions, and generalized perfect recall Generalized Perfect Equilibria exist in EFAE under an additional mild technical condition Needed to deal with the problem of incredible self restraint Nicholas Butler (Princeton) EFAE and Generalized Perfect Recall July 30, / 1