Extensive Form Abstract Economies and Generalized Perfect Recall

Transcription

1 Extensive Form Abstract Economies and Generalized Perfect Recall Nicholas Butler : Abstract I introduce the notion of an extensive form abstract economy (EFAE), a framework for modeling dynamic settings where players feasible strategies depend on those chosen by others. Extensive form abstract economies are motivated by models of limited strategic complexity, and nest a variety of existing frameworks including extensive form games, abstract economies, games played by finite automata, games with coupled constraints, and games with rationally inattentive players. I determine sufficient conditions for existence of an equilibrium in behavioral strategies in finite extensive form abstract economies, the most salient of which is a generalized convexity condition that is equivalent to perfect recall in extensive form games. I demonstrate that in extensive form abstract economies players may engage in incredible self restraint, a novel type of incredible threat. To rule out these implausible equilibria, I propose a refinement related to perfect equilibrium, and show that such an equilibrium exists under slightly stronger conditions. Date: November 3, This research was supported by a grant from the National Science Foundation SES : Princeton University. nob@princeton.edu 1

2 1 Introduction In the theory of noncooperative games, the most basic framework for modeling strategic situations is a normal form game. Extensive form games model the timing of players decisions and information explicitly, and are the natural dynamic extension of normal form games. Another variant of normal form games, where the strategies available to players depend on those chosen by others, are abstract economies introduced by Debreu (1952). This paper develops a theory of extensive form abstract economies (EFAE), a class of models combining these features. In extensive form games, the actions and local strategies available to a player at a given moment are completely determined by the structure of the game tree and information partition. Implicit in this framework lie the following assumptions. First, the available actions and local strategies at a given node depend exclusively on the realized history of actions. Second, the only permissible restrictions on local strategies across nodes are those induced by the exogenously specified by information partition, and hence impose equality. Both assumptions are relaxed in extensive form abstract economies. In lieu of an information partition, each player is endowed with a correspondence associating a set of feasible strategies to each strategy profile of others. Feasibility correspondences facilitate general endogenous restrictions on players strategies, rather than the exogenous equality constraints, and may be used to model a variety of salient phenomena including physical constraints, bounded rationality, complexity constraints, nonstandard consistency conditions, and endogenous information acquisition. The relationship between extensive form abstract economies and abstract economies is analogous to the relationship between extensive form games and normal form games. Specifically, a normal form game is an abstract economy with trivial feasibility correspondences, and an extensive form game is an extensive form abstract economy with feasibility correspondences induced by players information partitions. Extensive form abstract economies are a general framework for modeling dynamic strategic setting, and encompass a variety of nonstandard game theoretic models. In particular, EFAE are the natural framework for modeling dynamic strategic settings with endogenous complexity restrictions on players strategies. Games played by finite automata (Rubinstein [1986]; Abreu and Rubinstein [1988]) can be naturally represented as extensive form abstract economies, with feasibility correspondences derived from the structural assumptions imposed on the automata. Multi-self models of decision making with imperfect recall (Gilboa [1997]; Aumann, Hart, and Perry [1997]) are extensive form abstract economies, where players feasibility correspondences are derived from the consistency restrictions imposed on the various 2

3 selves. Many equilibrium refinements, including perfect equilibrium and subgame perfect equilibrium, can also be obtained as equilibrium of appropriately defined extensive form abstract economies. Dynamic general equilibrium models can be formulated as (infinite) extensive form abstract economies, where players feasibility correspondences are derived from combining their information sets and budget sets. A primary motivation for this paper is the distinct suitability of extensive form abstract economies for modeling strategic situations with endogenous information: where players acquire information subject to constraints determined in equilibrium. In these settings, unlike in extensive form games, the strategies available to players may depend on the strategies chosen by others and not the realized history alone. 1 In particular, this paper is motivated by the growing literature on games with rational inattention (Sims [1998]), where the mutual information between players strategies and the history is limited. Martin (2013) considers a game where consumers are rationally inattentive towards the quality of a product, and producers take this into account. Yang (2012) considers optimal security design and a coordination games with rationally inattentive players. In both papers, players are rationally inattentive towards moves of nature but not the actions of other players. I show that in an extensive form abstract economy feasibility correspondences can be constructed so that players are rationally inattentive towards both. The focus of this paper is on determining sufficient conditions for existence of an equilibrium in behavioral strategies in finite extensive form abstract economies. In finite extensive form games, the analogous condition is perfect recall. Kuhn (1953) showed that under perfect recall mixed and behavioral strategies are equivalent. Since an equilibrium in mixed strategies exists in any finite game, perfect recall is sufficient for existence of an equilibrium in behavioral strategies. However, there are apparently weaker conditions than perfect recall that are also sufficient for existence of an equilibrium in behavioral strategies. For instance, an equilibrium in behavioral strategies exists if players do not exhibit absentmindedness (Piccione and Rubinstein [1997]) and players behavioral strategies are equivalent to a convex set of mixed strategies. 2 The first major result of this paper, Theorem 1, implies that is that these conditions are in fact equivalent. In particular, I show that in extensive form games perfect recall is equivalent to players sets of behavioral strategies satisfying behavioral convexity, a novel generalized convexity condition. In addition, I prove that behavioral convexity is the weakest restriction on players behavioral strategies ensuring players feasible mixed strategies are convex, a necessary condition for applying standard fixed point theorems. 1 Attempting to model such situations as standard extensive form games can introduce conceptual difficulties regarding existence of equilibria and the interpretation of players strategies. 2 This condition is weaker than perfect recall since under perfect recall mixed and behavioral strategies are equivalent, and the set of mixed strategies is convex by construction. 3

4 Since behavioral convexity is a geometric property, it can be applied to the feasibility correspondences of extensive form abstract economies unlike perfect recall. Accordingly, a feasibility correspondence is said to satisfy generalized perfect recall if it is behavioral convex valued. The second major result of this paper, Theorem 2, provides sufficient conditions for existence of an equilibrium in behavioral strategies in an extensive form abstract economy, the most salient of which is generalized perfect recall. The remaining hypotheses include a novel notion of continuity, and other standard technical conditions. An extensive form abstract economy satisfying these ancillary hypotheses is said to be canonical. Since generalized perfect recall is a global condition, verifying that an arbitrary feasibility correspondence possesses generalized perfect recall is not straightforward. In response, I introduce the notion of partitional convexity, and use it to construct a simple sufficient condition for generalized perfect recall. Various families of feasibility correspondences, including those describing rational inattention, are shown to satisfy this condition. To verify that generalized perfect recall is tight, I construct a sequence of canonical EFAE with no equilibrium in behavioral strategies that converge to an extensive form game with perfect recall. This example is related to that of Wichardt (2008); however, they differ in two critical ways. First, players in my example have perfect recall of their own actions, whereas those of Wichardt (2008) do not. Second, since there are natural notions of convergence for feasibility correspondences but not information partitions, the implications of my example are stronger. I conclude by demonstrating that in an EFAE players may engage in incredible self restraint, a novel type of incredible threat. Fixing the strategy profile of others, players choose their behavioral strategies from feasibility sets that do not necessarily have a product structure across nodes in an EFAE. As a result, a player s choice at one node may affect his feasible local strategies at another. Players can exploit this by selecting strategies that constrain their choices off the equilibrium path, enabling them to make seemingly incredible threats. I show that naive extensions of standard perfection concepts do not rule this out, and propose a generalization of perfect equilibrium (Selten [1975]) which precludes these types of implausible equilibria. Theorem 3 provides sufficient conditions for existence of a generalized perfect equilibrium, which include the hypotheses of Theorem 2, and a minor condition ensuring players feasibility correspondences permit trembling. 4

5 2 Extensive Form Abstract Economies An extensive form abstract economy is a tuple G xi, A, T, ρ, b 0, u, Qy. The set of players is given by I t0,..., Iu, where 0 denotes nature. The set of actions is A, and the set of all sequences with members in A is denoted by A. The tree T Ă A, whose elements are referred to as nodes, is a finite set of sequences with members in A such that φ P T and if pa 1,..., a k q P T then pa 1,..., a k 1 q P T. The set of available actions at a particular node w P T is given by Apwq ta P A pw, aq P T u. Nodes with no available actions are called terminal nodes, and the set of all terminal nodes is given by Z. The player function ρ : T z Z Ñ I maps nonterminal nodes to the player who acts at that node. The set of nodes where player i acts is denoted by T i tw P T ρpwq iu. The probability of nature selecting an action a P A at a node w P T 0 is given by b 0 pa wq. The utility index u : Z Ñ R I maps terminal nodes to a utility value for each player. A pure strategy assigns an available action to every node where a player acts. The set of pure strategies of player i is given by the following. S i ź wpt i Apwq A mixed strategy is a probability distribution over the pure strategies. The set of mixed strategies of player i is given by Σ i ΔpS i q. 3 A behavioral strategy of player i assigns a local strategy b i p wq P ΔpApwqq to every node w P T i. The set of behavioral strategies of player i is given by the following. 4 B i ź wpt i ΔpApwqq The final component of an extensive form abstract economy is a collection of feasibility correspondences Q tq 1,..., Q I u. This paper focuses on the case where players feasibility correspondences are defined over their behavioral strategies, since behavioral strategies are the most natural notion in dynamic settings. Analogous results where players feasibility correspondences are defined over their mixed strategies can be found in the appendix. Formally, a feasibility correspondence Q i : B i Ñ 2 B i is a mapping from the behavioral strategies of players other than i to the sets of behavioral strategies of player i. Their interpretation is if other players choose to play b i P B i, then player i is compelled to choose some behavioral strategy b i P Q i pb i q. For a given strategy profile b P B, b i is called 3 The set of probability distributions over a sample space X is denoted by ΔpXq. 4 Define B ś ipi B i and B i ś j i B j, and similarly for mixed and pure strategies. 5

6 feasible if b i P Q i pb i q, and the profile b is called jointly feasible if b i is feasible for all i P I. Generally, players in an EFAE are only required to choose strategies that are feasible. More restrictive models where players take joint feasibility into account when considering a deviation can also be formulated as EFAE. These types of models are equivalent to imposing a single constraint in the space of behavioral strategy profiles, and are the extensive form analogues of games with coupled constraints introduced by Rosen (1965). Coupled constraints are natural when modeling physical restrictions on players strategies; however, the general EFAE framework is more natural in other applications. In particular, there is no reason a deviation which results in a jointly infeasible profile in a model of endogenous information acquisition ought to be disallowed. Rather, in such models deviations force players to revise their conjectures regarding the strategies of others, which subsequently affects their own sets of feasible strategies. 2.1 Extensive Form Games Extensive form games can be formulated as extensive form abstract economies by imposing appropriate feasibility correspondences. Formally, an extensive form game is a tuple G xi, A, T, ρ, b 0, u, Hy where the first six components are identical to those of an extensive form abstract economy. The final component of an extensive form game is a set of information partitions H th 1,..., H I u. Each information partition H i is a partition of T i, the nodes where player i acts. Elements of H i are called the information sets of player i, and the information set containing w P T is designated by hpwq. The sets of actions available at nodes in the same information set are assumed to be identical. The interpretation of an information set h P H i is that player i cannot distinguish between any w, w 1 P h. Since the tree rather than the information partition is the domain strategies in an EFAE, players strategies must be restricted when formulating an extensive form game as an extensive form abstract economy. A pure strategy is called standard if it implements the same action at nodes in the same information set. Given an information partition H i, the set of standard pure strategies of player i is denoted by S i ph i q. A mixed strategy is called standard if its support is a subset of S i ph i q, and the set of standard mixed strategies is denoted by Σ i ph i q. Similarly, a behavioral strategy is called standard if it implements identical local strategies at nodes in the same information set. The set of standard behavioral strategies is given by B i ph i q. By construction, standard strategies are precisely the strategies available in extensive form games. Definition 1. A feasibility correspondence is behavioral standard if Q i p q B i ph i q, and mixed standard if Q i p q Σ i ph i q. 6

7 Any extensive form game can be formulated as an extensive form abstract economy using either behavioral standard or mixed standard feasibility correspondences. The choice between these two conventions is left to the modeler, and depends on the natural notion of a strategy in a particular setting. 3 Behavioral Convexity and Perfect Recall In dynamic settings, behavioral strategies are often more natural than mixed strategies, since nodes represent points of decision making under the former, rather than points of execution under the latter. However, standard fixed point arguments cannot be applied to behavioral strategy best responses, since they are not generally convex valued. Consequently, the natural approach to proving existence of an equilibrium in behavioral strategies involves determining sufficient conditions for players behavioral strategies to be equivalent to their mixed strategies. Since an equilibrium in mixed strategies exists in any finite game, such a condition is also sufficient for existence of an equilibrium in behavioral strategies. For a set of strategy profiles X, denote the probability of reaching a node w P T when players follow the strategy profile x P X by P pw xq. Definition 2. x i, x 1 i P X are equivalent, denoted by x i x 1 i, if P pz x i, x i q P pz x 1 i, x i q for every z P Z and every strategy profile of others x i P X i. Definition 3. X i and Y i are equivalent if for every x i P X i there exists y i P Y i such that x i y i and conversely. A node w P T is called a predecessor of w 1 P T if there exists an a P A such that a φ and w 1 pw, a q. The set of predecessors of w where player i acts is denoted by P i pwq. If w is a predecessor of w 1, the action taken at w on the path to w 1 is given by apw, w 1 q. The pure strategies of player i that can reach a node w P T are given by the following. R i pwq ts i P S i w 1 P P i pwq ñ s i pw 1 q apw 1, wqu Intuitively, if s i P R i pwq then player i chooses the action on the path to w wherever possible under s i. For an extensive form abstract economy with behavioral standard feasibility correspondences, a mixed strategy σ i P Σ i is called feasible if there exists a b i P B i ph i q such that σ i b i, and similarly for mixed standard feasibility correspondences. It is straightforward 7

8 to verify that if σ i P Σ i and b i P B i then σ i b i if and only if the following holds. s i PR i pwq σ i ps i q ą 0 ñ b i pa wq s i PR i pwq,s i pwq a σ ips i q s i PR i pwq σ ips i q (1) Kuhn (1953) showed that for extensive form games the sets of mixed and behavioral strategies are equivalent if and only if the players have perfect recall, defined as follows. Definition 4. A player has perfect recall if for every h P H i (R1) If w 1 P hpwq then w is not a predecessor of w 1 and conversely. (R2) If w 1 P hpwq and w P P i pwq, then there exists w 1 P hpw 1 q such that w 1 P P i pw 1 q and ap w, wq ap w 1, w 1 q Informally, a player has perfect recall if at every node he remembers his past actions and past knowledge. A player whose information partition violates (R1) is said to be absentminded, and is said to have imperfect recall if his information partition violates either (R1) or (R2). Bonanno (2004) shows that (R2) can be decomposed into separate conditions describing perfect recall of past actions and perfect recall of past knowledge, a relevant distinction in subsequent sections, 3.1 Imperfect Recall and Non-Convexity In this section, I examine three simple decision problems exhibiting different types of imperfect recall. In each example, players feasibility correspondences are assumed to be behavioral standard, and payoffs are omitted from the terminal nodes as they are unnecessary to illustrate non-convexity of players sets of feasible mixed strategies. Consider the extensive form depicted in Figure 1, where the decision maker cannot distinguish between his initial actions upon reaching his second information set. 1 L R 1 1 L R L R Figure 1: Imperfect Recall of Past Actions 8

9 Proposition 1. The set of feasible mixed strategies of Figure 1 is not convex. Proof. The pure strategies of the decision maker are S 1 tpl, Lq, pl, Rq, pr, Lq, pr, Rqu, where actions at unreached nodes have been omitted. The second information set requires that b 1 pl Lq b 1 pl Rq. Consider the following mixed strategies: σ 1 pl, Lq 1 2 σ 1pL, Rq 1 4 σ 1pR, Lq 1 6 σ 1pR, Rq 1 12 σ 1 1pL, Lq 1 4 σ1 1pL, Rq 1 12 σ1 1pR, Lq 1 2 σ1 1pR, Rq 1 6 Let b 1 σ 1 and b 1 1 σ 1 1. By (1) the following must hold 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 b 1 1pL Lq σ 1 pl, Lq σ 1 pl, Lq ` σ 1 pl, Rq 3 4 σ 1 1pR, Lq σ 1 1pR, Lq ` σ 1 1pR, Rq b1 1pL Rq So both mixed strategies are feasible. However, a convex combination of these mixed strategies may not be feasible. Let σ σ 1 ` 1 2 σ1 1 and b 2 1 σ 2 1. By (1) b 2 1pL Lq 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 b2 1pL Rq Non-convexity arises due to the conjunction of two factors. reaching nodes in the second information set differs under σ 1 and σ 1 1. First, the probability of Second, the local strategies induced by σ 1 and σ 1 1 at the second information set are different. Intuitively, σ 1 1 has more influence over the local strategy induced by σ 2 1 at R than σ 1 since the probability of reaching R is higher under σ 1 1 than σ 1. 1 R R D D Figure 2: Absentmindedness 9

10 Non-convexity is not limited to games with imperfect recall of past actions. Consider the extensive form of the well known Paradox of the Absentminded Driver depicted in Figure 2. This decision problem has two standard pure strategies, namely playing R at both nodes or D at both nodes. The player s set of feasible mixed strategies is not convex since under any mixed strategy placing nonzero probability on both, the player always chooses R at the second node but chooses D with positive probability at the first. Proposition 2. The set of feasible mixed strategies of Figure 2 is not convex. 0 1{2 1{2 1 1 Out In In Out L R L R Figure 3: Imperfect Recall of Past Knowledge In the decision problem depicted in Figure 3, the player has imperfect recall of his past knowledge. Intuitively, since the nodes l and r are members of distinct information sets, the player initially knows the action taken by nature. However, after choosing In and arriving at his subsequent information set he forgets the action of nature, but remembers his initial action. Non-convexity arises in this example for similar reasons as in the first. Specifically, if two feasible mixed strategies select In with different probabilities at l and r, and induce different local strategies at the subsequent information set, a nontrivial convex combination of the two is not feasible. Proposition 3. The set of feasible mixed strategies of Figure 3 is not convex. 3.2 Behavioral Convexity The preceding examples are suggestive of a relationship between convexity and perfect recall in extensive form games. In this section, I demonstrate that this relationship holds generally. The primary result of this section, Theorem 1, states that in essentially any extensive 10

11 form game a player has perfect recall if and only if his set of standard behavioral strategies is behavioral convex. Moreover, I prove that behavioral convexity of players standard behavioral strategies is equivalent to convexity of their sets of feasible mixed strategies. Definition 5. A tree is irreducible if it contains no nodes with only one available action. An extensive form game or extensive form abstract economy is called irreducible if its constituent tree is irreducible. Intuitively, a node with only one available action can be pruned from the tree without any loss of generality since the action at that node is predetermined. The probability of reaching a node w P T given that the other players choose the action on the path to w with unit probability wherever possible is given by the following. P i pw b i q ź w 1 PP i pwq b i papw 1, wq w 1 q (2) The node dependent mixing coefficient of player i, denoted by Λ i : B i ˆ B i ˆ r0, 1s ˆ T i Ñ r0, 1s, is defined whenever P i pw b i q ą 0 or P i pw b 1 iq ą 0 as follows. Λ i pb i, b 1 i, λ, wq λp i pw b i q λp i pw b i q ` p1 λqp i pw b 1 i q (3) If both P i pw b i q 0 and P i pw b 1 iq 0, then Λ i pb i, b 1 i, λ, wq λ. The behavioral mixing function of player i, denoted by B i : B i ˆ B i ˆ r0, 1s Ñ B i, is defined componentwise using (3) as follows. B i pb i, b 1 i, λqpa wq Λ i pb i, b 1 i, λ, wqb i pa wq ` p1 Λ i pb i, b 1 i, λ, wqqb 1 ipa wq (4) A behavioral convex combination of b i, b 1 i P B i is a behavioral strategy b 2 i B i pb i, b 1 i, λq for some λ P r0, 1s. By equation (4) each component of the behavioral mixture B i pb i, b 1 i, λq is a convex combination of components of b i and b 1 i, with weights given by Λ i. In general, B i pb i, b 1 i, λq is not a convex combination of b i and b 1 i, since the mixing coefficient Λ i is node dependent, and adjusts for the relative probability of reaching w under b i versus b 1 i. Definition 6. V i Ă B i is behavioral convex if for every v i, v 1 i P V i and λ P r0, 1s there exists a behavioral convex combination of v i and v 1 i that is an element of V i. Proposition 4. The set of feasible mixed strategies is convex if and only if the set of behavioral standard strategies is behavioral convex. This result suggests behavioral convexity is a natural notion of generalized convexity for behavioral strategies, and implies the sets of standard behavioral strategies of the preceding 11

12 examples are not behavioral convex. Moreover, the relationship between convexity and perfect recall observed in the previous examples holds generally, as the following theorem demonstrates. Theorem 1. If G is irreducible then the players have perfect recall if and only if their sets of behavioral standard strategies are behavioral convex. The proof of sufficiency is straightforward. Perfect recall implies the set of feasible mixed strategies is equivalent to the set of mixed strategies, which is convex by construction. To complete the proof, I show that the set of feasible mixed strategies is convex and apply proposition 4. The proof of necessity mirrors the examples of the previous section. For each type of imperfect recall: absentmindedness, imperfect memory of past actions, and imperfect memory of past knowledge, feasible behavioral strategies that have infeasible behavioral convex combinations are constructed. The primary significance of Theorem 1 is it establishes a geometric characterization of perfect recall, which can be extended to the more general EFAE framework. By elucidating its underlying geometry, these results imply that perfect recall is the weakest condition which ensures players feasible mixed strategy best response correspondences are convex valued. In this sense, perfect recall is tight regarding existence of equilibrium in behavioral strategies in extensive form games, since convex valuedness is a necessary condition for applying many standard fixed point theorems. 4 Equilibrium in Extensive Form Abstract Economies In this section I introduce the notion of an equilibrium in behavioral strategies for extensive form abstract economies, and determine sufficient conditions for existence of such an equilibrium. Analogous results for mixed strategies can be found in the appendix. The probability of reaching a node w under the behavioral strategy profile b is given by. P pw bq ź ź b i papw 1, wq w 1 q ipi w 1 PP i pwq The expected utility of player i under b is given by. U i pbq zpz P pz bqu i pzq. 12

13 Definition 7. An equilibrium in behavior strategies is a b P B such that for every i P I b i P arg max bi PQ i pb i q U i p b i, b i q An equilibrium in behavioral strategies in extensive form abstract economies is the natural analogue of a Nash equilibrium in behavioral strategies in extensive form games. In equilibrium, players take the strategy profiles of the others as given and choose feasible behavioral strategies to maximize their expected utility. 4.1 Existence of Equilibrium Let X Ă R n and Y Ă R m endowed with the usual topology, C : Y Ñ 2 X be a correspondence, and f : X ˆ Y Ñ R. Definition 8. A 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 fpx nk, y nk q Ñ fpx, yq. This definition is identical to the sequential characterization of lower hemicontinuity, except it requires that fpx nk, y nk q Ñ fpx, yq, rather than x nk Ñ x. Notice that if C is lower hemicontinuous and f is jointly continuous then C is f-lower hemicontinuous. Intuitively, if C is f-lower hemicontinuous then the values of f that are achievable in the graph of C do not change discontinuously in the limit. This weaker notion of continuity is useful since some interesting feasibility correspondences, including those describing rational inattention, are U i -lower hemicontinuous but not lower hemicontinuous. Definition 9. A compact valued 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. A correspondence is called f-continuous if it is both upper hemicontinuous and f-lower hemicontinuous. The following lemma establishes a generalization of the maximum theorem for maximizing f over a f-continuous feasibility correspondence. Lemma 1. 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 fpx, yq and C pyq arg max xpcpyq fpx, yq. If C is f-continuous then f is continuous, and C is non-empty, compact valued, and upper hemicontinuous. The proof of Lemma 1 is similar to standard proofs of the maximum theorem. Many of its hypotheses could be weakened; however, the form above is sufficient for the purposes of this paper. In particular, Lemma 1 remains true if f-continuity is weakened to apply only to the constrained maximizers of f. 13

14 Definition 10. A feasibility correspondence Q i is canonical if it satisfies the following, (Q1) Q i is compact and nonempty valued (Q2) Q i is U i -continuous (Q3) If b j b 1 j for all j i then Q i pb i q Q i pb 1 iq An extensive form abstract economy is called canonical if each players feasibility correspondence is canonical. (Q1) is a standard technical condition ensuring that players best response correspondences are well defined. Since U i is continuous, (Q2) is weaker than standard continuity. By Lemma 1, (Q1) and (Q2) ensure that players best response correspondences are upper hemicontinuous. While these conditions are primarily technical, they do have natural interpretations. (Q1) has the natural interpretation that regardless of what the others do, players always have an option to do something. (Q2) has the interpretation that players feasible behavioral strategies and achievable utilities do not change discontinuously when other players change their strategies. (Q3) imposes equivalence classes on Q i which correspond to the equivalence classes of B i induced by. This restriction is rather weak, since equivalent behavioral strategies can only differ at nodes that are never reached under strategy profile of other players. Intuitively, (Q3) requires that a player s claim regarding his actions at nodes which cannot be reached based on his strategy alone has no effect on the feasible strategies of other players. In the absence of (Q3), players could costlessly manipulate other players sets of feasible strategies. 5 Definition 11. A player has generalized perfect recall if his feasibility correspondence is behavioral convex valued. An extensive form abstract economy satisfies generalized perfect recall if each player has generalized perfect recall, and if the feasibility correspondence of a player is not behavioral convex valued then he is said to have generalized imperfect recall. These definitions are inspired by Theorem 1, which implies that an extensive form game has perfect recall if and only if players behavioral standard feasibility correspondences are behavioral convex valued. In addition, generalized perfect recall in EFAE plays the same role as perfect recall in extensive form games regarding existence of an equilibrium in behavioral strategies, as it ensures players feasible mixed strategy best response correspondences are convex valued. Theorem 2. If an EFAE is finite, canonical, and satisfies generalized perfect recall then it has an equilibrium in behavioral strategies. 5 These manipulations are not only costless holding the strategy profile of the other players fixed, but also costless for any strategy profile of the other players. 14

15 The proof of Theorem 2 is relatively straightforward. I begin by associating a normal form abstract economy with the EFAE by transforming players feasibility correspondences into the space of mixed strategies. Players best response correspondences are well defined since their feasibility correspondences are canonical, and convex valued since they have generalized perfect recall. Care is required to ensure that players transformed feasibility correspondences satisfy the hypotheses of Lemma 1, which in turn implies that players mixed strategy best response correspondences are upper hemicontinuous. Applying a standard fixed point theorem to players mixed strategy best response correspondences, and confirming that the fixed point is feasible, yields an equilibrium of the normal form abstract economy. The primary difficulty of this approach lies in ensuring an equilibrium in mixed strategies of the associated normal form abstract economy yields an equilibrium in behavioral strategies in the EFAE, since the mapping between equivalent mixed and behavioral strategies is not bijective. (Q3) is sufficient to show this is indeed the case. A second approach involves applying Lemma 1 to players behavioral strategy best responses, and subsequently mapping these correspondences into the space of mixed strategies. The primary difficulty of this method lies in ensuring the mapping preserves upper hemicontinuity. and that the resulting mixed strategy best response correspondences is convex valued. The latter is a consequence of the following lemma. Lemma 2. U i pbpb i, b 1 i, λq, b i q λu i pb i, b i q ` p1 λqu i pb 1 i, b i q Intuitively, lemma 2 implies utility is linear in behavioral convex combinations of behavioral strategies, and suggests behavioral convexity is the natural notion of generalized convexity for behavioral strategies. Lemma 2 also ensures players behavioral strategy best responses are behavioral convex valued, which by proposition 4 implies their mixed strategy best response is convex valued. Finally, applying a fixed point theorem and transforming back into the space of behavioral strategies yields the desired equilibrium. 5 Tightness of Generalized Perfect Recall Since Theorem 2 provides sufficient conditions for existence of an equilibrium, a natural question is whether these conditions are tight. In this section, I examine the tightness generalized perfect recall. Fixing an extensive form, I construct a sequence of canonical EFAE with generalized imperfect recall that have no equilibria in behavioral strategies, which converge to an EFAE with generalized perfect recall. Consider the extensive form abstract economy depicted in Figure 4. Dashed lines indicate equality constraints, and the dashed enclosure indicates a looser constraint specified below. 15

16 P1 L R l r P2 l r l r l r P2 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 Figure 4: Tightness of Generalized Perfect Recall The feasibility correspondence of player 2 requires the following. 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, The feasibility correspondence of player 1 requires the following. 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 For ɛ ě 0 denote the EFAE depicted in Figure 4 by G ɛ. These EFAE describe type of sequential matching pennies. Player 2 wins if his second action matches the first action of player 1, and his first action matches the second action of player 1. The feasibility correspondences of both players are canonical; however, for ɛ ą 0 player 2 has generalized imperfect recall. Player 2 effectively forgets his past knowledge, since he is capable of (partially) conditioning his first action on the first action of player 1, but is unable to do so at subsequent nodes. Since player 2 seeks to match his second action to the first action of player 1, he can use his first action to signal to his future self; however, this may come at the cost of poorly predicting the second action of player 1. Fixing a strategy of player 1, player 2 best responds by encoding his partial knowledge of player 1 s first action by choosing different local strategies at L and R. Since he remembers his first action when choosing his second, player 2 is capable of partially inferring the first action of player 1 at his second node. However, given a fixed signalling scheme, player 1 16

17 has an incentive to deviate so that his second action is predicted poorly. For this reason, an equilibrium in behavioral strategies does not exist unless this unusual signalling capability is completely suppressed. Proposition 5. G ɛ does not have an equilibrium in behavioral strategies for any ɛ ą 0. The proof of proposition 5 proceeds in two steps. It is straightforward to show that for any ɛ ą 0 and any behavioral strategy of player 1, player 2 can secure a payoff of strictly more than 1/4. The more challenging step, which relies on an inductive argument, is to show that player 1 can achieve a payoff of at least 3/4 for any strategy of player 2. This implies it is impossible for both players to be simultaneously best responding, since the total payoff is always 1. Proposition 5 is striking as it is stronger than previously known examples of extensive form games without equilibria in behavioral strategies. While there is no natural notion of convergence for information partitions, there are for feasibility correspondences. Consider the following notion of distance, DpQ i, Q i q inf Q i PQ i sup d H pq i pb i q, Q i pb i qq (5) b i PB i where Q i is the set of feasibility correspondences of player i which satisfy generalized perfect recall, d is a metric inducing the usual topology on B i, and d H given by the following. d H px, Y q maxtsup inf xpx ypy dpx, yq, sup ypy inf dpx, yqu xpx is the Hausdorff distance The distance function D measures the degree to which Q i departs from generalized perfect recall. Since G 0 has generalized perfect recall it is straightforward to verify that if ɛ Ñ 0 then DpQ ɛ 2, Q 2 q Ñ 0. This implies there are canonical EFAE that are arbitrarily close to satisfying generalized perfect recall that have no equilibrium in behavioral strategies, and in this sense generalized perfect recall is tight. This example also demonstrates that care must be exercised when analyzing EFAE that do not satisfy generalized perfect recall, since even arbitrarily small departures may lead to non-existence of equilibria. 6 Partitional Convexity and Endogenous Information Since feasibility correspondences impose global constraints on players behavioral strategies, determining whether an arbitrary EFAE satisfies generalized perfect recall may be 17

18 difficult, whereas verifying that an information partition satisfies perfect recall is relatively simple. In response, I provide a simple sufficient condition for generalized perfect recall which parallels the definition of perfect recall, based on the notion of partitional convexity. Like perfect recall in extensive form games, this sufficient condition imposes structure on the partitions restricting players feasible strategies. However, this condition does not require that players choose the same local strategy at nodes in the same partition element, but instead requires the sets of feasible local strategies within each partition element to be convex. Using this sufficient condition, various families of natural feasibility correspondences, including those describing rational inattention, are shown to satisfy the hypotheses of Theorem Partitional Convexity Let Z Ă R m, I t1,..., mu and P tp 1,..., P K u be a partition of I. If i, j P I are contained the same partition element denote this by i» j. Definition 12. A P -partitional convex combination of z, z 1 P Z is a z 2 P Z such that z 2 i λ i z i ` p1 λ i qz 1 i for some λ P r0, 1s m satisfying if i» j then λ i λ j. A P -partitional convex combination is formed by taking convex combinations of the components of z and z 1 at every equivalence class specified by P. A set Z Ă R m is called P -partitionally convex if z, z 1 P Z implies every P -partitional convex combination of z and z 1 is an element of Z. Partitional convexity imposes significant structure on the form of Z, as the following lemma demonstrates. Lemma 3. A set Z Ă R m is P partitional convex if and only if Z ś K k 1 Zk where each Z k Ă R P k is convex. Intuitively, a P -partitionally convex set must have a product structure that respects the partition P. The proof of necessity is immediate, and sufficiency is obtained through an inductive argument on K. Notice that any behavioral standard feasibility correspondences is H i -partitionally convex valued by construction. 6.2 Partitional Convexity and Generalized Perfect Recall A partitional decomposition of a feasibility correspondence Q i is a pair p Q i, H i q satisfying Q i Q i X B i ph i q. Partitional decompositions are not generally unique and exist for any Q i, since H i can be taken to be the set singletons. An extensive form abstract economy with a nontrivial partitional decomposition can be interpreted as a behavioral standard 18

19 extensive form game with residual constraints specified by Q i. In settings with endogenous information acquisition, H i describes the nodes that are indistinguishable a priori, whereas Q i describes the process by which players can partially differentiate certain nodes upon acquiring information. Perfect recall is the relevant restriction on players information partitions for ensuring existence of an equilibrium in extensive form games. An analogous restriction for extensive form abstract economies is generated by the following partition. R i tw Ă T i w, w 1 P W ô R i pwq X S i ph i q R i pw 1 q X S i ph i qu (6) Each element of R i is a maximal set of nodes which can be reached under the same standard pure strategies of player i. Intuitively, player i cannot distinguish between nodes contained in an element of R i based on his past actions or past knowledge. However, he may be able to partially distinguish between these nodes based on his current knowledge or information he gathers. This suggests a relationship between R i and perfect recall, which is given by the following proposition. Proposition 6. If G is irreducible then H i satisfies perfect recall if and only if H i is finer than R i. Proposition 6 is analogous to Theorem 1, and implies that R i characterizes the coarsest information partitions of player i which satisfy perfect recall. Intuitively, a residual constraint is R i -partitional convex valued if local strategies chosen at one node have no effect on the feasible local strategies at another node unless the information sets and actions taken along the path to both nodes are identical. Moreover, any restriction on the local behavioral strategies must be convex within each partition element. The relationship between R i - partitional convexity and generalized perfect recall is as follows. Proposition 7. If H i satisfies perfect recall and Q i is R i -partitionally convex valued then Q i Q i X B i ph i q satisfies generalized perfect recall. 6.3 Endogenous Information Constraints Extensive form abstract economies are the natural framework for modeling strategic situations in which players devote a fixed quantity of resources or attention towards gathering information. Players fitting this description include government agencies and divisions of companies with a fixed budget, and individuals with a fixed attention capacity. In this section, I propose two natural families of feasibility correspondences and show that they satisfy the hypotheses of Theorem 2. For both types, players are generally not free to 19

20 choose an arbitrary behavioral strategy. Rather, the degree to which players local strategies may depend on the realized node is limited and dependent on the strategies of others. For this reason both types of constraints can be viewed as describing endogenous information acquisition or processing. The constraint partition of player i is a partition of T i denoted by W i tw 1 i,..., W K i i u, with elements called constraint sets. A partition trio is a triple pr i, W i, H i q. A partition X is said to be finer than a partition Y, if for any x P X there is a y P Y such that x Ă y. Definition 13. A partition trio is standard if W i is both finer than R i and coarser than H i. By Proposition 6, H i satisfies perfect recall in any partition trio, since H i is finer than R i. The set of local behavioral strategies of player i at constraint set Wi k following. B k i ź wpw k i ΔpA i pwqq is given by the A local feasibility correspondence is a Q k i : B i Ñ Bi k. For the remainder of this section feasibility correspondences are assumed to be of the following form. Q i p q Q źk i i p q X B i ph i q Q k i p q X B i ph i q A feasibility correspondences of this form can be interpreted as follows. Upon arriving at a node, a player cannot immediately determine which node in the corresponding constraint set has realized. Moreover, his behavioral strategy at nodes outside the constraint set do not influence the local behavioral strategies available to him at nodes in the constraint set and conversely. If the local feasibility correspondence were to impose equality then the constraint set is identical to an information set. However, imposing a nontrivial local feasibility correspondence can allow the player s local behavioral strategies to depend on the realized history to some extent. This can be interpreted as describing either information acquisition or information processing. Under the former interpretation the realized history is hidden from the player, but he has some capacity to investigate. Under the latter, information regarding the realized history is readily available; however, he is unable to map this information into actions with arbitrary precision due to limited cognitive capability, bounded rationality, or limitations in the channel of transmission. k 1 20

21 6.3.1 Rational Inattention Constraints The model of player i at constraint set Wi k, denoted by p k i, is the distribution over nodes in Wi k induced by the conjectured strategy profile conditional on everything player i knows upon arriving at Wi k. The mutual information between a constraint set Wi k and the corresponding local behavioral strategy of player i is given by the following. 6 IpW k i, ApW k i qq wpw k i b i pa wqp k i pwqlogp apapwq wpwi k b i pa wq b i pa wqp k i pwqq A constraint set Wi k is said to be on the path of play of b if there exists a node w P Wi k such that P pw bq ą 0, and on the potential path of play of b i if there exists a b i P B i such that it is on the path of play of pb i, b i q. The constraint sets of player i that are on the potential path of b i are denoted by Π i pb i q. Definition 14. A rational inattention constraint is a vector of capacities c i P R K i ` feasibility correspondence Q i satisfying, $ & tb k Q k i P Bi k i pb i q % B k i IpW k i, ApW k i qq ď c ik u, if W k i P Π i pb i q. otherwise. and a For each constraint set, the capacity specifies the maximum allowable mutual information between players local behavioral strategy and the history of actions up to that point. The assumption that players are unconstrained off the path of play ensures that Q i is upper hemicontinuous, but it also has an intuitive interpretation. Since players models are not pinned down by Baye s rule off the path of play, any model is permissible. It is straightforward to verify that if p k i pwq 1 for some w P Wi k, then the mutual information between any local behavioral strategy and Wi k is zero. Since the capacities are assumed to be weakly positive any local behavioral strategy is feasible, and therefore no behavioral strategy can be excluded off the potential path of play without making additional assumptions regarding players off path models. Proposition 8. If G is irreducible and pr i, W i, H i q is standard then rational inattention constraints satisfy the hypotheses of Theorem 2. The proof of Proposition 8 relies on Proposition 7 and the fact that mutual information is a convex function of the conditional distribution. Rational inattention constraints are not lower hemicontinuous since they are unconstrained off the path of play. However, rational 6 See the appendix for derivations of I and p k i. 21

22 inattention constraints are U i -lower hemicontinuous since they only change discontinuously at nodes that are never realized in the limit. Rational inattention constraints have the following interpretation inspired by information theory. When a player arrives at a constraint set, he has a model in mind which governs the nodes in the constraint set that is consistent with everything he knows, including equilibrium strategy profile. He then observes the realized history via an optimally designed channel, subject to his exogenously specified capacity, and acts accordingly. Under the information acquisition interpretation, the capacity represents the limited resources available for acquiring the hidden information. Under the information processing interpretation, the capacity represents the limited computational and reasoning faculties of the player. In either case, in equilibrium each players model coincides with the conditional distribution over constraint sets induced by the equilibrium strategy profile, and players design their channels optimally given their models and capacities Continuous Diameter Constraints While rational inattention constraints are intuitively appealing due to their information theoretic foundations, their functional form may not be applicable or tractable in all situations. A variant on the notion of rational inattention is the following. Definition 15. A continuous diameter constraint is a Q i satisfying (CD1) Q i pb i q tb k i P B k i b k i p wq b k i p w 1 q ď d k i pb i qu (CD2) d k i : B i Ñ R` is continuous (CD3) If b i b 1 i then d k i pb i q d k i pb i q Proposition 9. If G is irreducible and pr i, W i, H i q is standard then continuous diameter constraints satisfy the hypotheses of Theorem 2. The proof of Proposition 9 is similar to that of Proposition 8. However, unlike rational inattention constraints, continuous diameter constraints are lower hemicontinuous. Continuous diameter constraints have the following intuitive interpretation. At a constraint set W k i, the local strategies of player i are required to fit inside a closed ball of diameter d k i. This can be interpreted as a two step procedure where players first specify a status quo for each constraint set, given by the center of the closed ball, and subsequently choose their local strategies to fit inside. If d k i 0 then the constraint set is identical to an information set. At the other extreme, if d k i ě 1 then any local behavioral strategy is feasible. 22