extensive form abstract economies and generalized perfect recall · generalized perfect recall ......

Extensive Form Abstract Economies and

Generalized Perfect Recall˚

Nicholas Butler:

Abstract

I introduce the notion of an extensive form abstract economy (EFAE), a frame-

work for modeling dynamic settings where players’ feasible strategies depend on those

chosen by others. Extensive form abstract economies are motivated by models of lim-

ited 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 con-

ditions 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, 2015. This research was supported by a grant from the National Science FoundationSES-1060073.

:Princeton University. Email: [email protected]

1

1 Introduction

In the theory of noncooperative games, the most basic framework for modeling strategic sit-

uations 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 par-

tition. 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 in-

formation 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 condi-

tions, and endogenous information acquisition. The relationship between extensive form

abstract economies and abstract economies is analogous to the relationship between exten-

sive 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 exten-

sive form abstract economy with feasibility correspondences induced by players’ information

partitions.

Extensive form abstract economies are a general framework for modeling dynamic strate-

gic 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’ fea-

sibility correspondences are derived from the consistency restrictions imposed on the various

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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) ex-

tensive 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 ac-

quire 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 co-

ordination 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 equilib-

rium 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 per-

fect 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 (Pic-

cione 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 con-

vexity, 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.

1Attempting to model such situations as standard extensive form games can introduce conceptual diffi-culties regarding existence of equilibria and the interpretation of players’ strategies.

2This condition is weaker than perfect recall since under perfect recall mixed and behavioral strategiesare equivalent, and the set of mixed strategies is convex by construction.

3

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 feasibil-

ity 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 con-

dition 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.

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2 Extensive Form Abstract Economies

An extensive form abstract economy is a tuple G “ xI, A, T, ρ, b0, 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

pa1, ..., akq P T then pa1, ..., ak´1q 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 Ti “ tw P T | ρpwq “ iu. The

probability of nature selecting an action a P A at a node w P T0 is given by b0pa|wq. The

utility index u : Z Ñ RI 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.

Si “ź

wPTi

Apwq

A mixed strategy is a probability distribution over the pure strategies. The set of mixed

strategies of player i is given by Σi “ ΔpSiq.3 A behavioral strategy of player i assigns a local

strategy bip¨|wq P ΔpApwqq to every node w P Ti. The set of behavioral strategies of player i

is given by the following.4

Bi “ź

wPTi

ΔpApwqq

The final component of an extensive form abstract economy is a collection of feasibility

correspondences Q “ tQ1, ..., QIu. 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 Qi : Bí Ñ 2Bi 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 bí P Bí, then player i is compelled to

choose some behavioral strategy bi P Qipbíq. For a given strategy profile b P B, bi is called

3The set of probability distributions over a sample space X is denoted by ΔpXq.4Define B “

śiPI Bi and Bí “

śj‰i Bj , and similarly for mixed and pure strategies.

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feasible if bi P Qipb´iq, and the profile b is called jointly feasible if bi 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 ana-

logues 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 im-

posing appropriate feasibility correspondences. Formally, an extensive form game is a tuple

G “ xI, A, T, ρ, b0, 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 infor-

mation partitions H “ tH1, ..., HIu. Each information partition Hi is a partition of Ti, the

nodes where player i acts. Elements of Hi 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 Hi is that player i cannot distinguish between any w,w1 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 exten-

sive 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 Hi, the set of

standard pure strategies of player i is denoted by SipHiq. A mixed strategy is called standard

if its support is a subset of SipHiq, and the set of standard mixed strategies is denoted by

ΣipHiq. 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 BipHiq. By construction, standard strategies are precisely the strategies available

in extensive form games.

Definition 1. A feasibility correspondence is behavioral standard if Qip¨q “ BipHiq, and

mixed standard if Qip¨q “ ΣipHiq.

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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. xi, x1i P X are equivalent, denoted by xi „ x

1i, if P pz|xi, xíq “ P pz|x

1i, xíq

for every z P Z and every strategy profile of others xí P Xí.

Definition 3. Xi and Yi are equivalent if for every xi P Xi there exists yi P Yi such that

xi „ yi and conversely.

A node w P T is called a predecessor of w1 P T if there exists an a˚ P A˚ such that a˚ ‰ φ

and w1 “ pw, a˚q. The set of predecessors of w where player i acts is denoted by Pipwq. If

w is a predecessor of w1, the action taken at w on the path to w1 is given by apw,w1q. The

pure strategies of player i that can reach a node w P T are given by the following.

Ripwq “ tsi P Si | w1 P Pipwq ñ sipw

1q “ apw1, wqu

Intuitively, if si P Ripwq then player i chooses the action on the path to w wherever possible

under si. For an extensive form abstract economy with behavioral standard feasibility corre-

spondences, a mixed strategy σi P Σi is called feasible if there exists a bi P BipHiq such that

σi „ bi, and similarly for mixed standard feasibility correspondences. It is straightforward

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to verify that if σi P Σi and bi P Bi then σi „ bi if and only if the following holds.

ÿ

siPRipwq

σipsiq ą 0 “ñ bipa|wq “

řsiPRipwq,sipwq“a

σipsiqř

siPRipwqσipsiq

(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 Hi

(R1) If w1 P hpwq then w is not a predecessor of w1 and conversely.

(R2) If w1 P hpwq and w̃ P Pipwq, then there exists w̃1 P hpw1q such that w̃1 P Pipw1q and

apw̃, wq “ apw̃1, w1q

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.

L R

L

L R

R

1

1 1

Figure 1: Imperfect Recall of Past Actions

8

Proposition 1. The set of feasible mixed strategies of Figure 1 is not convex.

Proof. The pure strategies of the decision maker are S1 “ tpL,Lq, pL,Rq, pR,Lq, pR,Rqu,

where actions at unreached nodes have been omitted. The second information set requires

that b1pL|Lq “ b1pL|Rq. Consider the following mixed strategies:

σ1pL,Lq “1

2σ1pL,Rq “

1

4σ1pR,Lq “

1

6σ1pR,Rq “

1

12

σ11pL,Lq “1

4σ11pL,Rq “

1

12σ11pR,Lq “

1

2σ11pR,Rq “

1

6

Let b1 „ σ1 and b11 „ σ11. By (1) the following must hold

b1pL|Lq “σ1pL,Lq

σ1pL,Lq ` σ1pL,Rq“

2

3“

σ1pR,Lqσ1pR,Lq ` σ1pR,Rq

“ b1pL|Rq

b11pL|Lq “σ1pL,Lq


3

4“

σ11pR,Lqσ11pR,Lq ` σ

11pR,Rq

“ b11pL|Rq

So both mixed strategies are feasible. However, a convex combination of these mixed strate-

gies may not be feasible. Let σ21 “12σ1 ` 12σ

11 and b

21 „ σ

21 . By (1)

b21pL|Lq “σ2pL,Lq


9

13ă

8

11“

σ2pR,Lqσ2pR,Lq ` σ2pR,Rq

“ b21pL|Rq

Non-convexity arises due to the conjunction of two factors. First, the probability of

reaching nodes in the second information set differs under σ1 and σ11. Second, the local

strategies induced by σ1 and σ11 at the second information set are different. Intuitively, σ

11

has more influence over the local strategy induced by σ21 at R than σ1 since the probability

of reaching R is higher under σ11 than σ1.

D D

RR

1

Figure 2: Absentmindedness

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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.


Out

L R

In

1{2

L R

In Out

1{2

0

1 1

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.


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 gen-

erally. The primary result of this section, Theorem 1, states that in essentially any extensive

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form game a player has perfect recall if and only if his set of standard behavioral strate-

gies 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 con-

stituent 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 ipw|biq “ź

w1PPipwq

bipapw1, wq|w1q (2)

The node dependent mixing coefficient of player i, denoted by Λi : Bi ˆ Bi ˆ r0, 1s ˆ Ti Ñ́

r0, 1s, is defined whenever P ipw|biq ą 0 or P ipw|b1iq ą 0 as follows.

Λipbi, b1i, λ, wq “

λP ipw|biqλP ipw|biq ` p1 ´ λqP ipw|b1iq

(3)

If both P ipw|biq “ 0 and P ipw|b1iq “ 0, then Λipbi, b1i, λ, wq “ λ . The behavioral mixing

function of player i, denoted by Bi : Bi ˆ Bi ˆ r0, 1s Ñ́ Bi, is defined componentwise using

(3) as follows.

Bipbi, b1i, λqpa|wq “ Λipbi, b

1i, λ, wqbipa|wq ` p1 ´ Λipbi, b

1i, λ, wqqb

1ipa|wq (4)

A behavioral convex combination of bi, b1i P Bi is a behavioral strategy b

2i „ Bipbi, b

1i, λq

for some λ P r0, 1s. By equation (4) each component of the behavioral mixture Bipbi, b1i, λq

is a convex combination of components of bi and b1i, with weights given by Λi. In general,

Bipbi, b1i, λq is not a convex combination of bi and b1i, since the mixing coefficient Λi is node

dependent, and adjusts for the relative probability of reaching w under bi versus b1i.

Definition 6. Vi Ă Bi is behavioral convex if for every vi, v1i P Vi and λ P r0, 1s there exists

a behavioral convex combination of vi and v1i that is an element of Vi.

Proposition 4. The set of feasible mixed strategies is convex if and only if the set of behav-

ioral 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

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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 ex-

tensive 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

iPI

ź

w1PPipwq

bipapw1, wq|w1q

The expected utility of player i under b is given by.

Uipbq “ÿ

zPZ

P pz|bquipzq.

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Definition 7. An equilibrium in behavior strategies is a b P B such that for every i P I

bi P arg maxb̃iPQipb´iq

Uipb̃i, b´iq

An equilibrium in behavioral strategies in extensive form abstract economies is the nat-

ural 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 Rn and Y Ă Rm endowed with the usual topology, C : Y Ñ 2X be a correspon-

dence, and f : X ˆ Y Ñ R.

Definition 8. A correspondence C is f -lower hemicontinuous if for any yn Ñ y and x P Cpyq,

there exists tynku a subsequence of tynu, and xnk P Cpynkq such that fpxnk , ynkq Ñ fpx,yq.

This definition is identical to the sequential characterization of lower hemicontinuity,

except it requires that fpxnk , ynkq Ñ fpx,yq, rather than xnk 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 Ui-lower hemicontinuous but not lower hemicontinuous.

Definition 9. A compact valued correspondence C is upper hemicontinuous if for any yn Ñ y

with xn P Cpynq, there exists a convergent subsequence of txnu 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 2X be a compact valued

correspondence. Let f˚pyq “ maxxPCpyq fpx, yq and C˚pyq “ arg maxxPCpyq 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

Definition 10. A feasibility correspondence Qi is canonical if it satisfies the following,

(Q1) Qi is compact and nonempty valued

(Q2) Qi is Ui-continuous

(Q3) If bj „ b1j for all j ‰ i then Qipb´iq “ Qipb1´iq

An extensive form abstract economy is called canonical if each players’ feasibility corre-

spondence is canonical. (Q1) is a standard technical condition ensuring that players’ best

response correspondences are well defined. Since Ui is continuous, (Q2) is weaker than

standard continuity. By Lemma 1, (Q1) and (Q2) ensure that players’ best response corre-

spondences 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 discontin-

uously when other players change their strategies. (Q3) imposes equivalence classes on Qi

which correspond to the equivalence classes of Bi 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 regard-

ing 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.

5These manipulations are not only costless holding the strategy profile of the other players fixed, but alsocostless for any strategy profile of the other players.

14

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 re-

sponses, and subsequently mapping these correspondences into the space of mixed strategies.

The primary difficulty of this method lies in ensuring the mapping preserves upper hemi-

continuity. and that the resulting mixed strategy best response correspondences is convex

valued. The latter is a consequence of the following lemma.

Lemma 2. UipBpbi, b1i, λq, bíq “ λUipbi, bíq ` p1 ´ λqUipb1i, bíq

Intuitively, lemma 2 implies utility is “linear” in behavioral convex combinations of be-

havioral 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

0, 1

L

1, 0

R

l

1,0

L

1,0

R

r

l

1,0

L

0,1

R

l

1,0

L

1,0

R

r

r

L

1,0

L

1,0

R

l

0,1

L

1,0

R

r

l

1,0

L

1,0

R

l

1,0

L

0,1

R

r

r

R

P2

P2

P1

P1

Figure 4: Tightness of Generalized Perfect Recall

The feasibility correspondence of player 2 requires the following.

b2pl|Llq “ b2pl|Rlq b2pl|Lrq “ b2pl|Rrq |b2pl|Lq ´ b2pl|Rq| ď �,

The feasibility correspondence of player 1 requires the following.

b1pL|Lllq “ b1pL|Llrq “ b1pL|Lrlq “ b1pL|Lrrq

b1pL|Rllq “ b1pL|Rlrq “ b1pL|Rrlq “ b1pL|Rrrq

For � ě 0 denote the EFAE depicted in Figure 4 by G�. 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 correspon-

dences of both players are canonical; however, for � ą 0 player 2 has generalized imperfect

recall. Player 2 effectively forgets his past knowledge, since he is capable of (partially) con-

ditioning 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

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� does not have an equilibrium in behavioral strategies for any � ą 0.

The proof of proposition 5 proceeds in two steps. It is straightforward to show that for

any � 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,

DpQi,Qiq “ infQ̃iPQi

supbíPBí

dHpQipbíq, Q̃ipbíqq (5)

where Qi is the set of feasibility correspondences of player i which satisfy generalized perfect

recall, d is a metric inducing the usual topology on Bi, and dH is the Hausdorff distance

given by the following.

dHpX,Y q “ maxtsupxPX

infyPY

dpx, yq, supyPY

infxPX

dpx, yqu

The distance function D measures the degree to which Qi departs from generalized perfect

recall. Since G0 has generalized perfect recall it is straightforward to verify that if � Ñ 0

then DpQ�2,Q2q 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 strate-

gies, determining whether an arbitrary EFAE satisfies generalized perfect recall may be

17

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 correspon-

dences, including those describing rational inattention, are shown to satisfy the hypotheses

of Theorem 2.

6.1 Partitional Convexity

Let Z Ă Rm, I “ t1, ...,mu and P “ tP1, ..., PKu 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, z1 P Z is a z2 P Z such that

z2i “ λizi ` p1 ´ λiqz1i for some λ P r0, 1s

m satisfying if i » j then λi “ λj .

A P -partitional convex combination is formed by taking convex combinations of the

components of z and z1 at every equivalence class specified by P . A set Z Ă Rm is called

P -partitionally convex if z, z1 P Z implies every P -partitional convex combination of z and

z1 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 Rm is P´partitional convex if and only if Z “śK

k“1 Zk where each

Zk Ă R|Pk| 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 Hi-partitionally convex valued by construction.

6.2 Partitional Convexity and Generalized Perfect Recall

A partitional decomposition of a feasibility correspondence Qi is a pair pQ̃i, Hiq satisfying

Qi “ Q̃i X BipHiq. Partitional decompositions are not generally unique and exist for any

Qi, since Hi 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

extensive form game with residual constraints specified by Q̃i. In settings with endogenous

information acquisition, Hi describes the nodes that are indistinguishable a priori, whereas

Q̃i 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.

Ri “ tW Ă Ti | w,w1 P W ô Ripwq X SipHiq “ Ripw

1q X SipHiqu (6)

Each element of Ri 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 Ri 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 Ri and perfect recall, which is given by the

following proposition.

Proposition 6. If G is irreducible then Hi satisfies perfect recall if and only if Hi is finer

than Ri.

Proposition 6 is analogous to Theorem 1, and implies that Ri characterizes the coarsest

information partitions of player i which satisfy perfect recall. Intuitively, a residual constraint

is Ri-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 Ri-

partitional convexity and generalized perfect recall is as follows.

Proposition 7. If Hi satisfies perfect recall and Q̃i is Ri-partitionally convex valued then

Qi “ Q̃i X BipHiq 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

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 Ti denoted by Wi “ tW 1i , ...,WKii u,

with elements called constraint sets. A partition trio is a triple pRi,Wi, Hiq. 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 Wi is both finer than Ri and coarser than Hi.

By Proposition 6, Hi satisfies perfect recall in any partition trio, since Hi is finer than

Ri. The set of local behavioral strategies of player i at constraint set W ki is given by the

following.

Bki “ź

wPW ki

ΔpAipwqq

A local feasibility correspondence is a Qki : B´i Ñ Bki . For the remainder of this section

feasibility correspondences are assumed to be of the following form.

Qip¨q “ Q̃ip¨q X BipHiq “Kiź

k“1

Qki p¨q X BipHiq

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 corre-

spondence 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 in-

formation 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.

20

6.3.1 Rational Inattention Constraints

The model of player i at constraint set W ki , denoted by pki , is the distribution over

nodes in W ki induced by the conjectured strategy profile conditional on everything player i

knows upon arriving at W ki . The mutual information between a constraint set Wki and the

corresponding local behavioral strategy of player i is given by the following.6

IpW ki , ApWki qq “

ÿ

wPW ki

ÿ

aPApwq

bipa|wqpki pwqlogp

bipa|wqřwPW ki

bipa|wqpki pwqq

A constraint set W ki is said to be on the path of play of b if there exists a node w P Wki

such that P pw|bq ą 0, and on the potential path of play of bí if there exists a bi P Bi such

that it is on the path of play of pbi, bíq. The constraint sets of player i that are on the

potential path of bí are denoted by Πipbíq.

Definition 14. A rational inattention constraint is a vector of capacities ci P RKi` and a

feasibility correspondence Qi satisfying,

Qki pbíq “

$&

%

tbki P Bki | IpW

ki , ApW

ki qq ď ciku, if W

ki P Πipbíq.

Bki otherwise.

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 Qi 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 pki pwq “ 1 for some w P Wki , then the mutual information between any local

behavioral strategy and W ki 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 pRi,Wi, Hiq 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

6See the appendix for derivations of I and pki .

21

inattention constraints are Ui-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 acquir-

ing 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.

6.3.2 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 situa-

tions. A variant on the notion of rational inattention is the following.

Definition 15. A continuous diameter constraint is a Qi satisfying

(CD1) Qipbíq “ tbki P Bki | |b

ki p¨|wq ´ b

ki p¨|w

1q| ď dki pbíqu

(CD2) dki : Bí Ñ R` is continuous

(CD3) If bí „ b1í then dki pbíq “ d

ki pbiq

Proposition 9. If G is irreducible and pRi,Wi, Hiq 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. Continu-

ous diameter constraints have the following intuitive interpretation. At a constraint set W ki ,

the local strategies of player i are required to fit inside a closed ball of diameter dki . 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 dki “ 0 then the constraint set is identical to an information set.

At the other extreme, if dki ě 1 then any local behavioral strategy is feasible.

22

Any behavioral strategy composed of local strategies that are equal at nodes in the same

constraint set is feasible under either rational inattention or continuous diameter constraints

This is natural since such a strategy does not utilize any information beyond that of the

information partition. These constraints also have the natural property that a behavioral

strategy constructed by randomizing independently over feasible local behavioral strategies

is feasible.

7 Incredible Self Restraint

In addition to the usual types of incredible threats that may occur in equilibria of ex-

tensive form games, equilibria in extensive form abstract economies may exhibit additional

implausible properties. Specifically, a player may use his local strategy at one node to limit

his feasible actions at nodes off the equilibrium path to support more favorable play in equi-

librium. I refer to this novel phenomenon as incredible self restraint. Naive extensions of

standard perfection concepts do not rule this out, so in this section I propose a notion of

generalized perfect equilibrium which precludes these types of unintuitive equilibria.

Consider the extensive form abstract economy depicted in Figure 5 below. The feasibility

correspondence of player 2 requires that |b2pL|Mq ´ b2pL|Rq| ď 1{2. Intuitively, player 2 is

able to distinguish L from the other two nodes, but can only partially distinguish the nodes

M and R from each other.

-4,-4

L

0,4

R

L

2,-1

L

-4,-5

R

M

2,-1

L

-4,-1

R

R

1

2 2 2

Figure 5: Incredible Self Restraint

Proposition 10. The following strategy profile is an equilibrium of Figure 5

b1pL|�q “ 1 b2pR|Lq “ 1 b2pR|Mq “ 1{2 b2pR|Rq “ 1

The equilibrium above is peculiar in the following sense. Player 1 does not play M

because at the subsequent node player 2 chooses R half the time, to the detriment of both

23

players. While the local strategy of player 2 at M may appear to be incredible, he cannot

choose an arbitrary local strategy M due to his feasibility correspondence. Fixing his local

strategy at R, player 2 is in fact best responding at M subject to the induced constraint

b2pL|Mq ď 1{2. Since player 2 is indifferent between his actions at R and is clearly best

responding at L, his strategy is credible at these nodes as well. Player 2 is in fact best

responding at each of his decision nodes subject to the constraints generated by fixing his

behavioral strategy outside the node. In this sense, the equilibrium described in proposition

10 is “subgame perfect”. This suggests that naive extensions of subgame perfection are too

permissive, as the following thought experiment confirms.

Suppose player 2 was informed that play had arrived at either M or R, with positive

probability on each. Fixing his local strategy at L, he is afforded the opportunity to deviate at

his other decision nodes simultaneously provided the resulting behavioral strategy is feasible.

The best response player 2 is to set b2pL|Mq “ 1 and b2pL|Rq ě 1{2, which implies the

equilibrium strategy of player 2 is not credible in this more general sense. Intuitively, the

strategy of player 2 exhibits incredible self restraint because his local strategy at R constrains

his set of feasible local strategies at M , which deters player 1 from choosing M initially.

Moreover, player 2 would choose to deviate simultaneously upon arriving at the off path

nodes, if given the option.

Standard notions of perfection are not suitable for extensive form abstract economies,

since they only address continuation play beginning at individual nodes (subgame perfec-

tion) or at information sets (sequential equilibrium). While sequential equilibrium is often

sufficient for the purposes of extensive form games, players in an EFAE face more general

constraints, which can generate equilibria exhibiting incredible self restraint. The notion of

generalized perfect equilibrium addresses this issue by requiring players’ strategies to be the

limit of equilibrium strategy profiles of �-modified EFAE. Since every node is reached with

positive probability in an �-modified EFAE, in equilibrium players’ must be best responding

conditional on arriving at any set of nodes. This remains true in the limit, hence equilibria

of this form rule out standard incredible threats as well as incredible self restraint

For a given feasibility correspondence Qi, denote its �-modification by Qipb´i, �q “

Qipb´iqXΓip�q, where Γip�q is the set of behavioral strategies of player i satisfying bipa|wq ě �

for any w P Ti and a P Apwq. Denote by G� the EFAE obtained by replacing the feasibility

correspondences of G with their �-modifications.

Definition 16. A profile b P B is a generalized perfect equilibrium of G if b is an equilibrium

of G, and there exists �n Ñ 0 and bn Ñ b such that bn is an equilibrium of G�n .

Definition 17. Qi admits trembling if there exists a δ ą 0 such that for any � ď δ its

�-modification Qip¨, ¨q is nonempty valued and Ui-lower hemicontinuous.

24

An extensive form abstract economy admits trembling if every player’s feasibility cor-

respondence admits trembling. This restriction is relatively weak, but it is not without

consequence. First, it requires that if the probability of trembling is sufficiently small then

the �-modified feasibility correspondence is well defined. Second, since the intersection of

Ui-lower hemicontinuous correspondences is not necessarily Ui lower hemicontinuous it is

needed to ensure existence of an equilibrium in the �-modified EFAE.7

Theorem 3. If an EFAE is finite, canonical, admits trembling, and satisfies generalized

perfect recall then it has generalized perfect equilibrium.

The proof of Theorem 3 resembles the standard existence proofs for perfect equilibria in

extensive form games. The primary difficulty lies in showing that equilibria necessarily exist

in the �-modified EFAE. This result is significant since it guarantees existence of equilibria

that do not rely on incredible threats, under only modestly stronger conditions than those

of Theorem 2. To see that the equilibrium of proposition 10 is not totally perfect, notice

that for any full support strategy of player 1, any best response of player 2 in G� satisfies

b2pR|Mq “ �, hence any totally perfect equilibrium must have b2pR|Mq “ 0.

8 Conclusion

This paper lays the groundwork for the theory of extensive form abstract economies, a

framework for modeling dynamic situations with nonstandard strategic constraints. Exten-

sive form abstract economies generalize the notion of extensive form games, and contain

as special cases a variety of existing game theoretic models. This paper demonstrates that

in extensive form games perfect recall is equivalent to a notion of behavioral convexity. In

addition, sufficient conditions for existence of an equilibrium in behavioral strategies were

provided, the most important being generalized perfect recall. Concerning existence of equi-

libria in behavioral strategies, perfect recall was shown to be tight in extensive form games,

and generalized perfect recall was shown to be tight in extensive form abstract economies.

Using the notion of partitional convexity, a simple sufficient condition for generalized perfect

recall was provided, and specific feasibility correspondences describing endogenous informa-

tion acquisition were shown to satisfy this condition. Finally, the notion of a generalized

perfect equilibrium was introduced to rule out equilibria exhibiting incredible self restraint,

and sufficient conditions for existence of such an equilibrium were determined.

There are a multitude of possibilities for further inquiry in both the theory and ap-

plications of extensive form abstract economies. Regarding theory, many of the results of

7It is straightforward to show that the correspondences considered in the previous section admit trembling.

25

this paper could be extended to infinite extensive form abstract economies under additional

assumptions. Another possible theoretical direction would be to examine extensive form

abstract economies with generalized imperfect recall. In addition, extensive form abstract

economies may be used to construct equilibrium refinements for standard extensive form

games using perturbations of players’ feasibility correspondences. Different notions of per-

fection could also be extended to extensive form abstract economies. In any case, while this

paper lays a workable foundation, many interesting questions remain regarding the theory

and applications of extensive form abstract economies.

9 Appendix

9.1 Definitions

The set of terminal nodes is given by Z “ tw P T | Apwq “ Hu. The nodes where player

i acts are denoted by Ti “ tw P T | ρpwq “ iu. For w P Pipw1q let apw,w1q be the action

satisfying either pw, apw,w1qq P Pipw1q or pw, apw,w1qq “ w1. For a node w “ paw1 aw2 ...a

wk ),

define inductively for n ď k, w0 “ φ, and wn “ pwn´1, awn q. Let Rpwq be the set of pure

strategy profiles s such that si P Ripwq for every i P I. The probability of reaching a node

w under a mixed strategy profile σ is given by the following.

P pw|σq “ÿ

sPRpwq

σpsqź

w1PP0pwq

b0papw1, wq|w1q

The expected utility of player i under σ is given by

Uipσq “ÿ

zPLT

P pz|σquipzq

The probability of reaching w if other players choose the actions on the path to w is

P ipw|σiq “ÿ

siPRipwq

σipsiq

This implies for any σ P Σ and b P B that if σ „ b then Uipσq “ Uipbq. The set of standard

pure strategies is given by

SipHiq “ tsi P Si | w1 P hpwq ñ sipwq “ sipw

1qu

26

The set of standard mixed strategies is given by

ΣipHiq “ tσi P Σi | si R Ssi pHiq ñ σipsiq “ 0u

The set of standard behavioral strategies is denoted by

BipHiq “ tbi P Bi | w1 P hpwq ñ bip¨|wq “ bip¨|w

1qu

For a behavioral standard game with information partition H, denote the set of feasible

mixed strategies of player i by Σ˚i , and the feasible behavioral strategies of a mixed standard

game by B˚i .

9.2 Imperfect recall and non-convexity


Proof. Omitting actions at unreached nodes, the pure strategies of this game are

S1 “ tpDq, pR,Dq, pR,Rqu

Consider the mixed strategies σ1pDq “ 1 and σ11pR,Rq “ 1. Notice that σ1 „ b1 where

b1pD|φq “ b1pD|Rq “ 1, and σ11 „ b11 where b1pR|φq “ b1pR|Rq “ 1, so they are feasible.

However σ21 “12σ1 ` 12σ

11 „ b

21 where b

21pR|�q “

12

and b21pR|Rq “ 1, so σ21 is not feasible.


Proof. Omitting actions at unreached nodes the pure strategies are given by

S1 “ tpOutq, pIn, Lq, pIn,Rqu2

Consider the following mixed strategies.

σ1pIn, L|lq “ σ1pOut|rq “ 1

σ11pOut|lq “ σ11pIn,R|rq “ 1

So σ1 „ b1 where b1pL|l, Inq “ b1pL|r, Inq “ 1, and σ11 „ b11 where b1pR|l, Inq “

b1pR|r, Inq “ 1; however, σ21 “12σ1 ` 12σ

11 „ b

21 where b

21pL|l, Inq “ b

21pR|r, Inq “ 1. So

σ21 is not feasible.

27

9.3 Behavioral Convexity

Proposition 4. The set of feasible mixed strategies is convex if and only if the set of

behavioral standard strategies is behavioral convex

Proof. I first establish the following claim.

Claim. If bi, b1i P Bi and σi, σ

1i P Σi such that bi „ σi and b

1i „ σ

1i, then Bipbi, b

1i, λq „

λσi ` p1 ´ λqσ1i.

Suppose the hypotheses of the claim above hold, this implies that

P ipw|σiq ą 0 “ñ bipai|wq “

řsiPRipwq,sipwq“ai

σipsiqř

siPRipwqσipsiq

And similarly for σ1i. Let λ P r0, 1s, σ2i “ λσi ` p1 ´ λqσ

1i, and b

2i „ σ

2i .

If P ipw|σ2i q ą 0 then it follows that

b2i pai|wq “


σ2i psiqřsiPRipwq

σ2i psiq“


pλσipsiq ` p1 ´ λqσ1ipsiqqřsiPRipwq

pλσipsiq ` p1 ´ λqσ1ipsiqq“

λpř

siPRipwqσipsiqq

řsiPRipwq

pλσipsiq ` p1 ´ λqσ1ipsiqqbipai|wq `

p1 ´ λqpř

siPRipwqσ1ipsiqqř

siPRipwqpλσipsiq ` p1 ´ λqσ1ipsiqq

b1ipai|wq “

λP ipw|σiqλP ipw|σiq ` p1 ´ λqP ipw|σ1iq

bipai|wq `p1 ´ λqP ipw|σ1iq

λP ipw|σiq ` p1 ´ λqP ipw|σ1iqb1ipai|wq “

λP ipw|biqλP ipw|biq ` p1 ´ λqP ipw|b1iq

bipai|wq `p1 ´ λqP ipw|b1iq

λP ipw|biq ` p1 ´ λqP ipw|b1iqb1ipai|wq “ Bpb, b

1, λqpai|wq

So λσi ` p1 ´ λqσ2i „ Bpbi, b1i, λq for every λ P r0, 1s, which proves the claim.

Let bi, b1i P BipHiq and σi, σ

1i P Σ

˚i such that bi „ σi and b

1i „ σ

1i, and suppose Σ

˚i

is convex. Let λ P r0, 1s. Since Σ˚i is convex λσi ` p1 ´ λqσ1i P Σ

˚i , and by the claim

λσi ` p1 ´ λqσ1i „ Bpbi, b1i, λq. By the definition of feasibility there exists a b

2i P BipHiq

such that λσi ` p1 ´ λqσ1i „ b2i . This implies b

2i „ Bpbi, b

1i, λq. so b

2i is a behavioral convex

combination of bi and b1i, which implies BipHiq is behavioral convex.

Now suppose BipHiq is behavioral convex. By the claim above for any λ P r0, 1s λσi `

p1 ´ λqσ1i „ Bpbi, b1i, λq. Since BipHiq is behavioral convex there exists a b

2i „ Bpbi, b

1i, λq such

that b2i P BipHiq so by the definition of feasibility λσi ` p1 ´ λqσ1i P Σ

˚i , so Σ

˚i is convex.

28

Lemma 2: UipBpbi, b1i, λq, bíq “ λUipbi, bíq ` p1 ´ λqUipb1i, bíq

Proof. By the claim in Proposition 4 if bi, b1i P Bi and σi, σ

1i P Σi such that bi „ σi and

b1i „ σ1i, then Bipbi, b

1i, λq „ λσi ` p1 ´ λqσ

1i. Clearly if b „ σ then Uipbq “ Uipσq.

Let b “ pbi, bíq „ σ and b1 “ pb1i, bíq „ σ1 “ pσ1i, σíq, and σ

2 “ λσ ` p1 ´ λqσ1. Since Uiis linear in mixed strategies, and since Bipbi, b1i, λq „ σ

2i , it follows that.

UipBipbi, b1i, λq, bíq “ Uipσ

2i , σíq “ λUipσi, σíq ` p1 ´ λqUipσ

1i, σíq

“ λUipbi, bíq ` p1 ´ λqUipb1i, bíq

Theorem 1. If an extensive form game is irreducible then players have perfect recall if and

only if their sets of behavioral standard strategies are behavioral convex.

Proof. Suppose by way of contradiction that player i does not have perfect recall.

Step 1: Absentmindedness

Suppose player i is absentminded. Then there exists w,w1 P Ti such that w P Pipw1q

and w1 P hpwq. By irreducibility there exists si P SipHiq such that si P Ripwq and

sipwq ‰ apw,w1q. Let s1i P Ripw1q, bi „ si and b1i „ s

1i. By construction bi, b

1i P BipHiq,

and bipapw,w1q|wq “ 0 and b1ipapw,w1q|wq “ 1. Let b2i “ Bpbi, b

1i,

12q. Notice that

b2i papw,w1q|wq “ 1

2and that b2i papw,w

1q|w1q “ 1 so b2i R BipHiq.

Step 2: Imperfect Recall of Past Information

Suppose without loss of generality that player i is not absentminded. Let w1 P hpwq and

suppose there is a w̃ P Pipwq such that there exists no w̃1 P Pipw1q such that w̃1 P hpw̃q.

Construct the behavioral strategies bi and b1i as follows.

For x P Pipwq such that Dx1 P Pipw1q satisfying x P hpx1q, let b1pa|xq “ bpa|xq “

1{|Apxq| @a P Apxq. For x P Pipwq such that @x1 P Pipw1q satisfying x P hpx1q, let

bpapx,wq|xq “ 1 and b1papx,wq|xq “ 1 if x ‰ w̃ and let b1papw̃, wq|w̃q “ 0. For x1 P Pipw1q

such that @x P Pipwq satisfying x P hpx1q, let b1papx1, w1q|x1q “ bpapx1, w1q|x1q “ 1. This

completely characterizes bi and b1i at the predecessors of w and w

1.

By construction bi, b1i P BipHiq. Since player i is not absentminded there is a bijec-

tion between the predecessors of w and w1 that are in the same information set, hence

P ipw|bq “ P ipw1|bq “ P ipw1|b1q ą 0, and by construction P ipw|b1q “ 0. Since G is

irreducible there are unique a1, a2 P Apwq “ Apw1q. Let bpa1|wq “ bpa1|w1q “ 1, and let

b1pa2|wq “ b1pa2|w1q “ 1. Consider b2i “ Bpb, b1, 1

2q. Since P ipw|b1q “ 0 and P ipw1|bq ą 0

29

it follows that b2pa1|wq “ bpa1|wq “ 1. Moreover, since P ipw1|bq “ P ipw1|b1q ą 0 it fol-

lows that b2pa1|w1q “ 12bpa1|w1q` 1

2b1pa1|wq “ 12 . Since w

1 P hpwq this implies that b2i R BipHiq.

Step 3: Imperfect Recall of Past Actions

Without loss of generality assume that player i is not absentminded, and does not have

imperfect recall of past information. So there exists a w1 P hpwq such that there is a w̃ P Pipwq

and w̃1 P Pipw̃q, such that w̃ P hpw̃1q, and apw̃, wq ‰ apw̃1, w1q.

By irreducibility there are unique actions a1, a2 P Apwq. Since apw̃, wq ‰ apw̃1, w1q

there exists si P Ripwq such that sipwq “ sipw1q “ a1, and si R Ripw1q. Similarly there

exists s1i P Ripw1q such that s1ipwq “ s

1ipw

1q “ a2, and s1i R Ripwq. Let bi „ si and

b1i „ s1i. By construction P

ipw1|b1iq “ Pipw|biq “ 1. Consider b2i “ Bipbi, b

1i,

12q. Notice that

b2i pa1|wq “ b2i pa2|w

1q “ 1, so b2i R BipHiq since w1 P hpwq.

The steps above imply that if player i has imperfect recall then BipHiq is not behavioral

convex. Therefore if BipHiq is behavioral convex then player i has perfect recall. I now

proceed to prove the converse.

Suppose player i has perfect recall. Kuhn’s Theorem implies that ΣipHiq is equivalent to

BipHiq. By construction BipHiq is equivalent to Σ˚i and therefore ΣipHiq is equivalent to Σ˚i .

By construction ΣipHiq is convex. Suppose σi, σ1i P ΣipHiq and ςi, ς1i P Σ

˚i such that σi „ ςi

and σ1i „ ς1i. By construction if P

ipw|σiq ą 0 or P ipw|ςiq ą 0 then P ipw|σiq “ P ipw|ςiq and

for all a P Apwq


σipsiqř

siPRipwqσipsiq

“


ςipsiqř

siPRipwqςipsiq

And similarly for σ1i and ς1i. Let σ

2i “ λσi ` p1 ´ λqσi and ς

2i “ λςi ` p1 ´ λqςi. Notice

that if P ipw|σ2i q ą 0 or Pipw|ς2i q ą 0 then P

ipw|σ2i q “ Pipw|ς2i q and for all a P Apwq



pλσipsiq ` p1 ´ λqσ1ipsiqq“



pλσipsiq ` p1 ´ λqσ1ipsiqq

This implies σ2i „ ς2i . Notice that σ

2i P ΣipHiq since ΣipHiq is convex. Since ΣipHiq

is equivalent to Σ˚i there exists σ˚i P Σ

˚i such that σ

˚i „ σ

2i . Since σ

˚i P Σ

˚i there exists

bi P BipHiq such that bi „ σ˚i , and therefore bi „ σ˚i „ σ

2i „ ς

2i which implies that ς

2i P Σ

˚i ,

hence Σ˚i is convex, so BipHiq is behavioral convex by proposition 4.

30

9.4 Properties of Correspondences

Definition 18. A correspondence C : Y Ñ 2X is compact/closed valued if Cpyq is com-

pact/closed for every y P Y .

Definition 19. A correspondence C is closed if xn Ñ x, yn Ñ y, and xn P Cpynq then

y P Cpxq.

Fact 1. If C is closed valued, and Y is compact then C is upper hemicontinuous if and only

if C is closed.

Proof. See Border (1985) Proposition 11.9 (a)(b)

The image of K Ă Y under C is given by

CpKq “ď

kPK

Cpkq

Fact 2. If C is upper hemicontinuous and compact valued and K Ă Y is compact then

CpKq is compact.

Proof. See Border (1985) Proposition 11.16

Definition 20. A correspondence C is lower hemicontinuous if for any yn Ñ y and x P Cpyq,

there exists tynku a subsequence of tynu, and xnk P Cpynkq such that xnk Ñ x.

The composition of two correspondences C1 : A Ñ 2B and C2 : B Ñ 2C written as

C1 ˝ C2 : A Ñ 2C is given by

pC2 ˝ C1qpaq “ď

bPC1paq

C2pbq

Fact 3. Let C1 : A Ñ 2B, and C2 : B Ñ 2C . If C1 and C2 are upper hemicontinuous (resp

lower hemicontinuous) then C2 ˝ C1 is upper hemicontinuous (resp lower hemicontinuous).

Proof. See Border (1985) Proposition 11.23 (a)(b)

Fact 4. Let C1 : Y Ñ 2X , and C2 : Y Ñ 2X be closed valued. If C1 and C2 are upper hemi-

continuous at x and C1pxq X C2pxq is nonempty then C1pxq X C2pxq is upper hemicontinuous

at x.

Proof. See Border (1985) Proposition 11.21(a)

31

For a collection of correspondences Ci : Y Ñ 2Xi indexed by i “ 1, ..., k their product

correspondence C : Y Ñ 2X , where X “śk

i“1 Xi is given by

kź

i“1

Ci : y Ñkź

i“1

Cipyq

Fact 5. If each Ci is upper hemicontinuous and compact valued then their product cor-

respondence is upper hemicontinuous and compact valued. Moreover, if each Ci is lower

hemicontinuous then their product correspondence is lower hemicontinuous.

Proof. See Border (1985) Proposition 11.25

9.5 Equivalence Correspondences

For bi P Bi, let Ψipbiq “ tσi P Σi | σi „ biu. By equation (1) this can be expressed as

Ψipbiq “ tσi P Σi |ÿ

siPRipwq

σipsiq ą 0 “ñ bipai|wq “


σipsiqř

siPRipwqσipsiq

u

For σi P Σi, let Φipσiq “ tbi P Bi | bi „ σiu. By equation (1) this can be expressed as

Φipσiq “ tbi P Bi |ÿ

siPRipwq

σipsiq ą 0 “ñ bipai|wq “


σipsiqř

siPRipwqσipsiq

u

Let Ψ “ś

iPIzt0u Ψi and Φ “ś

iPIzt0u Φi.

Remark 1. Ψi and Ψ are continuous, non-empty valued, and compact valued.

Proof. Upper Hemicontinuity: Let bni Ñ bi, σni Ñ σi, with σ

ni P Ψipb

ni q. By construction this

implies that

P ipw|σni q ą 0 “ñ bni pai|wq “


σni psiqřsiPRipwq

σni psiq

IfP ipw|σiq ą 0 there exists an integer N such that if n ą N then P ipw|σni q ą 0. So there is

an open neighborhood around σi such that bni pai|wq is a continuous function of σ

ni . Hence

bipai|wq “ limnÑ8

bni pai|wq “ limnÑ8


σni psiqřsiPRipwq

σni psiq“


σipsiqř

siPRipwqσipsiq

32

Which implies that σi P Ψipbiq.

Proof. Lower Hemicontinuity: To prove that Ψi is lower hemicontinuous I first establish the

following claim.

Claim. If σi „ bi and |bi´b̃i| ă �, then there exists a σ̃i „ b̃i and δp�q such that |σi´σ̃i| ă δp�q.

Moreover δp�q Ñ 0 as � Ñ 0.

Let |bi ´ b̃i| ă �, and let w1, ..., wN be an ordering of the elements of Ti such that if i ą j

then wi is not a predecessor of wj . Let Tni “ tw1, ..., wnu be the restriction of Ti to the first

n nodes. I proceed by induction on the number of nodes of Ti.

For the base case assume n “ 1. Set σ̃ipaiq “ b̃ipai|w1q. By construction σ̃i „ b̃i, and

notice that |σi ´ σ̃i| “ |bi ´ b̃i| ă �. So let δp�q “ δ1p�q “ � proving the base case.

Assume the result holds for n ă N , and consider mixed strategies defined over TN´1i .

By the inductive hypothesis there exists a σ̃i and δN´1p�q, such that |σi ´ σ̃i| ă δN´1p�q and

δN´1p�q Ñ 0, as � Ñ 0. Let psi, aq be the pure strategy defined over TNi where player i

follows si at the first N ´ 1 nodes and chooses a at wN , so that by construction σipsiq “ř

aiPApwN qσipsi, aiq. Construct a candidate mixed strategy σ1i „ b

1i as follows.

σ1ipsi, aiq “

$&

%

σipsi, aiqσ̃ipsiqσipsiq

if σipsiq ą 0

σipsiqb̃ipai|wNq if σipsiq “ 0

In both cases σ1ipsiq “ř

aiPApwN qσ1ipsi, aiq “ σ̃ipsiq so b

1ipai|wnq “ b̃ipai|wnq for n ă N.

Case 1: Suppose that P ipwN |σiq “ 0 and P ipwN |σ̃iq “ 0

Let σ̃ipsi, aiq “ σ1ipsi, aiq. Since PipwN |σ̃iq “ 0 it follows that σ̃i „ b̃i. If si P RipwNq then

σ̃ipsi, aiq “ σipsi, aiq “ 0, so |σ̃ipsi, aiq ´ σipsi, aiq| “ 0.

If si R RipwN q and σipsiq ą 0, then σ̃ipsi, aiq “ σ1ipsi, aiq “ σipsi, aiqσ̃ipsiqσipsiq

by construction.

Therefore,

|σ̃ipsi, aiq ´ σipsi, aiq| “ σipsi, aiq|1 ´σ̃ipsiqσipsiq

| ă maxsiPSi,θPr´δN´1p�q,δN´1p�qs

|1 ´σipsiq ` θ

σipsiq| “ δ1N´1p�q.

Notice that as � Ñ 0, δ1N´1p�q Ñ 0.

If si R RipwNq and σipsiq “ 0, then σipsi, aiq “ 0 and σ̃ipsi, aiq “ σ1ipsi, aiq “ σ̃ipsiqb̃ipai|wNq

by construction. Therefore,

|σ̃ipsi, aiq ´ σipsi, aiq| “ σ̃ipsi, aiq ď σ̃ipsiq “ |σ̃ipsiq ´ σipsiq| ă δN´1p�q

33

Case 2: Suppose that P ipwN |σiq “ 0 and P ipwN |σ̃iq ą 0

Let σ̃ipsi, aiq “ σ1ipsi, aiq. If si P RipwNq then σipsiq “ 0, so σ̃ipsi, aiq “ σ1ipsi, aiq “

σ̃ipsiqb̃ipai|wNq. By construction

řsiPRipwq

σ̃ipsi, aiqř

siPRipwqσ̃ipsiq

“

řsiPRipwq

σ̃psiqb̃pai|wNqř

siPRipwqσ̃psiq

“ b̃pai|wN q

So σ̃i „ b̃i Since σpsiq “ 0 it follows that

|σ̃ipsi, aiq ´ σipsi, aiq| “ σ̃ipsi, aiq ď σ̃ipsiq “ |σ̃ipsiq ´ σipsiq| ă δN´1p�q

If si R RipwNq then the same argument from Case 1 applies.

Case 3: Suppose that P ipwN |σiq ą 0 and P ipwN |σ̃iq “ 0

Let σ̃ipsi, aiq “ σ1ipsi, aiq. Since PipwN |σ̃iq “ 0 it follows that σ̃i „ b̃i. If si P RipwNq then

σ̃ipsiq “ř

aiPApwN qσ̃ipsi, aiq “ 0, so σipsiq “ |σ̃ipsiq ´ σipsiq| ă δN´1p�q. Since σipsiq “

řaiPApwN q

σipsi, aiq, it follows that

|σ̃ipsi, aiq ´ σipsi, aiq| “ σipsi, aiq ď σipsiq ă δN´1p�q

For si R RipwNq the same argument from Case 1 applies.

Case 4: Suppose that P ipwN |σiq ą 0 and P ipwN |σ̃iq ą 0

For si R RipwNq let σ̃ipsi, aiq “ σ1ipsi, aiq, and apply the argument in Case 1. For si P RipwN q,

look for a solution of the form σ̃ipsi, aiq “ σ1ipsi, aiq ` γpsi, aiq, satisfying the following

properties.

(C1)ř

aiPApwN qγpsi, aiq “ 0

(C2)ř

siPRipwN qγpsi, aiq “ pb̃ipai|wNq ´ b1ipai|wNqq

řsiPRipwN q

σ1ipsiq

(C3) γpsi, aiq ą 0 “ñ γps1i, aiq ą 0

(C4) γpsi, aiq ă 0 “ñ γps1i, aiq ă 0

34

If such a γ exists, (C1) implies that σ̃ipsiq “ σ1ipsiq, and (C2) implies that

řsiPRipwN q

σ̃ipsi, aiqř

siPRipwN qσ̃ipsiq

“

řsiPRipwN q

pσ1ipsi, aiq ` γpsi, aiqqřsiPRipwN q

σ1ipsiq

“

řsiPRipwN q

σ1ipsi, aiqřsiPRipwN q

σ1ipsiq`

pb̃pai|wNq ´ b1pai|wNqqř

siPRipwN qσ1ipsiqř

siPRipwN qσ1ipsiq

“ b1ipai|wNq ` b̃ipai|wNq ´ b1ipai|wNq “ b̃ipai|wNq

So σ̃i „ b̃i. Let A´ “ tai P ApwN q : pb̃ipai|wNq ´ b1ipai|wNqq ă 0u, and let A` “ tai P

ApwNq : pb̃ipai|wNq ´ b1ipai|wN qq ě 0u. For ai P A´ the following holds.

ÿ

siPRipwN q

σ̃ipsi, aiq “ÿ

siPRipwN q

σ1ipsi, aiq `ÿ

siPRipwN q

σ1ipsiqpb̃ipai|wNq ´ b1ipai|wNqq ě

ÿ

siPRipwN q

σ1ipsi, aiq ´ÿ

siPRipwN q

σ1ipsiqb1ipai|wNq “

ÿ

siPRipwN q

σ1ipsi, aiq ´ÿ

siPRipwN q

σ1ipsi, aiq “ 0

Therefore for ai P A´ there exists a γpsi, aiq satisfying pC1q, pC2q, and pC4q such that

σ̃ipsi, aiq “ σ1ipsi, aiq ` γpsi, aiq ě 0. Fix such a solution for ai P A´. Notice that if pC3q and

pC2q hold it follows that.

σ̃ipsi, aiq “ σ1ipsi, aiq ` γpsi, aiq ď σ

1ipsi, aiq `

ÿ

siPRipwN q

σ1ipsiqpb̃ipai|wNq ´ b1ipai|wNqq “

σ1ipsi, aiq `ÿ

siPRipwN q

σ̃ipsi, aiq ´ÿ

siPRipwN q

σ1ipsi, aiq ďÿ

siPRipwN q

σ̃ipsi, aiq ď 1

Define the following coefficients for each ai P A`

φai “b̃ipai|wNq ´ b1ipai|wNqř

aiPA`pb̃ipai|wNq ´ b1ipai|wNqq

For ai P A` let γpsi, aiq “ φaip´ř

aiPA´γpsi, aiqq ě 0, so pC3q is satisfied. By construction

pC1q is satisfied since.

ÿ

aiPA`

γpsi, aiq “ ´ÿ

aiPA´

γpsi, aiq “ñÿ

aiPApwN q

γpsi, aiq “ 0

35

claim.

Let σi P Ψipbiq, and bni Ñ bi. By the claim if |bni ´ bi| ď �, then there exists a σ

ni P Ψipb

ni q

such that |σni ´ σi| ď δp�q. Moreover δp�q Ñ 0 as � Ñ 0, which implies that σni Ñ σi, so Ψi

is indeed lower hemicontinuous.

Proof. Compact Valued: Let σni P Ψipbiq for all n and let σni Ñ σi. Since σ

ni Ñ σi, there

exists an integer N such that for n ě N then

ÿ

siPRipwq

σni psi

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