Games and Economic Behavior 88 (2014) 1–15

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Games and Economic Behavior

www.elsevier.com/locate/geb

Unpredictability of complex (pure) strategies

Tai-Wei Hu ∗,1

Northwestern University, United States

a r t i c l e i n f o a b s t r a c t

Article history:Received 11 September 2013Available online 15 August 2014

JEL classification:D01D80

Keywords:Kolmogorov complexityObjective probabilityFrequency theory of probabilityMixed strategyZero-sum gameAlgorithmic randomness

Unpredictable behavior is central to optimal play in many strategic situations because predictable patterns leave players vulnerable to exploitation. A theory of unpredictable behavior based on differential complexity constraints is presented in the context of repeated two-person zero-sum games. Each player’s complexity constraint is represented by an endowed oracle and a strategy is feasible if it can be implemented with an oracle machine using that oracle. When one player’s oracle is sufficiently more complex than the other player’s, an equilibrium exists with one player fully exploiting the other. If each player has an incompressible sequence (relative to the opponent’s oracle) according to Kolmogorov complexity, an equilibrium exists in which equilibrium payoffs are equal to those of the stage game and all equilibrium strategies are unpredictable. A full characterization of history-independent equilibrium strategies is also obtained.

© 2014 Elsevier Inc. All rights reserved.

1. Introduction

Unpredictable behavior is central to optimal play in many strategic situations, especially in social interactions with con-flicts of interest. There are many illustrative examples from competitive sports, such as the direction of tennis serves and penalty kicks in soccer. Other relevant examples include secrecy in military affairs, bluffing behavior in poker, and tax audit-ing. In these situations, von Neumann and Morgenstern (1944) suggest that, even against a moderately intelligent opponent, a player should aim at being unpredictable. Schelling (1980) argues that the essence of such unpredictable behavior is to avoid simple regularities, which may allow the opponent to predict one’s behavior for exploitation. These arguments seem to be based on an intuitive connection between complexity and unpredictability, and this paper formalizes this connection in the context of a repeated two-person zero-sum game. Specifically, I propose a new model of strategic complexity in which unpredictable behavior emerges in equilibrium due to complexity considerations.

The proposed framework has two basic ingredients. First, each player i is endowed with an infinite binary sequence, θ i , which represents the “sources” of player i’s complicated or unpredictable behavior. Second, each player i has to choose an oracle machine to implement his strategy, using θ i as the oracle. Except for being able to ask queries and obtain answers from the oracle, an oracle machine behaves exactly the same as a Turing machine. In particular, each oracle machine has only finitely many instructions, and any successful computation has to terminate within finitely many steps. Thus, the set of strategies that player i can implement is a countable set. Moreover, the oracle θ i also represents the complexity of player i’s strategies. An oracle θ j is more complex than another oracle θ i if there is an oracle machine that computes θ i using θ j

* Corresponding address: 2001 Sheridan Road, Jacobs Center 548, Evanston, IL 60208-0001, United States.E-mail address: t-hu@kellogg.northwestern.edu.

1 I am very grateful to Neil Wallace, Kalyan Chatterjee, and Edward Green for their guidance and support. I am also indebted to Nabil Al-Najjar, Lance Fortnow, Ehud Kalai, Mamoru Kaneko, Brian Rogers, Eran Shmaya, Stephen Simpson, Jakub Steiner for useful discussions and inputs.

http://dx.doi.org/10.1016/j.geb.2014.08.0020899-8256/© 2014 Elsevier Inc. All rights reserved.

2 T.-W. Hu / Games and Economic Behavior 88 (2014) 1–15

as the oracle. On the one hand, if θ i is computable from θ j , then any of player i’s strategies is simple in the sense that it can be simulated by player j. On the other hand, if θ i is not computable from θ j and vice versa, each player can implement some strategy that is complex relative to the other player’s complexity constraint.

The existence of mutually uncomputable oracles distinguishes my approach from most of the previous literature on repeated games with complexity constraints. Many papers in the literature model strategic complexity using finite automata and use the number of states to measure complexity. In the context of repeated zero-sum games, Ben-Porath (1993) obtains a precise bound on the difference between the sizes of the automata available to the two players so that one can fully exploit the other. Intuitively, when one player can use sufficiently larger automata than the other, the weaker player becomes perfectly predictable. However, to obtain an equilibrium whose value equals to that of the stage game and hence with no player exploiting the other, randomization is necessary and the sizes of the automata available to the two players have to be close to each other.

In contrast, no randomization is allowed in my framework, and the oracles are the only sources of unpredictability. While a randomized strategy is assumed to be unpredictable for any player (even for the one who is using it), in my approach unpredictability is a relative notion and is connected to complexity. A strategy can be unpredictable for one player but perfectly predictable for the other. This feature brings about new results that show an intimate relationship between complexity and unpredictability in the context of repeated zero-sum games. It also formalizes the intuition that equilibrium only requires players’ behavior to be unpredictable to its opponents. The main results highlight a precise unpredictability requirement in terms of both complexity and statistical patterns for a strategy to be optimal.

This paper contains three main results, which can be classified according to the value obtained in equilibrium. The first result (Theorem 3.1) is concerned with the case where one player fully exploit the other, which resembles previous results in the literature. I show that an equilibrium exists in which one player can fully exploit the other when the former player’s oracle is sufficiently more complex than the other player’s. The second and the third results are concerned with obtaining a value equal to the value of the stage-game mixed equilibrium. To obtain such a value, the two oracles are necessarily mutually uncomputable (Proposition 3.1), and any of the equilibrium strategies is uncomputable from the other player’s perspective. This result, while intuitive, is novel because it shows the necessity of strategic complexity for optimality, and suggests that the role of randomization in the previous works is to provide strategic complexity that cannot be modeled by finite automata.

More specifically, the second result (Theorem 3.2) gives a sufficient condition, called mutual complexity, on the two oracles for equilibrium existence with the stage-game mixed-equilibrium value. Mutual complexity is a stronger require-ment than mutual uncomputability, and it requires the two players’ oracles to be sufficiently complex relative to each other according to Kolmogorov complexity. This result extends the well-known connection between Kolmogorov complexity and unpredictability in the algorithmic randomness literature (see Downey et al., 2006 for a survey) to a game-theoretical setting. It shows that complexity can be a source for unpredictability in repeated games. The third result (Theorem 3.3) gives a full characterization of equilibrium history-independent strategies under mutual complexity. As mentioned earlier, any equilib-rium strategy has to be uncomputable from the other player’s perspective. The characterization result shows that optimality also requires unpredictable properties in terms of statistical patterns. Thus, for a sequence of plays to be an equilibrium strategy, it has to not only be complex relative to the other player’s oracle but also satisfy frequency requirements.

1.1. Related literature

Conceptually, the model here is close to repeated-game models with finite automata initiated by Aumann (1981), as well as models with bounded recall (Lehrer, 1988). My first result (Theorem 3.1) is similar to Theorem 1 in Ben-Porath(1993), and to many others which consider two-person repeated zero-sum games and show that the “smarter” player exploits his opponent in equilibrium.2 The main difference between my framework and the existing literature, however, is manifested in my second result (Theorem 3.2) and third result (Theorem 3.3). Without invoking randomization (although, as discussed in Section 4, my results are robust to the introduction of randomization), my results show that unpredictable behavior may emerge in equilibrium due to complexity constraints, but only when both players are “smart” relative to each other in the sense that each player has a strategy that the other player cannot simulate. Moreover, they give a precise requirement of unpredictability in terms of both complexity and statistical patterns (without the introduction of random variables, however). This mutual “smartness” requirement is not present in the existing literature, but my results suggest that it is the key to understand the connection between complexity and unpredictability in repeated games.

There are also models that use Turing machines to measure strategic complexity, including Anderlini and Sabourian(1995) and Hu and Shmaya (2013). Anderlini and Sabourian (1995) consider games with common interests in which only computable strategies are allowed and hence have a very different focus. Hu and Shmaya (2013) also employ oracle ma-chines to model forecasting strategies in the context of expert testing, but the focus there is on when the expert can “outsmart” the test but not on the connection between complexity and unpredictability.

2 Neyman (1998) obtains asymptotic results, including the exploitation result, in finitely repeated games with finite automata. There are many other papers considering settings with an unconstrained player and a restricted player. Neyman and Okada (1999, 2000) study the value of the repeated game as a function of the constraint on the restricted player’s strategic entropy. Gossner and Vieille (2002) consider a situation where one player can only condition his actions on a biased coin and show that bounded strategic entropy emerges endogenously.

T.-W. Hu / Games and Economic Behavior 88 (2014) 1–15 3

2. Relative computability and complexity

The framework in this paper is built on relative computability based on oracle machines and Kolmogorov complexity. Due to space constraints, I refer readers to Nies (2009) and Downey and Hirschfeldt (2010) for detailed formal definitions, while this section serves two purposes: the first is to introduce concepts and notations necessary for the main text; the second is to introduce two results, the Enumeration Theorem from computability theory, and Lemma 2.1, which shows unpredictable properties of sequences with high Kolmogorov complexities. Those two results will be useful for my equilibrium analysis. I begin with some informal discussion about Turing computability.

Intuitively, a function f with arguments and values in natural numbers is computable if there exists a computer program that computes f , i.e., if n ∈ N is in the domain of f then the program halts on input n and produces output f (n). Two remarks useful for later purposes are in order. First, the domain of a computable function can be a strict subset of N: an algorithm may run into an infinite loop and never produce an output for some inputs. The notation f :⊂ N → N is used to emphasize that the domain of f , denoted by dom( f ), may be a strict subset of N. When dom( f ) = N, f is said to be total. Second, the notion of computability can be extended to other sets of mathematical objects such as strings over a finite set. Indeed, the set of finite strings over a finite set X , denoted by X∗ = ⋃

n∈N Xn (where X0 = {ε} and ε being the empty string), can be effectively identified with N by encoding σ = (σ0, ..., σn−1) as

∑n−1t=0 σt2n−1−t + 2n − 1. Given the coding,

computable functions to and from X∗ are well-defined.A fundamental result in computability theory, the Enumeration Theorem (Odifreddi, 1989, Theorem II.1.5), states the

existence of a binary computable function U :⊂ N2 →N, called the universal Turing machine, such that for every computable function f there is an s for which f (·) ∼= U (s, ·), that is, f (t) is defined if and only if U (s, t) is defined, and when both are defined their values coincide. This theorem also provides an explicit example of an uncomputable function: the characteristic function for the set H = {t ∈ N : (t, t) ∈ dom(U )}, denoted by κ = (κ0, κ1, ...), with κt = 1 if t ∈ H and κt = 0 otherwise.

Now I turn to the notion of an oracle machine, which is a Turing machine with access to a black box, called an oracle. An oracle machine is a Turing machine with an additional instruction: it may “call” oracles during its computation. Calling an oracle is similar to calling another program, and an oracle machine may give queries to an oracle which returns answers that are not directly computed by the machine. In general, an oracle can be any function from N. However, here I define an oracle as an infinite binary sequence θ = (θ0, θ1, ...) ∈ {0, 1}N , and a function f is said to be computable relative to θ , or θ -computable, if it can be computed with an oracle machine using θ as the oracle. Restricting oracles to binary sequences is without loss of generality: for an arbitrary oracle θ ′ , there exists a binary sequence θ such that the set of θ ′-computable functions coincides with the set of θ -computable functions.

Many results in Turing-computability can be relativized to θ -computability (Nies, 2009, page 10). One instance of such results is the relativized Enumeration Theorem for θ -computability: for any oracle θ , there exists a binary θ -computable function U θ :⊂ N2 → N such that for every θ -computable function f there is an s for which f (·) ∼= U θ (s, ·). As a corol-lary, only countably many functions are θ -computable. Another relativization result states that the characteristic function, denoted by κθ , for the set Hθ = {t : (t, t) ∈ dom(U θ )}, the halting problem relative to θ , is not θ -computable (Nies, 2009, Proposition 2.1.12). The sequence κθ is used to obtain a sufficient condition for equilibrium existence with one player fully exploiting the other in Theorem 3.1.

An oracle can be regarded as a function over N and hence, given two oracles θ and η, it is legitimate to ask whether θ is η-computable or not. It is easy to show that if θ is η-computable and if a function f is θ -computable, then f is also η-computable. The computability relation then gives a partition over oracles called Turing degrees: θ and η belong to the same degree if and only if θ is η-computable and vice versa. For any oracle θ , θ and κθ do not belong to the same degree. Crucial to my analysis, there exist mutually uncomputable oracles θ and θ ′ such that θ is not θ ′-computable and vice versa(Nies, 2009, Theorem 1.6.8).

Turing degrees give a qualitative measure of complexity for oracles. Another concept, Kolmogorov complexity (Kolmogorov, 1965), gives a quantitative measure. Its definition is based on the minimum description length with respect to a given (oracle) machine. Given a machine L : {0, 1}∗ → {0, 1}∗ , the Kolmogorov complexity of a string σ ∈ {0, 1}∗ is defined as

K L(σ ) = min{|τ | : τ ∈ {0,1}∗, L(τ ) = σ

},

where |τ | denotes the length of the string τ . It is fruitful to focus on prefix-free machines only.3 For any oracle θ , because of the relativized Enumeration Theorem, there exists a θ -computable prefix-free machine for θ , denoted by L∗

θ , that is optimal among prefix-free machines in terms of asymptotic description lengths: for any θ -computable prefix-free machine L, there exists a constant d such that for all σ ∈ {0, 1}∗ , K L∗

θ(σ ) ≤ K L(σ ) + d (Nies, 2009, Proposition 2.2.7). I use Kθ to denote K L∗

θ.

Moreover, for any oracle θ , there is a constant d ∈ N such that Kθ (σ ) ≤ |σ | + 2 log2 |σ | + d for all σ ∈ {0, 1}∗ . Given an oracle θ , another oracle η is said to be θ -incompressible, if for some d ∈ N, Kθ (η[t]) > t − d for all t ∈ N, where η[t] is the initial segment of η with length t . Incompressibility relative to θ says that, intuitively, η cannot be expressed in a shorter way than η itself w.r.t. the best θ -computable machine.

3 A machine L is prefix-free if for any two strings σ , τ ∈ dom(L), σ is not an initial segment of τ . See Li and Vitányi (1997) for a detailed discussion about reasons to restrict attention to prefix-free machines.

4 T.-W. Hu / Games and Economic Behavior 88 (2014) 1–15

An explicit example of a θ -incompressible sequence is Chaitin’s (Chaitin, 1975) Omega, defined as Ωθ = ∑σ∈ dom(L∗

θ ) 2−|σ | . The binary expansion of Ωθ , denoted by ωθ = (ωθ

0, ωθ1, ...), is θ -incompressible and its Turing degree is below or equal to

that of κθ (Downey and Hirschfeldt, 2010, Theorems 15.2.1 and 15.2.2). The set of θ -incompressible sequences has measure 1 under the uniform measure4 over {0, 1}N (Downey and Hirschfeldt, 2010, Corollary 6.2.6). The sequence ω, which denotes ωθ for a Turing computable θ , will be used to construct oracles that satisfy the sufficient condition for my existence result.

Here I present some unpredictable properties of incompressible sequences that are useful for my analysis. Following Schnorr (1971), these properties are defined through computable martingales. I begin with notation about concatenation: I use στ to denote the string that joins two strings σ and τ end to end. A martingale over {0, 1} is a nonnegative function B : {0, 1}∗ → R+ that satisfies the following property:

B(σ ) = 1

2B(σ0) + 1

2B(σ1) for all σ ∈ {0,1}∗. (1)

A martingale B corresponds to a betting strategy that begins with capital B(ε) (recall that ε is the empty string) and de-scribes the evolution of the amount of capital resulting from the corresponding betting strategy for different realization of outcomes described by σ . Condition (1) ensures that the gamble is “fair.” The corresponding betting strategy is a contin-gency plan—at period t , depending on the partial realization σ = (σ0, ..., σt−1) before t , the betting strategy chooses the amount to be bet on the outcomes 0 and 1, subject to the amount of capital available, B(σ ).

A martingale B is said to succeed over an infinite sequence ξ = (ξ0, ξ1, .ξ2, ...) ∈ {0, 1}N if B reaches unbounded payoffs by betting against ξ , that is, lim supt→∞ B(ξ [t]) = ∞. On the one hand, if the sequence ξ has a simple pattern, then it is easy to devise a simple martingale to succeed over it. For example, consider a sequence ξ sp (sp stands for “simple”) satisfying that ξ

sp2t+1 = 1 for all t . To succeed over ξ sp , one can simply bet all the available capital on the outcome 1 at odd periods and bet

nothing at even periods. This corresponds to the martingale Bsp defined as Bsp(ε) = 1, Bsp(σ1) = Bsp(σ0) = Bsp(σ ) for all σ with even lengths (including ε), and Bsp(σ1) = 2Bsp(σ ) and Bsp(σ0) = 0 for all σ with odd lengths. It is straightforward to verify that Bsp(ξ sp[2t]) = 2t−1 for all t ≥ 1 and hence Bsp succeeds over ξ sp . On the other hand, it seems intuitive that any martingale would not succeed over a random sequence following an i.i.d. ( 1

2 , 12 ) process. In fact, for any martingale B , the set of infinite sequences over which B succeeds, denoted succ(B), has zero probability under an i.i.d. ( 1

2 , 12 ) process (Nies, 2009, Proposition 7.1.15).

Instead of discussing stochastic processes and all martingales, however, the algorithmic randomness literature considers martingales with bounded complexity and defines randomness with respect to those martingales. Given an oracle θ , an infinite binary sequence ξ ∈ {0, 1}N is said to be computably random relative to θ , or θ -computably-random, if ξ /∈ succ(B)

for any θ -computable martingale B (here I restrict the martingales to take values only in rational numbers).5 It follows that sequences with simple patterns are not computably random. For example, because Bsp is computable and it succeeds over the sequence ξ sp , it disqualifies the sequence ξ sp as a computably random sequence. Similarly, if ξ is a θ -computable sequence, then ξ is not θ -computably-random. However, incompressible sequences are computably random, as the following lemma shows.

Lemma 2.1. Let θ be an oracle. If ξ ∈ {0, 1}N is θ -incompressible, then ξ is θ -computably-random.

This lemma is a direct implication of the relativized versions of Nies (2009), Theorem 3.2.9 and Proposition 7.2.6. The in-tuition for the proof is that if there is a martingale that succeeds over ξ , one can use that strategy to devise a θ -computable prefix-machine L such that for any d, there is an initial segment of ξ whose Kolmogorov complexity w.r.t. L is less than its length minus d and hence ξ cannot be θ -incompressible. Because θ -computable sequences are not θ -computably-random, it follows that any θ -incompressible sequence ξ is not θ -computable.

3. Strategic complexity in repeated zero-sum games

3.1. Strategic complexity

Here I propose a model of repeated two-person zero-sum games with complexity constraints. The stage game is a finite two-person zero-sum game g = 〈X1, X2, h1, h2〉, where for i = 1, 2, Xi is the set of player i’s actions and hi : X1 × X2 → Q

is the payoff function for player i (Q is the set of rational numbers).6 The game g is zero-sum: h1(x1, x2) + h2(x1, x2) = 0for all (x1, x2) ∈ X1 × X2. It is well-known that if the payoff functions are rational-valued, then g has a rational-valued mixed equilibrium (p∗

1, p∗2), i.e., p∗

i ∈ �(Xi) = {p ∈ ([0, 1] ∩ Q)Xi : ∑x∈Xi

p[x] = 1} for both i = 1, 2. I focus on repeated games whose stage games have no pure equilibrium.7 The following two payoff levels are used later: v∗

i (g), the (mixed)

4 The uniform measure is the unique countably additive measure μ over {0, 1}N such that μ([σ ]) = 2−|σ | for all σ ∈ {0, 1}∗ , where [σ ] is the set of sequences that have σ as their initial segments.

5 Notice that in this definition the sequence ξ is a deterministic sequence of 0’s and 1’s. This terminology is borrowed from the algorithmic randomness literature (see Downey and Hirschfeldt, 2010 or Nies, 2009 for a survey).

6 I assume rational-valued payoffs for computability issues.7 If the stage game has a pure equilibrium, there are obvious equilibria in the repeated game as well where unpredictability plays no role.

T.-W. Hu / Games and Economic Behavior 88 (2014) 1–15 5

equilibrium payoff for i, and v̄ i(g) = minx−i∈X−i maxxi∈Xi hi(xi; x−i), the maximum payoff that player i can achieve if he can “predict” his opponent’s actions. It is well-known that v∗

1(g) = −v∗2(g), and g has no pure equilibrium (for both players) if

and only if v̄1(g) > v∗1(g) > −v̄2(g).

Each player i is endowed with an oracle θ i , and is restricted to use an oracle machine to implement his strategy. Hence, a strategy is feasible for player i if and only if it is θ i -computable. The set of all θ i -computable functions that are total is denoted by C(θ i). Notice that, by the relative Enumeration Theorem, the set C(θ i) is countable and, hence, the set of feasible strategies for each player is a countable set. In particular, any strategy more powerful, i.e., has a higher Turing degree, than κθ i

, the halting problem relative to θ i , is not feasible for player i.The long-run average criterion is adopted to evaluate payoffs.8 However, because not all sequences of average payoffs

converge, it is necessary to extend the criterion to nonconvergent sequences. Formally, let ϕi : ∞ → R be a mapping from the set of all bounded sequences of real numbers to real numbers for each i = 1, 2. The function ϕi extends the notion of limits to nonconvergent sequences, and I require three properties on ϕ1, ϕ2: for any bounded sequences {at}∞t=0 and {bt}∞t=0,

(C1) for each i = 1, 2, if {at} has a limit a, then ϕi({at}) = a;(C2) for each i = 1, 2, if at ≤ bt for all t , then ϕi({at}) ≤ ϕi({bt});(C3) if at + bt = c for all t , then ϕ1({at}) + ϕ2({bt}) = c.

The conditions (C1) and (C2) are rather weak and natural, and they are concerned with individual ϕi ’s only. (C3) requires ϕ1 and ϕ2 to be connected in a way that preserves zero-sum. There are many examples of (ϕ1, ϕ2) that satisfy (C1)–(C3), including (ϕ1, ϕ2) = (lim sup, lim inf) and ϕ1 = ϕ2 being a Banach limit. From now on, unless otherwise specified, I fix a pair (ϕ1, ϕ2) that satisfies (C1)–(C3).

Definition 3.1. Let g = 〈X1, X2, h1, h2〉 be a finite zero-sum game and let θ1, θ2 be two oracles. The repeated game with oracles θ1, θ2 based on the stage game g , denoted by RG(g, θ1, θ2) = 〈A1, A2, u1, u2〉, is defined as follows:

(a) Ai = {αi : X∗−i → Xi : αi ∈ C(θ i)} is the set of player i’s strategies;

(b) ui :A1 ×A2 →R is player i’s payoff function defined as

ui(α1,α2) = ϕi

({T −1∑t=0

hi(ξα,1t , ξ

α,2t

)/T

}∞

T =1

), (2)

where (ξα,1t , ξα,2

t ) is the outcome of period t for the strategy profile α = (α1, α2) defined by ξα, j0 = α j(ε) and for any t ≥ 0,

ξα, jt+1 = α j(ξ

α,− j0 , ξα,− j

1 , ..., ξα,− jt ), for both j = 1, 2.

Because (ϕ1, ϕ2) satisfies (C3), RG(g, θ1, θ2) is also a two-person zero-sum game. Therefore, if a Nash equilibrium exists, the equilibrium payoff is unique. The equilibrium payoff for player 1 is called the value. Note that there may not exist any Nash equilibrium (in pure strategies).9 Following the literature, I will explore the relationship between complexity constraints (modeled by θ1 and θ2) and the value. In contrast to the literature, I focus on unpredictable properties of the equilibrium behavior that result from the complexity constraints.

Let g be a zero-sum game without equilibrium in pure strategies. I focus on two payoff levels for player 1: v∗1(g), the

equilibrium payoff of g in mixed strategies, and v̄1(g) = minx2∈X2 maxx1∈X1 h1(x1, x2). As in most papers on repeated zero-sum games, it is easier to obtain clean results for these two cases, and, specifically for my approach, they highlight the connection between complexity and unpredictability. In Section 3.2, I give a sufficient condition on (θ1, θ2) for equilibrium existence with equilibrium payoff v̄1(g), which requires θ1 to be sufficiently more complex than θ2 according to the Turing degrees. In this case, player 2’s behavior is perfectly predictable from player 1’s perspective. Although I use a different com-plexity measure, this result resembles previous results in the literature on repeated games with finite-automata. Section 3.3deals with value v∗

1(g), and the results obtained there are novel. Section 3.3.1 has two results regarding existence. The first is a necessary condition, called mutual uncomputability, to achieve v∗

1(g). It requires that, for both i = 1, 2, θ i is not θ−i -computable. This result has implications for unpredictability of equilibrium behavior as well. The second result is a sufficient condition for equilibrium existence with value v∗

1(g), called mutual complexity. The latter implies mutual un-computability. Section 3.3.2 gives a characterization of history-independent equilibrium strategies under mutual complexity, which gives the precise unpredictability requirement for equilibrium behavior.

8 See Hu (2013) for an axiomatic justification of the long-run average criterion, and Aumann (1997) for an argument to adopt this criterion in models of bounded rationality.

9 As the strategy set is countably infinite for both players, a mixed Nash equilibrium may not exist either. However, I focus on pure equilibria only because of the purposes of the paper.

6 T.-W. Hu / Games and Economic Behavior 88 (2014) 1–15

3.2. Equilibrium when one player is predictable

Here I give a sufficient condition for equilibrium existence with value v̄1(g). Recall that for any oracle θ , κθ is the halting problem relative to θ ; θ is κθ -computable but κθ is not θ -computable.

Theorem 3.1. Suppose that κθ2is θ1-computable. Then, an equilibrium in RG(g, θ1, θ2) exists with value v̄1(g).

The condition in Theorem 3.1 is not vacuous: for any oracle θ , (θ1, θ2) = (κθ , θ) satisfies the condition. Moreover, for any oracle θ , there are uncountably many oracles θ ′ such that κθ is θ ′-computable, and hence (θ1, θ2) = (θ ′, θ) satisfies the condition as well.

The value in Theorem 3.1 implies that player 1 has a strategy α∗1 such that for any strategy α2 of player 2, α∗

1 guarantees payoff v̄1(g) against that strategy and hence respond optimally to player 2’s actions for almost all periods. Thus, Theorem 3.1shows that when player 1’s oracle is sufficiently stronger, he can fully exploit player 2 and player 2’s strategies are almost perfectly predictable from player 1’s perspective. This result is very similar to Theorem 1 in Ben-Porath (1993), which shows that for any set of strategies that can be implemented with automata of a fixed size that player 2 can use, there is a sufficiently large automaton that player 1 can use to obtain payoff v̄1(g) in equilibrium. The intuition here is also similar to that in Ben-Porath (1993): the stronger player can devise a strategy that first finds out the strategy used by the weaker player and then fully exploits the weaker henceforth.

However, while in Ben-Porath (1993) for a fixed size n there are only finitely many strategies that can be implemented with an automaton with n states, countably many strategies are θ2-computable. Thus, it is necessary to use the Enumeration Theorem to construct the strategy α∗

1 to ensure that it is κθ2-computable. The following is a sketch of the proof. Let

φ0, φ1, ...φk, ... be an effective enumeration of θ2-computable functions (including partial ones). Because the enumeration is effective, α∗

1 , against any strategy α2 of player 2, can find its index and then fully exploit it. However, the enumeration does not tell which ones are total and hence qualify as strategies. The assumption that κθ2

is θ1-computable ensures that when α∗

1 finds a potential index k for player 2’s strategy, player 1 can check whether φk is a valid strategy in the sense that it gives an action for the next period.

3.3. Equilibrium when both players are unpredictable

3.3.1. Equilibrium existenceFirst I give a necessary condition for equilibrium existence with value v∗

1(g). Let g be a zero-sum game without equi-librium in pure strategies. Intuitively, to have an equilibrium with value v∗

1(g), it is necessary for each player to have a strategy that is “unpredictable” to the opponent. The following proposition shows that, in the repeated game RG(g, θ1, θ2)

with complexity constraints, this unpredictability requirement is translated into a complexity requirement.

Proposition 3.1 (Mutual uncomputability). Let g be a zero-sum game with v̄1(g) > v∗1(g) > −v̄2(g). Suppose that an equilibrium

exists in RG(g, θ1, θ2) with value v∗1(g). Then, θ i is not θ−i -computable for both i = 1, 2.

The mutual uncomputability condition in Proposition 3.1 is not vacuous in my framework because there exist incom-parable Turing degrees (Nies, 2009, Theorem 1.6.8). The proof of Proposition 3.1 implies that, if an equilibrium exists with value v∗

1(g) in RG(g, θ1, θ2), then for both i = 1, 2, and for any equilibrium strategy α∗i , α∗

i is not θ−i -computable. Thus, equilibrium behavior is necessarily unpredictable in a relative sense: each player cannot fully cope with the other player’s strategy in equilibrium. While Theorem 3.1 resembles previous findings, Proposition 3.1 is novel.

First, the condition in Proposition 3.1 requires the two oracles to be incomparable. Theorem 3.2 below gives a sufficient condition for existence, and, as stated there, the Chaitin’s Omega provides an example of oracles that satisfy the sufficient condition. In contrast, for models with finite automata, because complexity is measured by number of states and hence the complexity measure is a linear order, results like Proposition 3.1 and Theorem 3.2 are not obtainable. Second, Proposition 3.1has implications for the complexity of equilibrium behavior. As mentioned earlier, any equilibrium strategy for player i is necessarily θ−i -uncomputable. Section 3.3.2 gives a precise characterization of history-independent equilibrium strategies in terms of their unpredictable properties. These implications are different from findings in the literature. With finite automata, Ben-Porath (1993) obtains the value v∗

1(g) (asymptotically) with randomization, but the equilibrium strategies employed by the two players are necessarily similar in terms of state-complexity. In contrast, Proposition 3.1 implies that equilibrium strategies in my framework have quite different complexities in terms of Turing degrees.

Proposition 3.1 shows that, for equilibrium existence with value v∗1(g), a minimum complexity requirement on (θ1, θ2)

is mutual uncomputability. Now I give a sufficient condition based on incompressible sequences introduced in Section 2. Recall that an oracle ν is incompressible relative to another oracle θ (called θ -incompressible hereafter) if for some d, Kθ (ν[t]) > t − d for all t ∈N.

Definition 3.2. Two oracles θ1 and θ2 are mutually complex if for both i = 1, 2, there exists a θ i -computable oracle ν i that is θ−i -incompressible.

T.-W. Hu / Games and Economic Behavior 88 (2014) 1–15 7

Because for oracle θ , θ -incompressible sequences are not θ -computable, mutual complexity implies mutual uncom-putability. Here I give a concrete example of mutually complex oracles. Van Lambalgen’s Theorem (Nies, 2009, Theo-rem 3.4.6) shows that if θ is incompressible (relative to the Turing computability), then θodd is θ even-incompressible and θ even is θodd-incompressible, where θ even

t = θ2t and θoddt = θ2t+1 for all t ∈ N. Hence, if θ is incompressible, then θodd and

θ even are mutually complex. Because ω, the Chaitin’s Omega, is incompressible, this implies that (θ1, θ2) = (ωeven, ωodd)

are mutually complex. This also shows that (ωeven, ωodd) satisfies mutual uncomputability. Moreover, because the set of incompressible sequences has measure 1 under the uniform measure, Van Lambalgen’s Theorem also implies that there are uncountably many pairs of oracles satisfying mutual complexity. The following proposition summarizes these find-ings.

Proposition 3.2. There are uncountably many pairs of oracles that satisfy mutual complexity.

The following theorem states that mutual complexity is a sufficient condition for the existence of an equilibrium with value equal to that of the stage game.

Theorem 3.2 (Existence). Suppose that θ1 and θ2 are mutually complex. Then there exists an equilibrium in RG(g, θ1, θ2) with value v∗

1(g).

The existence result does not depend on (C3), but the uniqueness of the equilibrium payoff does. Without (C3), an equilibrium still exists with equilibrium payoff for player i equal to v∗

i (g), but there may be other equilibria with different equilibrium payoffs.

The proof of Theorem 3.2 is based on the close connection between incompressibility and unpredictability as stated in Lemma 2.1. Intuitively, player i could guarantee the payoff v∗

i (g) by playing a sequence that has unpredictable proper-ties similar to a θ−i -computably-random sequence (as introduced in Section 2). However, because the appropriate mixture of different actions needed to guarantee v∗

1(g) may not be uniform across different actions, I generalize the notions of θ -computable martingales and θ -computable-randomness to sequences over an arbitrary finite set X and an arbitrary dis-tribution p ∈ �(X), where �(X) is the set of probability distributions over X with rational probability values. After defining θ -computable-randomness w.r.t. an arbitrary distribution, I give two key lemmas—one relating θ -incompressibility to gen-eral θ -computably-random sequences and the other gives some unpredictable properties of general θ -computably-random sequences—that are the main engines for the proof.

Given a finite set X and a distribution p ∈ �(X), a martingale over X for p is a nonnegative function B : X∗ → R+ such that for all σ ∈ X∗ ,

B(σ ) =∑x∈X

p[x]B(σ x).

As before, a martingale B for p is said to succeed over an infinite sequence ξ ∈ XN if lim supt→∞ B(ξ [t]) = ∞. A sequence ξ ∈ XN is said to be θ -computably-random for p if there is no θ -computable martingale for p ∈ �(X) that succeeds over ξ . The following lemma extends Lemma 2.1 and shows that θ -incompressible sequences can be used to generate general θ -computably-random sequences.

Lemma 3.1. Let θ be an oracle and let ν ∈ {0, 1}N be a θ -incompressible sequence. For any finite set X and any p ∈ �(X), there is a ν-computable sequence ξ ∈ XN that is θ -computably-random for p.

Computable randomness is a useful concept because it is preserved under many computable operations, as shown in the proof of Lemma 3.1. However, for the game-theoretical analysis, I need to use another concept called stochastic-ity, which is a weaker unpredictable property based on computable schemes of choosing subsequences. Such schemes are formalized by selection functions. Given a finite set X , a selection function for X is a total function r : X∗ → {0, 1}. Given a sequence ξ = (ξ0, ξ1, ξ2, ...) ∈ XN , the function r is used to obtain a subsequence ξ r of ξ in the following man-ner. For each t = 0, 1, 2, ... , the function r decides whether ξt enters the subsequence ξ r based on the initial history ξ [t] = (ξ0, ξ1, ..., ξt−1), that is, ξt is selected to form ξ r if and only if r(ξ [t]) = 1. Formally, for all t ∈ N, ξ r

t = ξg(t) , where g(0) = min{t : r(ξ [t]) = 1}, and g(t) = min{s : r(ξ [s]) = 1, s > g(t − 1)} for t > 0.10 For example, take the selection function rsp such that rsp(σ ) = 1 if |σ | is odd and rsp(σ ) = 0 if |σ | is even; for any sequence ξ = (ξ0, ξ1, ξ2, ...), ξ rsp = (ξ1, ξ3, ξ5, ...). In general, a selection function might not produce an infinite subsequence, but only a finite initial segment. The fol-lowing lemma, which generalizes Theorem 7.4.2 in Downey and Hirschfeldt (2010), shows that θ -computably-random sequences for p have relative frequencies according to p on all subsequences selected by θ -computable selection func-tions.

10 If at some t , there is no s > g(t − 1) such that r(ξ [s]) = 1, then g(t) is undefined.

8 T.-W. Hu / Games and Economic Behavior 88 (2014) 1–15

Lemma 3.2. Let X be a finite set and let p ∈ �(X). If ξ ∈ XN is θ -computably-random for p, then

limT →∞

T −1∑t=0

cx(ξrt )

T= p[x] for all x ∈ X (3)

for any θ -computable selection function r for X such that ξ r is an infinite sequence.

A sequence ξ ∈ XN is said to be a θ -stochastic sequence for p if (3) holds for all θ -computable selection function r for Xsuch that ξ r is an infinite sequence. Sequences with simple patterns cannot be stochastic for non-degenerate distributions. For example, for any p ∈ �({0, 1}) with p[0] ∈ (0, 1), a sequence ξ sp ∈ {0, 1}N satisfying ξ sp

2t+1 = 1 for all t is not stochastic for p, because under the selection function rsp , the limit frequency of 1’s in (ξ sp)rsp

is 1 �= p[1].Given Lemmas 3.1 and 3.2, the logic to prove Theorem 3.2 is the following. Let (p∗

1, p∗2) be a mixed equilibrium of the

game g . By Lemma 3.1, for each i, there exists a θ i -computable sequence ξ i that is θ−i -computably-random for p∗i . Then,

by Lemma 3.2, ξ i is θ−i -stochastic for p∗i . The main part of the proof shows that if ξ i is θ−i -stochastic for p∗

i , then for any α−i ∈A−i , ui(ξ

i; α−i) ≥ v∗i (g). This implies that (ξ1, ξ2) is an equilibrium with payoff for player 1 equal to v∗

1(g).11

3.3.2. Equilibrium characterizationTheorem 3.2 shows that, if (θ1, θ2) are mutually complex, then the game RG(g, θ1, θ2) has value v∗

1(g). Moreover, it shows that if ξ i is a θ−i -stochastic sequence for p∗

i with p∗i being a mixed equilibrium strategy in g , then ξ i is an equilib-

rium strategy in the repeated game. Thus, θ−i -stochasticity is a sufficient unpredictability requirement for optimality. For a necessary unpredictability requirement, Proposition 3.1 implies that any equilibrium strategy α∗

i is not θ−i -computable for both i = 1, 2. Here I give a stronger requirement, which is very close to stochasticity, for equilibrium history-independent strategies. As is typical in repeated games, a full characterization of equilibrium (history-dependent) strategies will be diffi-cult and I do not pursue it here.

Let θ be an oracle and let p ∈ �(X) for a finite set X . A sequence ξ ∈ XN is said to be an essentially θ -stochastic sequence for p if (3) holds for all θ -computable selection functions r for X such that lim infT →∞

∑T −1t=0

r(ξ [t])T > 0.

Theorem 3.3 (Equilibrium characterization). Suppose that (θ1, θ2) are mutually complex. Let g be a zero-sum game with a unique mixed equilibrium (p∗

1, p∗2) ∈ �(X1) × �(X2).

(a) If ξ ∈ XN

i is both θ i -computable and is a θ−i -stochastic sequence for p∗i , then ξ is an equilibrium strategy in RG(g, θ1, θ2) for any

(ϕ1, ϕ2) satisfying (C1)–(C3).(b) If a θ i -computable sequence ξ ∈ XN

i is an equilibrium strategy in RG(g, θ1, θ2) for any (ϕ1, ϕ2) satisfying (C1)–(C3), then ξ is an essentially θ−i -stochastic sequence for p∗

i .

In both parts of Theorem 3.3, the equilibrium strategies are required to be robust to how the limits are extended to non-convergent sequences in evaluating the payoffs, given by (ϕ1, ϕ2). This robustness is necessary for part (b) because the convergence of relative frequencies required by essential stochasticity may not hold for equilibrium strategies with respect to an arbitrary pair (ϕ1, ϕ2). Theorem 3.3 can be easily extended to repeated games with (θ1, θ2) in which g has multiple equilibria, but the characterization is slightly more complicated as equilibrium strategies are closed under convex combinations in g and hence different subsequences in ξ i may have different limit frequencies (although all of them are mixed equilibrium strategies of g).

Similar to Proposition 3.1, Theorem 3.3 shows that equilibrium (history-independent) strategies necessarily satisfy certain unpredictable properties. Unlike Proposition 3.1, however, the property identified here, (essential) stochasticity, is concerned both with statistical patterns and with complexity. In particular, taking r to be the selection function that selects all el-ements, then Theorem 3.3 implies that the limit relative frequency of any equilibrium history-independent strategy is an equilibrium mixed strategy of the stage-game. Nonetheless, notice that the unpredictability requirement is still concerned with complexity—the equilibrium (history-independent) strategy has to be essentially stochastic relative to the other player’s oracle.

The stochasticity requirement also resembles some empirical tests used to test the equilibrium hypothesis in repeated zero-sum games. Let (p∗

1, p∗2) be the unique equilibrium in g . Given an equilibrium (ξ1, ξ2)12 and an action x ∈ X1 such

that p∗1[x] > 0, let rx be the selection function such that rx(σ ) = 1 if and only if ξ1|σ | = x. Then, by Theorem 3.3, in the

subsequence of ξ2 selected by rx , the limit frequency is p∗2 and hence the average payoff along that subsequence is v∗

2(g). Thus, for any x, y with p∗

1[x] > 0 and p∗2[y] > 0, the average payoffs for player 2 along the two subsequences where player 1

uses x and y, respectively, are the same. Such an implication, which basically compares one player’s average payoffs along

11 This last step requires (C3). However, I give a stronger claim (Claim 2) in the proof so that (C3) is not necessary for (ξ1, ξ2) to be an equilibrium.12 Here I assume that (ξ1, ξ2) use history-independent strategies and they satisfy the necessary condition identified in Theorem 3.3.

T.-W. Hu / Games and Economic Behavior 88 (2014) 1–15 9

subsequences where the other player uses particular actions, is exactly the same kind of statistical patterns that are used in many empirical papers, such as Walker and Wooders (2001) and Palacios-Huerta (2003), to test the equilibrium hypothesis in repeated zero-sum games.

Finally, I show that equilibrium strategies for player i under mutual complexity may violate some properties that seem intuitive for a “random” sequence. In particular, the following proposition shows that, if (θ1, θ2) are mutually complex and g has a unique non-degenerate mixed equilibrium (p∗

1, p∗2), then there is an equilibrium strategy ξ i that is θ−i -stochastic

but violates a particular statistical pattern, the Law of the Iterated Logarithm (LIL), that holds for an i.i.d. process w.r.t. p∗i .

Formally, for a finite set X and a distribution p ∈ �(X), a sequence ξ ∈ XN satisfies LIL for p if

lim supT →∞

|∑Tt=0 cx(ξt) − T p[x]|√

2p[x](1 − p[x])T log log T= 1. (4)

This law gives the exact convergence rate of the frequency that is satisfied almost surely if ξ follows an i.i.d. process. It also can be shown to be satisfied deterministically by all θ -incompressible sequences for any oracle θ w.r.t. the uniform distribution (see Lemma 3.1 in the Supplemental Material of Hu, 2012). The proof of the following proposition, due to the technicalities, is given in the Supplemental Material (Hu, 2012), Section 3.

Proposition 3.3. Suppose that (θ1, θ2) are mutually complex. Let g be a zero-sum game with a unique non-degenerate mixed equi-librium (p∗

1, p∗2) ∈ �(X1) × �(X2). There exists an equilibrium strategy ξ ∈ XN

i that is θ−i -stochastic for p∗i but for some x ∈ Xi ,

limT →∞

∑Tn=0 cx(ξ

in) − T p∗

i [x]√2p∗

i [x](1 − p∗i [x])T log log T

= ∞. (5)

4. Concluding remarks

Interpretation of the complexity constraints. Conceptually, there are at least two aspects of complexity in repeated games: (a) the complexity of strategies used by the players (complexity of strategy implementation), and (b) for given strategy sets of all players, the complexity of finding the best responses or the optimal strategies (complexity of optimization). Papadimitriou (1992) shows that restricting players’ complexity w.r.t. aspect (a) may increase their burden in terms of complexity w.r.t. aspect (b) in the context of models with finite automata. Because I use Nash equilibrium as the solution concept without taking aspect (b) into account, I implicitly assume that players are unbounded w.r.t. aspect (b), as most models of repeated games with finite automata do. However, the extent of these difficulties depends on the interpretation of Nash equilibrium, as discussed in Aumann (1997), which also argues that the evolution/learning approach to bounded rationality may alleviate these difficulties.

Regarding aspect (a), another difficulty arises as the sufficient conditions in Theorems 3.1 and 3.2 for equilibrium ex-istence require players’ oracles to be as difficult as the halting problem, and hence uncomputable. As such, these results are more relevant for theoretical considerations than for practical uses. However, as discussed later in this section, those results can be extended to finitely repeated games if we introduce time-/resource-constraints on strategy-implementation. The oracles required to obtain existence will then be of finite lengths and hence computable. In such an extension, the sufficient conditions in the existence results may be reinterpreted as requirements on particular combinations of complexity requirements on players’ oracles and resource-/time-constraints on their machines, but all within Turing computability. Of course, such results require techniques and analysis beyond the scope of the current paper.

Interpretation of the oracles. In my model the oracles endowed to the players are exogenously given, without postulating a theory about their origins. As such, the ultimate sources for the endowed oracle may be recorded outcomes from a physical device, or may directly come from the player’s intellectual ability.13 However, as discussed above, here players are assumed to be unbounded in terms of the complexity of optimization (aspect (b) above), and hence they are assumed to know the relative strengths of the two oracles. But each has to rely on his own oracle to implement his strategies.

In any case, the oracle is not a random variable in the sense of probability theory, and there are no probabilistic beliefs in my model. However, under mutual complexity, equilibrium history-independent strategies exhibit statistical properties that are intuitively related to unpredictability, but such properties are generated from the complexity of the oracle. Never-theless, even though those statistical properties are akin to unpredictable properties in the standard model, the (essential) stochasticity requirement in Theorem 3.3 is still a complexity requirement in that only frequencies along subsequences that can be effectively selected by the other oracle are considered.

The role of randomization. Randomization is allowed in the literature of repeated games with finite-automata, and a mixed strategy in those models is a probability distribution over the set of strategies with bounded complexity w.r.t. the size of

13 The use of uncomputable oracles does not contradict the Church–Turing thesis; that thesis only asserts that all finite procedures can be captured by Turing machines but does not imply that all human insights are bounded by computable ones. See Goldin and Wegner (2008) for an argument that refutes the “strong Church–Turing thesis” which asserts the equivalence between Turing computability and all forms of intelligence.

10 T.-W. Hu / Games and Economic Behavior 88 (2014) 1–15

the finite automata. In my model, a corresponding formulation would define a mixed strategy for player i as a distribu-tion over Ai . Allowing such randomization has no effect on Theorems 3.1, 3.2, and 3.3, because a pure equilibrium is still an equilibrium in mixed strategies under expected utilities. However, Proposition 3.1 may not hold if randomization is al-lowed. Indeed, Theorem 2 in Ben-Porath (1993) suggests that v∗

1(g) may be achievable with mixed strategies when θ1 = θ2. However, if this is true, it would suggest that randomization is a substitute for complexity.

Equilibrium existence without mutual complexity. In this paper the equilibrium analysis focused on two polar cases: either one player’s oracle is stronger than the halting problem relative to the other or the two players’ oracles are mutually com-plex, but intermediate cases, potentially interesting, are not discussed. One such case could be when players have access only to “limited randomness” due to complexity constraints and hope for results similar to those in Gossner and Vieille (2002). However, the current framework cannot capture this idea: it is known (Downey and Hirschfeldt, 2010, Theorem 6.12.9) that if one oracle can compute a non-degenerate computably random sequence relative to another, then it can also compute an incompressible sequence relative to that oracle. Nevertheless, if resource-/time-constraints are introduced, even under mutual complexity not all distributions may be computed as some may require “long” computations.14

Another interesting, but difficult, case is to consider oracles that are mutually uncomputable but not mutually complex. One particular example of such oracles comes from minimal degrees. An oracle θ is of minimal degree if for any other oracle η that is θ -computable, either η and θ belong to the same Turing degree or η is computable. It is known that minimal degrees exist (Downey and Hirschfeldt, 2010, Theorem 2.18.7), and that any pair of oracles belonging to two distinct minimal degrees are not incompressible (and hence not mutually complex), but they are mutually uncomputable. Analysis of such pairs seems difficult because such oracles lack statistical patterns that exist in incompressible sequences, but they also lack computable patterns. It remains an open question whether a Nash equilibrium exists for such pairs.

Extension to finitely repeated games. All the main results can be extended to finitely repeated games, with two modi-fications. First, because in a finitely repeated game any strategy is computable, the finite-repetition version also imposes restrictions on the size of oracle machines that players can use. Second, precise equilibrium is no longer possible, and I con-sider ε-equilibria. In the Supplemental Material (Hu, 2012), Section 1, I formulate a finitely repeated game with restrictions on the oracle-machine sizes (which can be motivated by time-/resource-constraints). I prove an asymptotic version of The-orem 3.1, which states that if θ2 is θ1-computable, then for any ε > 0 and for any fixed machine-size for player 2, there is an ε-equilibrium with payoffs close to v̄1(g) for player 1 in the finitely repeated game with (θ1, θ2), given that the ma-chine size for player 1 is sufficiently large and that the repetition is sufficiently long. I also prove an asymptotic version of Theorem 3.2, which states that if (θ1, θ2) are mutually complex, then for any sufficiently large but fixed machine sizes for the two players, there is an ε-equilibrium with equilibrium payoffs close to (v∗

1(g), v∗2(g)) for sufficiently long repetitions.

Finally, an asymptotical version of Theorem 3.3 is also obtained.

Extension to N-person games. Theorem 3.2 can be extended to general stage games with N players. The extension meets two difficulties that are overcome in the Supplemental Material (Hu, 2012). First, I extend the notion of mutual complexity to the N-person case: N oracles (θ1, ..., θ N ) are mutually complex if for each i, θ i is incompressible relative to the oracle that combines all the other oracles (θ j) j �=i . Unlike the two-person case, an independence result is necessary and is shown to hold under this extension of mutual complexity (Lemma 2.1 in Hu, 2012). Second, while there is always a mixed equilibrium that is rational-valued in two-person zero-sum games (given that the payoffs are rational-valued), this is not true for general N-person games. However, Prasad (2009) shows that there always exists a computable mixed equilibrium in any finite N-person games. To generate a sequence with nice unpredictability properties for a distribution with irrational probability values requires many technical treatments that are beyond the scope of this paper, but those treatments can be found in Hu (2012), Section 2. The main result there states that if (θ1, ..., θ N ) are mutually complex, then for any computable mixed equilibrium of g , there is an equilibrium in RG(g, θ1, ..., θ N ) with the same equilibrium payoffs and the equilibrium strategies have frequencies corresponding to the mixed equilibrium. In this extension I only require the payoff criteria to satisfy (C1) and (C2).

5. Proofs

Proof of Theorem 3.1. Let {φ0, φ1, ..., φk, ...} be a θ2-computable enumeration of (partial) functions from X∗1 to X2 which are

computable relative to θ2. Such an enumeration exists by the relativized Enumeration Theorem. Therefore, for any strategy α2 ∈A2, there exists some k so that α2 = φk; for any k, φk ∈A2 if and only if φk is total.

Define functions g1 : ⋃∞n=0(Xn

1 × Xn2) →N and g2 : ⋃∞

n=0(Xn1 × Xn

2) → X1 as follows.

g1(ε, ε) = 0; (6)

g1(σ 1,σ 2) = min

{k : φk

(σ 1) is defined and φk

(σ 1[t]) = σ 2

t for all t = 0, ...,n − 1}

for all n > 0 and for all(σ 1,σ 2) ∈ Xn

1 × Xn2;

14 Analysis following similar ideas is conducted in Budinich and Fortnow (2011). However, they consider players who are much more constrained in a setting where there are only n rounds of plays and players have access to only γ n (pseudo-)random bids, γ < 1.

T.-W. Hu / Games and Economic Behavior 88 (2014) 1–15 11

g2(ε, ε) = arg maxx∈X1

h1(x, φ0(ε)

); (7)

g2(σ 1,σ 2) = arg max

x∈X1h1

(x, φg1(σ 1,σ 2)

(σ 1)) for all n > 0 and for all

(σ 1,σ 2) ∈ Xn

1 × Xn2.

For any history of actions (σ 1, σ 2) ∈ Xn1 × Xn

2 from periods 0 to n − 1 with n > 0, g1(σ1, σ 2) finds the minimum index

k such that φk as player 2’s strategy is consistent with player 2’s actions so far. Such k exists because the enumeration includes all θ2-computable strategies, and for any finite history, there is always a computable strategy consistent with that history. Thus, g1 is a total function. The function g1 is κθ2

-computable because the enumeration is θ2-computable and the halting problem relative to θ2 is used in defining g1. Because κθ2

is θ1-computable, g1 is θ1-computable. When defining g2it is assumed that the maximizer for x is unique; if not, the x with the minimum index can be used. g2 is θ1-computable because g1 is.

Define the strategy α∗1 inductively as follows:

(1) The induction basis: α∗1(ε) = g2(ε, ε).

(2) Suppose that α∗1(τ 2) has already been defined for all τ 2 ∈ Xm

2 with m < n, n > 0. For σ 2 ∈ Xn2 , define α∗

1(σ 2) =g2(σ

1, σ 2), where σ 1t = α∗

1(σ 2[t]) for all t = 0, ..., |σ | − 1.

α∗1 is θ1-computable because g1 and g2 are. Now I show that for any strategy α2 ∈ A2, u1(α

∗1 , α2) ≥ v̄1(g) =

minx2∈X2 maxx1∈X1 h1(x1, x2). Fix an arbitrary strategy α2, and let α2 = φk . Let (ξα,1, ξα,2) be the sequence of ac-tions induced by (α∗

1 , α2) as defined in (2). First I show that there exist T̄ ∈ N and l ≤ k such that for all T ≥ T̄ , g1(ξ

α,1[T ], ξα,2[T ]) = l and α2(ξα,1[t]) = φl(ξ

α,1[t]) for all t ∈ N. Let l be the smallest index such that φl(ξα,1[t]) = ξ

α,2t

for all t ∈ N. Such l exists because α2 = φk; hence, l ≤ k. Moreover, for each l′ < l, there exists some Tl′ such that either φl′(ξα,1[Tl′ ]) is undefined or φl′ (ξα,1[Tl′ ]) �= ξ

α,2Tl′ . So for T ≥ T̄ = max{Tl′ : l′ < l} + 1, g1(ξ

α,1[T ], ξα,2[T ]) = l. Moreover, for

all t ≥ T̄ , h1(ξα,1t , ξα,2

t ) ≥ v̄1(g) by construction. Hence, u1(α∗1, α2) ≥ v̄1(g) by (C2).

Finally, let α∗2 be such that α∗

2(σ ) ∈ arg maxx2∈X2 minx1∈X1 h2(x1, x2) for all σ ∈ X∗1 . Then, for any α1 ∈ A1, u2(α1, α∗

2) ≥−v̄1(g) by (C2). Therefore, (α∗

1, α∗2) forms an equilibrium with value v̄1(g). �

Proof of Proposition 3.1. Suppose, by contradiction, that θ2 is θ1-computable and that there is an equilibrium with value v∗

1(g). I first show that for any α2 ∈ A2, there is strategy α̂1 ∈ A1 such that u1(α̂1, α2) ≥ v̄1(g) = minx2∈X2 maxx1∈X1 h1(x1,

x2). Given α2, α̂1 is constructed as follows. For all σ ∈ Xt2, α̂1(σ ) = ζt , where ζ is defined by

ζ0 = arg maxx∈X1

h1(x,α2(ε)

), and for t > 0, ζt = arg max

x∈X1h1

(x,α2

(ζ [t])).

Here it is assumed that the maximizer for x is unique; if not, the x with the minimum index can be used. ζ is θ2-computable because α2 is and hence α̂1 is θ1-computable because θ2 is θ1-computable. By construction, for all t ∈N,

h1(ζt,α2

(ζ [t])) ≥ v̄1(g).

Hence, by (C2), u1(α̂1, α2) ≥ v̄1(g).Thus, if (α∗

1 , α∗2) is an equilibrium, then u1(α

∗1 , α∗

2) ≥ v̄1(g), a contradiction to the fact that the value is v∗1(g) < v̄1(g).

Therefore, θ2 cannot be θ1-computable. A symmetric argument shows that θ1 cannot be θ2-computable. �The proof of Theorem 3.2 will rely on Lemmas 3.1 and 3.2. Hence, I give the proofs of those two lemmas first.

Proof of Lemma 3.1. Let X = {x1, ..., xK } and let p[xi] = liL , where l1, ..., lK , L ∈ N. Let m be such that 2m−1 < L ≤ 2m . The

construction takes three steps: The first step transforms ν into ζ , which is θ -computably-random over a set of 2m actions for the uniform distribution; The second step transforms ζ into η by dropping 2m − L actions form ζ and makes η a θ -computably-random sequence over a set of L actions for the uniform distribution; The third step transforms η into ξ by combining li actions in η into a single action xi in ξ and makes ξ a θ -computably-random sequence over X for p.(Step 1). Let W = {w1, ..., w2m }. Enumerate the set {0, 1}m as {ρ1, ..., ρ2m }. Construct ζ ∈ W N from ν as follows:

For each n ∈N, ζn = wi if (νnm, νnm+1, ..., ν(n+1)m−1) = ρ i .

I show that ζ is θ -computably-random for (2−m, 2−m, ..., 2−m) by contradiction. Suppose that there exists a θ -computable martingale B for (2−m, 2−m, ..., 2−m) that succeeds over ζ . I devise a martingale C that succeeds over ν , using the fact that B succeeds over ζ .

Define the mapping Γ1 : ⋃∞N=0{0, 1}Nm → W ∗ by setting Γ1(σ ) = τ with τn = wi if |σ | = Nm and (σnm, σnm+1, ...,

σ(n+1)m−1) = ρ i for each n = 0, 1, ..., N − 1. Construct C as follows.

12 T.-W. Hu / Games and Economic Behavior 88 (2014) 1–15

(1.1) For all σ with |σ | = Nm for some N ∈ N, C(σ ) = B(Γ1(σ )).(1.2) Suppose that C is defined over all strings σ ’s with length Nm − (k − 1), 1 ≤ k < m − 1. Consider a string σ with

|σ | = Nm − k. Take C(σ ) = 12 C(σ0) + 1

2 C(σ1).

C is a martingale. By construction, C(σ ) = 12 C(σ0) + 1

2 C(σ1) holds for all σ with |σ | �= Nm for any N . Moreover, if |σ | = Nm − k for some k between 1 and m − 1, then C(σ ) = ∑

ρ∈{0,1}k 2−kC(σρ). Thus, if |σ | = Nm, then C(σ0) =∑ρ∈{0,1}m−1

12m−1 C(σ0ρ) and C(σ1) = ∑

ρ∈{0,1}m−11

2m−1 C(σ1ρ), and hence

1

2C(σ0) + 1

2C(σ1) =

∑ρ∈{0,1}m

1

2mC(σρ) =

∑w∈W

1

2mB(Γ1(σ )w

) = B(Γ1(σ )

) = C(σ ).

The martingale C is θ -computable because B is θ -computable and Γ1 is computable. Moreover, for any n, B(ζ [n]) =C(Γ −1

1 (ζ [n])) = C(ν[nm]). C succeeds over ν because B succeeds over ζ , a contradiction. Thus, ζ is θ -computably-random for (2−m, 2−m, ..., 2−m).(Step 2). Because ζ is θ -computably-random for the uniform distribution, there are infinitely many k’s such that ζk = w1. Otherwise, one can easily devise a martingale that bets all the wealth equally on outcomes other than w1 and obtainsunbounded payoffs.

Let Z = {w1, ..., w L} ⊂ W . Construct η ∈ ZN from ζ as follows. First, define g : N → N by (a) g(0) = min{k ∈ N : ζk ∈ Z}; (b) for n ≥ 0, g(n + 1) = min{k > g(n) : ζk ∈ Z}. g is total. Then, define η by setting ηn = ζg(n) for all n ∈ N.

The sequence η is θ -computably-random for the uniform distribution ( 1L , ...., 1L ) over Z . To show this, suppose, by con-

tradiction, that a θ -computable martingale B for ( 1L , ...., 1L ) succeeds over η. Construct a θ -computable martingale C for

(2−m, 2−m, ..., 2−m) as follows.

(2.1) Define Γ2 : W ∗ → Z∗ by setting Γ2(σ ) = τ , where τ is obtained from σ by eliminating all the occurrences of w L+1, ..., w2m in σ .

(2.2) Define C by setting C(σ ) = B(Γ2(σ )) for all σ ∈ W ∗ .

By construction, for any σ ∈ W ∗ , C(σ wi) = B(Γ2(σ )wi) if i ≤ L and C(σ wi) = B(Γ2(σ )) if i > L. C is a martingale for (2−m, ..., 2−m): let σ ∈ W ∗ ,

2m∑i=1

2−mC(σ wi) =[∑

i≤L

L−1 B(Γ2(σ )wi

)] L

2m+

[∑j>L

2−m B(Γ2(σ )

)]

= L

2mB(Γ 2(σ )

) + 2m − L

2mB(Γ 2(σ )

) = B(Γ 2(σ )

) = C(σ ).

C is θ -computable because B is θ -computable and Γ2 is computable. Finally, C(ζ [g(n)]) = B(η[n]) for all n ∈ N. So Csucceeds over ζ because B succeeds over η, a contradiction.(Step 3). Let X = {x1, ..., xK }. Construct ξ ∈ XN from η as follows. Let L0 = 0; for k ≥ 0, let Lk+1 = Lk + lk+1. Define ξ by setting ξn = xk if ηn ∈ {w Lk−1+1, ..., w Lk } for all n ∈N. Now I show that ξ is θ -computably-random for p = ( l1

L , ..., lKL ).

Suppose, by contradiction, that a θ -computable martingale B over X for p succeeds over ξ . Construct a θ -effective martingale C over Z as follows.

(3.1) Define Γ3 : Z∗ → X∗ by setting Γ3(σ ) = τ with τn = xk if σn ∈ {w Lk−1+1, ..., w Lk } for all n = 0, ..., |σ | − 1.(3.2) Define C by setting C(σ ) = B(Γ3(σ )) for all σ ∈ Z∗ .

Thus, for any σ ∈ Z∗ , C(σ wi) = B(Γ3(σ )xk) if i ∈ {Lk−1 + 1, ..., Lk}. C is a martingale for ( 1L , ...., 1L ): for any σ ∈ Z∗ ,

L∑i=1

1

LC(σ wi) =

K∑k=1

lkL

B(Γ3(σ )xk

) =K∑

k=1

p[xk]B(Γ3(σ )xk

) = B(Γ3(σ )

) = C(σ ).

C is θ -computable because B is θ -computable and Γ3 is computable. Finally, C(η[n]) = B(ξ [n]) for all n ∈ N. So C succeeds over η because B succeeds over ξ , a contradiction.

Therefore, ξ is θ -computably-random for p. ξ is ν-computable because both ζ and η are. �Proof of Lemma 3.2. I prove the lemma by two claims.

Claim 1. If ξ ∈ XN is θ -computably-random for p ∈ �(X), and if r : X∗ → {0, 1} is a θ -computable selection function for Xsuch that ξ r is an infinite sequence, then ξ r is also θ -computably-random for p.

T.-W. Hu / Games and Economic Behavior 88 (2014) 1–15 13

Proof. Suppose, by contradiction, that a θ -computable martingale B succeeds over ξ r . Construct a martingale C for p as follows: C(σ ) = B(σ r) for all σ ∈ X∗ . C is also a martingale for p: let σ ∈ X∗ be given; if r(σ ) = 0, then (σ x)r = σ r

and hence C(σ x) = C(σ ) for all x, which implies that ∑

x∈X p[x]C(σ x) = C(σ ); if r(σ ) = 1, then (σ x)r = σ r x and hence C(σ x) = B(σ r x) for all x, which implies that

∑x∈X p[x]C(σ x) = ∑

x∈X p[x]B(σ r x) = B(σ r) = C(σ ). C is θ -computable be-cause B is θ -computable and r is θ -computable. Finally, for any n ∈ N, there exists mn ≥ n such that (ξ [mn])r = ξ r[n]. Thus, for any n, C(ξ [mn]) = B(ξ r[n]). Thus, C succeeds over ξ because B succeeds over ξ r , a contradiction. Therefore, ξ r is θ -computably-random for p. �Claim 2. If ξ ∈ XN is θ -computably-random for p ∈ �(X), then limT →∞

∑T −1t=0 cx(ξt )

T = p[x] for each x ∈ X .

Proof. If p[x] = 0 and if lim supT →∞∑T −1

t=0 cx(ξt )

T > 0, then the computable martingale Bx such that Bx(σ x) = 2Bx(σ ) and Bx(σ y) = Bx(σ ) for all σ and for all y �= x will succeed over ξ . Thus, to prove the claim by contradiction, assume that for some y ∈ X with p[y] ∈ (0, 1) and some ε > 0, there is an infinite sequence T0 < T1 < ... < T j < ... such that

limT j→∞∑T j−1

t=0 c y(ξt )

T j= qy < p[y] − 2ε. Let d ∈ (0, 1/2(1 − p[y])) be so small that

A(d) ≡ [1 − d

(1 − p[y])]p[y]−ε[

1 + dp[y]]1−p[y]+ε> 1.

Such d exists because A(0) = 1 and A′(0) > 0. I construct a computable martingale B for p as follows.Take B(ε) = 1; B(σ y) = (1 −d(1 − p[y]))B(σ ) and B(σ x) = (1 +dp[y])B(σ ) for all x �= y. B is a martingale for p because∑

x∈X

p[x]B(σ x) = p[y](1 − d(1 − p[y]))B(σ ) +

∑x�=y

p[x](1 + dp[y])B(σ ) = B(σ ).

Let J be so large that j > J implies that ∑T j−1

t=0 c y(ξt )

T j< p[y] − ε. For all j > J ,

B(ξ [T j]

) = [(1 − d

(1 − p[y]))

∑T j−1t=0 c y (ξt )

T j(1 + dp[y])1−

∑T j−1t=0 c y (ξt )

T j]T j

≥ [(1 − d

(1 − p[y]))p[y]−ε(

1 + dp[y])1−p[y]+ε]T j = AT j .

Thus, lim j→∞ B(ξ [T j]) ≥ lim j→∞ AT j = ∞, that is, B succeeds over ξ , a contradiction. �The lemma follows from Claims 1 and 2: if ξ is θ -computably-random for p and if r is a θ -computable selection function

for X , then, by Claim 1, ξ r is also θ -computably-random for p, given that it is an infinite sequence, and by Claim 2,

limT →∞∑T −1

t=0 cx(ξrt )

T = p[x] for each x ∈ X . �Proof of Theorem 3.2. I first give two claims for the proof proper.

Claim 1. Suppose that ϕi satisfies (C1)–(C2). Then, for any {at}∞t=0 ∈ ∞ , lim inft→∞ at ≤ ϕi({at}) ≤ lim supt→∞ at .

Proof. Let bt = inft′≥t at′ and let ct = supt′≥t at′ . Then, bt ≤ at ≤ ct for all t ∈ N. By (C1), ϕi({bt}) = lim inft→∞ at and ϕi({ct}) = lim supt→∞ at and by (C2) ϕi({bt}) ≤ ϕi({at}) ≤ ϕi({ct}). �Claim 2. Let p∗

i ∈ �(Xi) be an equilibrium strategy for player i in g . Suppose that ξ i ∈ XN

i is a sequence such that for any θ−i -computable selection function r for Xi for which (ξ i)r is an infinite sequence,

limT →∞

∑T −1t=0 cxi ((ξ

i)rt )

T= p∗

i [xi] for each xi ∈ Xi . (8)

Then, for all α−i ∈A−i ,

ui(ξ i;α−i

) ≥ lim infT →∞

T −1∑t=0

hi(ξit ;α−i(ξ

i[t]))T

≥ v∗i (g). (9)

Proof. Let α−i ∈ A−i be given. For each y ∈ X−i , let r y : X∗i → {0, 1} be the selection function for Xi such that r y(σ ) = 1

if α2(σ ) = y, and r y(σ ) = 0 otherwise. Define L y(T ) = #{t ∈ N : 0 ≤ t ≤ T − 1, r y(ξ i[t]) = 1} and ξ y = (ξ i)r y. r y is

θ−i -computable because α−i is. Let

14 T.-W. Hu / Games and Economic Behavior 88 (2014) 1–15

E1 ={

y ∈ X−i : limT →∞ L y(T ) = ∞

}and E2 =

{y ∈ X−i : lim

T →∞ L y(T ) < ∞}.

For each y ∈ E2, let B y = limT →∞ L y(T ) and let C y = ∑B y−1t=0 hi(ξ

yt ; y). For any y ∈ E1, because ξ i satisfies (8) and because

r y is a θ−i -computable selection function,

limT →∞

T −1∑t=0

cx(ξy

t )

T= p∗

i [x], and hence

limT →∞

T −1∑t=0

hi(ξy

t ; y)

T= lim

T →∞∑x∈Xi

T −1∑t=0

cx(ξy

t )hi(x; y)

T=

∑x∈Xi

p∗i [x]hi(x; y) ≥ v∗

i (g), (10)

where the last inequality holds because p∗i is an equilibrium mixed strategy of g .

In what follows I show that for any ε > 0, there is some T ′ such that if T > T ′ , then

T −1∑t=0

hi(ξit ;α−i(ξ

i[t]))T

≥ v∗i (g) − ε. (11)

Fix some ε > 0. Because of (10), T1 exists such that, if T > T1, then

T −1∑t=0

hi(ξy

t ; y)

T≥ v∗

1(g) − ε

2|X−i | for all y ∈ E1, andC y

T> − ε

2|X−i| for all y ∈ E2. (12)

Let T ′ > T1 be so large that, for all y ∈ E1, L y(T ′) > T1 and v∗i (g)

∑y∈E1

L y(T )

T ≥ v∗i (g) − ε

2 for all T > T ′ . If T > T ′ , then

T −1∑t=0

hi(ξit ;α−i(ξ

i[t]))T

=∑y∈E1

L y(T )

T

L y(T )−1∑t=0

hi(ξy

t ; y)

L y(T )+

∑y∈E2

L y(T )−1∑t=0

hi(ξy

t ; y)

T

≥∑y∈E1

L y(T )

T

(v∗

1(g) − ε

2|X−i|)

−∑y∈E2

ε

2|X−i| ≥ v∗1(g) − ε.

This proves (11), which implies the second inequality in (9). The first inequality in (9) follows from Claim 1. �Proof of Theorem 3.2. By mutual complexity, there is a θ i -computable sequence ν i that is θ−i -incompressible. By Lemma 2.1, ν i is θ−i -computably-random. Let p∗

i be an equilibrium mixed strategy of g and let v∗i (g) be the value of

g for player i. By Lemma 3.1, there is a ν i -computable sequence ξ i that is θ−i -computably-random for p∗i . By Lemma 3.2,

for both i = 1, 2, ξ i satisfies (8) for all θ−i -computable selection function r for Xi such that (ξ i)r is an infinite sequence and hence, by Claim 2,

ui(ξ i;α−i

) ≥ v∗i (g) for all α−i ∈ A−i . (13)

The symmetric statement holds for player −i as well. By Claims 1 and 2 (taking i as −i), for any αi ∈Ai ,

ui(αi; ξ−i) ≤ lim sup

T →∞

T −1∑t=0

hi(αi(ξ−i[t]); ξ−i

t )

T= − lim inf

T →∞

T −1∑t=0

h−i(ξ−it ;αi(ξ

−i[t]))T

≤ −v∗−i(g) = v∗

i (g), (14)

where the first inequality follows from Claim 1 and the second inequality follows from Claim 2. Now I show that (ξ1, ξ2) is an equilibrium. By (13), u1(ξ

1, ξ2) ≥ v∗1(g). By (14), u1(ξ

1, ξ2) ≤ v∗1(g). Thus, u1(ξ

1, ξ2) = v∗1(g). By (14), for any α1 ∈ A1,

u1(α1, ξ2) ≤ u1(ξ1, ξ2). A symmetric argument shows that for any α2 ∈ A2, u2(ξ

1, α2) ≤ v∗2(g) = u2(ξ

1, ξ2). This implies that (ξ1, ξ2) is an equilibrium. �Proof of Theorem 3.3. (a) Using the arguments in Proof of Theorem 3.2, if ξ i is θ i -computable and is θ−i -stochastic for p∗

ifor both i = 1, 2, then (ξ1, ξ2) is an equilibrium.

(b) I will prove by contradiction. Consider (ϕ1, ϕ2) = (lim inf, lim sup). Let ξ be an equilibrium history-independent strategy for player 1 but not essentially θ2-stochastic for p∗

1, and let r be the selection function that witnesses this. Be-

cause lim infT →∞∑T −1 r(ξ [t])

> 0, there exist some ε0 > 0 and some T̄ ∈ N such that for all T ≥ T̄ , ∑T −1 r(ξ [t])

> 2ε0.

t=0 T t=0 T

T.-W. Hu / Games and Economic Behavior 88 (2014) 1–15 15

Because ξ is not essentially θ2-stochastic for p∗1 w.r.t. r, for some infinite sequence {T 0

0 < T 01 < ... < T 0

j < ...} there ex-

ists p1 �= p∗1 such that for all x ∈ X1, lim j→∞

∑T 0j −1

t=0 r(ξ [t])cx(ξt )∑T 0j −1

t=0 r(ξ [t])= p1[x]. Because p1 �= p∗

1, there exists some δ > 0 so that

minx2∈X2 h1(p1, x2) < v∗1(g) − δ. Consider two cases:

(b.1) limT 0j →∞

∑T 0j −1

t=0r(ξ [t])

T 0j

= 1. Define a strategy α2 : X∗1 → X2 as follows: α2(σ ) = y1 ∈ arg miny∈X2 h1(p1, y) if r(σ ) = 1

and α2(σ ) = y2 for some arbitrary y2 ∈ X2 otherwise. Then, lim j→∞∑T 0

j −1

t=0h1(ξt ,α2(ξ [t]))

T 0j

= h1(p1, y1) < v∗1(g) − δ <

v∗1(g).

(b.2) For some ε2 > 0, lim infT 0j →∞

∑T 0j −1

t=0r(ξ [t])

T 0j

< 1 − ε2. Now, let {T 1j }∞j=0 be a subsequence of {T 0

j }∞j=0 such that

(1) lim j→∞∑T 1

j −1

t=0r(ξ [t])

T 1j

= a ∈ (0, 1);

(2) for each x ∈ X1, lim j→∞∑T 1

j −1

t=0 (1−r(ξ [t]))cx(ξt )∑T 1j −1

t=0 (1−r(ξ [t]))= p2[x] for some p2 ∈ �(X1).

Define a strategy α2 : X∗1 → X2 as follows: α2(σ ) ∈ arg miny∈X2 h1(p1, y) if r(σ ) = 1 and α2(σ ) ∈ arg miny∈X2 h1(p2, y)

otherwise. In particular, let y1 and y2 be the actions chosen for r(σ ) = 1 and r(σ ) = 0, respectively. Then,

limj→∞

T 1j −1∑

t=0

h1(ξt,α2(ξ [t]))T 1

j

= limj→∞

[1

T 1j

T 1j −1∑

t=0

r(ξ [t])

] ∑x∈X1

[∑T 1j −1

t=0 r(ξ [t])cx(ξt)∑T 1j −1

t=0 r(ξ [t])

]h1

(x, y1)

+ limj→∞

[1

T 1j

T 1j −1∑

t=0

(1 − r

(ξ [t]))

] ∑x∈X1

[∑T 1j −1

t=0 (1 − r(ξ [t]))cx(ξt)∑T 1j −1

t=0 (1 − r(ξ [t]))

]h1

(x, y2)

= ah1(

p1, y1) + (1 − a)h1(

p2, y2) < a(

v∗1(g) − δ

) + (1 − a)v∗1(g)

= v∗1(g) − aδ < v∗

1(g).

Thus, in either case, the strategy α2 is such that u1(ξ, α2) < v∗1(g), a contradiction to the optimality of ξ . �

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