sign in sign up
<<
Home , Ariel Rubinstein, Determinacy, Melvin Dresher, Nash equilibrium, Ordinal number

of Nonzero-sum

Wenzhang Zhang 1

April 4, 2008

1School of , Shanghai University of Finance and Economics, Shanghai 200433, China. Email: [email protected] Abstract

In this paper we study a of infinite games with and define a new that we call determinacy for these games. We start by motivat- ing and formalizing three simple behavioral about how rational players should behave in games, pure games and pure games, respectively. Then we say that a general is determined if it can be solved by repeatedly applying these axioms to its . This solution concept has nice properties and important implications for perfect and iter- ated weak dominance. In particular, the given by determinacy is a unique payoff vector that corresponds to the value of a subgame perfect Nash equilibrium. Thus determinacy can be regarded as a unique refinement of subgame perfect Nash equilibrium.

Keywords: Determinacy, Backward induction, Nonzero-sum games, Infinite games of perfect information. JEL Classification: C72, C73. 1 Introduction

In their pioneering study, Gale and Stewart (1953) investigate the following class of infinite games with perfect information. Two players, 1 and 2, alternate choosing elements of a Y . First player 1 chooses y0, then player 2 chooses y1, then player 1 chooses y2, etc., so that an infinite of choices (y0, y1, y2,... ) is specified. The payoffs of the players are defined by a pair of A1 and A2 that form a partition of the total space of such (y0, y1, y2,... ): The sets A1 and A2 are called the payoff sets of players 1 and 2, respectively, and a player wins just in case the sequence

(y0, y1, y2,... ) is a member of his payoff set. So one and only one player wins in each play and these games are called zero-sum games. A winning for a player is a strategy following which he always wins, re- gardless of what his opponent does. A game is called determined if one of the players has a winning strategy. Gale and Stewart prove the fundamental result that if one of the two payoff sets is closed in the natural product of the total space, then the game is determined. They further ask whether this is true if the payoff sets are Borel sets. This question inspires many years of study, e.g., Wolfe (1955), Davis (1964), and the seminal work of Martin (1975), which provides the final affirmative answer. These games are certainly deep and elegant objects of study. For example, they play a key role in the foundation of (see, e.g., Chapter 6 of Kanamori (2000)). Nevertheless, the requirement that the games be zero-sum is too restrictive for practical applications. For example, the games in economics are exclusively nonzero-sum games, with even more general payoff and information structures. The aim of this paper is to provide a general study of nonzero-sum games, that is, games in which the pair of payoff sets A1 and A2 need not be a partition of the total space. Unlike for zero-sum games, there is no obvious notion of determinacy for nonzero-sum games. Thus our first task is to define a notion of determinacy for nonzero-sum games. Our approach is axiomatic and the main steps are summarized as follows.

1 (i) We call a game trivial if each payoff set Ai is either the or the total space. Thus a player in a trivial game either wins or loses for sure. We start with the observation that a trivial game can be regarded as determined since no matter how the game is played the payoffs to the players are the same. (ii) Next we consider several simple games in which it is clear what rational players should follow. We propose several axioms requiring that in these games the players should follow these rational strategies. If the players follow these strategies, then each such game is equivalent to a trivial game that has the same outcome. So we let each take the form that a game satisfying certain conditions can be reduced to a trivial game. (iii) Finally, we define the determinacy of a general game G by trying to reduce it to a trivial game using these axioms. Start with a subgame of G that satisfies the conditions of an axiom. We obtain a new game G1 from G by replacing this subgame by the equivalent, trivial game suggested by the axiom. Similarly we can apply the axioms to a subgame of G1 to obtain another game G2. Continuing in this manner we obtain a chain of games hG, G1,G2,...,Gni. If at certain stage we reach a trivial game Gn, then we stop and say that the game G is determined. Intuitively, a game is determined if it can be “solved” by repeatedly solving its subgames using the axioms. The focus of this paper is to describe this class of games, motivate and formalize the axioms, define determinacy, and illustrate the working of this definition. Building on these, in two companion papers (Zhang (2008a) and Zhang (2008b)), we show that the axioms are consistent, complete and independent. Specifically, we prove that: (i) If we have two such chains for a given game, then the trivial games at the end of these two chains must be the same. Therefore the axioms do not contradict each other and the definition of determinacy does not depend on the the axioms are applied. (ii) All games with closed payoff sets are determined. Thus the axioms are complete in the sense that they are able to solve the class of games with closed payoff sets. This result is also an extension of the fundamental theorem of Gale and Stewart to nonzero-sum games. (iii) If we drop any of these axioms, then there will be a game indetermined.

2 So none of the axioms is redundant. These results have important implications for subgame perfect Nash equilibrium and iterated weak dominance. In another companion paper (Zhang (2008c)), we show that these results also enhance our understanding of these two solution concepts. For subgame perfect Nash equilibrium, we establish the following results. (i) We investigate why most games have a large of subgame perfect Nash equilibria, and show that this multiplicity arises because Nash equilibrium is too weak. The most severe case arises in pure cooperation games in which there are common interests that require the players to cooperate to realize these common interests. Nash equilibrium always fails to give the cooperative outcomes as the only predictions. (ii) We show that the outcome given by determinacy is a unique payoff vector that corresponds to the value of a subgame perfect Nash equilibrium, and so determinacy can be regarded as a unique refinement of subgame perfect Nash equilibrium. (iii) By combining these results we are able to prove that for the class of closed games a subgame perfect Nash equilibrium always exists. For iterated weak dominance, we investigate why it is in general order dependent. We find that one prediction of weak dominance on backward induction games is too strong and this leads to the order dependence. The facts that our axioms do not have this prediction and that determinacy is order independent suggest that in order to have an order independent solution concept we have to abandon this strong and inconsistent prediction of weak dominance. The rest of this paper is organized as follows. In Section 2 we review the related literature and point out directions for further work. In Section 3 we introduce the axioms. In Section 4 we define determinacy and summarize the main results in the companion papers. In Section 5 we interpret the outcome given by determinacy, and illustrate the working of determinacy by an example and by proving that finite games are determined. For a technical reason we have to use a transfinite version of determinacy that allows us to apply the axioms infinitely many times. This is important for a general analysis,

3 but the main ideas of the definitions and the main results can be understood using the finite version. For this reason in Section 4 we use the finite version for exposition. We motivate the transfinite version of determinacy and restate the main definitions and results using this version in Section 6. We also illustrate the working of this transfinite version by solving an example. Section 7 concludes.

2 Related Literature and Further Work

2.1 Related literature

In this section we briefly review the literature that are related to the results of this paper and the results mentioned in this paper but studied in detail in the companion papers. Related literature on equilibrium refinements. The notion of subgame perfect Nash equilibrium is defined in Selten (1965) and Selten (1975). Maskin and Tirole (2001) study a widely applied refinement of subgame perfect Nash equilibrium in which the players’ strategies depend only on the state variables (Markov perfect equilibrium). Our papers contribute to this literature since determinacy is a unique refinement of subgame perfect Nash equilibrium, and so it can be regarded as a realization of the refinement program in the case of infinite two-person perfect-information games with characteristic payoff functions. Moreover, the analysis in these papers improves our understanding about why most games have multiple equilibria. Related literature on the existence of subgame perfect Nash equilibrium. The exis- tence of subgame perfect Nash equilibrium in games with perfect information and with infinite horizon has been studied in several influential papers, for example, Fudenberg and Levine (1983), Harris (1985a), Harris (1985b), Harris, Reny, and Robson (1995), Hellwig and Leininger (1987), and Hellwig, Leininger, Reny, and Robson (1990). Fu- denberg and Levine (1983) prove the existence of subgame perfect Nash equilibrium under the assumption that the payoff functions are continuous and the action sets are

4 finite. Harris (1985b) and Hellwig and Leininger (1987) generalize this result to the case where the payoff functions are still continuous but the action sets can be infinite and compact. The games that we consider in these papers have characteristic payoff functions, which are discontinuous in general. The action sets are also general, need not be finite or compact. Indeed, we do not need a topological structure on the action sets. Thus the existence of subgame perfect Nash equilibria for this class of games does not follow from the existence theorems in the literature. Our papers contribute to this literature in two ways. First, we provide an existence theorem for games with discontinuous payoff functions and with arbitrary action sets. Second, we prove it by combining a completeness theorem of determinacy and a refine- ment theorem, and so we also provide a new technique to prove existence theorems. It remains to be explored what are the key ingredients of this proof that guarantee the existence. This is important since it may allow us to prove more general existence theorems for games with discontinuous payoff functions. Related literature on games with common interests. One of our axioms deals with games in which the players have essentially identical payoff functions. These games are closely related to the games with common interests studied in Gale (1995), Gale (2001), Rubinstein and Wolinsky (1995), Lagunoff and Matsui (1997), Dutta (2003), and Takahashi (2005). The main concern of this literature is when the subgame perfect Nash equilibria of games with common interests are efficient. Lagunoff and Matsui (1997), in the context of asynchronously repeated games, and Gale (2001), in the general context, show that if the players have identical continuous payoff functions, then any subgame perfect Nash equilibrium is efficient. Takahashi (2005) obtains a more general efficiency theorem for a class of K-cordination games, extending all the previous results. Takahashi (2005) also mentions that one can construct an inefficient subgame pefect Nash equilibrium in games with common interests by appealing to the folk theorem for stochastic games with generic payoff functions in Dutta (1995). We give several

5 examples (Section 3.3) showing that inefficient subgame perfect Nash equilibria can arise in very simple situations. Indeed, this inefficiency is quite general and is closely related to the multiplicity problem. Thus our papers also contribute to this literature by providing a general understanding of this inefficiency. Related literature on iterated dominance. Gilboa, Kalai, and Zemel (1990), Dufwen- berg and Stegeman (2003), and Chen, Long, and Luo (2007) study the order depen- dence of iterated strict dominance. In the case of iterated weak dominance, Gilboa, Kalai, and Zemel (1990) show that order does not matter for zero-sum games. Ewerhart (2002) shows that two-person zero-sum games with perfect information can be solved by weak dominance in finite steps, depending on the number of possible outcomes but not on the length of the game . Marx and Swinkels (1997) study iterated weak dominance in the general setting. They define a notion of nice weak dominance and use it to identify a class of games in which iterated weak dominance is order independent, including extensive form games with generic assignment of payoffs to terminal nodes. This result does not apply to the games that we study here, since the payoff functions of these games are characteristic functions and so are not generic by nature. Our papers contribute to this literature by analyzing why and when iterated weak dominance can be order dependent. It would be desirable to employ these insights to modify iterated weak dominance directly to obtain an order independent solution concept. This is important because iterated weak dominance is such a simple and widely applied solution concept. In the technical version of determinacy we have to use the technique of transfi- nite induction in order to apply the axioms infinitely many times. Chen, Long, and Luo (2007) also use this technique in defining a transfinite version of iterated strict dominance, extending the finite version of Dufwenberg and Stegeman (2003). Related mathematical literature. Mycielski (1992) and Chapter 6 of Kanamori (2000) survey the literature of zero-sum games in connection with the foundation of mathematics. In another direction, L¨owe (Forthcoming) extends the analysis of Gale and Stewart to n-person zero-sum games. It is an interesting open question whether

6 his method can be combined with ours to give an analysis for n-person nonzero-sum games. There has also been much progress on the determinacy of infinite imperfect- information games, that is, games in which the players simultaneously make their choices in each period. L¨owe (2007) surveys this literature. Our papers the existing mathematical literature by initiating an analysis of nonzero-sum games.

2.2 Further work

Much further work remains to be done. For example, are all games with Borel payoff sets determined, as is true in the zero-sum case (Martin (1975))? In another direction, can this analysis be extended to n-person games? A more ambitious goal would be to have a notion of determinacy for n-person games with general payoff functions rather than characteristic functions. This is important because most games in applications have general payoff functions rather than characteristic functions. The success of such a program will give us not only an appropriate solution concept for general infinite games of perfect information, but also a better understanding of the existing solution concepts. For example, it will allow us to identify the relative strengths of the basic behavioral principles underlying these solution concepts, and to clarify why and when they are appropriate and why and when they can be wrong, as we do for subgame perfect Nash equilibrium and iterated weak dominance in these papers. Results of this kind will shed light on many of the unsolved problems in applications, for instance, the problem of multiplicity of subgame perfect Nash equilibrium in the n-person alternating-offer game (see, e.g., Sutton (1986) and Osborne and Rubinstein (1990)).

3 The Axioms

3.1 Notation and definitions

Notation 1. 1. Let ω denote the set of natural . That is, ω = {0, 1, 2,... }.

2. Let Y denote an arbitrary set with at least two elements. For example, Y =

7 {L, R}, Y = {continue, stop}, and Y = ω.

3. A path is an infinite sequence of elements of Y . Let Y ω denote the set of all paths. For example, if Y = {continue, stop}, then the infinite sequence

(continue, continue, continue,... )

is a path in Y ω.

ω Definition 3.1. A game G is a pair hA1,A2i, where A1 and A2 are subsets of Y . The sets A1 and A2 are called the payoff sets of players 1 and 2, respectively.

Definition 3.2. Let G = hA1,A2i. Then G is called a trivial game if each Ai is either

ω ω Y or ∅. Suppose that G is trivial. Then we say that player i wins G if Ai = Y ; otherwise we say that he loses G.

Remark 3.3. So there are four trivial games: h∅, ∅i, h∅,Y ωi, hY ω, ∅i, and hY ω,Y ωi.

Notation 2. 1. A position is a finite sequence of elements of Y . Let Y <ω denote the set of all positions. For example, if Y = {L, R}, then the sequence (L, L, R) is a position in Y <ω.

2. The empty sequence, denoted by ∅, is in Y <ω. A sequence (y) of length 1 will be identified with the y.

3. Let p and q be two positions with p = (y0, y1, . . . , yk) and q = (z0, z1, . . . , zm).

Let y be an element of Y . Then pay denotes the concatenation (y0, y1, . . . , yk, y),

and paq denotes the concatenation (y0, y1, . . . , yk, z0, z1, . . . , zm).

4. Suppose that p = (y0, y1, . . . , ym), then Np denotes the set

ω { (z0, z1,... ) ∈ Y | z0 = y0, . . . , zm = ym }.

Definition 3.4. Let p = (y0, y1, . . . , ym). Say that p is a position for player 1 to move if m is an odd number or p is the empty sequence.

8 Remark 3.5. Throughout this paper if a definition is to be given for each of the two players, we shall only state it for player 1.

Definition 3.6. Let G = hA1,A2i and p = (y0, y1, . . . , ym). For each i define the set

ω Ai,p by letting, for each path (z0, z1,... ) ∈ Y ,

(z0, z1,... ) ∈ Ai,p if and only if (y0, y1, . . . , ym, z0, z1,... ) ∈ Ai.

Then the subgame Gp is defined to be hA1,p,A2,pi.

Remark 3.7. Thus the subgame Gp is the remaining game starting from the position p. The player that moves first in Gp is the player that moves at p in G.

The following definition gives the usual notion of finite games in the infinite setting.

Definition 3.8. A game hA1,A2i is called a finite game if there exists a finite number n such that for each position p of length n and for each i, either Ai ∩ Np = Np or

Ai ∩ Np = ∅.

Remark 3.9. That is, for each position p of length n, the subgame Gp is a trivial game.

3.2 The refined backward induction axiom

An axiom has the form that a game G satisfying certain conditions can be reduced to a trivial game G∗. The reason why each axiom takes this form is this: Essentially what we have is a behavioral axiom instructing how the players should play in the game G. If the players follow this behavioral axiom then we know what the outcome of this game is. So the original game, following the axiom, is equivalent to the trivial game G∗ that has the same outcome. Our first axiom refines the usual backward induction. For an illustration, consider the two-period game in Figure 1. Player 2’ payoffs of choosing L and R in the second period are the same, so he is indifferent about which one to choose and both are possible. In particular, he may choose L so that player 1 loses the game. So player 1

9 player 1 @ LR b @ @ @@ player 2 1, 0 @ r LR rr @ @ @@ 0,r0 1,r0 Figure 1: A game in which player 1 cannot guarantee a win if he chooses R. cannot guarantee that he will win if he chooses R in the first period. But if instead, he chooses L in the first period, then he is sure to win the game. So L is a better choice for player 1 because it guarantees a win. Roughly speaking, the refined backward induction axiom we propose below strength- ens the usual backward induction by requiring that, when there are choices that lead to guaranteed wins and choices that do not, the players should always choose those that guarantee a win. For instance, in the above example, the axiom requires that player 1 choose L in the first period. To be more precise, we need to distinguish between the following types of choices. A choice can lead to the following three possible outcomes for the player that makes this choice, in the subgame after the choice.

1. The player is sure to win the subgame. For instance, player 1’s choice L in the first period.

2. The player is sure to lose the subgame. For instance, player 2’s choices L and R in the second period.

3. It is both possible that the player can win the subgame and can lose the subgame. Which case happens depends on the choice of the other player, who is indifferent about this. For instance, player 1’s choice R in the first period.

Between the first and the second types of choices, the usual backward induction is able to suggest that the players should play the first type of choices. However, between

10 the first and the third types of choices, the usual backward induction is not able to single out the first types of choices. That is, a choice of the third type can also be part of a subgame perfect Nash equilibrium. So the refined backward induction axiom strengthens the usual backward induction by requiring, that between the first and the third types of choices, the players should choose that of the first type. To formalize this axiom, we require that a player “formally regard” a subgame in which he is unable to guarantee a win as a subgame he loses. For example, in the game of Figure 1, if player 1 “thinks” that the subgame after he plays R is a game “he will lose”, then he will choose L in the first period and will not choose R. This is only a “formal” requirement since we do not require that, say in this example, player 2 have to actually choose L in the second period so that player 1 for sure loses. Player 2 is still free to make either choice.

Let G be a game defined by the pair hA1,A2i. Suppose that for each player i and each choice y ∈ Y , either Ai ∩ Ny = Ny or Ai ∩ Ny = ∅. That is, the subgame Gy is trivial for each y. Now player 1, who moves first, can guarantee a win in the game G if and only if there exists some y such that if he chooses this y then he wins for sure.

That is, there exists y such that A1 ∩ Ny = Ny. So there are two possibilities for player 2.

1. There exists y such that A1 ∩ Ny = Ny. If there are more than one such y, then player 1 can choose any of them. So if player 2 wants to guarantee a win in G,

he has to be sure that he wins Gy for all such y. That is, for each y such that

A1 ∩ Ny = Ny we must also have A2 ∩ Ny = Ny. This implies that A1 ⊆ A2.

2. Otherwise, for each y we have A1 ∩ Ny = ∅. Then A1 = ∅ and player 1 will for sure lose the game G. So player 1 is indifferent about which y to choose, and any y is possible. Now if player 2 wants to guarantee a win in this case, he has to be

sure that he wins Gy for each y. That is, for each y ∈ Y we have A2 ∩ Ny = Ny.

ω This implies that A2 = Y .

To summarize, player 2 can guarantee a win only when the following conditions

11 hold,

If A1 ∩ Ny = Ny for some y then A1 ⊆ A2, (1)

ω and if A1 = ∅ then A2 = Y . (2)

For example, if there exists y such that

A1 ∩ Ny = Ny and A2 ∩ Ny = ∅, (3) then the condition in (1) is violated and player 2 cannot guarantee a win. If

A1 = ∅ and there exists y such that A2 ∩ Ny = ∅, (4) then the condition in (2) is violated and player 2 cannot guarantee a win.

Axiom 1 (Axiom B1). Let G = hA1,A2i. Suppose that for each y ∈ Y the subgame

∗ ∗ Gy is trivial. Then G can be reduced to the trivial game hA1,A2i, where

∗ ω ∗ 1. A1 = Y if there exists y such that A1 ∩ Ny = Ny, and A1 = ∅ otherwise;

∗ ω ∗ 2. A2 = Y if the conditions in (1) and (2) hold, and A2 = ∅ otherwise.

Axiom 2 (Axiom B2). Let G = hA1,A2i. Suppose that there exists y such that the condition in (3) holds. Then G can be reduced to the game hY ω, ∅i.

Axiom 3 (Axiom B3). Let G = hA1,A2i. Suppose that there exists y such that the condition in (4) holds. Then G can be reduced to the game h∅, ∅i.

Remark 3.10. Note that Axioms B2 and B3 do not require that the subgame Gy be trivial for each y. If the subgame Gy is trivial for each y, then Axioms B2 and B3 are special cases of Axiom B1. Axioms B1, B2 and B3 together will be referred to as Axiom B.

3.3 The cooperation axiom

The idea of this axiom can be illustrated by the following example. The players al- ternate saying “stop” or “continue”. If either player says “stop”, then the game ends

12 immediately and both players get nothing. Otherwise the game continues forever and the payoff to each player is 1.

1 c 2 c 1 c 2 c 1 c 2 c ··· 1, 1 s b s rr s rr s rr s rr s rr r

0,r0 0,r0 0,r0 0,r0 0,r0 0,r0

Figure 2: A game in which cooperation is the only reasonable outcome.

The main feature of this example is that the players have common interests: They have identical payoff functions. Moreover, in this context coordination does not seem to be a problem: The players move alternately rather than simultaneously, and the actions are revealed to the other player immediately. They can cooperate by simply “not giving up”, that is, not to say “stop” to lose the game for sure. So it seems reasonable to conclude that rational players will cooperate in this context. Thus it is natural to propose an axiom requiring the players to cooperate whenever they have identical payoff sets. The game in Figure 3 shows that the requirement that the players have exactly the same payoff sets may be too strong, it suffices that the payoff sets are “essentially” identical.

1 c 2 c 1 c 2 c 1 c 2 c ··· 1, 1 s b s rr s rr s rr s rr s rr r

0,r1 1,r0 0,r1 1,r0 0,r1 1,r0

Figure 3: Cooperation is still the only reasonable outcome in this game, but the players have different payoff sets.

If a player says “stop” at any stage, then he is sure to lose the game. But if they both avoid such moves, then they still have identical payoff sets: the set that consists of the single path in which the players always say “continue”. So it is reasonable to have the following more general behavioral axiom: If modulo the sure-to-lose moves,

13 the players have the same payoff set which is nonempty, then they should cooperate to win the game (by simply avoiding the sure-to-lose moves).

To formalize this axiom, let G be a game defined by the pair hA1,A2i. Say that player i is sure to lose the game after a choice y at the position p if Ai ∩ Npay = ∅. So if the players avoid the sure-to-lose moves then the paths in following set will not be played.

K(G) = ∪{ Npay | the subgame Gp is not trivial, the subgame Gpay is trivial,

and Ai ∩ Npay = ∅, where i is the player that moves at p }.

Remark 3.11. Indeed, we do not need the requirements that the subgame Gp is not trivial and the subgame Gpay is trivial. But adding these regularity conditions only weakens our axioms and strengthens our results.

For each i, define ˜ Ai = Ai \ K(G).

˜ Then Ai is the new payoff set of player i if both players avoid the sure-to-lose moves. The cooperation axiom says that the players should cooperate to win the game if

˜ ˜ A1 = A2 6= ∅. (5)

That is, if the players have the same nonempty payoff set modulo the sure-to-lose moves, then they should cooperate to win the game.

Axiom 4 (Axiom C). Let G = hA1,A2i. Suppose that the condition in (5) holds. Then G can be reduced to the game hY ω,Y ωi.

3.4 The noncooperation axiom

Definition 3.12. Let M1 be the set of positions for player 1 to move. A strategy S1 for player 1 is a mapping from M1 to Y .

14 Definition 3.13. Let S1 and S2 be strategies for players 1 and 2, respectively. The play according to S1 and S2, denoted by S1 ∗ S2, is the sequence (y0, y1, y2,... ) where

y0 = S1(∅), y1 = S2(y0), y2 = S1((y0, y1)), etc.

Definition 3.14. Say that S1 is a winning strategy for player 1 in the game hA1,A2i if for each strategy S2 for player 2 we have S1 ∗ S2 ∈ A1.

ω Axiom 5 (Axiom N). 1. If player 1 has a winning strategy in the game hA1,Y i,

ω ω ω then hA1,Y i can be reduced to hY ,Y i.

2. If player 1 has a winning strategy in the game hA1, ∅i, then hA1, ∅i can be reduced to hY ω, ∅i.

Remark 3.15. The first and second items will be referred to as Axioms N1 and N2, respectively.

4 Definition of Determinacy

Definition 4.1 (Applying an axiom). Let G = hA1,A2i and let p be a position.

Suppose that the subgame Gp can be reduced to a trivial game hB1,B2i by an axiom.

∗ ∗ ∗ Let G = hA1,A2i, where for each i,  A ∪ N , if B = Y ω; ∗  i p i Ai =  Ai \ Np, if Bi = ∅.

We say that G∗ is the game obtained from G by applying this axiom to the subgame

∗ Gp, or simply say that G is reduced to G by this axiom.

Definition 4.2. A reduction chain is a sequence of games hG0,G1,...,Gni such that for each k < n the game Gk+1 is obtained from Gk by applying an axiom to a subgame of Gk.

Definition 4.3. A game G is called determined if there exists a reduction chain hG0,G1,...,Gni such that G0 = G and Gn is trivial.

15 Remark 4.4. Definition 4.3 is incomplete for a technical reason, but it suffices for the purpose of exposition of the main ideas. The technically complete version, which allows the chain to go infinitely long, is given in Definitions 6.4 and 6.6. It will not be needed until Section 6.

In the following we summarize the related results in the companion papers (Zhang (2008a), Zhang (2008b), and Zhang (2008c)) that demonstrate the main properties of determinacy. The proofs of these results can be found in these papers.

Theorem 1. Let hG0,G1,...,Gni and hH0,H1,...,Hmi be two reduction chains such that G0 = H0 and the games Gn and Hm are trivial. Then Gn = Hm.

Remark 4.5. This result can be viewed in several ways. First, since there can be many reduction chains, this result guarantees that determinacy is well-defined. In particular, it is order independent. Second, it says that the outcome given by determinacy is unique. Third, it shows that the axioms are consistent in the sense that they do not lead to contradicting outcomes.

Recall that a player wins a trivial game just in case his payoff set is Y ω.

Definition 4.6. Let G be a determined game with a reduction chain hG0,G1,...,Gni such that G0 = G and Gn is trivial. We say that player i wins G if he wins Gn. Otherwise we say that he loses G.

Let Y ω be given the natural product topology with Y discrete. That is, for each

ω position p the set Np is a basic open neighborhood. A A of Y is called an open set if it is the of some basic open neighborhoods. A subset A of Y ω is called a if its complement Y ω \ A is an open set.

Definition 4.7. The game hA1,A2i is called a closed game if each Ai is a closed set.

Theorem 2. All closed games are determined.

Remark 4.8. This result can also be viewed as saying that the axioms are complete in the sense that they are sufficient to solve the class of closed games.

16 Theorem 3. Axioms B2, B3 and C together imply Axiom B1.

Remark 4.9. Thus, strictly speaking, we do not need Axiom B1. We also treat it as an axiom in this paper for two reasons. First, it is so natural that it may be of independent interest. Second, it is not clear that Axiom B1 is still independent when we extend this study to the more general n-person games.

Theorem 4. If any of Axioms B2, B3, C, N1 and N2 is dropped then there exists a closed game that is not determined.

Remark 4.10. So none of Axioms B2, B3, C, N1 or N2 is redundant.

Remark 4.11. Thus, Theorems 1, 2, 3 and 4 show that the axiom system consisting of Axioms B2, B3, C, N1 and N2 is consistent, complete and independent.

The following theorem relates determinacy to subgame perfect Nash equilibrium.

<ω Theorem 5. Let G = hA1,A2i. Suppose that for each p ∈ Y the subgame Gp is determined. Then there exists a subgame perfect Nash equilibrium (S1,S2) such that for each i,

S1 ∗ S2 ∈ Ai if and only if player i wins the game G.

Remark 4.12. The theorem shows that the outcome of determinacy can be supported by a subgame perfect Nash equilibrium. Since the outcome given by determinacy is unique, determinacy can thus be viewed as a unique refinement of subgame perfect Nash equilibrium.

By combining Theorems 2 and 5 we are able to show the following result.

Theorem 6. All closed games have a subgame perfect Nash equilibrium.

Finally we relate our notion of determinacy to the original notion of determinacy defined for zero-sum games. Let G = hA1,A2i. Say that G is a zero-sum game if the

ω ω payoff sets A1 and A2 form a partition of Y . That is, A1 ∩ A2 = ∅ and A1 ∪ A2 = Y . Say that player i wins G in the sense of Gale and Stewart (1953) if and only if he has a winning strategy.

17 Theorem 7. Suppose that G is a closed, zero-sum game. Then a player wins G in the sense of Definition 4.6 if and only if he wins G in the sense of Gale and Stewart (1953).

5 Illustrations

5.1 Interpretation of determinacy

The outcome given by determinacy requires an interpretation different from the usual one. The reason is that in Axiom B we use the expression “a player loses a game” not only to mean that this player will for sure lose the game, but also to mean that he is unable to guarantee a win in the game. Suppose that G is reduced by a reduction

∗ ∗ ∗ ω chain to a trivial game hA1,A2i. If player i wins the game G, that is, Ai = Y , then indeed player i can win the game if both players play rationally. But if player i loses

∗ the game G, that is, Ai = ∅, then there are two possibilities. The first possibility is that indeed player i is for sure to lose the game G. The second possibility is that at certain stage of the reduction chain Axiom B is applied to a subgame of G, and “player i loses the game G” only means that he cannot guarantee a win in that subgame. So in this situation it is indeed possible for player i to lose the game G by losing that subgame; but it is also possible for him to win G. Which case happens depends entirely on the other player, who is indifferent about this.

5.2 An example

We illustrate the working of determinacy by showing that finite games are determined. We consider an example first. Let G denote the game in Figure 4. Since player 1 wins the subgames GL and G(R,R) and player 2 wins only the subgame G(R,R,R), they have the payoff sets NL ∪ N(R,R) and N(R,R,R), respectively. So G = hNL ∪ N(R,R),N(R,R,R)i.

The subgame G(R,R) in the third period, after player 1 chooses R in the first period

ω and player 2 chooses R in the second period, is hY ,NRi. Since player 1 moves at the

18 player 1 @ LR b @ @ @@ player 2 1, 0 @ r LR rr @ @ @@ player 1 0, 0 @ r LR rr @ @ @@ 1,r0 1,r1

Figure 4: A finite game G defined by the pair hNL ∪ N(R,R),N(R,R,R)i.

player 1 @ player 1 LR b @ @ @ b @@ player 2 LR @ @ 1, 0 @ r LR rr @ @ @ @ @@ 1,r0 0,r0 0,r0 1,r0

Figure 5: Two games in the reduction chain: G1 = hNL ∪ N(R,R), ∅i and G2 = hNL, ∅i. position (R,R) and he can choose either L or R, player 2 cannot guarantee a win in this subgame. So by Axiom B1 the subgame G(R,R) can be reduced to the trivial game

ω hY , ∅i. Thus the game G can be reduced to the game hNL ∪ N(R,R), ∅i by applying

Axiom B1 to the subgame G(R,R). Let G1 = hNL ∪ N(R,R), ∅i. The game G1 is depicted in Figure 5.

The subgame G1,R of G1 in the second period, after player 1 chooses R in the first period, is hNR, ∅i. Since player 2 moves at the position R and he can choose either L or R, player 1 cannot guarantee a win in this subgame. So by Axiom B1 the subgame

G1,R can be reduced to the trivial game h∅, ∅i. Thus the game G1 can be reduced to the game hNL, ∅i by applying Axiom B1 to the subgame G1,R. Let G2 = hNL, ∅i. The game G2 is depicted in Figure 5.

Finally, player 1 wins the game G2 since player 1 moves in this game and he can

19 choose L. Player 2 cannot guarantee a win in this game. So by Axiom B1 the game

ω ω G2 can be reduced to the trivial game hY , ∅i. Let G3 = hY , ∅i. Thus we obtain a reduction chain hG, G1,G2,G3i for the game G and G3 is trivial. So G is determined, player 1 wins G and player 2 loses G.

5.3 Determinacy of finite games

In general, we have the following theorem.

Theorem 8. Finite games are determined.

Proof. We only show the case Y is a finite set. The case Y is infinite is a special case of

Theorem 2. Let G be a finite game of length n. That is, the subgame Gp is trivial for each position p of length n. Define by induction on k a reduction chain h Gk | k ≤ m i, where the number m is to be determined.

Base case. Let G0 = G.

Induction step. Assume that Gk−1 has been defined. If Gk−1 is trivial, then let m = k − 1 and we are done. Otherwise there exist positions p such that the subgames

Gk−1,p are not trivial. Then by our assumption that for each position p of length less than or equal to n the subgame Gp is trivial, there must exist p such that Gk−1,p is not trivial but Gk−1,pay is trivial for each y. So the subgame Gk−1,p satisfies the conditions of Axiom B1. Let Gk be obtained from Gk−1 by applying Axiom B1 to the subgame

Gk−1,p. Since Y is finite and G is finite, this process terminates at some finite stage and the number m exists. Thus we obtain a finite reduction chain h Gk | k ≤ m i such that

G0 = G and Gm is trivial. So G is determined.

20 6 The Complete Version of Determinacy

6.1 Ordinal numbers

In the version of determinacy we gave in Definition 4.3 the reduction chain is finite. However, there are games that we need to apply the axioms infinitely many times in order to reduce it to a trivial game. Consider, for instance, the game in Figure 6. Each subgame of the form

G(c, c, . . . , c,d)

| n copies{z } is an n-period game that we need to apply Axiom B1 or B3 n times in order to reduce it to a trivial game. Moreover, the other axioms are not applicable to any subgame. Since n can be arbitrarily large, we cannot reduce the entire game to a trivial game in finite steps. Thus we need to allow for infinite reduction chains in the definition of determinacy. The techniques of definition and proof by induction on ordinal numbers are the appropriate tools for this purpose.

1 c 2 c 1 c 2 c ··· 0, 0 d rb d r d r d r r 1 c 0, 1 2 c 1, 0 1 c 0, 1 0, 0 rr d r r d r r d r r 1 c 0, 1 2 c 1, 0 0, 0 r d r r d r r 1 c 0, 1 0, 0 r d r r

0,r0 Figure 6: A game that cannot be reduced to a trivial game by a finite reduction chain.

The notion of ordinal numbers was introduced by .1 After the natural numbers 0, 1, . . . , n, . . . comes the first infinite ω, followed by ω + 1, ω + 2, . . . , ω + ω, . . . , and so forth. Each ordinal number α is usually identified with the set { ν | ν < α } of its predecessors. Thus 0 = ∅, α + 1 = α ∪ {α}, etc. And the ∈

1See, e.g, Hrbacek and Jech (1999) for an introduction.

21 on ordinals coincides with <. An ordinal number α 6= 0 is called a successor ordinal if α = β + 1 for some β; otherwise it is called a . So ω, which is identified with the set of natural numbers, is the first limit ordinal as it is neither 0 nor a successor ordinal. In the following, Greek letters α, β, γ and δ denote general ordinal numbers, θ and λ denote limit ordinals. Definition and proof by induction on ordinal numbers are natural extensions of definition and proof by induction on natural numbers. For example, if we want to

ω define a sequence h Aα | α ≤ γ i of subsets of Y indexed by ordinal numbers less than or equal to γ, it suffices to proceed in the following steps. First define A0 in the base case. Then in the successor stage, assume that Aα has been defined and proceed to define Aα+1. Finally in the limit stage where θ is a limit ordinal less than or equal to

γ, assume that Aα has been defined for each α < θ, and proceed to define Aθ. Similarly if we want to prove that a proposition P (α) holds for each ordinal number α less than or equal to γ, it suffices to proceed in the following steps. First prove that P (0) is true in the base case. Then in the successor stage, assume that P (α) is true and proceed to prove that P (α + 1) is also true. Finally in the limit stage where θ is a limit ordinal less than or equal to γ, assume that P (α) is true for each α < θ and proceed to prove that P (θ) is true.

6.2 The complete version of determinacy

Definition 6.1. Let θ be a limit ordinal and let h Aα | α < θ i be a sequence of subsets

ω of Y . Then the set limα<θ Aα is defined to be

ω { f ∈ Y | there exists α0 < θ such that f ∈ Aα for each α > α0 }.

Remark 6.2. That is, the path f belongs to the limit set limα<θ Aα if and only if it belongs to Aα for all but a bounded initial segment of h α | α < θ i.

Definition 6.3. Let θ be a limit ordinal and let h Gα | α < θ i be a sequence of games, where Gα = hA1,α,A2,αi for each α. Then the game limα<θ Gα is defined to be hlimα<θ A1,α, limα<θ A2,αi.

22 Definition 6.4. A reduction chain is a sequence of games h Gα | α ≤ γ i such that each Gα+1 is obtained from Gα by applying one of Axioms B1, B2, B3, C, N1 and

N2 to a subgame of Gα, and for each limit ordinal θ less than or equal to γ, we have

Gθ = limα<θ Gα.

Remark 6.5. Thus if Gα = hA1,α,A2,αi for each α, then for each limit orindal θ and each i we have Ai,θ = limα<θ Ai,α.

We can now state the complete version of the definition of determinacy.

Definition 6.6. A game G is called determined if there exists a reduction chain h Gα |

α ≤ γ i such that G0 = G and Gγ is trivial.

Theorem 9. Let h Gα | α ≤ γ i and h Hα | α ≤ δ i be two reduction chains such that

G0 = H0 and the games Gγ and Hδ are trivial. Then Gγ = Hδ.

Definition 6.7. Let G be a determined game with a reduction chain h Gα | α ≤ γ i such that Gγ is trivial. Say that player i wins the game G if he wins Gγ; otherwise we say that he loses G.

Remark 6.8. These are the corresponding versions of Theorem 1 and Definition 4.6.

6.3 An illustration

Here we illustrate the working of the complete version of determinacy in Definition 6.6 by showing that the game in Figure 6 is determined. Indeed, we construct a reduction chain

hG0,G1,...,Gm,Gm+1,...,Gωi for this game such that Gω is trivial.

We denote this game by G = hA1,A2i. Let n and k be natural numbers and let pn,k denote the following position

pn,k = (c, c, . . . , c, d, d, . . . , d). | {z } | {z } n copies k copies

23 The payoff structure of G can be summarized as follows. It consists of three parts. First, for each n we have

Gpn,n+1 = h∅, ∅i.

Second, for each k such that 1 ≤ k ≤ n and pn,k is a position for player 1 to move we have

ω Gpn,kac = h∅,Y i, where

pn,kac = (c, c, . . . , c, d, d, . . . , d, c); | {z } | {z } n copies k copies and for each k such that 1 ≤ k ≤ n and pn,k is a position for player 2 to move we have

ω Gpn,kac = hY , ∅i.

Finally,

(c, c, c, . . . ) ∈/ A1 and (c, c, c, . . . ) ∈/ A2.

Thus after c is played in the first n periods and d in the (n+1)-th period (the game

starts from period 0), there is a subgame Gpn,1 of length n. For example, after player

1 chooses c in period 0 and player 2 chooses d in period 1, there is a subgame Gp1,1 , which is G(c,d), for player 1 to choose between c and d, and either choice leads to the end of the game. To reduce each Gpn,1 into a trivial game we need to apply Axiom

B1 or B3 n times. In the following we define the reduction chain h Gm | m < ω i that systematically reduces each Gpn,1 into a trivial game. Note that for each m greater than 0, there is a unique pair n and k such that m = n(n + 1)/2 + k and 1 ≤ k ≤ n + 1. (6)

For small m, the n and k are listed in Table 1.

We now define the sequence of games hGm | m < ω i, where Gm = hA1,m,A2,mi for each m, by induction on m such that for each m > 0, the game Gm is obtained from

Gm−1 by applying Axiom B1 to the subgame Gm−1,pn+1,n+1−k+1 and

Gm,pn+1,n+1−k+1 = h∅, ∅i,

24 m n k pn+1,n+1−k+1

1 0 1 p1,1

2 1 1 p2,2

3 1 2 p2,1

4 2 1 p3,3

5 2 2 p3,2

6 2 3 p3,1

Table 1: The first few cases of m, n, k and pn+1,n+1−k+1.

where n and k is the pair that satisfies (6). For small m, the positions pn+1,n+1−k+1 are listed in Table 1.

Base case. Let G0 = G.

Successor stage. Assume that Gm has been defined and

Gm,pn+1,n+1−k+1 = h∅, ∅i, where n and k is the unique pair that satisfies (6). Case 1. Suppose that k < n + 1. Then k + 1 ≤ n + 1. So we can write

m + 1 = n(n + 1)/2 + (k + 1), and

pn+1,n+1−(k+1)+1 = pn+1,n+1−k.

Without loss of generality, let pn+1,n+1−k be a position for player 1 to move. Since

ω Gm,pn+1,n+1−kac = Gpn+1,n+1−kac and Gpn+1,n+1−kac = h∅,Y i, we have

ω Gm,pn+1,n+1−kac = h∅,Y i.

Since Gm,pn+1,n+1−kad = Gm,pn+1,n+1−k+1 , by the induction hypothesis that Gm,pn+1,n+1−k+1 = h∅, ∅i, we have

Gm,pn+1,n+1−kad = h∅, ∅i.

25 Thus Axiom B1 can be used to reduce the subgame Gm,pn+1,n+1−k to the trivial game h∅, ∅i. Let Gm+1 be obtained from Gm by applying Axiom B1 to Gm,pn+1,n+1−k . Then

Gm+1,pn+1,n+1−k = h∅, ∅i.

Case 2. Suppose that k = n + 1. Then we can write

m + 1 = n(n + 1)/2 + k + 1

= n(n + 1)/2 + (n + 1) + 1

= (n + 1)(n + 2)/2 + 1, and

p(n+1)+1,[(n+1)+1]−1+1 = pn+2,n+2.

Without loss of generality, let pn+2,n+2 be a position for player 1 to move. Since

ω Gm,pn+2,n+2ac = Gpn+2,n+2ac and Gpn+2,n+2ac = h∅,Y i, we have

ω Gm,pn+2,n+2ac = h∅,Y i.

Since Gm,pn+2,n+2ad = Gm,pn+2,n+3 , Gm,pn+2,n+3 = Gpn+2,n+3 , and Gpn+2,n+3 = h∅, ∅i, we have

Gm,pn+2,n+2ad = h∅, ∅i.

So Axiom B1 can be used to reduce the subgame Gm,pn+2,n+2 to the trivial game h∅, ∅i.

Let Gm+1 be obtained from Gm by applying Axiom B1 to Gm,pn+2,n+2 . Thus

Gm+1,pn+2,n+2 = h∅, ∅i.

The induction is now complete. For the first few cases of Gm, see Figures 7–9.

Now we compute the game Gω, which is defined to be limm<ω Gm. Note that for each n, we have Gm,pn,1 = h∅, ∅i for all sufficiently large m. Indeed, for each m let n(m) and k(m) be the unique pair that satisfies (6). Let m0 be such that n(m0) = n and k(m0) = 1. Then Gm,pn,1 = h∅, ∅i for all m ≥ m0. Thus in the limit we have

Gω,p = lim Gm,p = h∅, ∅i. (7) n,1 m<ω n,1

26 1 c 2 c 1 c 2 c ··· 0, 0 d rb d r d r d r r 2 c 1, 0 1 c 0, 1 0, 0 0, 0 rr rr d r r d r r 1 c 0, 1 2 c 1, 0 d r r d r r 1 c 0, 1 0, 0 r d r r

0,r0

Figure 7: The game G1 is obtained from G by applying Axiom B1 to the subgame

G(c,d) and G1,(c,d) = h∅, ∅i.

1 c 2 c 1 c 2 c ··· 0, 0 d rb d r d r d r r 2 c 1, 0 1 c 0, 1 0, 0 0, 0 rr rr d r r d r r 2 c 1, 0 0, 0 r d r r 1 c 0, 1 d r r

0,r0

Figure 8: The game G2 is obtained from G1 by applying Axiom B1 to the subgame

G1,(c,c,d,d) and G2,(c,c,d,d) = h∅, ∅i.

Since (c, c, c, . . . ) ∈/ A1 and (c, c, c, . . . ) ∈/ A2, and no axiom is applied to subgames of the form Gm,pn,0 , we have (c, c, c, . . . ) ∈/ Am,1 and (c, c, c, . . . ) ∈/ Am,2 for each m < ω. Therefore

(c, c, c, . . . ) ∈/ Aω,1 and (c, c, c, . . . ) ∈/ Aω,2. (8)

Combining the results in (7) and (8) we have

Gω = lim Gm = h∅, ∅i. m<ω

Thus we obtain a reduction chain h Gm | m ≤ ω i such that Gω is trivial and Gω = h∅, ∅i. So G is determined and no player is able to guarantee a win in this game.

27 1 c 2 c 1 c 2 c ··· 0, 0 d rb d r d r d r r 1 c 0, 1 0, 0 0, 0 0, 0 rr rr rr d r r 2 c 1, 0 d r r 1 c 0, 1 d r r

0,r0

Figure 9: The game G3 is obtained from G2 by applying Axiom B1 to the subgame

G2,(c,c,d) and G3,(c,c,d) = h∅, ∅i.

7 Conclusion

This is the first of a of papers in which we study the class of two-person infinite games with perfect information and nonzero-sum characteristic payoff functions. In this paper we focus on describing the games, delivering the definition of determinacy, and illustrating the working of this definition. We start by motivating and formalizing three axioms that deal with backward induction games, pure cooperation games and pure competition games, respectively. Then we define a game to be determined if it can be solved by repeatedly applying these three axioms. In the companion papers we show that these axioms are consistent, complete and independent. Thus determinacy is a well-defined, consistent solution concept and a whole class of games are solvable by determinacy. This analysis also has important implications for subgame perfect Nash equilibrium and iterated weak dominance.

References

Chen, Yi-Chun, Ngo Van Long, and Xiao Luo, 2007, Iterated strict dominance in general games, Games and Economic Behavior 61, 299–315.

Davis, Morton, 1964, Infinite games of perfect information, in , Lloyd S.

28 Shapley, and Alan W. Tuchker, ed.: Advances in , Annals of Mathe- matical Studies, vol. 52 . pp. 85–101 (Princeton University Press: Princeton).

Dufwenberg, Martin, and Mark Stegeman, 2003, Existence and uniqueness of maximal reductions under iterated strict dominance, 70, 2007–2024.

Dutta, Prajit K., 1995, A folk theorem for stochastic games, Journal of Economic Theory 66, 1–32.

, 2003, Coordination need not be a problem, Unpublished manuscript.

Ewerhart, Christian, 2002, Iterated weak dominance in strictly competitive games of perfect information, Journal of Economic Theory 107, 474–482.

Fudenberg, Drew, and David K. Levine, 1983, Subgame-perfect equilibria of finite and infinite horizon games, Journal of Economic Theory 31, 251–268.

Gale, Douglas, 1995, Dynamic coordination games, Economic Theory 5, 1–18.

, 2001, Monotone games with positive spillovers, Games and Economic Behavior 37, 295–320.

Gale, David, and F.M. Stewart, 1953, Infinite games with perfect information, in Ad- vances in Game Theory, Annals of Mathematical Studies, vol. 28 . pp. 245–266 (Princeton University Press: Princeton).

Gilboa, Itzhak, , and Eitan Zemel, 1990, On the order of eliminating dom- inated strategies, Operations Research Letters 9, 85–89.

Harris, Christopher, 1985a, A characterization of the perfect equilibria of infinite- horizon games, Journal of Economic Theory 37, 99–125.

, 1985b, Existence and characterization of perfect equilibrium in games of per- fect information, Econometrica 53, 613–628.

29 , Philip Reny, and Arthur Robson, 1995, The existence of subgame-perfect equilibrium in continuous games with almost perfect information: A case for public randomization, Econometrica 63, 507–544.

Hellwig, M., and W. Leininger, 1987, On the existence of subgame-perfect equilibrium in infinite-action games of perfect information, Journal of Economic Theory 43, 55– 75.

, Philip Reny, and Arthur Robson, 1990, Subgame-perfect equilibrium in con- tinuous games of perfect information: An elementary approach to existence and approximation by discrete games, Journal of Economic Theory 52, 406–422.

Hrbacek, Karel, and , 1999, Introduction to (Marcel Dekker: New York).

Kanamori, Akihiro, 2000, The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings (Springer: Berlin).

Lagunoff, Roger, and Akihiko Matsui, 1997, Asynchronous choice in repeated coordi- nation games, Econometrica 65, 1467–1477.

L¨owe, Benedikt, 2007, Set theory of infinite imperfect information games, in Alessandro Andretta, ed.: Set Theory: Recent Trends and Applications, Quaderni di Matemat- ica, vol. 17 pp. 137–181.

, Forthcoming, Determinacy for infinite games with more than two players with preferences, Journal of Applied Logic.

Martin, Donald A., 1975, Borel determinacy, Annals of Mathematics 102, 263–371.

Marx, Leslie M., and Jeroen M. Swinkels, 1997, Order independence for iterated weak dominance, Games and Economic Behavior 18, 219–245.

Maskin, Eric, and , 2001, Markov perfect equilibrium: I. observable actions, Journal of Economic Theory 100, 191–219.

30 Mycielski, Jan, 1992, Games with perfect information, in Robert J. Aumann, and Sergiu Hart, ed.: Handbook of Game Theory with Economic Applications . , vol. 1 (North-Holland: Amsterdam).

Osborne, Martin J., and , 1990, Bargaining and Markets (Academic Press: San Diego).

Rubinstein, Ariel, and Asher Wolinsky, 1995, Remarks on infinitely repeated extensive- form games, Games and Economic Behavior 9, 110–115.

Selten, Reinhard, 1965, Spieltheoretische behandlung eines oligopolmodels mit nach- fragetragheit, Zeitschrift fur die Gesamte Staatswissenschaft 121, 301–324.

, 1975, Reexamination of the perfectness concept for equilibrium points in extensive games, International Journal of Game Theory 4, 25–55.

Sutton, John, 1986, Non- theory: An introduction, Review of Economic Studies 53, 709–724.

Takahashi, Satoru, 2005, Infinite horizon common interest games with perfect informa- tion, Games and Economic Behavior 53, 231–247.

Wolfe, Philip, 1955, The strict determinacy of certain infinite games, Pacific Journal of Mathematics 5, 841–847.

Zhang, Wenzhang, 2008a, Determinacy of nonzero-sum games: Completeness, Unpub- lished manuscript.

, 2008b, Determinacy of nonzero-sum games: Consistency and independence, Unpublished manuscript.

, 2008c, Determinacy of nonzero-sum games: Implications, Unpublished manu- script.

31