Determinacy of Nonzero-sum Games Wenzhang Zhang 1 April 4, 2008 1School of Economics, Shanghai University of Finance and Economics, Shanghai 200433, China. Email: [email protected] Abstract In this paper we study a class of infinite games with perfect information and define a new solution concept that we call determinacy for these games. We start by motivat- ing and formalizing three simple behavioral axioms about how rational players should behave in backward induction games, pure cooperation games and pure competition games, respectively. Then we say that a general game is determined if it can be solved by repeatedly applying these axioms to its subgames. This solution concept has nice properties and important implications for subgame perfect Nash equilibrium and iter- ated weak dominance. In particular, the outcome 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 set Y . First player 1 chooses y0, then player 2 chooses y1, then player 1 chooses y2, etc., so that an infinite sequence of choices (y0, y1, y2,... ) is specified. The payoffs of the players are defined by a pair of subsets A1 and A2 that form a partition of the total space of such sequences (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 strategy 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 topology 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 mathematics (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 almost 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 empty set 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 strategies 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 axiom 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 order 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 number 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).