Finite Normal Form Games Research Articles

Correlated equilibrium under costly disobedience

We extend Aumann’s (1974) model of correlated equilibrium by requiring each player to bear an exogenously given cost if s/he disobeys the recommendation of the mediator. Calling the modified solution the costly correlated equilibrium (costly-CE), we show that in any finite normal-form game that has an unpure Nash equilibrium, the set of costly-CE strictly expands even with an arbitrarily small increase in the disobedience cost provided that the game is non-trivial and there is room for expansion. We also study the effects of the disobedience cost on the total welfare of players and the value of mediation.

Game-Theoretic Models of Moral and Other-Regarding Agents (extended abstract)

We investigate Kantian equilibria in finite normal form games, a class of non-Nashian, morally motivated courses of action that was recently proposed in the economics literature. We highlight a number of problems with such equilibria, including computational intractability, a high price of miscoordination, and problematic extension to general normal form games. We give such a generalization based on concept of program equilibria, and point out that that a practically relevant generalization may not exist. To remedy this we propose some general, intuitive, computationally tractable, other-regarding equilibria that are special cases Kantian equilibria, as well as a class of courses of action that interpolates between purely self-regarding and Kantian behavior.

Every normal-form game has a Pareto-optimal nonmyopic equilibrium

It is well known that Nash equilibria may not be Pareto-optimal; worse, a unique Nash equilibrium may be Pareto-dominated, as in Prisoners’ Dilemma. By contrast, we prove a previously conjectured result: every finite normal-form game of complete information and common knowledge has at least one Pareto-optimal nonmyopic equilibrium (NME) in pure strategies, which we define and illustrate. The outcome it gives, which depends on where play starts, may or may not coincide with that given by a Nash equilibrium. We use some simple examples to illustrate properties of NMEs—for instance, that NME outcomes are usually, though not always, maximin—and seem likely to foster cooperation in many games. Other approaches for analyzing farsighted strategic behavior in games are compared with the NME analysis.

Correlated Equilibrium Under Costly Disobedience

A recent experimental literature focusing on overcoming coordination problems in strategic games points out to the need for incentivizing players to follow third-party recommendations to coordinate on better outcomes. In this paper, we consider finite normal-form games to strengthen the incentives in the Aumann's (1974) well-known coordination model under correlated equilibrium by introducing a non-negative cost of disobedience for all players. Calling the new solution costly correlated equilibrium (CCE), we show that in any game that is non-trivial, the set of CCE is larger in the presence of even an arbitrarily small cost of disobedience than in its absence if and only if the game has an unpure Nash equilibrium. We also illustrate that there are games in which strictly better social outcomes can be attained, and the value of mediation increases, even with an arbitrarily small increase in the cost level. In addition, we illustrate how our model can be integrated with a voluntary cost-selection game where players non-cooperatively choose their costs of disobedience before mediation occurs. We show that there exist cost-selection games in which setting the cost level at zero is a strictly dominated strategy for each player as well as games this strategy becomes weakly dominant for everyone.

Farsightedness in Games: Stabilizing Cooperation in International Conflict

We show that a cooperative outcome—one that is at least next-best for the players—is not a Nash equilibrium (NE) in 19 of the 57 2 x 2 strict ordinal conflict games (33%), including Prisoners’ Dilemma and Chicken. Auspiciously, in 16 of these games (84%), cooperative outcomes are nonmyopic equilibria (NMEs) when the players make farsighted calculations, based on backward induction; in the other three games, credible threats induce cooperation. More generally, in all finite normal-form games, if players’ preferences are strict, farsighted calculations stabilize at least one Pareto-optimal NME. We illustrate the choice of NMEs that are not NEs by two cases in international relations: (i) no first use of nuclear weapons, chosen by the protagonists in the 1962 Cuban missile crisis and since adopted by some nuclear powers; and (ii) the 2015 agreement between Iran, and a coalition of the United States and other countries, that has been abrogated by the United States but has forestalled Iran’s possible development of nuclear weapons.

Strict Incentives and Strategic Uncertainty

This paper proposes a comprehensive perspective on the question of self-enforcing solutions for normal form games. While this question has been widely discussed in the literature, the focus is usually either on strict incentives for players to stay within the proposed solution or on strategic uncertainty, i.e. robustness to trembles. The present approach combines both requirements in proposing the concept of robust sets, i.e. sets of strategy profiles which satisfy both strict incentives and robustness to strategic uncertainty. The result is a set valued solution, a variant of which is shown to exist for all finite normal form games.

Stabilizing Cooperative Outcomes in Two-Person Games: Theory and Cases

We show that a cooperative outcome—one that is at least next-best for the players—is not a Nash equilibrium (NE) in 19 of the 57 2 x 2 strict ordinal conflict games (33%), including Prisoners’ Dilemma and Chicken. Auspiciously, in 16 of these games (84%), cooperative outcomes are nonmyopic equilibria (NMEs) when the players make farsighted calculations, based on backward induction; in the other three games, credible threats induce cooperation. More generally, in all finite normal-form games, if players’ preferences are strict, farsighted calculations stabilize at least one Pareto-optimal NME. We illustrate the choice of NMEs that are not NEs by two cases in international relations: (i) no first use of nuclear weapons, chosen by the protagonists in the 1962 Cuban missile crisis and since adopted by some nuclear powers; and (ii) the 2015 agreement between Iran, and a coalition of the United States and other countries, that has been abrogated by the United States but has forestalled Iran’s possible development of nuclear weapons.

Selecting efficient correlated equilibria through distributed learning

A learning rule is completely uncoupled if each player's behavior is conditioned only on his own realized payoffs, and does not need to know the actions or payoffs of anyone else. We demonstrate a simple, completely uncoupled learning rule such that, in any finite normal form game with generic payoffs, the players' realized strategies implements a socially optimal coarse correlated (Hannan) equilibrium a very high proportion of the time. That is, the empirical frequency associated with the players' collective behavior will be consistent with a socially optimal coarse correlated equilibrium. A variant of the rule implements a socially optimal correlated equilibrium a very high proportion of the time.

Epistemically Robust Strategy Subsets

We define a concept of epistemic robustness in the context of an epistemic model of a finite normal-form game where a player type corresponds to a belief over the profiles of opponent strategies and types. A Cartesian product X of pure-strategy subsets is epistemically robust if there is a Cartesian product Y of player type subsets with X as the associated set of best reply profiles such that the set Y i contains all player types that believe with sufficient probability that the others are of types in Y − i and play best replies. This robustness concept provides epistemic foundations for set-valued generalizations of strict Nash equilibrium, applicable also to games without strict Nash equilibria. We relate our concept to closedness under rational behavior and thus to strategic stability and to the best reply property and thus to rationalizability.

Equilibrium Notions for Agents with Cumulative Prospect Theory Preferences

Relying on the experimental findings that actual choice behavior often violates the axioms of expected utility theory (EUT), we study non-EUT preferences in a noncooperative game-theoretic framework. In particular, agents’ preferences are represented by the pair of functions suggested in cumulative prospect theory (CPT). Accordingly, three key aspects of CPT are incorporated: subjective probability weighting, loss aversion, and reference dependence. We introduce a correlated equilibrium and two mixed strategy equilibria for agents with CPT preferences. We prove the existence of equilibria for finite normal form games and investigate the sets of equilibria in some examples.

When Can Limited Randomness Be Used in Repeated Games?

The central result of classical game theory states that every finite normal form game has a Nash equilibrium, provided that players are allowed to use randomized (mixed) strategies. However, in practice, humans are known to be bad at generating sequences, and true random bits may be unavailable. Even if the players have access to enough random bits for a single instance of the game their randomness might be insufficient if the game is played many times. In this work, we ask whether randomness is necessary for equilibria to exist in finitely repeated games. We show that for a large class of games containing arbitrary two-player zero-sum games, approximate Nash equilibria of the n-stage repeated version of the game exist if and only if both players have Ω(n) random bits. In contrast, we show that there exists a class of games for which no equilibrium exists in pure strategies, yet the n-stage repeated version of the game has an exact Nash equilibrium in which each player uses only a constant number of random bits. When the players are assumed to be computationally bounded, if cryptographic pseudorandom generators (or, equivalently, one-way functions) exist, then the players can base their strategies on random-like sequences derived from only a small number of truly random bits. We show that, in contrast, in repeated two-player zero-sum games, if pseudorandom generators do not exist, then Ω(n) random bits remain necessary for equilibria to exist.

MIXED EQUILIBRIUM IN 2 × 2 NORMAL FORM GAMES: WHEN BURNING MONEY IS RATIONAL

We discuss the rationality of burning money behavior from a new perspective: the mixed Nash equilibrium in 2 × 2 normal form games. We support our argument analyzing the first-order derivatives of the players' mixed equilibrium expected utility payoffs with respect to their own utility payoffs. We establish necessary and sufficient conditions that guarantee the existence of negative derivatives. In particular, games with negative derivatives are the ones that create incentives for burning money behavior since such behavior in these games improves the player's mixed equilibrium expected utility payoff. We show that a negative derivative of some player i 's mixed equilibrium expected utility payoff occurs if, and only if, he has a strict preference for one of the strategies of the other player. Moreover, negative derivatives always occur when they are taken with respect to player i 's highest and lowest game utility payoffs. We also present sufficient conditions that ensure that such derivatives are always non-negative in finite normal form games.

Tenable Strategy Blocks and Settled Equilibria

When people interact in familiar settings, social conventions usually develop so that people tend to disregard alternatives outside the convention. For rational players to usually restrict attention to a block of conventional strategies, no player should prefer to deviate from the block when others are likely to act conventionally and rationally inside the block. We explore two set-valued concepts, coarsely and finely tenable blocks, that formalize this notion for finite normal-form games. We then identify settled equilibria, which are Nash equilibria with support in minimal tenable blocks. For a generic class of normal-form games, our coarse and fine concepts are equivalent, and yet they differ from standard solution concepts on open sets of games. We demonstrate the nature and power of the solutions by way of examples. Settled equilibria are closely related to persistent equilibria but are strictly more selective on an open set of games. With fine tenability, we obtain invariance under the insertion of a subgame with a unique totally mixed payoff-equivalent equilibrium, a property that other related concepts have not satisfied.

Mixed Strategy Equilibrium for Agents with Cumulative Prospect Theory Preferences

Relying on the experimental findings that actual choice behavior often violates the axioms of expected utility theory (EUT), we study non-EUT preferences in a non-cooperative game-theoretic framework. In particular, agents' preferences are represented by the pair of functions suggested in cumulative prospect theory (CPT). Accordingly, three key aspects of CPT are incorporated: subjective probability weighting, loss aversion, and reference dependence. We present two types of mixed strategy equilibrium for agents with CPT preferences and prove the existence of these equilibria for finite normal form games.

Correlated Equilibrium for Agents with Cumulative Prospect Theory Preferences

We study correlated equilibrium for finite normal form games in which agents’ preferences are represented by the pair of functions suggested in cumulative prospect theory (CPT). Accordingly, three key aspects of CPT are incorporated: subjective probability weighting, loss aversion, and reference dependence. We show the existence of equilibrium and investigate the set of equilibria under CPT preferences. JEL Classification: C72, D81

Lipschitz Games

The Lipschitz constant of a finite normal-form game is the maximal change in some player's payoff when a single opponent changes his strategy. We prove that games with small Lipschitz constant admit pure ϵ-equilibria, and pinpoint the maximal Lipschitz constant that is sufficient to imply existence of a pure ϵ-equilibrium as a function of the number of players in the game and the number of strategies of each player. Our proofs use the probabilistic method.

Existence of perfect equilibria: a direct proof

We provide a direct proof of the existence of perfect equilibria in finite normal form games and extensive games with perfect recall. It is done by constructing a correspondence whose fixed points are precisely the perfect equilibria of a given finite game. Existence of a fixed point is secured by a generalization of Kakutani theorem, which is proved in this paper. This work offers a new approach to perfect equilibria, which would hopefully facilitate further study on this topic. We also hope our direct proof would be the first step toward building an algorithm to find the set of all perfect equilibria of a strategic game.

Stochastic Evolution of Rules for Playing Finite Normal Form Games

The evolution of boundedly rational rules for playing normal form games is studied within stationary environments of stochastically changing games. Rules are viewed as algorithms prescribing strategies for the different normal form games that arise. It is shown that many of the “folk results” of evolutionary game theory, typically obtained with a fixed game and fixed strategies, carry over to the present environments. The results are also related to some recent experiments on rules and games.

Iterated weak dominance and subgame dominance

In this paper, we consider finite normal form games satisfying transference of decisionmaker indifference. We show that any set of strategies surviving k rounds of elimination of some weakly dominated strategies can be reduced to a set of strategies equivalent to the set of strategies surviving k rounds of elimination of all weakly dominated strategies in every round by (at most k) further rounds of elimination of weakly dominated strategies. The result develops work by Gretlein [Gretlein, R., 1983. Dominance elimination procedures on finite alternative games. International Journal of Game Theory 12, 107–113]. We then consider applications and demonstrate how we may obtain a unified approach to the work by Gretlein and recent results by Ewerhart [Ewerhart, C., 2002. Iterated weak dominance in strictly competitive games of perfect information. Journal of Economic Theory 107, 474-482] and Marx and Swinkels [Marx, L.M., Swinkels, J.M., 1997. Order independence for iterated weak dominance. Games and Economic Behavior 18, 219-245].

Individually rational pure strategies in large games

In “randomly selected” large finite normal form games, pure strategy “strongly individually rational” outcomes (all payoffs exceeding pure strategy minimax payoffs) exist with high probability. As a corollary, pure strategy Nash equilibria also exist with high probability in certain infinitely repeated games. Specifically, the pure strategy Folk Theorem, which can be true in a vacuous sense, has nontrivial substance with probability approaching one as pure strategy sets increase in cardinality, provided strategy set growth rates are not too different. To prove this, and noting a possible “bounded rationality” interpretation, we begin with a study of “approximate equilibria” in finite normal form games. Given a positive integer l, these are pure strategy outcomes where each player's payoff is among the l largest available given strategy choices of opponents. When pure strategy sets are large, such outcomes are highly probable and usually have payoffs dominating pure minimax payoffs.