Equilibria for far-sighted players

A new equilibrium concept for non-cooperative games, based on the assumptions that players are rational and far-sighted, is examined. An outcome is extended non-myopically (XNM) stable for a player if that player is assured that no movecountermove sequence he could initiate by departing unilaterally from that outcome would benefit him. The extended non-myopic (XNM) equilibria of a game, the outcomes which are XNM stable for each player, therefore model permanent (enduring) equilibria in an ongoing conflict.Algorithms for the identification of XNM equilibria in a 2 × 2 game are presented. The XNM concepts are then applied to three special classes of games (no-conflict games, games of complete opposition, and strict ordinal games) to compare their predictions of long-term stability with the known properties of games in these classes.Research supported by Natural Sciences and Engineering Research Council of Canada Grant No. A8974.     

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.      

LEXICOGRAPHIC ADDITIVITY FOR MULTI-ATTRIBUTE PREFERENCES ON FINITE SETS

Payoff dominance, a criterion for choosing between equilibrium points in games, is intuitively compelling, especially in matching games and other games of common interests, but it has not been justified from standard game-theoretic rationality assumptions. A psychological explanation of it is offered in terms of a form of reasoning that we call the Stackelberg heuristic in which players assume that their strategic thinking will be anticipated by their co-player(s). Two-person games are called Stackelberg-soluble if the players' strategies that maximize against their co-players' best replies intersect in a Nash equilibrium. Proofs are given that every game of common interests is Stackelberg-soluble, that a Stackelberg solution is always a payoff-dominant outcome, and that in every game with multiple Nash equilibria a Stackelberg solution is a payoff-dominant equilibrium point. It is argued that the Stackelberg heuristic may be justified by evidentialist reasoning.     

Belief system foundations of backward induction

Two justifications of backward induction (BI) in generic perfect information games are formulated using Bonanno's (1992; Theory and Decision 33, 153) belief systems. The first justification concerns the BI strategy profile and is based on selecting a set of rational belief systems from which players have to choose their belief functions. The second justification concerns the BI path of play and is based on a sequential deletion of nodes that are inconsistent with the choice of rational belief functions.     

Staying power in sequential games

Staying power is the ability of a player to hold off choosing a strategy in a two-person game until the other player has selected his, after which the players are assumed to be able to move and countermove sequentially to ensure their best possible outcomes before the process cycles back to the initial outcome and then repeats itself (rational termination). These rules of sequential play induce a determinate, Paretosuperior outcome in all two-person, finite, sequential games in which the preferences of the players are strict.In 57 of the 78 distinct 2 × 2 ordinal games (73 percent), it makes no difference who the (second-moving) player with staying power is, but in the other 21 games the outcome is power-dependent. In all but one of these games, staying power benefits the player who possesses it.If no player has staying power, the outcomes that result from sequential play and rational termination are called terminal; they coincide with staying power outcomes if they are Pareto-superior. Normative implications of the analysis for rationally justifying cooperation in such games as Prisoners' Dilemma and Chicken, and implementing Pareto-superior outcomes generally, are also discussed.We are grateful to D. Marc Kilgour for very valuable comments on an earlier version of this paper, causing us to rethink and redefine staying power. The earlier version was presented at the Seventeenth North American Conference, Peace Science Society (International), University of Pennsylvania, November 9–11, 1981.     

EVOLUTION AND ULTIMATUM BARGAINING

Empirical research has discovered that experimental subjects in ultimatum bargaining situations generally fail to play the decision-theoretic optimum strategy, and instead play something between that strategy and a fair split. In evolutionary dynamics, fair division and nearly fair division strategies often go to fixation and weakly dominated strategies can do quite well. Computer simulations were done using three different ultimatum bargaining games as determinates of fitness. (1) No tendency toward the elimination of weakly dominated strategies was observed, with or without mutation. (2) Strategies making fair demands had sizable basins of attraction. (3) In a system where five different demands can be made, demands closest to (approximately) 91% had the largest basins of attraction. (4) If the strategies have thresholds for acceptable demands, rather than individuated responses to each demand, the apparent optimum demand may be quite low: 64% for one set of trials.     

Endogenizing the order of moves in matrix games

Players often have flexibility in when they move and thus whether a game is played simultaneously or sequentially may be endogenously determined. For 2 × 2 games, we analyze this using an extended game. In a stage prior to actual play, players choose in which of two periods to move. A player moving at the first opportunity knows when his opponent will move. A player moving at the second turn learns the first mover's action. If both select the same turn, they play a simultaneous move subgame.If both players have dominant strategies in the basic game, equilibrium payoffs in the basic and extended games are identical. If only one player has a dominant strategy or if the unique equilibrium in the basic game is in mixed strategies, then the extended game equilibrium payoffs differ if and only if some pair of pure strategies Pareto dominates the basic game simultaneous play payoffs. If so, sequential play attains the Pareto dominating payoffs. The mixed strategy equilibrium occurs only when it is not Pareto dominated by some pair of pure strategies.In an alternative extended game, players cannot observe delay by opponents at the first turn. Results for 2×2 games are essentially the same as with observable delay, differing only when only one player has a dominant strategy.     

Outsourcing with identical suppliers and shortest-first policy: a laboratory experiment

We study experimentally in the laboratory two 2-player games that mimic a decentralized decision-making situation in which firms repeatedly outsource production orders to multiple identical suppliers. The first game has a unique (inefficient) equilibrium in mixed strategies, while the second game has two (efficient) equilibria in pure strategies and an infinite number of (inefficient) equilibria in mixed strategies. In both games, the optimal social costs can also be obtained via dominated strategies. We find that only in the second game subjects manage to reach an efficient outcome more often when matched in fixed pairs than when randomly rematched each round. Surprisingly, this is because subjects coordinate on dominated strategies (and not an efficient pure strategy equilibrium). We show theoretically that preferences for efficiency cannot explain our experimental results. Inequality aversion, on the other hand, cannot be rejected.     

On Hobbes’s state of nature and game theory

Hobbes’s state of nature is often analyzed in two-person two-action non-cooperative games. By definition, this literature only focuses on duels. Yet, if we consider general games, i.e., with more than two agents, analyzing Hobbes’s state of nature in terms of duel is not completely satisfactory, since it is a very specific interpretation of the war of all against all. Therefore, we propose a definition of the state of nature for games with an arbitrary number of players. We show that this definition coincides with the strategy profile considered as the state of nature in two-person games. Furthermore, we study what we call rational states of nature (that is, strategy profiles which are both states of nature and Nash equilibria). We show that in rational states of nature, the utility level of any agent is equal to his maximin payoff. We also show that rational states of nature always exist in inessential games. Finally, we prove the existence of states of nature in a class of (not necessarily inessential) symmetric games.     

No Switchbacks: Rethinking Aspiration-Based Dynamics in the Ultimatum Game

Aspiration-based evolutionary dynamics have recently been used to model the evolution of fair play in the ultimatum game showing that incredible threats to reject low offers persist in equilibrium. We focus on two extensions of this analysis: we experimentally test whether assumptions about agent motivations (aspiration levels) and the structure of the game (binary strategy space) reflect actual play, and we examine the problematic assumption embedded in the standard replicator dynamic that unhappy agents who switch strategies may return to a rejected strategy without exploring other options. We find that the resulting “no switchback” dynamic predicts the evolution of play better than the standard dynamic and that aspirations are a significant motivator for our participants. In the process, we also construct and analyze a variant of the ultimatum game in which players can adopt conditional (on their induced aspirations) strategies.     

Two-Speed Evolution of Strategies and Preferences In Symmetric Games

Agents in a large population are randomly matched to play a certain game, payoffs in which represent fitness. Agents may have preferences that are different from fitness. They learn strategies according to their preferences, and evolution changes the preference distribution in the population according to fitness. When agents know the preferences of the opponent in a match, only efficient symmetric strategy profiles of the fitness game can be stable. When agents do not know the preferences of the opponent, only Nash equilibria of the fitness game can be stable. For 2 × 2 symmetric games I characterize preferences that are stable.Jel Codes: C72, A13     

When learning meets salience

Behavior in one-shot coordination games with common knowledge labels can be described by theories of salience and focal points. Behavior in repeated games, including coordination games, can be explained by theories of learning. This paper considers games in which both theories apply, repeated coordination games with common knowledge labels. The research question asks how players combine the two sources of information—salience and the history of play—when making their choices. We specifically ask whether salience, normally considered as a one-shot strategy, continues to influence players’ actions beyond the first round, even while the player might learn from the history of play. We explore two possible mechanisms for such a continuing effect of salience: via an influence on prior beliefs, and/or via a bias, given beliefs. Regression analysis of individual-level choices shows that salience, normally considered only in the context of one-shot games, does exert a lasting effect, with the precise mechanism depending on the details of the game.     

Ellsberg games

In the standard formulation of game theory, agents use mixed strategies in the form of objective and probabilistically precise devices to conceal their actions. We introduce the larger set of probabilistically imprecise devices and study the consequences for the basic results on normal form games. While Nash equilibria remain equilibria in the extended game, there arise new Ellsberg equilibria with distinct outcomes, as we illustrate by negotiation games with three players. We characterize Ellsberg equilibria in two-person conflict and coordination games. These equilibria turn out to be related to experimental deviations from Nash equilibrium play.     

Focal points in pure coordination games: An experimental investigation

This paper reports an experimental investigation of the hypothesis that in coordination games, players draw on shared concepts of salience to identify focal points on which they can coordinate. The experiment involves games in which equilibria can be distinguished from one another only in terms of the way strategies are labelled. The games are designed to test a number of specific hypotheses about the determinants of salience. These hypotheses are generally confirmed by the results of the experiment.     

Endogenous correlated equilibria in noncooperative games

Most of the results of modern game theory presuppose that the choices rational agents make in noncooperative games are probabilistically independent. In this paper I argue that there is noa priori reason for rational agents to assume probabilistic independence. I introduce a solution concept for noncooperative games called anendogenous correlated equilibrium, which generalizes the Nash equilibrium concept by dropping probabilistic independence. I contrast the endogenous correlated equilibrium with the correlated equilibrium defined by Aumann (1974, 1987). I conclude that in general the endogenous correlated equilibrium concept is a more appropriate solution concept for noncooperative game theory than the less general Nash equilibrium concept. I close by discussing the relationship between endogenous correlated equilibrium and the game solution concept calledrationalizability introduced by Bernheim (1984) and Pearce (1984).     

On Loss Aversion in Bimatrix Games

In this article three different types of loss aversion equilibria in bimatrix games are studied. Loss aversion equilibria are Nash equilibria of games where players are loss averse and where the reference points—points below which they consider payoffs to be losses—are endogenous to the equilibrium calculation. The first type is the fixed point loss aversion equilibrium, introduced in Shalev (2000; Int. J. Game Theory 29(2):269) under the name of ‘myopic loss aversion equilibrium.’ There, the players’ reference points depend on the beliefs about their opponents’ strategies. The second type, the maximin loss aversion equilibrium, differs from the fixed point loss aversion equilibrium in that the reference points are only based on the carriers of the strategies, not on the exact probabilities. In the third type, the safety level loss aversion equilibrium, the reference points depend on the values of the own payoff matrices. Finally, a comparative statics analysis is carried out of all three equilibrium concepts in 2 × 2 bimatrix games. It is established when a player benefits from his opponent falsely believing that he is loss averse.     

Dynamic focal points in N-person coordination games

To understand how groups coordinate, we study infinitely repeated N-player coordination games in the context of strategic uncertainty. In a situation where players share no common language or culture, ambiguity is always present. However, finding an adequate principle for a common language is not easy: a tradeoff between simplicity and efficiency has to be made. All these points are illustrated on repeated N-player coordination games on m loci. In particular, we demonstrate how a common principle can accelerate coordination. We present very simple rules that are optimal in the space of all languages for m (number of coordination loci) from 2 to 5 and for all N, the number of players. We also show that when more memory is used in the language (strategies), players may not coordinate, whereas this is never the case when players remember only the previous period.     

The Backward Induction Argument

The backward induction argument purports to show that rational and suitably informed players will defect throughout a finite sequence of prisoner's dilemmas. It is supposed to be a useful argument for predicting how rational players will behave in a variety of interesting decision situations. Here, I lay out a set of assumptions defining a class of finite sequences of prisoner's dilemmas. Given these assumptions, I suggest how it might appear that backward induction succeeds and why it is actually fallacious. Then, I go on to consider the consequences of adopting a stronger set of assumptions. Focusing my attention on stronger sets that, like the original, obey the informedness condition, I show that any supplementation of the original set that preserves informedness does so at the expense of forcing rational participants in prisoner's dilemma situations to have unexpected beliefs, ones that threaten the usefulness of backward induction.