Limited-move equilibria In 2 × 2 games

Summary and Conclusion

This article introduces the concept of a limited-move equilibrium, an outcome that is stable when players are able to make more than the minimum (i.e., one) but less than the maximum (i.e., four) number of logically possible moves and countermoves in a 2 × 2 game. It defines two types of limitedmove equilibria and their associated decision rules, a Type II equilibrium where the total number of strategy switches permitted in a game is two (Rule II), and a Type III equilibrium where three total moves and countermoves (Rule III) from an initial outcome or status quo point are possible. When games that contain a limited-move equilibrium were compared to games that contain a Nash (Type I) or nonmyopic (Type IV) equilibrium, the 78 distinct 2 × 2 ordinal games identified by Rapoport and Guyer were shown to fall into three mutually exclusive categories:

• 1. Rule-neutral games (29 games — 37%). In this class of games, a single outcome in each game is rendered stable regardless of the number of moves and countermoves assumed to be available to the players.

• 2. Rule-dependent games (34 games — 44%). This is the largest of the three categories. Within this class are the games in which all decision rules associated with a stable outcome are associated with the same outcome. The congruence of decision rules and stable outcomes range from the case in which all decision rules except one (i.e., Rule IV) induce the same equilibrium outcome to the case in which a game has an equilibrium only when one of the four decision rules is assumed.

• 3. Rule-variable games (15 games — 19%). In this category, different decision rules imply different stable outcomes. In rule-variable games, then, the impact of the operative decision rule is potentially the most significant, especially since players in these games are not indifferent towards which of the various outcomes emerge in a stable state. Consequently, an important dimension of conflict in these games might very well concern adjudication of the rules governing the ability of the players to make moves and countermoves.

Perhaps the most interesting observation that can be gleaned from this analysis is that at least one outcome in each of the 78 distinct 2 × 2 games is stable under one of the four postulated decision rules. Consequently, there is an equilibrium of some type in every strict ordinal 2 × 2 game. In nine of these games — six rule-dependent games, and three rule-variable games — only limited-move equilibria exist. Moreover, there are four games without a Type IV equilibrium that have a limited-move equilibrium different from a unique Type I equilibrium.

In general, limitations on the ability of players to make moves and countermoves tend to foster stability rather than destroy it in 2 × 2 games. Of the 78 games of this type, 69(88%) contain one or more Type I and Type II equilibria, 57(73%) have Type III equilibria, while only 37(47%) have outcomes that exhibit Type IV or nonmyopic stability. Environmental limitations on the number of moves and countermoves players can make, then, may help explain much of the stability observed in ongoing, real life games that do not contain Type IV equilibria.

In this context, it is worth pointing out that one of the underlying premises of this work has been that the concept of a limited-move equilibrium constitutes a useful and necessary adjunct to the concept of a nonmyopic equilibrium and of the dynamic conception of a game it implies, especially for empirical studies where ideosyncratic features of real life games might enter into the calculus of a player considering the long-term consequences of a strategy change. For while it is probably true that the assumption, defining a Nash equilibrium, that players are able to make only a single unilateral deviation from an outcome, is overly restrictive and not representative of the fluid nature of many actual conflict-of-interest situations, it is also probably true that the assumption, defining a nonmyopic equilibrium, that players can make an unlimited number of moves and countermoves, is frequently not satisfied in many empirical settings. Consequently, the two intermediate equilibrium concepts identified in this essay should provide a firmer theoretical base for, and enhance the explanatory power of, applications of simple game-theoretic models to a world characterized by a wide variety of actual decision rules.