(Strategic) Equilibrium

Profile of plans leading to an outcome of a game that is stable in the sense that given the other players adhere to the equilibrium prescription, no single player wants to deviate from the prescription. Any outcome that is reached by the play of strategies which do not form an equilibrium is an implausible way of playing the game, because at least one player could improve by selecting another strategy.

The concept of strategic equilibrium is completely unrelated to (Pareto) efficiency. Correspondigly, infinitely many games have (only) inefficient strategic equilibria; for a striking example, see the Prisoners' Dilemma game. As a strategic equilibrium is a profile of strategies that is unilaterally unimprovable given that all (other) players conform to their equilibrium strategies, the concept is weak and very general, but on the other hand most games possess several strategic equilibria. One of the major achievements of game theory accordingly has been the refinement of the concept of strategic equilibrium to allow for sharper predictions.

Two major achievements in refining the concept of equilibrium center around the 'time consistency' of strategically stable plans for sequential games, and on making precise the role of the players' beliefs about other players' plans of actions and information. A more general definition of strategic equilibrium is the following: an equilibrium is a profile of strategies and a profile of beliefs such that given the beliefs, the strategies are unilaterally unimprovable given equilibrium behavior, and such that the beliefs are consistent with the actual courses of action prescribed by the equilibrium strategies.

Nash Equilibrium: A profile of strategies such that given the other players conform to the (hypothesized) equilibrium strategies, no player has an incentive to unilaterally deviate from his (hypothesized) equilibrium strategy. The self-reference in this definition can be made more explicit by saying that a Nash equilibrium is a profile of strategies that form 'best responses' to one another, or a profile of strategies which are 'optimal reactions' to 'optimal reactions'. Nash equilibrium is the pure form of the basic concept of strategic equilibrium; as such, it is useful mainly in normal form games with complete information. When allowing for randomized strategies, at least one Nash equilibrium exists in any game(unless the players' payoff functions are irregular); for an example, see the game of matching pennies in the entry on game theory. Typically, a game possess several Nash equilibria, and the number of these is odd.

Refinement: Either a sharpening of the concept of strategic (or, Nash) equilibrium, or another criterion to discard implausible and to select plausible equilibria when a game exhibits multiple equilibria. For example, symmetric or Pareto efficient equilibria may more plausibly be played by the players in favor of asymmetric or inefficient equilibria. Likewise, equilibrium outcomes that are 'focal' in the cultural and psychological context in which the game is played might be more plausible than those which lack such salient features. Preferring symmetric outcomes in many games leads to the selection of an equilibrium in mixed strategies. In the following, we give an idea of the basic modifications of Nash equilibrium in more complex games.

Subgame perfect equilibrium: In extensive-form games with complete information, many strategy profiles that form best responses to one another imply incredible threats or promises that a player actually does not want to carry out anymore once he must face an (unexpected) off-equilibrium move of an opponent. If the profile of strategies is such that no player wants to amend his strategy whatever decision node can be reached during the play of the game, an equilibrium profile of strategies is called subgame perfect. In this sense, a subgame-perfect strategy profile is 'time consistent' in that it remains an equilibrium in whatever truncation of the original game (subgame) the players may find themselves.

Bayes-Nash equilibrium: In normal form games of incomplete information, the players have no possibility to update their prior beliefs about their opponents payoff-relevant characteristics, called their types. All that a player knows, except from the game itself (and the priors), is his own type, and the fact that the other players do not know his own type as well. As their best responses, however, depend on the players' actual types, a player must see himself through his opponents' eyes and plan an optimal reaction against the possible strategies of his opponents for each potential type of his own. Thus, a strategy in a Bayesian game of incomplete information must map each possible type of each player into a plan of actions. Then, since the other players' types are unknown, each player forms a best response against the expected strategy of each opponent, where he averages over the (well-specified) reactions of all possible types of an opponent, using his prior probability measure on the type space. Such a profile of type-dependent strategies which are unilaterally unimprovable in expectations over the competing types' strategies forms a Bayes Nash equilibrium. Basically, a Bayes Nash equilibrium is thus a Nash equilibrium 'at the interim stage' where each player selects a best response against the average best responses of the competing players.

Perfect Bayesian Nash equilibrium: Parallel to the extension of Nash equilibrium to subgame perfect equilibrium in games of complete information, the concept of Bayesian Nash equilibrium loses much of its bite in extensive form games and is accordingly refined to 'Perfect Bayesian' equilibrium. In a sequential game, it is often the threats about certain reactions 'off the equilibrium path' that force the players' actions to be best responses to one another 'onto the equilibrium path'. In sequential games with incomplete information, where the players hold beliefs about their opponents' types and optimize given their beliefs, a player then effectively 'threatens by the beliefs' he holds about his opponents' types after moves that deviate from the equilibrium path. Different beliefs about other players' types after deviations typically yield different reactions, some of which force the players back on the (candidate) equilibrium path, some of which lead them even farer away. In the first case, the plans of actions are confirmed by the beliefs about them, and the crucial self-confirming property of equilibrium beliefs and equilibrium strategies is met. The concept of Perfect Bayesian equilibrium makes precise this self-confirming 'interaction' of beliefs about types selecting certain actions and their 'actual' strategies. First, it requires that players forms a complete system of beliefs about the opponents' types at each decision node that can be reached. Next, this system of beliefs is updated according to Bayes' rule whenever possible (in particular, 'along the equilibrium path'), and finally, given each player's system of beliefs, the strategies from best responses to one another in the sense of ordinary Bayesian Nash equilibrium. A Bayesian equilibrium thus is a profile of complete strategies and a profile of complete beliefs such that (i) given the beliefs, the strategies are unilaterally unimprovable at each potential decision node that might be reached, and such that (ii) the beliefs are consistent with the actual evolution of play as prescribed by the equilibrium strategies.

Sequential equilibrium: Kind of refinement of Perfect Bayesian Equilibrium that puts sharper requirements on the beliefs which cannot be formed by Bayes' rule, but which are hold after moves off the equilibrium path. These beliefs have to be formed in a 'continuous' way from the information available in the extensive form of the game. Further refinements of Perfect Bayesian equilibrium restrict the players' beliefs about moves off the equilibrium path to the set of those types only for which the observed off-equilibrium move could have been worthwhile at all.

See also: equilibrium (in economics), competitive market equilibrium

Entry by: Jan Vleugels

December 1, 1997
Direct questions and comments to: Glossary master