Game Theory with Translucent Players

TL;DR

This paper introduces a framework for game theory with translucent players, where players believe their strategy changes can influence others, leading to more efficient outcomes and a characterization of rational strategies under counterfactual beliefs.

Contribution

It develops a new model for reasoning about games with translucent players and characterizes strategies consistent with counterfactual rationality beliefs.

Findings

Translucent players can achieve more efficient outcomes.

Strategies surviving iterated minimax domination are characterized by CCBR.

Counterfactual beliefs influence strategic stability and outcomes.

Abstract

A traditional assumption in game theory is that players are opaque to one another---if a player changes strategies, then this change in strategies does not affect the choice of other players' strategies. In many situations this is an unrealistic assumption. We develop a framework for reasoning about games where the players may be translucent to one another; in particular, a player may believe that if she were to change strategies, then the other player would also change strategies. Translucent players may achieve significantly more efficient outcomes than opaque ones. Our main result is a characterization of strategies consistent with appropriate analogues of common belief of rationality. Common Counterfactual Belief of Rationality (CCBR) holds if (1) everyone is rational, (2) everyone counterfactually believes that everyone else is rational (i.e., all players i believe that everyone else would still be rational even if $i$ were to switch strategies), (3) everyone counterfactually believes that everyone else is rational, and counterfactually believes that everyone else is rational, and so on. CCBR characterizes the set of strategies surviving iterated removal of minimax dominated strategies, where a strategy s for player i is minimax dominated by s' if the worst-case payoff for i using s' is better than the best possible payoff using s.