Limited-trust equilibria

We propose and analyze a new type of equilibrium, in which limited-trust exists between players with long-term interactions. We assume heterogeneous interactions: players will engage in several games over an undetermined period of time with payoffs for each game drawn from a distribution. As such, players may not engage in the same game more than once. We define a Limited-Trust equilibrium to address these heterogeneous games, show its existence in all finite simultaneous games, and analyze it in general and in several common classes of games. We provide several interpretations of this equilibrium in leader-follower games. We then numerically compare the social utility generated from these equilibria in both simultaneous and leader-follower games to that generated by Nash and Stackelberg equilibria in the same games: when players display a similar level of trust δ, each sees an average gain of approximately δ in its utility each game over what it would achieve in traditional competitive/rational games, meaning for each game a player loses δ, there is another game it gains 3δ. Thus while players appear to play “non-rationally” by giving something up, they actually gain more and are each able to come out ahead of what they would have received if playing rationally as in a Nash equilibrium.