Abstract
In nature and society, problems that arise when different interests are difficult to reconcile are modeled in game theory. While most applications assume that the players make decisions based only on the payoff matrix, a more detailed modeling is necessary if we also want to consider the influence of correlations on the decisions of the players. We therefore extend here the existing framework of correlated strategies by giving the players the freedom to respond to the instructions of the correlation device by probabilistically following or not following its suggestions. This creates a new type of games that we call “correlated games”. The associated response strategies that can solve these games turn out to have a rich structure of Nash equilibria that goes beyond the correlated equilibrium and pure or mixed-strategy solutions and also gives better payoffs in certain cases. We here determine these Nash equilibria for all possible correlated Snowdrift games and we find these solutions to be describable by Ising models in thermal equilibrium. We believe that our approach paves the way to a study of correlations in games that uncovers the existence of interesting underlying interaction mechanisms, without compromising the independence of the players.
Highlights
Game theory[1] has been used as a powerful tool to model problems from diverse research areas
Is it possible for each player to obtain a better payoff than in correlated equilibrium by having a different strategy over the correlations, other than always complying with the instructions from the correlation device, and that is in a Nash equilibrium? Secondly, in case the initially agreed upon or externally imposed correlations do not allow for a correlated equilibrium, is it still possible for the players to use the correlation device to find a better solution other than returning to the case without correlations, by ignoring the correlation device and using the pure or mixed strategies solutions of the game? To answer these questions, we introduce a new kind of strategy, which consists of a probability distribution over following or not following the correlation device
With the ultimate goal to further advance the application of games to real-life situations and to explain group behavior from the pair-wise interactions, we focus in this paper on two-player games that are described by a payoff matrix and a correlation device, both of which are specified beforehand and remain unchanged during the game
Summary
Game theory[1] has been used as a powerful tool to model problems from diverse research areas. When we are dealing with a system of only two players that are solely informed by a payoff matrix the solutions are very well understood, but when several players are involved it becomes, in spite of that, hard to predict the final equilibrium only from the pair-wise interactions[4] This has found many applications in biology[5], economics[6,7,8], politics[9], and social sciences[10,11]. If we consider that the players play a Snowdrift game instead, the best individual payoff can sometimes be to cooperate This explains the occurrence of cooperation as part of a mixed strategy solution[2], but explaining how it becomes an emergent behavior for a large number of players is still complicated[21].
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