Multi-games and a double game extension of the Prisoner's Dilemma

We propose a new class of games, called Multi-Games (MG), in which a given number of players play a fixed number of basic games simultaneously. In each round of the MG, each player will have a specific set of weights, one for each basic game, which add up to one and represent the fraction of the player's investment in each basic game. The total payoff for each player is then the convex combination, with the corresponding weights, of the payoffs it obtains in the basic games. The basic games in a MG can be regarded as different environments for the players. When the players' weights for the different games in MG are private information or types with given conditional probability distributions, we obtain a particular class of Bayesian games. We show that for the class of so-called completely pure regular Double Game (DG) with finite sets of types, the Nash equilibria (NE) of the basic games can be used to compute a Bayesian Nash equilibrium of the DG in linear time with respect to the number of types of the players. We study a DG for the Prisoner's Dilemma (PD) by extending the PD with a second so-called Social Game (SG), generalising the notion of altruistic extension of a game in which players have different altruistic levels (or social coefficients). We study two different examples of Bayesian games in this context in which the social coefficients have a finite set of values and each player only knows the probability distribution of the opponent's social coefficient. In the first case we have a completely pure regular DG for which we deduce a Bayesian NE. Finally, we use the second example to compare various strategies in a round-robin tournament of the DG for PD, in which the players can change their social coefficients incrementally from one round to the next.