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We propose a new class of games, called Multi-Games (MG), in which a givennumber of players play a fixed number of basic games simultaneously. In eachround of the MG, each player will have a specific set of weights, one for eachbasic game, which add up to one and represent the fraction of the player'sinvestment in each basic game. The total payoff for each player is then theconvex combination, with the corresponding weights, of the payoffs it obtainsin the basic games. The basic games in a MG can be regarded as differentenvironments for the players. When the players' weights for the different gamesin MG are private information or types with given conditional probabilitydistributions, we obtain a particular class of Bayesian games. We show that forthe class of so-called completely pure regular Double Game (DG) with finitesets of types, the Nash equilibria (NE) of the basic games can be used tocompute a Bayesian Nash equilibrium of the DG in linear time with respect tothe 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), generalisingthe notion of altruistic extension of a game in which players have differentaltruistic levels (or social coefficients). We study two different examples ofBayesian games in this context in which the social coefficients have a finiteset of values and each player only knows the probability distribution of theopponent's social coefficient. In the first case we have a completely pureregular DG for which we deduce a Bayesian NE. Finally, we use the secondexample to compare various strategies in a round-robin tournament of the DG forPD, in which the players can change their social coefficients incrementallyfrom one round to the next.

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