[Paper Review] Multi-games and a double game extension of the Prisoner's Dilemma

Abbas Edalat, Ali Ghoroghi+1 more|arXiv (Cornell University)|May 22, 2012

Evolutionary Game Theory and Cooperation8 references3 citations

TL;DR

This paper introduces Multi-Games (MG), a framework where players simultaneously engage in multiple basic games with weighted investments, enabling modeling of strategic resource allocation across different markets. It proposes a Double Game (DG) extension of the Prisoner's Dilemma with a Social Game to model prosocial behavior, showing that for completely pure regular DGs, Bayesian Nash equilibria can be computed in linear time relative to the number of types.

ABSTRACT

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.

Motivation & Objective

• To model strategic resource allocation across multiple economic environments using a new class of games called Multi-Games (MG).
• To extend the Prisoner's Dilemma with a Social Game to capture prosocial behavior through variable social coefficients.
• To develop a computationally efficient method for identifying Bayesian Nash equilibria in a subclass of MGs, specifically completely pure regular Double Games.
• To evaluate the performance of adaptive strategies in repeated Double Games through a round-robin tournament framework.

Proposed method

• Define Multi-Games (MG) as a convex combination of payoffs from multiple basic games, weighted by players' investment proportions.
• Introduce Double Games (DG) as a two-player MG where each player has the same strategy set in both basic games and private weights (types) representing investment or social preference.
• Formulate the DG as a Bayesian game when weights are private information with known conditional probability distributions.
• Establish the concept of 'completely pure regular' DGs, where each player's optimal strategy depends only on its own type, enabling linear-time verification of equilibrium conditions.
• Derive a test for complete purity and regularity in linear time relative to the number of types, directly yielding a Bayesian Nash equilibrium.
• Design and simulate a round-robin tournament using a novel strategy (SEG) that dynamically adjusts social coefficients based on Nash equilibria of the $(\lambda,\gamma)$ diagram.

Experimental results

Research questions

• RQ1 Can a class of games be constructed where players allocate resources across multiple simultaneous games with distinct strategy sets and payoff structures?
• RQ2 Under what conditions can Bayesian Nash equilibria in such games be computed efficiently, specifically in linear time?
• RQ3 How can prosocial behavior in repeated interactions be modeled through a second game layer (Social Game) that captures altruism or social preferences?
• RQ4 What strategies emerge as effective in repeated Double Games when players adapt their social coefficients based on opponents' behavior?
• RQ5 Can a strategy be designed that dynamically leverages Nash equilibria of the type-space to achieve robust cooperation and retaliation?

Key findings

• For the class of completely pure regular Double Games with finite types, a Bayesian Nash equilibrium can be computed in linear time with respect to the number of types.
• The proposed test for complete purity and regularity is both necessary and sufficient and can be executed in linear time, enabling efficient equilibrium computation.
• In a repeated Double Game tournament, the SEG strategy—based on dynamic adjustment of social coefficients using Nash equilibrium predictions—emerged as highly effective, balancing cooperation, retaliation, and exploitation.
• SEG successfully exploited non-retaliating strategies like ALLC by initially defecting and later cooperating when retaliation was absent, maximizing payoff without long-term loss.
• SEG demonstrated robustness by quickly shifting from defection to cooperation when facing reciprocal strategies like TFT, achieving mutual cooperation after initial defection.
• The framework enables a convex combination of material and social payoffs at the end of a game sequence, allowing societal-level weighting of outcomes, enhancing realism in modeling human behavior.