Stag Hunt with unknown outside options

European Journal of Operational Research, 1999

… of the Royal …, 2007

Voluntary participation in public goods games (PGGs) has turned out to be a simple but effective mechanism for promoting cooperation under full anonymity. Voluntary participation allows individuals to adopt a riskaversion strategy, termed loner. A loner refuses to participate in unpromising public enterprises and instead relies on a small but fixed pay-off. This system leads to a cyclic dominance of three pure strategies, cooperators, defectors and loners, but at the same time, there remain two considerable restrictions: the addition of loners cannot stabilize the dynamics and the time average pay-off for each strategy remains equal to the pay-off of loners. Here, we introduce probabilistic participation in PGGs from the standpoint of diversification of risk, namely simple mixed strategies with loners, and prove the existence of a dynamical regime in which the restrictions no longer hold. Considering two kinds of mixed strategies associated with participants (cooperators or defectors) and non-participants (loners), we can recover all basic evolutionary dynamics of the two strategies: dominance; coexistence; bistability; and neutrality, as special cases depending on pairs of probabilities. Of special interest is that the expected pay-off of each mixed strategy exceeds the pay-off of loners at some interior equilibrium in the coexistence region.

arXiv (Cornell University), 2012

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. Each player can have different sets of strategies for the different basic games. 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, and, in particular, we submit that MG can be used to model investment in a global economy with different national or continental markets. 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 show that, with respect to the SG we choose, the payoffs for PD give rise to two different types of DG's. 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.

Theoretical Population Biology

Many mathematical frameworks of evolutionary game dynamics assume that the total population size is constant and that selection affects only the relative frequency of strategies. Here, we consider evolutionary game dynamics in an extended Wright-Fisher process with variable population size. In such a scenario, it is possible that the entire population becomes extinct. Survival of the population may depend on which strategy prevails in the game dynamics. Studying cooperative dilemmas, it is a natural feature of such a model that cooperators enable survival, while defectors drive extinction. Although defectors are favored for any mixed population, random drift could lead to their elimination and the resulting pure-cooperator population could survive. On the other hand, if the defectors remain, then the population will quickly go extinct because the frequency of cooperators steadily declines and defectors alone cannot survive. In a mutation-selection model, we find that (i) a steady supply of cooperators can enable long-term population survival, provided selection is sufficiently strong, and (ii) selection can increase the abundance of cooperators but reduce their relative frequency. Thus, evolutionary game dynamics in populations with variable size generate a multifaceted notion of what constitutes a trait's long-term success.

International Journal of Game Theory, 2006

Two-person noncooperative games with finitely many pure strategies and ordinal preferences over pure outcomes are considered, in which probability distributions resulting from mixed strategies are evaluated according to t-degree stochastic dominance. A t-best reply is a strategy that induces a t-degree stochastically undominated distribution, and a t-equilibrium is a pair of t-best replies. The paper provides a characterization and existence proofs of t-equilibria in terms of representing utility functions, and shows that for t becoming large-which can be interpreted as the players becoming more risk averse-behavior converges to a specific form of max-min play. More precisely, this means that in the limit each player puts all weight on a strategy that maximizes the worst outcome for the opponent, within the supports of the strategies in the limiting sequence of t-equilibria.

Evolution, 2011

not volunteer can benefit from the public good produced by the contributions of others. Therefore it is generally believed that public goods can be produced only in the presence of repeated interactions (which allow reciprocation, reputation effects and punishment) or relatedness (kin selection). Cooperation, however, often occurs in the absence of iterations and relatedness. We show that when the production of a public good is a Volunteer's Dilemma, in which a fixed number of cooperators is necessary to produce the public good, cooperators and defectors persist in a mixed equilibrium, without iterations and without relatedness.

Studies in Economic Theory, 1991

Shafer-Sonnenschein [23] or Yannelis-Prabhakar [24] instead of assigning each player a random payoff or utility function, we equip each player with a random preference correspondence which need not be representable by a random utility function. It should be noted, however, that our random equilibrium results, provide as corollaries random versions of the theorems of Nash, Fan, and Browder. Moreover, we show that these random equilibrium theorems can be used to obtain equilibrium existence results for games with incomplete information, and in particular, for Bayesian games. The main reference for the latter type of games appears to be Harsanyi's [10] seminal paper. Recently there is a growing literature on this subject. In particular, Bayesian games have found very interesting applications in economic theory, e.g., Aumann [2], Myerson [17], Palfrey-Srivastava [19, 20], Peck-Shell [21] and Postlewaite-Schmeidler 2 [22] among others.

Journal of Mathematical Economics, 1999

We present an example of a two-person game of private information in which there is no equilibrium in pure strategies. Our example satisfies all the hypotheses of the existence theorems present in the literature on the subject of pure strategy equilibria, except for the w x fact that the action set of each player is given by the interval y1,1. As such, it illustrates the limitations that pertain to the purification of equilibria in a standard setting.

The research question of this Mathematics Extended Essay is “How can designing an iterative tournament of the Prisoner’s Dilemma to determine the most effective strategy teach us about the value of cooperation and defection?” on the topic of the application of Game Theory. I have decided to use the context of the applicability of mathematics to solve both real and abstract problems, as this would enlighten me in the idea of decision-making in a mathematical sense, whether it would be beneficial to aim for cooperating to a mutual benefit or opting for a solo victory.

Proceedings of the fourth international joint conference on Autonomous agents and multiagent systems, 2005