Chris Cannings: A Life in Games

Chris Cannings was one of the pioneers of evolutionary game theory. His early work was inspired by the formulations of John Maynard Smith, Geoff Parker and Geoff Price; Chris recognized the need for a strong mathematical foundation both to validate stated results and to give a basis for extensions of the models. He was responsible for fundamental results on matrix games, as well as much of the theory of the important war of attrition game, patterns of evolutionarily stable strategies, multiplayer games and games on networks. In this paper we describe his work, key insights and their influence on research by others in this increasingly important field. Chris made substantial contributions to other areas such as population genetics and segregation analysis, but it was to games that he always returned. This review is written by three of his students from different stages of his career

Persistence of power: Repeated multilateral bargaining with endogenous agenda setting authority

We extend a simple repeated, multilateral bargaining model to allow successful agenda setters to hold on to power as long as they maintain the support of a majority of other committee members. Theoretically and experimentally, we compare this Endogenous Power environment with a standard Random Power environment in which agenda setters are appointed randomly each period. Although the theoretical analysis predicts that the two environments are outcome equivalent, the experimental analysis shows substantial differences in behavior and outcomes across the games. The Endogenous Power environment results in the formation of more stable coalitions, less-equitable budget allocations, the persistence of power across periods, and higher long-run inequality than the Random Power environment. We present evidence that the stationary equilibrium refinements traditionally used in the literature fail to predict behavior in either game

Two-population replicator dynamics and number of Nash equilibria in random matrix games

We study the connection between the evolutionary replicator dynamics and the number of Nash equilibria in large random bi-matrix games. Using techniques of disordered systems theory we compute the statistical properties of both, the fixed points of the dynamics and the Nash equilibria. Except for the special case of zero-sum games one finds a transition as a function of the so-called co-operation pressure between a phase in which there is a unique stable fixed point of the dynamics coinciding with a unique Nash equilibrium, and an unstable phase in which there are exponentially many Nash equilibria with statistical properties different from the stationary state of the replicator equations. Our analytical results are confirmed by numerical simulations of the replicator dynamics, and by explicit enumeration of Nash equilibria.Comment: 9 pages, 2x2 figure

Evolutionarily stable strategies of random games, and the vertices of random polygons

An evolutionarily stable strategy (ESS) is an equilibrium strategy that is immune to invasions by rare alternative (``mutant'') strategies. Unlike Nash equilibria, ESS do not always exist in finite games. In this paper we address the question of what happens when the size of the game increases: does an ESS exist for ``almost every large'' game? Letting the entries in the $n\times n$ game matrix be independently randomly chosen according to a distribution $F$, we study the number of ESS with support of size $2.$ In particular, we show that, as $n\to \infty$, the probability of having such an ESS: (i) converges to 1 for distributions $F$ with ``exponential and faster decreasing tails'' (e.g., uniform, normal, exponential); and (ii) converges to $1-1/\sqrt{e}$ for distributions $F$ with ``slower than exponential decreasing tails'' (e.g., lognormal, Pareto, Cauchy). Our results also imply that the expected number of vertices of the convex hull of $n$ random points in the plane converges to infinity for the distributions in (i), and to 4 for the distributions in (ii).Comment: Published in at http://dx.doi.org/10.1214/07-AAP455 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Evolution of Coordination in Social Networks: A Numerical Study

Coordination games are important to explain efficient and desirable social behavior. Here we study these games by extensive numerical simulation on networked social structures using an evolutionary approach. We show that local network effects may promote selection of efficient equilibria in both pure and general coordination games and may explain social polarization. These results are put into perspective with respect to known theoretical results. The main insight we obtain is that clustering, and especially community structure in social networks has a positive role in promoting socially efficient outcomes.Comment: preprint submitted to IJMP

Fixation and escape times in stochastic game learning

Evolutionary dynamics in finite populations is known to fixate eventually in the absence of mutation. We here show that a similar phenomenon can be found in stochastic game dynamical batch learning, and investigate fixation in learning processes in a simple 2x2 game, for two-player games with cyclic interaction, and in the context of the best-shot network game. The analogues of finite populations in evolution are here finite batches of observations between strategy updates. We study when and how such fixation can occur, and present results on the average time-to-fixation from numerical simulations. Simple cases are also amenable to analytical approaches and we provide estimates of the behaviour of so-called escape times as a function of the batch size. The differences and similarities with escape and fixation in evolutionary dynamics are discussed.Comment: 19 pages, 9 figure

Evolutionary games on graphs

Game theory is one of the key paradigms behind many scientific disciplines from biology to behavioral sciences to economics. In its evolutionary form and especially when the interacting agents are linked in a specific social network the underlying solution concepts and methods are very similar to those applied in non-equilibrium statistical physics. This review gives a tutorial-type overview of the field for physicists. The first three sections introduce the necessary background in classical and evolutionary game theory from the basic definitions to the most important results. The fourth section surveys the topological complications implied by non-mean-field-type social network structures in general. The last three sections discuss in detail the dynamic behavior of three prominent classes of models: the Prisoner's Dilemma, the Rock-Scissors-Paper game, and Competing Associations. The major theme of the review is in what sense and how the graph structure of interactions can modify and enrich the picture of long term behavioral patterns emerging in evolutionary games.Comment: Review, final version, 133 pages, 65 figure