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183
Stochastic evolutionary game dynamics
 Theoret. Population Biol
, 1990
"... The concept of an evolutionary stable strategy (ESS) is a useful tool for studying the dynamics of natural selection. One of its limitations, however, is that it does not capture the notion of longrun stability when the system is subjected to stochastic effects. We define the concept of stability i ..."
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Cited by 143 (7 self)
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The concept of an evolutionary stable strategy (ESS) is a useful tool for studying the dynamics of natural selection. One of its limitations, however, is that it does not capture the notion of longrun stability when the system is subjected to stochastic effects. We define the concept of stability in a stochastic dynamical system, and show that it differs from both the traditional ESS and the concept of an attractor in a dynamical system. The stochastically stable set may be computed analytically using recent advances in potential theory
Replicator Equations, Maximal Cliques, and Graph Isomorphism
, 1999
"... We present a new energyminimization framework for the graph isomorphism problem that is based on an equivalent maximum clique formulation. The approach is centered around a fundamental result proved by Motzkin and Straus in the mid1960s, and recently expanded in various ways, which allows us to fo ..."
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Cited by 62 (11 self)
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We present a new energyminimization framework for the graph isomorphism problem that is based on an equivalent maximum clique formulation. The approach is centered around a fundamental result proved by Motzkin and Straus in the mid1960s, and recently expanded in various ways, which allows us to formulate the maximum clique problem in terms of a standard quadratic program. The attractive feature of this formulation is that a clear onetoone correspondence exists between the solutions of the quadratic program and those in the original, combinatorial problem. To solve the program we use the socalled replicator equations—a class of straightforward continuous and discretetime dynamical systems developed in various branches of theoretical biology. We show how, despite their inherent inability to escape from local solutions, they nevertheless provide experimental results that are competitive with those obtained using more elaborate meanfield annealing heuristics.
Approximating the Maximum Weight Clique Using Replicator Dynamics
, 2000
"... Given an undirected graph with weights on the vertices, the maximum weight clique problem (MWCP) is to find a subset of mutually adjacent vertices (i.e., a clique) having largest total weight. This is a generalization of the classical problem of finding the maximum cardinality clique of an unweig ..."
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Cited by 34 (9 self)
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Given an undirected graph with weights on the vertices, the maximum weight clique problem (MWCP) is to find a subset of mutually adjacent vertices (i.e., a clique) having largest total weight. This is a generalization of the classical problem of finding the maximum cardinality clique of an unweighted graph, which arises as a special case of the MWCP when all the weights associated to the vertices are equal. The problem is known to be NP hard for arbitrary graphs and, according to recent theoretical results, so is the problem of approximating it within a constant factor. Although there has recently been much interest around neural network algorithms for the unweighted maximum clique problem, no effort has been directed so far towards its weighted counterpart. In this paper, we present a parallel, distributed heuristic for approximating the MWCP based on dynamics principles developed and studied in various branches of mathematical biology. The proposed framework centers aroun...
Evolutionary game theory: temporal and spatial effects beyond replicator dynamics
, 2009
"... Evolutionary game dynamics is one of the most fruitful frameworks for studying evolution in different disciplines, from Biology to Economics. Within this context, the approach of choice for many researchers is the socalled replicator equation, that describes mathematically the idea that those indiv ..."
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Cited by 28 (1 self)
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Evolutionary game dynamics is one of the most fruitful frameworks for studying evolution in different disciplines, from Biology to Economics. Within this context, the approach of choice for many researchers is the socalled replicator equation, that describes mathematically the idea that those individuals performing better have more offspring and thus their frequency in the population grows. While very many interesting results have been obtained with this equation in the three decades elapsed since it was first proposed, it is important to realize the limits of its applicability. One particularly relevant issue in this respect is that of nonmean field effects, that may arise from temporal fluctuations or from spatial correlations, both neglected in the replicator equation. This review discusses these temporal and spatial effects focusing on the nontrivial modifications they induce when compared to the outcome of replicator dynamics. Alongside this question, the hypothesis of linearity and its relation to the choice of the rule for strategy update is also analyzed. The discussion is presented in terms of the emergence of cooperation, as one of the current key problems in Biology and in other disciplines.
Pairwise comparison dynamics and evolutionary foundations for Nash equilibrium. working paper
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Stable Games and their Dynamics
, 2009
"... We study a class of population games called stable games. These games are characterized by selfdefeating externalities: when agents revise their strategies, the improvements in the payoffs of strategies to which revising agents are switching are always exceeded by the improvements in the payoffs of ..."
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Cited by 26 (4 self)
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We study a class of population games called stable games. These games are characterized by selfdefeating externalities: when agents revise their strategies, the improvements in the payoffs of strategies to which revising agents are switching are always exceeded by the improvements in the payoffs of strategies which revising agents are abandoning. We prove that the set of Nash equilibria of a stable game is globally asymptotically stable under a wide range of evolutionary dynamics. Convergence results for stable games are not as general as those for potential games: in addition to monotonicity of the dynamics, integrability of the agents’ revision protocols plays a key role.
Generalised weakened fictitious play
, 2004
"... A general class of adaptive processes in games is developed, which significantly generalises weakened fictitious play [Van der Genugten, B., 2000. A weakened form of fictitious play in twoperson zerosum games. Int. Game Theory Rev. 2, 307–328] and includes several interesting fictitiousplaylike ..."
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Cited by 26 (3 self)
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A general class of adaptive processes in games is developed, which significantly generalises weakened fictitious play [Van der Genugten, B., 2000. A weakened form of fictitious play in twoperson zerosum games. Int. Game Theory Rev. 2, 307–328] and includes several interesting fictitiousplaylike processes as special cases. The general model is rigorously analysed using the best response differential inclusion, and shown to converge in games with the fictitious play property. Furthermore, a new actor–critic process is introduced, in which the only information given to a player is the reward received as a result of selecting an action—a player need not even know they are playing a game. It is shown that this results in a generalised weakened fictitious play process, and can therefore be considered as a first step towards explaining how players might learn to play Nash equilibrium strategies without having any knowledge of the game, or even that they are playing a game.
Annealed Replication: A New Heuristic for the Maximum Clique Problem
 Discr. Appl. Math
, 2000
"... In this paper, a new heuristic for approximating the maximum clique problem is proposed, based on a detailed analysis of a class of continuous optimization models which yield a complete solution to this NPhard combinatorial problem. The idea is to alter a regularization parameter iteratively in suc ..."
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Cited by 23 (9 self)
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In this paper, a new heuristic for approximating the maximum clique problem is proposed, based on a detailed analysis of a class of continuous optimization models which yield a complete solution to this NPhard combinatorial problem. The idea is to alter a regularization parameter iteratively in such a way that an iterative procedure with the updated parameter value would avoid unwanted, inefficient local solutions, i.e., maximal cliques which contain less than the maximum possible number of vertices. The local search procedure is performed with the help of the replicator dynamics, and the regularization parameter is chosen deliberately as to render dynamical instability of the (formerly) stable solutions which we want to discard in order to get an improvement. In this respect, the proposed procedure differs from usual simulated annealing approaches which mostly use a "blackbox" cooling schedule. To demonstrate the validity of this approach, we report on the performance applied to sel...
The Projection Dynamic and the Geometry of Population Games
 GAMES ECON. BEHAV.
, 2008
"... The projection dynamic is an evolutionary dynamic for population games. It is derived from a model of individual choice in which agents abandon their current strategies at rates inversely proportional to the strategies ’ current levels of use. The dynamic admits a simple geometric definition, its re ..."
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Cited by 20 (4 self)
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The projection dynamic is an evolutionary dynamic for population games. It is derived from a model of individual choice in which agents abandon their current strategies at rates inversely proportional to the strategies ’ current levels of use. The dynamic admits a simple geometric definition, its rest points coincide with the Nash equilibria of the underlying game, and it converges globally to Nash equilibrium in potential games and in stable games.