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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 self-defeating 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 self-defeating 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.
On the existence of general equilibrium in finite games and general game dynamics
- ArXiv
"... A notion of incentive for agents is introduced which leads to a very general notion of an equilibrium for a finite game. Sufficient conditions for the existence of these equilibria are given. Known existence theorems are shown to be corollaries to the main theorem of this paper. Furthermore, conditi ..."
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Cited by 8 (6 self)
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A notion of incentive for agents is introduced which leads to a very general notion of an equilibrium for a finite game. Sufficient conditions for the existence of these equilibria are given. Known existence theorems are shown to be corollaries to the main theorem of this paper. Furthermore, conditions for the existence of equilibria in certain symmetric regions for games are also given. From the notion of general equilibrium, a general family of game dynamics are derived. This family incorporates all canonical examples of game dynamics. A proof is given for the full generality of this system. 1 Notation and Definitions We shall denote the finite set of agents by N = {1, 2,..., n} for some n ∈ N. Each agent i is endowed with a finite set of pure strategies, which will be denoted Si = {1, 2,..., si}, with si ∈ N as well. To allow the agents to mix their strategies, they may choose strategies from the simplex on si vertices, ∆i =
Stochastic Approximations with Constant Step Size and Differential Inclusions
, 2012
"... We consider stochastic approximation processes with constant step size whose as-sociated deterministic system is an upper semicontinous differential inclusion. We prove that over any finite time span, the sample paths of the stochastic process are closely approximated by a solution of the differenti ..."
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Cited by 6 (3 self)
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We consider stochastic approximation processes with constant step size whose as-sociated deterministic system is an upper semicontinous differential inclusion. We prove that over any finite time span, the sample paths of the stochastic process are closely approximated by a solution of the differential inclusion with high probability. We then analyze infinite horizon behavior, showing that if the process is Markov, its stationary measures must become concentrated on the Birkhoff center of the determin-istic system. Our results extend those of Benaı̈m for settings in which the deterministic system is Lipschitz continuous, and build on the work of Benaı̈m, Hofbauer, and Sorin for the case of decreasing step sizes. We apply our results to models of population dy-namics in games, obtaining new conclusions about the medium and long run behavior of myopic optimizing agents.
Dynamic network security deployment under partial information: Global analysis,” 2008, in preparation. APPENDIX We find the equilibrium points X0, X1, X2 by solving the system (4) in the three different regions of I specified in the function pSP
- I < I∗ − ɛ 2 , I∗ − ɛ 2 < I < I∗ + ɛ 2 , I > I∗ + ɛ 2 . We
"... Abstract—A network user’s decision to start and continue using security products is based on economic considerations. The cost of a security compromise (e.g., worm infection) is compared against the cost of deploying and maintaining a sufficient level of security. These costs are not necessarily the ..."
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Cited by 5 (1 self)
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Abstract—A network user’s decision to start and continue using security products is based on economic considerations. The cost of a security compromise (e.g., worm infection) is compared against the cost of deploying and maintaining a sufficient level of security. These costs are not necessarily the real ones, but rather the perceived costs, which depend on the amount of information available to a user at each time. Moreover, the costs (whether real or perceived) depend on the decisions of other users, too: The probability of a user getting infected depends on the security deployed by all the other users. In this paper, we combine an epidemic model for malware propagation in a network with a game theoretic model of the users ’ decisions to deploy security or not. Users can dynamically change their decision in order to maximize their currently perceived utility. We study the equilibrium points, and their dependence on the speed of the learning process through which the users learn the state of the network. We find that the faster the learning process, the higher the total network cost. I.
The Kullback-Liebler Divergence as a Lyapunov Function for Incentive Based Game Dynamics
, 2014
"... It has been shown that the Kullback-Leibler divergence is a Lyapunov function for the replicator equations at evolutionary stable states, or ESS. In this paper we extend the result to a more general class of game dy-namics. As a result, sufficient conditions can be given for the asymptotic stability ..."
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Cited by 4 (4 self)
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It has been shown that the Kullback-Leibler divergence is a Lyapunov function for the replicator equations at evolutionary stable states, or ESS. In this paper we extend the result to a more general class of game dy-namics. As a result, sufficient conditions can be given for the asymptotic stability of rest points for the entire class of incentive dynamics. The previous known results will be can be shown as corollaries to the main theorem. 1 Information Theory and The Replicator Dy-namics Information theory was originally developed by Claude Shannon and Warren Weaver [Sha01, SW49] as a mathematical framework to describe problems in communication including, but not limited to, data compression and storage. They introduced measures of information called entropy1. Shannon’s entropy, denoted H(P), is a measure of the average uncertainty in a random variable, P. It can be interpreted as the average number of bits needed to encode a message drawn i.i.d. from P. Maximizing the entropy can be used to give a lower bound on this average number of bits needed for encryption. For our purposes, the concepts of cross entropy and relative entropy will be of great use. The Kullback-Leibler divergence (KL divergence or DKL) [KL51], or relative entropy is a measure of information gain (loss) from one state to another. More precisely, it is an average measure of the additional bits needed 1In fact, the Shannon entropy is simply the Boltzmann entropy [Jay65] without the con-stants 1 ar
Population games and deterministic evolutionary dynamics
, 2014
"... Population games describe strategic interactions among large numbers of small, anonymous agents. Behavior in these games is typically modeled dynamically, with agents occasionally receiving opportunities to switch strategies, basing their choices on simple myopic rules called revision protocols. Ove ..."
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Cited by 3 (0 self)
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Population games describe strategic interactions among large numbers of small, anonymous agents. Behavior in these games is typically modeled dynamically, with agents occasionally receiving opportunities to switch strategies, basing their choices on simple myopic rules called revision protocols. Over finite time spans the evolution of aggregate behavior is well approximated by the solution of a differential equation. From a different point of view, every revision protocol defines a map—a deterministic evolutionary dynamic—that assigns each population game a differential equation describing the evolution of aggregate behavior in that game. In this chapter, we provide an overview of the theory of population games and deterministic evolutionary dynamics. We introduce population games through a series of examples and illustrate their basic geometric properties. We formally derive deter-ministic evolutionary dynamics from revision protocols, introduce the main families of dynamics—imitative/biological, best response, comparison to average payoffs, and pairwise comparison—and discuss their basic properties. Combining these streams, we consider classes of population games in which members of these families of dynamics converge to equilibrium; these classes include potential games, contractive games, games solvable by iterative solution concepts, and supermodular games. We relate these classes to the classical notion of an evolutionarily stable state (ESS) and to recent work on deterministic equilibrium selection. We present a variety of examples of cycling and chaos under evolutionary dynamics, as well as a general result on survival of strictly dominated strategies. Finally, we provide connections to other approaches to game dynamics, and indicate applications of evolutionary game dynamics to economics and social science.
2014: On the robustness of learning in games with stochastically perturbed payoff observations
"... HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte p ..."
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Cited by 1 (1 self)
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HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Mean evolutionary dynamics for stochastically switching environments
, 2013
"... Abstract. Populations of replicating entities frequently experience sudden or cyclical changes in environment. We explore the implications of this phenomenon via a environ-mental switching parameter in several common evolutionary dynamics models including the replicator dynamic for linear symmetric ..."
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Cited by 1 (1 self)
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Abstract. Populations of replicating entities frequently experience sudden or cyclical changes in environment. We explore the implications of this phenomenon via a environ-mental switching parameter in several common evolutionary dynamics models including the replicator dynamic for linear symmetric and asymmetric landscapes, the Moran process, and incentive dynamics. We give a simple relationship between the probability of environmental switching, the relative fitness gain, and the effect on long term behavior in terms of fixation probabilities and long term outcomes for deterministic dynamics. We also discuss cases where the dynamic changes, for instance a population evolving under a replicator dynamic switching to a best-reply dynamic and vice-versa, giving Lyapunov stability results. 1.
