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Monotone Imitation Dynamics in Large Populations
We analyze a class of imitation dynamics with mutations for games with any finite number of actions, and give conditions for the selection of a unique equilibrium as the mutation rate becomes small and the population becomes large. Our results cover the multiple-action extensions of the aspiration-and-imitation process of Binmore and Samuelson [Muddling through: noisy equilibrium selection, J. Econ. Theory 74 (1997) 235–265] and the related processes proposed by Benaı¨m and Weibull [Deterministic approximation of stochastic evolution in games, Econometrica 71 (2003) 873–903] and Traulsen et al. [Coevolutionary dynamics: from finite to infinite populations, Phys. Rev. Lett. 95 (2005) 238701], as well as the frequency-dependent Moran process studied by Fudenberg et al. [Evolutionary game dynamics in finite populations with strong selection and weak mutation, Theoretical Population Biol. 70 (2006) 352–363]. We illustrate our results by considering the effect of the number of periods of repetition on the selected equilibrium in repeated play of the prisoner's dilemma when players are restricted to a small set of simple strategies.Economic
Let Cognitive Radios Imitate: Imitation-based Spectrum Access for Cognitive Radio Networks
In this paper, we tackle the problem of opportunistic spectrum access in
large-scale cognitive radio networks, where the unlicensed Secondary Users (SU)
access the frequency channels partially occupied by the licensed Primary Users
(PU). Each channel is characterized by an availability probability unknown to
the SUs. We apply evolutionary game theory to model the spectrum access problem
and develop distributed spectrum access policies based on imitation, a behavior
rule widely applied in human societies consisting of imitating successful
behavior. We first develop two imitation-based spectrum access policies based
on the basic Proportional Imitation (PI) rule and the more advanced Double
Imitation (DI) rule given that a SU can imitate any other SUs. We then adapt
the proposed policies to a more practical scenario where a SU can only imitate
the other SUs operating on the same channel. A systematic theoretical analysis
is presented for both scenarios on the induced imitation dynamics and the
convergence properties of the proposed policies to an imitation-stable
equilibrium, which is also the -optimum of the system. Simple,
natural and incentive-compatible, the proposed imitation-based spectrum access
policies can be implemented distributedly based on solely local interactions
and thus is especially suited in decentralized adaptive learning environments
as cognitive radio networks
Strategy abundance in 2x2 games for arbitrary mutation rates
We study evolutionary game dynamics in a well-mixed populations of finite
size, N. A well-mixed population means that any two individuals are equally
likely to interact. In particular we consider the average abundances of two
strategies, A and B, under mutation and selection. The game dynamical
interaction between the two strategies is given by the 2x2 payoff matrix
[(a,b), (c,d)]. It has previously been shown that A is more abundant than B, if
(N-2)a+Nb>Nc+(N-2)d. This result has been derived for particular stochastic
processes that operate either in the limit of asymptotically small mutation
rates or in the limit of weak selection. Here we show that this result holds in
fact for a wide class of stochastic birth-death processes for arbitrary
mutation rate and for any intensity of selection.Comment: version 2 is the final published version that contains minor changes
in response to referee comment
Convergence in Models with Bounded Expected Relative Hazard Rates
We provide a general framework to study stochastic sequences related to
individual learning in economics, learning automata in computer sciences,
social learning in marketing, and other applications. More precisely, we study
the asymptotic properties of a class of stochastic sequences that take values
in and satisfy a property called "bounded expected relative hazard
rates." Sequences that satisfy this property and feature "small step-size" or
"shrinking step-size" converge to 1 with high probability or almost surely,
respectively. These convergence results yield conditions for the learning
models in B\"orgers, Morales, and Sarin (2004), Erev and Roth (1998), and
Schlag (1998) to choose expected payoff maximizing actions with probability one
in the long run.Comment: After revision. Accepted for publication by Journal of Economic
Theor
Robust stochastic stability
A strategy profile of a game is called robustly stochastically stable if it is stochastically stable for a given behavioral model independently of the specification of revision opportunities and tie-breaking assumptions in the dynamics. We provide a simple radius-coradius result for robust stochastic stability and examine several applications. For the logit-response dynamics, the selection of potential maximizers is robust for the subclass of supermodular symmetric binary-action games. For the mistakes model, the weaker property of strategic complementarity suffices for robustness in this class of games. We also investigate the robustness of the selection of risk-dominant strategies in coordination games under best-reply and the selection of Walrasian strategies in aggregative games under imitation.Learning in games, stochastic stability, radius-coradius theorems, logit-response dynamics, mutations, imitation
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