Large deviations and stochastic stability in the small noise double limit

We consider a model of stochastic evolution under general noisy best response protocols, allowing the probabilities of suboptimal choices to depend on their payoff consequences. Our analysis focuses on behavior in the small noise double limit: we first take the noise level in agents' decisions to zero, and then take the population size to infinity. We show that in this double limit, escape from and transitions between equilibria can be described in terms of solutions to continuous optimal control problems. These are used in turn to characterize the asymptotics of the the stationary distribution, and so to determine the stochastically stable states. We use these results to perform a complete analysis of evolution in three-strategy coordination games that satisfy the marginal bandwagon property and that have an interior equilibrium, with agents following the logit choice rule.

[1]  V. Boltyanskii.  Sufficient Conditions for Optimality and the Justification of the Dynamic Programming Method , 1966 .

[2]  K Sigmund,et al.  The theory of evolution and dynamic systems mathematical aspects of selection , 1984 .

[3]  M. Freidlin,et al.  Random Perturbations of Dynamical Systems , 1984 .

[4]  J. Hofbauer.  The selection mutation equation , 1985, Journal of mathematical biology.

[5]  Josef Hofbauer,et al.  The theory of evolution and dynamical systems , 1988 .

[6]  P. Dupuis.  Large Deviations Analysis of Some Recursive Algorithms with State Dependent Noise , 1988 .

[7]  H. Peyton Young,et al.  Stochastic Evolutionary Game Dynamics , 1990 .

[8]  I. Gilboa,et al.  Social Stability and Equilibrium , 1991 .

[9]  H. Young,et al.  The Evolution of Conventions , 1993 .

[10]  R. Rob,et al.  Learning, Mutation, and Long Run Equilibria in Games , 1993 .

[11]  L. Blume.  The Statistical Mechanics of Strategic Interaction , 1993 .

[12]  Ken Binmore,et al.  Muddling Through: Noisy Equilibrium Selection☆ , 1997 .

[13]  L. Samuelson,et al.  Musical Chairs: Modeling Noisy Evolution , 1995 .

[14]  L. Blume,et al.  POPULATION GAMES , 1995 .

[15]  Glenn Ellison,et al.  Basins of Attraction, Long-Run Equilibria, and the Speed of Step-by-Step Evolution , 1995 .

[16]  Michihiro Kandori,et al.  Evolution of Equilibria in the Long Run: A General Theory and Applications , 1995 .

[17]  André de Palma,et al.  Discrete Choice Theory of Product Differentiation , 1995 .

[18]  J. Lynch,et al.  A weak convergence approach to the theory of large deviations , 1997 .

[19]  R. Rob,et al.  Bandwagon Effects and Long Run Technology Choice , 2010 .

[20]  H. Young.  Individual Strategy and Social Structure , 2020 .

[21]  O. Catoni.  Simulated annealing algorithms and Markov chains with rare transitions , 1999 .

[22]  Glenn Ellison.  Basins of Attraction, Long-Run Stochastic Stability, and the Speed of Step-by-Step Evolution , 2000 .

[23]  T. Sargent,et al.  Escaping Nash Inflation , 2000, SSRN Electronic Journal.

[24]  Héctor J. Sussmann,et al.  Regular Synthesis and Sufficiency Conditions for Optimality , 2000, SIAM J. Control. Optim..

[25]  William H. Sandholm,et al.  Potential Games with Continuous Player Sets , 2001, J. Econ. Theory.

[26]  William H. Sandholm,et al.  ON THE GLOBAL CONVERGENCE OF STOCHASTIC FICTITIOUS PLAY , 2002 .

[27]  David P. Myatt,et al.  A multinomial probit model of stochastic evolution , 2003 .

[28]  Lawrence E. Blume,et al.  How noise matters , 2003, Games Econ. Behav..

[29]  H. Peyton Young,et al.  Strategic Learning and Its Limits , 2004 .

[30]  Drew Fudenberg,et al.  Imitation Processes with Small Mutations , 2004, J. Econ. Theory.

[31]  William H. Sandholm,et al.  Simple formulas for stationary distributions and stochastically stable states , 2007, Games Econ. Behav..

[32]  Drew Fudenberg,et al.  Monotone Imitation Dynamics in Large Populations , 2007, J. Econ. Theory.

[33]  William H. Sandholm,et al.  Large population potential games , 2009, J. Econ. Theory.

[34]  William H. Sandholm,et al.  Decompositions and potentials for normal form games , 2010, Games Econ. Behav..

[35]  William H. Sandholm,et al.  Orders of limits for stationary distributions, stochastic dominance, and stochastic stability , 2010 .

[36]  William H. Sandholm,et al.  Population Games And Evolutionary Dynamics , 2010, Economic learning and social evolution.

[37]  Nick Netzer,et al.  The logit-response dynamics , 2010, Games Econ. Behav..

[38]  William H. Sandholm,et al.  Large deviations and multinomial probit choice , 2011, J. Econ. Theory.

[39]  H. Schättler,et al.  Geometric Optimal Control , 2012 .

[40]  Mathias Staudigl,et al.  Stochastic stability in asymmetric binary choice coordination games , 2012, Games Econ. Behav..

[41]  H. Schättler,et al.  Geometric Optimal Control: Theory, Methods and Examples , 2012 .

[42]  William H. Sandholm.  Stochastic imitative game dynamics with committed agents , 2012, J. Econ. Theory.

[43]  William H. Sandholm,et al.  Stochastic Approximations with Constant Step Size and Differential Inclusions , 2013, SIAM J. Control. Optim..

[44]  W. Sandholm,et al.  A Sample Path Large Deviation Principle for a Class of Population Processes , 2016 .