Equilibrium Selection under the Bayes-Based Strategy Updating Rules

In this paper, first, an evolutionary game model for Bayes-based strategy updating rules was constructed, in which players can only observe a signal that reveals a strategy type instead of the strategy type directly, which deviates from the strategy type of players. Then, the equilibrium selection of populations in the case of the asymmetric game, the Battle of the Sexes (BoS), and the case of a symmetric coordination game was studied where individuals make decisions based on the signals released by each player. Finally, it was concluded that in the BoS game, when the accuracy of the signal is low, the population eventually reaches an incompatible state. If the accuracy of the signal is improved, the population finally reaches a coordinated state. In a coordination game, when the accuracy of the signal is low, the population will eventually choose a payoff-dominated equilibrium. With the improvement of signal accuracy, the equilibrium of the final selection of the population depends on its initial state.

[1]  C. Hauert,et al.  Replicator dynamics for optional public good games. , 2002, Journal of theoretical biology.

[2]  M. Apostolov,et al.  Shapley-Folkman-Lyapunov theorem and Asymmetric First price auctions , 2019, Applied Mathematics and Nonlinear Sciences.

[3]  Ramón Alonso-Sanz,et al.  Self-organization in the spatial battle of the sexes with probabilistic updating , 2011 .

[4]  Mangalagama Dewasurendra,et al.  On the Method of Inverse Mapping for Solutions of Coupled Systems of Nonlinear Differential Equations Arising in Nanofluid Flow, Heat and Mass Transfer , 2018, Applied Mathematics and Nonlinear Sciences.

[5]  J. Hofbauer,et al.  Time averages, recurrence and transience in the stochastic replicator dynamics , 2009, 0908.4467.

[6]  L. Imhof The long-run behavior of the stochastic replicator dynamics , 2005, math/0503529.

[7]  Yunshyong Chow,et al.  Equilibrium Selection in Evolutionary Games with Imperfect Monitoring , 2008, Journal of Applied Probability.

[8]  A. Cabrales Stochastic replicator dynamics , 2000 .

[9]  Ferenc Szidarovszky,et al.  n-person Battle of sexes games—a simulation study , 2008 .

[10]  André Orléan Bayesian interactions and collective dynamics of opinion: Herd behavior and mimetic contagion , 1995 .

[11]  Xiang Li,et al.  Mixed evolutionary strategies imply coexisting opinions on networks. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[13]  M. Nowak,et al.  Evolutionary game dynamics in a Wright-Fisher process , 2006, Journal of mathematical biology.

[14]  Chaitanya S. Gokhale,et al.  How small are small mutation rates? , 2011, Journal of Mathematical Biology.

[15]  Hisashi Ohtsuki,et al.  Fixation Probabilities of Strategies for Bimatrix Games in Finite Populations , 2015, Dynamic Games and Applications.

[16]  Ola Olsson,et al.  Bayes' theorem and its applications in animal behaviour , 2006 .

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

[18]  Long Wang,et al.  Aspiration dynamics of multi-player games in finite populations , 2014, Journal of The Royal Society Interface.

[19]  Naoki Masuda,et al.  A collective opinion formation model under Bayesian updating and confirmation bias , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[20]  S. Mylius,et al.  What pair formation can do to the battle of the sexes: towards more realistic game dynamics. , 1999, Journal of theoretical biology.

[21]  M. Nowak,et al.  Evolutionary games on cycles with strong selection. , 2017, Physical review. E.

[22]  D. Fudenberg,et al.  Emergence of cooperation and evolutionary stability in finite populations , 2004, Nature.

[23]  Yunshyong Chow,et al.  Boundedly rational quasi-Bayesian learning in coordination games with imperfect monitoring , 2006, Journal of Applied Probability.

[24]  Carl Veller,et al.  Finite-population evolution with rare mutations in asymmetric games , 2015, J. Econ. Theory.

[25]  D. Fudenberg,et al.  Evolutionary Dynamics with Aggregate Shocks , 1992 .

[26]  A Houston,et al.  The application of statistical decision theory to animal behaviour. , 1980, Journal of theoretical biology.

[27]  Martin A. Nowak,et al.  Public Goods With Punishment and Abstaining in Finite and Infinite Populations , 2008, Biological theory.

[28]  Sabin Lessard,et al.  Long-term stability from fixation probabilities in finite populations: new perspectives for ESS theory. , 2005, Theoretical population biology.

[29]  H. Çitil Important Notes for a Fuzzy Boundary Value Problem , 2019, Applied Mathematics and Nonlinear Sciences.