# Orders of limits for stationary distributions, stochastic dominance, and stochastic stability

## Author

Listed:
• Sandholm, William H.

() (Department of Economics, University of Wisconsin)

## Abstract

A population of agents recurrently plays a two-strategy population game. When an agent receives a revision opportunity, he chooses a new strategy using a noisy best response rule that satisfies mild regularity conditions; best response with mutations, logit choice, and probit choice are all permitted. We study the long run behavior of the resulting Markov process when the noise level $\eta$ is small and the population size $N$ is large. We obtain a precise characterization of the asymptotics of the stationary distributions $\mu^{N,\eta}$ as $\eta$ approaches zero and $N$ approaches infinity, and we establish that these asymptotics are the same for either order of limits and for all simultaneous limits. In general, different noisy best response rules can generate different stochastically stable states. To obtain a robust selection result, we introduce a refinement of risk dominance called \emph{stochastic dominance}, and we prove that coordination on a given strategy is stochastically stable under every noisy best response rule if and only if that strategy is stochastically dominant.

## Suggested Citation

• Sandholm, William H., 2010. "Orders of limits for stationary distributions, stochastic dominance, and stochastic stability," Theoretical Economics, Econometric Society, vol. 5(1), January.
• Handle: RePEc:the:publsh:554
as

File URL: http://econtheory.org/ojs/index.php/te/article/viewFile/20100001/3302/135

## Citations

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Cited by:

1. Robert Molzon, 2012. "Large Population Limits for Evolutionary Dynamics with Random Matching," Dynamic Games and Applications, Springer, vol. 2(1), pages 146-159, March.
2. Staudigl, Mathias, 2012. "Stochastic stability in asymmetric binary choice coordination games," Games and Economic Behavior, Elsevier, vol. 75(1), pages 372-401.
3. repec:eee:thpobi:v:89:y:2013:i:c:p:75-87 is not listed on IDEAS
4. Sandholm, William H., 2012. "Stochastic imitative game dynamics with committed agents," Journal of Economic Theory, Elsevier, vol. 147(5), pages 2056-2071.
5. Carlos Alós-Ferrer & Nick Netzer, 2015. "Robust stochastic stability," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 58(1), pages 31-57, January.
6. Sandholm, William H. & Staudigl, Mathias, 2016. "Large Deviations and Stochastic Stability in the Small Noise Double Limit, I: Theory," Center for Mathematical Economics Working Papers 505, Center for Mathematical Economics, Bielefeld University.
7. Kevin Hasker, 2014. "The Emergent Seed: A Representation Theorem for Models of Stochastic Evolution and two formulas for Waiting Time," Levine's Working Paper Archive 786969000000000954, David K. Levine.
8. Sandholm, William H. & Staudigl, Mathias, 2016. "Large Deviations and Stochastic Stability in the Small Noise Double Limit, II: The Logit Model," Center for Mathematical Economics Working Papers 506, Center for Mathematical Economics, Bielefeld University.
9. repec:eee:thpobi:v:103:y:2015:i:c:p:2-26 is not listed on IDEAS
10. Ryoji Sawa, 2012. "Mutation rates and equilibrium selection under stochastic evolutionary dynamics," International Journal of Game Theory, Springer;Game Theory Society, vol. 41(3), pages 489-496, August.

### Keywords

Evolutionary game theory; stochastic stability; equilibrium selection;

### JEL classification:

• C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games
• C73 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Stochastic and Dynamic Games; Evolutionary Games

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