# Fixation Probabilities of Strategies for Bimatrix Games in Finite Populations

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## Abstract

Recent developments in stochastic evolutionary game theory in finite populations yield insights that complement the conventional deterministic evolutionary game theory in infinite populations. However, most studies of stochastic evolutionary game theory have investigated dynamics of symmetric games, although not all social and biological phenomena are described by symmetric games, e.g., social interactions between individuals having conflicting preferences or different roles. In this paper, we describe the stochastic evolutionary dynamics of two-player \(2 \times 2\) bimatrix games in finite populations. The stochastic process is modeled by a frequency-dependent Moran process without mutation. We obtained the fixation probability that the evolutionary dynamics starting from a given initial state converges to a specific absorbing state. Applying the formula to the ultimatum game, we show that evolutionary dynamics favors fairness. Furthermore, we present two novel concepts of stability for bimatrix games, based on our formula for the fixation probability, and demonstrate that one of the two serves as a criterion for equilibrium selection.

## Keywords

Bimatrix games Equilibrium selection Finite population Fixation probability Stability Stochastic evolution## Notes

### Acknowledgments

The authors thank the two anonymous reviewers for their helpful comments. This work was supported by the Grant-in-Aid for JSPS Fellows Grant Number 13J05358 (TS) and JSPS KAKENHI Grant Number 25118006 (HO).

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