Abstract
We present examples of symmetric two-player games admitting a unique Nash equilibrium and a unique evolutionarily stable strategy (ESS) and such that the ESS payoff is strictly higher than the Nash payoff. In this sense, we show that spiteful behavior can make everybody better off.
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Notes
The original ESS concept introduced by Maynard Smith and Price (1973) is only suitable for infinite populations.
The definition can be extended to mixed strategies in the obvious way.
Proof. If \(s^E\) is evolutionarily stable, we have \((u_1-u_2)(s, s^E, \ldots , s^E) \le 0\) for all \(s \in X\) and, by symmetry, \((u_1-u_2)(s^E, \ldots , s^E) = 0\). Hence, \(s^E\) satisfies (1). If, however, \(s^E\) is not evolutionarily stable, there exists a strategy s satisfying \((u_1-u_2)(s, s^E, \ldots , s^E) >0\), so that \(s^E\) does not satisfy (1). \(\square \)
In the context of tax competition, this was recently proved under various assumptions by Sano (2012), Wagener (2013) and Philipowski (2015). It seems that this question has not been addressed explicitly in other branches of economics. However, Schaffer (1989), Vega-Redondo (1997) and Alós-Ferrer and Ania (2005) showed under various assumptions that in Cournot oligopoly evolutionarily stable behavior leads to the Walrasian outcome and hence to lower profits than Nash behavior. Moreover, Ania (2008) and Hehenkamp at al. (2010) investigated conditions under which Nash and evolutionarily stable equilibrium coincide.
References
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I would like to thank two anonymous referees for their very useful comments.
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Philipowski, R. Spiteful behavior can make everybody better off. Evolut Inst Econ Rev 13, 113–116 (2016). https://doi.org/10.1007/s40844-016-0035-4
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DOI: https://doi.org/10.1007/s40844-016-0035-4