Abstract
Ania (J Econ Behav Organ 65:472–488, 2008) shows that in the class of symmetric games with weak payoff externalities, symmetric Nash equilibria are equivalent to symmetric evolutionary equilibria (Schaffer in J Econ Behav Organ 12:29–45, 1989). We introduce a notion of a game with partial weak payoff externalities. We show that the class of games with partial weak payoff externalities includes most of previously known classes of games in which the equivalence prevails. We also establish a number of pure strategy Nash equilibrium existence results for a game with weak payoff externalities, and for a class of games that includes games with partial weak payoff externalities. The results include, in particular, the existence of pure strategy Nash equilibrium in some finite games.
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Notes
A game is finite if there are only finitely many strategies. We always assume that the number of players is finite.
Moulin (1986, pp. 115–116) is an early example of such a treatment.
One might also call this condition weak payoff externalities with TDI (transference of decisionmaker indifference; Marx and Swinkels 1997).
The partial WPE is always with TDI, allowing payoff ties.
The original definition of weakly unilaterally competitive game by Kats and Thisse (1992) accommodates asymmetric games.
In particular, the class includes two-person symmetric zero-sum games, since any such game satisfies (WUC).
Iimura and Watanabe (2016, Fig. 7) contains a game satisfying (WC) with no SNE that fails to satisfy (pWPE).
If \(n=2\) then set \(\zeta \) to be the empty string.
In a game \(G=(S^n,u)\) satisfying (\(\hbox {WPE}_0\)), assume that u is continuous in own strategy. Then one can prove that it is continuous in any other’s strategy.
One can show that in a partial WPE game the relation is acyclic if the strategy set is totally ordered. It follows that such a game has an SNE if it is finite, or it does with some additional topological conditions as in Proposition 4.5. If the strategy set is only partially ordered, the relation need not be acyclic. We shall consider such games in Sect. 4.3.1.
Our existence results apply to any n-person game that has a pairwise solvable two-person reduction. Hence they may apply to a WC game if its two-person reduction is pairwise solvable.
Recall that \(x\in {\text {co}}Y\) if and only if there are \(x_i\in Y\), \(\alpha _i\ge 0\), and \(\sum _i\alpha _i=1\), \(i=1,\dots ,K\), such that \(x=\sum _i \alpha _ix_i\).
The topological vector space need not be Hausdorff. On this, see, for example, Yuan (1998, p. 6).
We can see this as well by reversing the argument in the proof of Proposition 4.11.
But see footnote 10.
Actually, one can show that the symmetric equilibrium is Pareto-efficient: no asymmetric profile Pareto-dominates it either.
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The authors thank valuable comments from an anonymous referee and an associate editor. An earlier version of this paper was presented at UECE Lisbon Meetings in Game Theory and Applications 2016. This work is supported by JSPS Grant-in-Aid for Scientific Research (C) (KAKENHI) 25380233, 16K0355301, and 17K03631.
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Iimura, T., Maruta, T. & Watanabe, T. Equilibria in games with weak payoff externalities. Econ Theory Bull 7, 245–258 (2019). https://doi.org/10.1007/s40505-018-0157-4
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DOI: https://doi.org/10.1007/s40505-018-0157-4
Keywords
- Existence of equilibrium
- Evolutionary equilibrium
- Weak payoff externalities
- Weakly unilaterally competitive games
- Weakly competitive games
- Potential games