Pure-strategy Nash equilibria in large games: characterization and existence

In this paper, we first characterize pure-strategy Nash equilibria in large games restricted with countable actions or countable payoffs. Then, we provide a counterexample to show that there is no such characterization when the agent space is an arbitrary atomless probability space (in particular, Lebesgue unit interval) and both actions and payoffs are uncountable. Nevertheless, if the agent space is a saturated probability space, the characterization result is still valid. Next, we show that the characterizing distributions for the equilibria exist in a quite general framework. This leads to the existence of pure-strategy Nash equilibria in three different settings of large games. Finally, we notice that our characterization result can also be used to characterize saturated probability spaces.

1. Interested readers can refer to (Khan and Sun 1999, 2002), Kalai (2004), Khan et al. (2013).

2. Blonski (1999) provides a characterization result for the case of two actions and the result of Blonski (2005) is confined to a finite action space. Our paper works on both countable and uncountable action spaces.

3. The large game discussed here is a generalization to the large non-anonymous games discussed in (Khan and Sun 1999, 2002).

4. The proof of our first result uses Bollobas and Varopoulos (1975)’s extension of the famous marriage theorem (or the Hall’s theorem) and the proof of the third result relies on Keisler and Sun (2009)’s result on the distributional properties of correspondence on saturated probability spaces.

5. If initially, \((A_i)_{i\in I}\) are not disjoint, we can always introduce a disjoint set of action sets \((A_i')_{i\in I}\) by adding an index dimension to the original action sets while keeping the same topological structure. For example, if \(A_1=A_2=\{a,b\}\), we can let \(A_1'=\{(1,a),(1,b)\}\) and \(A_2'=\{(2,a),(2,b)\}\).

6. Unless otherwise specified, any topological space discussed in this paper is tacitly understood to be equipped with its Borel \(\sigma \)-algebra (the \(\sigma \)-algebra generated by the family of open sets) and measurability is defined in terms of it.

7. Such a large game is often called a large non-anonymous game in the literature. See e.g., Khan and Sun (2002).

8. This lemma was also used by Yu and Zhang (2007) to show the existence of pure-strategy equilibria in games with countable actions.

9. Throughout the paper, we refer to results previously available in the literature as “Lemma”.

10. This payoff function is similar to a payoff function used in Khan et al. (1997).

11. The concept of saturated was firstly introduced by Hoover and Keisler (1984). For a more detailed explanation on the properties of the space, see, e.g., Keisler and Sun (2009) and Sun and Zhang (2015).

12. Note that the map \(f_{\bar{\mu }}: A\times \fancyscript{U}\rightarrow R\) defined by \(f_{\bar{\mu }}(a,u)=u(a,({\bar{\mu }}_i)_{i\in I})\) is continuous (see Theorem 46.10 in Munkres (2000)).

13. See e.g., Lemma 16.4 in Aliprantis and Border (1999).

14. By the Measurable Maximum Theorem, \(B_i^{{\bar{\mu }}}\) admits a measurable selection \(g_i\) and hence, \({\bar{\eta }}=(\lambda _ig_i^{-1})_{i\in I}\) is a trivial element of \(\Phi ({\bar{\mu }})\).

15. Continuous real function on compact metric space is also uniformly continuous.

16. Just let \(u_m'\) be a little bit bigger than \(u_m\) around the area of \(a_m\).

17. See Theorem 10.7 in Aliprantis and Border (1999).

18. It is also straightforward to generalize this result to the case where \(I\) is any finite or countable set.

The authors wish to thank Haomiao Yu and Yongchao Zhang for stimulating conversation and correspondence. We are also indebted to anonymous referees for their careful reading and useful comments. Haifeng Fu also wish to thank Jolene Yi-Lin Tan for proofreading the paper. This paper was presented at the 2013 Asian Meeting of the Econometric Society, Singapore, August 2–4, 2013: we thank the participants for their constructive comments

Authors and Affiliations

1. International Business School Suzhou, Xi’an Jiaotong-Liverpool University, Jiangsu, 215103, China

   Haifeng Fu

2. Department of Mathematics, National University of Singapore, Singapore, Singapore

   Ying Xu & Luyi Zhang

Corresponding author

Correspondence to Haifeng Fu.

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