Abstract
We provide conditions guaranteeing the existence of Nash equilibrium in games in which players’ preferences can be arbitrary binary relations. Our main result generalizes Reny’s (Economic Theory, forthcoming) existence result for games with ordered preferences and He and Yannelis’ (Economic Theory, forthcoming) existence result for abstract economies with non-ordered preferences.
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Notes
For an example in the context our paper, see Sect. 6.
Actually, Theorem 1 includes Theorem 4.2 in Reny (2013) when correspondence security is assumed to hold with respect to the entire set of players. It is straightforward to define the notion of correspondence target security with respect to a subset of players and to obtain a corresponding extension of Theorem 1, so that Theorem 4.2 in Reny (2013) is covered without the above additional assumption; see Sect. 7.9.
In the terminology of He and Yannelis (2014), this means that \(P_i\cap A_i\) has the continuous inclusion property at x.
While our definition of the weak continuous inclusion property does not explicitly require such i to belong to I(x), this is of course a consequence of the definition.
Note that the definition of correspondence target security with respect to J does not explicitly require the player for whom conditions (a) and (b) in Definition 5 hold to belong to J. However, this may be shown to be a consequence of the definition.
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We thank Nicholas Yannelis and an anonymous referee for helpful comments. Any remaining errors are, of course, ours.
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Carmona, G., Podczeck, K. Existence of Nash equilibrium in ordinal games with discontinuous preferences. Econ Theory 61, 457–478 (2016). https://doi.org/10.1007/s00199-015-0901-z
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DOI: https://doi.org/10.1007/s00199-015-0901-z