On the purification of Nash equilibria of large games

Carmona, Guilherme

doi:10.1016/j.econlet.2004.04.008

Cited by 14 publications

(8 citation statements)

References 6 publications

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“…This last conclusion generalizes the result in Carmona (2004b): since for the nonequicontinuous game in our example there is no pure ε -equilibrium for all ε > 0 small enough, then it follows that no mixed strategy equilibrium can be ε -purified.…”

Section: On the Asymptotic Existence Resultssupporting

confidence: 59%

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On the Existence of Pure-Strategy Equilibria in Large Games

Carmona

Podczeck²

2008

SSRN Journal

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Over the years, several formalizations and existence results for games with a continuum of players have been given. These include those of Schmeidler (1973), Rashid (1983), Mas-Colell (1984), Khan and Sun (1999) and Podczeck (2007a).The level of generality of each of these existence results is typically regarded as a criterion to evaluate how appropriate is the corresponding formalization of large games.In contrast, we argue that such evaluation is pointless. In fact, we show that, in a precise sense, all the above existence results are equivalent. Thus, all of them are equally strong and therefore cannot rank the different formalizations of large games.

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Section: On the Asymptotic Existence Resultssupporting

confidence: 59%

“…As shown by Khan, Rath, and Sun (1997), Schmeidler's theorem does not extend to general games -in fact, one has to assume that either the action space or the family of payoff functions is denumerable in order to guarantee the existence of a pure strategy equilibrium (see Khan and Sun (1995b) and Carmona (2008)). …”

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confidence: 99%

On the Existence of Pure-Strategy Equilibria in Large Games

Carmona

Podczeck²

2008

SSRN Journal

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“…See Khan et al (2013b) for a recent discussion of related issues. 16 Approximated versions of the result in Schmeidler (1973) have been given for a large but finite number of players (Rashid 1983;Carmona 2004Carmona , 2008. are countable and compact, conditions for the existence of pure Nash equilibrium are given in Khan and Sun (1995) and then generalized in Yu and Zhang (2007).…”

Section: Discussionmentioning

confidence: 99%

Strict Nash equilibria in non-atomic games with strict single crossing in players (or types) and actions

Bilancini

Boncinelli

2015

Econ Theory Bull

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In this paper, we study games where the space of players (or types, if the game is one of incomplete information) is atomless and payoff functions satisfy the property of strict single crossing in players (types) and actions. Under an additional assumption of quasisupermodularity in actions of payoff functions and mild assumptions on the player (type) space-partially ordered and with sets of uncomparable players (types) having negligible size-and on the action space-lattice, second countable and satisfying a separation property with respect to the ordering of actions-we prove that every Nash equilibrium is essentially strict. Furthermore, we show how our result can be applied to incomplete information games, obtaining the existence of an evolutionary stable strategy, and to population games with heterogeneous players.

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“…The fact that the proof presented here, and the alternative one presented in [58] (see Remarks 1 and 4 below for the difference in approach), rely on the Shapley-Folkman theorem is of some substantive consequence for mathematical economics: the subject can be seen (admittedly in hindsight) to have evolved by providing asymptotic implementations, and rates of convergence, of the fundamental results obtained through Lyapunov's theorem or its nonstandard analogues (as in [39,34,35,36]), and thereby systematically replacing the tools for these idealized (continuum) objects by their counterparts for finite ones. For this trajectory, see [54,10,2] and their followers: in alphabetical order, [3,4,5,6,7,8,17,18,19,20,25,29,30,44,46,47,48,53,63]), as well as Remark 7 below. (An expression of this point of view is also available in [56,37,57].)…”

Section: Introductionmentioning

confidence: 99%

The Shapley–Folkman theorem and the range of a bounded measure: an elementary and unified treatment

Khan

Rath

2012

Positivity

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On the purification of Nash equilibria of large games

Cited by 14 publications

References 6 publications

On the Existence of Pure-Strategy Equilibria in Large Games

On the Existence of Pure-Strategy Equilibria in Large Games

Strict Nash equilibria in non-atomic games with strict single crossing in players (or types) and actions

The Shapley–Folkman theorem and the range of a bounded measure: an elementary and unified treatment

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