Modeling large societies: Why countable additivity is necessary

Khan, M. Ali; Qiao, Lei; Rath, Kali P.; Sun, Yeneng

doi:10.1016/j.jet.2020.105102

Cited by 8 publications

(14 citation statements)

References 61 publications

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“…The next example builds on Khan, Qiao, Rath, and Sun [19]. 21 Consider T = N and let λ be a natural density.…”

Section: Examplementioning

confidence: 99%

“…a nonatomic probability on the power set is consistent with Savage's [31] approach and equivalent to having a qualitative probability on the players' space, satisfying axiom P6'. However, modelling the players' space in this generality implies that Nash equilibria might fail to exist (see Example 1 based on Khan, Qiao, Rath, and Sun [19]).…”

Section: Introductionmentioning

confidence: 99%

“…This not only complicates the technical analysis, but generates a conceptual loss. In fact, in an independent paper, Khan, Qiao, Rath, and Sun [19] show that the existence of Nash equilibria for any game with players' space (T, T , λ) is equivalent to the countable additivity of λ. Since the existence of Nash equilibria cannot be guaranteed with mere finite additivity, they study the existence of ε-Nash equilibria, thus overlapping our Corollary 2.…”

Section: Introductionmentioning

confidence: 99%

“…where M is a metric space with distance d. 19 For each t in T , m t (a, x) represents the message player t receives when he chooses strategy a and the distribution of players' strategies is x. In games with finitely many players, typically the message function…”

mentioning

confidence: 99%

“…The complete proof is available upon request. 19 To simplify notation, we assume that the message space is the same for all players. This is without loss of generality.…”

mentioning

confidence: 99%

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Equilibria of nonatomic anonymous games

Cerreia‐Vioglio¹,

Maccheroni²,

Schmeidler³

2020

Preprint

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We add here another layer to the literature on nonatomic anonymous games started with the 1973 paper by Schmeidler. More specifically, we define a new notion of equilibrium which we call ε-estimated equilibrium and prove its existence for any positive ε. This notion encompasses and brings to nonatomic games recent concepts of equilibrium such as self-confirming, peer-confirming, and Berk-Nash. This augmented scope is our main motivation. At the same time, our approach also resolves some conceptual problems present in Schmeidler (1973), pointed out by Shapley. In that paper the existence of pure-strategy Nash equilibria has been proved for any nonatomic game with a continuum of players, endowed with an atomless countably additive probability. But, requiring Borel measurability of strategy profiles may impose some limitation on players' choices and introduce an exogenous dependence among players' actions, which clashes with the nature of noncooperative game theory. Our suggested solution is to consider every subset of players as measurable. This leads to a nontrivial purely finitely additive component which might prevent the existence of equilibria and requires a novel mathematical approach to prove the existence of ε-equilibria.

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“…The next example builds on Khan, Qiao, Rath, and Sun [19]. 21 Consider T = N and let λ be a natural density.…”

Section: Examplementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

“…The complete proof is available upon request. 19 To simplify notation, we assume that the message space is the same for all players. This is without loss of generality.…”

mentioning

confidence: 99%

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