Modeling infinitely many agents

He, Wei; Sun, Xiang; Sun, Yeneng

doi:10.3982/te1647

Cited by 29 publications

(33 citation statements)

References 42 publications

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“…In the following theorem, we show that the nowhere equivalence condition is indeed necessary and sufficient for the existence of pure strategy Nash equilibria in large games with any fixed uncountable compact metric space as the action space. The sufficiency part of this theorem is a special case of the corresponding result in Theorem 2 of [13]. The necessity part will be proved in Section 7.…”

Section: Definitionmentioning

confidence: 82%

“…The relationship that (iv) ⇒ (ii) was proved in Lemma 4.4 of [15]. See Lemmas 2 and 7 in [13] for the complete proof. Lemma 1.…”

Section: Preliminary Lemmasmentioning

confidence: 87%

“…Let F be a sub-σ-algebra of T , which can be understood as the σ-algebra generated by the function G that specifies the agents' characteristics. Such a probability space (T, F, λ) is called the characteristic type space in [13].…”

Section: Definitionmentioning

confidence: 99%

“…As noted in [13], the nowhere equivalence condition was motivated by the fact that various equilibrium properties in economics may require different agents with the same characteristic to choose different actions. Thus, one needs to distinguish the σ-algebra T in a probability space (T, T , λ) (modeling the space of agents) from the σ-algebra F generated by the mapping specifying the individual characteristics.…”

Section: Introductionmentioning

confidence: 99%

“…The action spaces for the large games in Example 9 of[13] consist of some parallel line segments. Since…”

mentioning

confidence: 99%

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The necessity of nowhere equivalence

Sun

2018

Advances in Mathematics

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We prove some regularity properties (convexity, closedness, compactness and preservation of upper hemicontinuity) for distribution and regular conditional distribution of correspondences under the nowhere equivalence condition. We show the necessity of such a condition for any of these properties to hold. As an application, we demonstrate that the nowhere equivalence condition is satisfied on the underlying agent space if and only if pure-strategy Nash equilibria exist in general large games with any fixed uncountable compact action space. 1 See [22] and [26] for the construction of Loeb spaces. 2 As noted in [15], atomless Loeb probability spaces are saturated. For some other applications of Loeb and saturated probability spaces, see, for example, [5], [7], [8], [10], [16], [19], [24], [34], [35], and [36]. 3 When the target space is a Banach space, one can also consider Bochner and Gelfand integration of correspondences. The same kind of regularity properties were shown to hold under Loeb/saturated probability spaces in [30], [38] and [39], while the necessity of saturation for these properties was indicated in [30] and [39]. A related issue on the purification of measure-valued maps on Loeb/saturated probability spaces was considered in [23], [25] and [31] with the necessity of saturation in [25] and [31]. 4 See Definition 1 in [13] and Definition 1 in Section 2 below.

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Section: Definitionmentioning

confidence: 82%

“…The relationship that (iv) ⇒ (ii) was proved in Lemma 4.4 of [15]. See Lemmas 2 and 7 in [13] for the complete proof. Lemma 1.…”

Section: Preliminary Lemmasmentioning

confidence: 87%

Section: Definitionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…The action spaces for the large games in Example 9 of[13] consist of some parallel line segments. Since…”

mentioning