A convexity result for the range of vector measures with applications to large economies

Abstract

On a Boolean algebra we consider the topology u induced by a finitely additive measure µ with values in a locally convex space and formulate a condition on u that is sufficient to guarantee the convexity and weak compactness of the range of µ. This resultà la Lyapunov extends those obtained in (Khan, Sagara 2013) to the finitely additive setting through a more direct and less involved proof. We will then give an economical interpretation of the topology u in the framework of coalitional large economies to tackle the problem of measuring the bargaining power of coalitions when the commodity space is infinite dimensional and locally convex. We will show that our condition on u plays the role of the "many more agents than commodities"condition introduced by Rustichini and Yannelis in (1991). As a consequence of the convexity theorem, we will obtain two straight generalizations of Schmeidler's and Vind's Theorems on the veto power of coalitions of arbitrary economic weight.

Bayesian social aggregation with almost‐objective uncertainty

“…Theorem 3 requires the GH representations of all agents to be "polytopic" because Lemma A.1 only applies to finite-dimensional vector-valued measures, since it relies on a version of Lyapunov's convexity theorem for R N -valued measures. There are versions of Lyapunov's theorem for V-valued measures where V is an infinitedimensional locally convex vector space Sagara (2013, 2015), Greinecker and Podczeck (2013), Urbinati (2019)). These yield corresponding versions of the Dubins-Spanier theorem (by the same proof as our Lemma A.1).…”

The Walrasian objection mechanism and Mas-Colell’s bargaining set in economies with many commodities

“…In this paragraph, we study the specific case in which the space of agents is saturated, a property that has been introduced in Hoover and Keisler (1984) and that is now central in the economic literature. For a broad description of this property and its applications we mention the survey Fajardo and Keisler (2004); Keisler and Sun (2009) and its references, similar and more general conditions are studied in Greinecker and Podczeck (2013), Khan and Sagara (2013) and Urbinati (2019).…”

“…Even in this case, it is possible to reproduce the same results in an infinite dimensional setting under the additional assumption that the measure space of agents is saturated. This is done in Khan and Sagara (2013), Greinecker and Podczeck (2013) and Urbinati (2019). It shall be stressed, however, that the veto mechanism that follows from limiting the size of objecting coalitions is too weak to be used in a two-step process as the one that defines the bargaining set.…”

Fuzzy Core Equivalence in Large Economies: A Role for the Infinite-Dimensional Lyapunov Theorem

“…weak compactness and convexity of the Bochner integral of a multifunction (see Podczeck (2008); Sun and Yannelis (2008)), the bang-bang principle (see Sagara (2014, 2016)), and Fatou's lemma (see ; Khan, Sagara and Suzuki (2016)). For a further generalization of Theorem 2.1 to nonseparable locally convex spaces, see Greinecker and Podczeck (2013); Sagara (2015, 2016); Sagara (2017); Urbinati (2019). Another intriguing characterization of saturation in terms of the existence of Nash equilibria in large games is found in Keisler and Sun (2009).…”