arXiv subjects
Publications and source records attributed to Youcef Askoura.
We define and investigate some preliminary properties of an integral of vector valued functions and its canonically associated indefinite capacity. More precisely, we adapt the Dunford integral to the Choquet integration setting for Banach-valued functions.
We prove a conditional expectation bang-bang principle. Based on properties of the conditional expectation vector measure, we establish that the conditional expectation of a set-valued mapping coincides with the conditional expectation of the set of selections of its extreme points part. As a by-product, we obtain straightforwardly a purification principle.
We prove a compactness criterion in $L^p(μ,X)$: a subset of $L^p(μ,X)$ is relatively norm compact iff the set of integrals of its functions over any measurable set is relatively norm compact, it satisfies the Fréchet oscillation restriction condition and it is p-uniformly integrable. The proof is elementary.
We consider the interim core of normal form cooperative games and exchange economies with incomplete information based on the partition model. We develop a solution concept that we can situate roughly between Wilson's coarse core and Yannelis's private core. We investigate the interim negotiation of contracts and address the two situations of contract delivery: interim and ex post. Our solution differs from Wilson's concept because the measurability of strategies in our solution is postponed until the consumption date (assumed with respect to the information that will be known by the players at the consumption date). For interim consumption, our concept differs from Yannelis's private core because players can negotiate conditional on proper common knowledge events in our solution, which strengthens the interim aspect of the game, as we will illustrate with examples.
We study the core of normal form games with a continuum of players and without side payments. We consider the weak-core concept, which is an approximation of the core, introduced by Weber, Shapley and Shubik. For payoffs depending on the players' strategy profile, we prove that the weak-core is nonempty. The existence result establishes a weak-core element as a limit of elements in weak-cores of appropriate finite games. We establish by examples that our regularity hypotheses are relevant in the continuum case and the weak-core can be strictly larger than the Aumann's $α$-core. For games where payoffs depend on the distribution of players' strategy profile, we prove that analogous regularity conditions ensuring the existence of pure strategy Nash equilibria are irrelevant for the non-vacuity of the weak-core.
We formulate a multi-valued version of the Tietze-Urysohn extension theorem. Precisely, we prove that any upper semicontinuous multi-valued map with nonempty closed convex values defined on a closed subset (resp. closed perfectly normal subset) of a completely normal (resp. of a normal) space $X$ into the unit interval $[0,1]$ can be extended to the whole space $X$. The extension is upper semicontinuous with nonempty closed convex values. We apply this result for the extension of real semicontinuous functions, the characterization of completely normal spaces, the existence of Gale-Mas-Colell and Shafer-Sonnenschein type fixed point theorems and the existence of equilibrium for qualitative games.