The G 0 -dichotomy due to Kechris, Solecki and Todorčević characterizes the analytic relations having a Borel-measurable countable coloring. We give a version of the G 0 -dichotomy for Σ 0 ξ -measurable countable colorings when ξ ≤ 3. A Σ 0 ξ -measurable countable coloring gives a covering of the diagonal consisting of countably many Σ 0 ξ squares. This leads to the study of countable unions of Σ 0 ξ rectangles. We also give a Hurewicz-like dichotomy for such countable unions when ξ ≤ 2.
Abstract. The main aim of this paper is to give a simpler proof of the following assertion. Let A be an analytic non-σ-porous subset of a locally compact metric space E. Then there exists a compact non-σ-porous subset of A. Moreover, we prove the above assertion also for σ-P-porous sets, where P is a porosity-like relation on E satisfying some additional conditions. Our result covers σ-g -porous sets, σ-porous sets, and σ-symmetrically porous sets. , where structural properties of σ-porosity are studied. Namely, we consider, for several types of porosity, the following question:
The Shapley value of a cooperative transferable utility game distributes the dividend of each coalition in the game equally among its members. Given exogenous weights for all players, the corresponding weighted Shapley value distributes the dividends proportionally to their weights. A proper Shapley value, introduced in Vorob'ev and Liapounov (Game Theory and Applications, vol IV. Nova Science, New York, pp 155-159, 1998), assigns weights to players such that the corresponding weighted Shapley value of each player is equal to her weight. In this contribution we investigate these proper Shapley values in the context of monotone games. We prove their existence for all monotone transferable utility games and discuss other properties of this solution.
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