Games of bounded deviation, additivizations of games and asymptotic martingales (Q1176593)

Let \({\mathcal B}\) be a Boolean algebra of coalitions, usually subsets of a player space, and let \({\mathcal B}_ f\) denote the set of finite subalgebras of \({\mathcal B}\) ordered by inclusion. A game on \({\mathcal B}\) is a real-valued function \(v\) on \({\mathcal B}\) with \(v(\emptyset)=0\). For any game \(v\) on \({\mathcal B}\), the collection \(\overline{M}(v):=\{E(v|{\mathcal A}):\;{\mathcal A}\in{\mathcal B}_ f\}\) is a finite additive stochastic process of additive parts of the game on \({\mathcal B}_ f\). Here the additive part \(E(v|{\mathcal A})\) of \(v\) on \({\mathcal A}\) is also called the \({\mathcal A}\)- additivization of the game \(v\). The first results state that the game \(v\) is (super)additive if and only if \(\overline{M}(v)\) is a (super)martingale. Furthermore, it is shown that \(\overline M\) is an isomorhism between Gilboa's Banach space \(BS({\mathcal B})\) of games on \({\mathcal B}\) [see \textit{I. Gilboa}, Math. Oper. Res. 14, No. 1, 1-17 (1989; Zbl 0672.90110)] and bounded processes of the author [Trans. Am. Math. Soc. 279, 271-295 (1983; Zbl 0521.60052)]; between Shapley's games of bounded deviation \(BD({\mathcal B})\) and bounded \(F\)-processes of Armstrong. Specifically \(BS({\mathcal B})\) is the order solid hull of \(BD({\mathcal B})\) in the ordered vector space \(\mathbb{R}^{\mathcal B}\). Finally it is shown that the space of games \(v\) with \(\overline{M}(v)\) an asymptotic martingale is the largest subspace of \(BD({\mathcal B})\) admitting a continuous additivization which is given by the assignment \(v\to E(v|{\mathcal B})=\lim_{{\mathcal A}\in{\mathcal B}_ r}E(v|{\mathcal A})\).