The inner core and the strictly inhibitive set
The relation between the inner core and the strictly inhibitive set is explored. Let \(G= (N, V)\) denote a game in characteristic function form where \(N\) is the (finite) set of agents and \(V\) is a map from the set of subsets \(S\) of \(N\) into \(R^ s\). \(V(S)\) stands for the set of (non transferable) payoff vectors members of the coalition \(S\) can achieve for themselves. A game \(G\) is compactly (resp. convexly) generated if it can be represented in a form \((N, F)\) where each \(F(S)\) is compact (resp. convex). It is also called polyhedral if each \(F(S)\) is a compact polyhedron. \(S\) is a blocking coalition against a payoff vector \(u\) in \(V(N)\) if there is \(u'\) in \(V(S)\) s.t. \(u_ i'> u_ i\), all \(i\) in \(S\). The core is the set of all payoff vectors in \(V(N)\) against which there is no blocking coalition. Given a point \(\lambda\) in the simplex in \(R^ N\), \(H^ S_ \lambda= \{u'\in R^ S\mid \lambda^ S\cdot u'= 0\}\) denotes the set of utility transfers possible according to the exchange rates in \(\lambda^ S\), the projection of \(\lambda\) in the coordinates of \(S\). \(G_ \lambda\) denotes the game with utility transfers according to \(\lambda\), derived from \(G\), denoted by \((N, V_ \lambda)\) where \(V_ \lambda\) is defined by \(V_ \lambda(S)= V(S)+ H^ S_ \lambda\) for each \(S\subseteq N\). The inner core of a compactly generated game \(G\) is defined as the set of payoff vectors \(u\) s.t., for some \(\lambda\), \(u\) is in the core of \(G_ \lambda\). A randomized blocking plan is denoted by a pair of functions \((p, w)\) defined on the set of all subsets of \(N\). \(p(S)\geq 0\) for all \(S\subseteq N\) and \(\sum_{S\subseteq N} p(S)= 1\). \(p(S)\) represents the probability that a mediator will make \(S\) a blocking coalition. \(W(S)\in V(S)\) for each \(S\) and is the payoff vector that members of \(S\) would get if they did form a blocking coalition. A payoff vector \(U\) is strictly blocked by a randomized blocking plan \((p,w)\) if for each \(i\in N\) the expected payoff to player \(i\), conditional on being invited to participate in the blocking coalition, and if everyone accepts the mediator's invitation, is at least as much as his payoff \(u_ i\) under \(u\), and is larger for some \(i\). i.e. \(\sum_{S\ni i} p(S) w_ i(S)/ \sum_{S\ni i} p(S)\geq u_ i\), for all \(i\in N\) and \(>\) for some \(i\). A payoff vector \(u\) is strictly inhibitive if it is not strictly blocked by any randomized blocking plan. The set of all strictly inhibitive payoff vectors which are in \(V(N)\) is called the strictly inhibitive set. The following theorems are proved: Theorem 2: For compactly generated games, the inner core is a subset of the strictly inhibitive set; Theorem 3: For polyhedral games, the inner core equals the strictly inhibitive set; Theorem 3: If \(G\) is a compactly and convexly generated game, \(u\) is in the strictly inhibitive set of \(G\), and \(V(N)\) has supporting hyperplanes with only strictly positive normal vectors at \(u\), then \(u\) is in the inner core of \(G\). A characterization of the strictly inhibitive set in the context of an abstract formulation of a dynamic matching process is also provided.
scientific article; zbMATH DE number 222134