Reconstructing a simple game from a uniparametric family of allocations (Q384514)

The author deals with the class \(SG\) of cooperative monotonic simple games \((N,v)\), and studies the class \(\{\Psi_{\alpha}\}_{0\leq \alpha\leq 1}\) of \(\alpha\)-binomial semivalues \(\Psi_{\alpha}=(\Psi_{\alpha})_{i\in N}\) defined on \textit{SG} by the formula: \[ (\Psi_{\alpha} [N,v])_i = \sum_{S\subset N, i\in S} \alpha^{s-1}(1-\alpha)^{n-s}[v(S)-v(S\setminus \{ i\})] \] for \(i\in N\) and \((N,v)\in SG\). For a simple game \((N,v)\), several notions are introduced like the vector of utility by size \(a_s(v)\) with \(s=1,\dots,n\), the \(\alpha\)-potential \(P_\alpha (N,v)\) of a semivalue \((N,v)\), the vector \(a_i(v)\) of utility by size for player \(i\), the size-by-size equivalence of two simple games, the size-by-size equivalence of two players in two different simple games, and the concept of equal \(\alpha\)-allocation by binomial semivalues to two simple games. In terms of these notions, the authors discuss different properties of simple games and \(\alpha\)-binomial semivalues. Besides, different mutual relationships between those notions are derived.