Cores of cooperative games, superdifferentials of functions, and the Minkowski difference of sets
How can one describe a core \(C(v)\) of a cooperative game \(v\), i.e., the set of all feasible payments which cannot be improved upon by any coalition of players? For a simple game \(v_S\), with \(v_S(S')=1\) if \(S'\supseteq S\) and \(v_S(S')=0\) otherwise, the core is an easily described set. It is known that a general game can be reduced to simple games; namely, every cooperative game with a characteristic function \(v\) can be represented as a linear combination \(v=\sum \lambda_Sv_S\) of simple games \(v_S\). The values \(\lambda_S\) form a Möbius transform of \(v\). It is desirable to use this representation to describe the core \(C(v)\). It is known that if all the coefficients \(\lambda_S\) are non-negative (such games are called monotonic), then the core of \(v\) is the Minkowski sum of cores of the corresponding simple games: \(C(v)=\oplus \lambda_SC(v_S)\). A non-monotonic game can be represented as a difference between two monotonic ones: \(v=v^+-v^-\), where \(v^+\) contains all positive \(v_S\), and \(v^-\) contains all negative ones. The authors prove that \(C(v)=C(v^+)\ominus C(v^-)\), where the Minkowski difference \(A\ominus B\) is defined as \(\{z |z\oplus B\subseteq A\}\).
scientific article; zbMATH DE number 1497285