How to Compute Von Neumann-morgenstern Solutions

Cooperative games and von Neumann-Morgenstern solution. Situations when all participants collaborate with each other is known as a cooperative game. One way to describe a cooperative game is to assign, to every subset S ⊆ N def = {1, . . . , n} of the set of all the participants, the value v(S) that describes what players from S can gain if they collaborate between themselves only. Such subsets S are called coalitions. We consider cooperative situations, so if two disjoint coalitions S and S′ collaborate, they should be able to gain not less than they would get on their own, i.e., we should have v(S ∪ S′) ≥ v(S) + v(S′). It always makes sense to consider only gains due to collaboration, so if v({i}) ̸= 0, then we can take v′(S) = v(S) − ∑