Dual cores and effectivity functions (Q1595503)

For a cooperative game, a core is defined as the set of all payoff vector \(u_i\) which are ``stable'' in the following sense: whatever set of players \(S\) decides to form a coalition, the total amount \(v(S)\) that the coalition \(S\) can guarantee to its participants does not exceed the amount \(\sum_{i\in S} u_i\) that these players get under the payoff \(u_i\). The author introduces a natural notion of a dual core as the set of all payoffs vector \(u_i\) for which, for any possible coalition \(S\), the adversary coalition \(\overline S\) (formed by all players outside the coalition \(S\)) can guarantee that at least one player \(i\in S\) gets a payoff smaller than \(u_i\). The author proves that for many known properties of a core, natural similar results can be proven for the new notion of a dual core.