Solution Concepts for Cooperative Games and Related Subjects

Abstract

With the characteristic function v at hand and supposing that some type of understanding is arrived at by the players, they have to divide the total savings v(N) of their grand coalition. A distribution of the amount v(N) among the players will be represented by a real-valued function x on the player set N satisfying the efficiency principle $\sum\limits_{j \in N} {x(j) = v(N)}$ . Here x(i) which is also denoted by x_i, represents the payoff to player i according to the i involved payoff function x. Because we generally suppose that the player set N = {1,2,...,n}, we usually identify a real-valued function x ∈ℝ^N on N with the n-tuple x = (x ₁, x₂,..., x_n) ∈ℝⁿ of real numbers. The vectors x ∈ℝⁿ which satisfy the efficiency principle x(N) = v(N) are called efficient payoff vectors or pre-imputations for the n-person game v. The nonempty set of all pre-imputations is denoted by I^* (y), i.e.

$$ {I^*}(v): = \{ x \in {^n}|x(N) = v(N)\} $$

((2.1))