Solutions Manual to Accompany Game Theory: An Introduction, 2nd Edition

Appendix F: Cooperative Games

Definition F.1   Let 2^N denote the set of all possible coalitions for the players N. If S = {i} is a coalition containing the single member i, we simply denote S by i.

Any function υ : 2^N → satisfying

is a characteristic function (of an n-person cooperative game).

Definition F.2   Let x_i be a real number for each i = 1, 2, . . ., n, with . A vector = (x₁, . . ., x_n) is an imputation if

• x_i ≥ υ(i) (individual rationality);
• (group rationality).

Each x_i represents the share of the value of υ (N) received by player i. The imputation is also called a payoff vector or an allocation, and we will use these words interchangeably.

Definition F.3   Let S ⊂ N be a coalition and let . The excess of coalition S ⊂ N for imputation is defined by

It is the amount by which ...