Chapter summary

Coalitional games model situations in which players may cooperate to achieve their goals. It is assumed that every set of players can form a coalition and engage in a binding agreement that yields them a certain amount of profit. The maximal amount that a coalition can generate through cooperation is called the worth of the coalition.

In this and subsequent chapters we ask which coalitions of players will form, and, if the players are partitioned into a certain collection of coalitions, how each coalition will divide its worth. Specifically, in this chapter we present the model of coalitional games, and introduce various classes of coalitional games: revenue games, cost games, simple games, weighted majority games, market games, sequencing games, spanning tree games, and cost-sharing games. We define the notion of strategic equivalence between games.

We then turn to define the notion of a solution concept. A single-valued solution concept is a function that assigns to each coalitional game a vector in ℝN indicating the amount each player receives. A set solution concept is a function that assigns to each coalitional game a set of vectors in ℝN. Single-valued solution concepts model a judge or an arbitrator who has to recommend to the players how to divide the worth of the coalition among its members. A set solution concept may indicate which divisions are more likely than others.

Finally, using barycentric coordinates, we introduce a graphic representation of three-player coalitional games.