A Course on Cooperative Game Theory

Introduction

Given a particular way of forming the grand coalition in a coalition form game, the marginal contribution of a player to the grand coalition is the amount by which the worth of the coalition increases when he joins the coalition of players that precede him. For instance, the marginal contribution of each share holder in a joint profit-making business is the additional amount of profit that he can guarantee to the coalition of players who have joined the business before him. The Weber set is the smallest convex set containing the set of marginal contribution vectors of the players.

In the next section of this chapter, we will introduce the Weber set of a game and show that it contains the core of the game (Weber 1988). In Section 6.3, we will study the properties of convex coalition form games. Convexity of a coalition form game may be interpreted as the condition where there are higher incentives for joining a coalition as the size of the coalition increases (Shapley 1971). This section also shows that for a convex game, the core is non-empty. It is then shown that the Weber set and the core of a game coincide if and only if the game is convex. It is also demonstrated that the bankruptcy game is an example of a convex game. Next, we show that the Shapley value for a convex game is an element of the core, which in turn demonstrates that the core of a convex game is non-empty. In Section 6.4, we will analyze random order values and their relations with the Weber set and the Shapley value. It is explicitly proven that O'Neill's (1982) random arrival rule, a solution to the bankruptcy problem, coincides with the Shapley value of the corresponding bankruptcy game.

The Weber Set and Core

Recall that given a particular arrangement or permutation of players in the grand coalition of a game, the corresponding marginal contribution vector gives each player his marginal contribution to the coalition formed by his entrance according to the specific permutation. Note that the set of all marginal contribution vectors in a game is a closed set.