Linear and TU Games

We will define a linear game as follows: The value to be distributed among the members of the coalition is a constant V. Let the members of the coalition be C = {i, j, . . . , k} and

where yt is the payout to member i and k is an idiosyncratic constant. For a transferable utility, TU, game

That is, 4i [ C, ki = 1. Clearly a linear game is a generalization of a TU game. In the TU game the coalition simply determines how its surplus is divided among the members of the coalition. We might think of a linear game instead as a mechanism in which the coalition does not determine the payoffs but determines the distribution of a resource, in limited supply, from which the agents produce their payoffs with productivities that vary among them but that are constant for each agent (that is, there are no diminishing returns). The productivity of i is k. Thus, by allocating the resource, the coalition indirectly determines the payoffs of the agents.

For a bargaining power game, the solution is characterized by McCain (2013, Chapter 6)

subject to appropriate constraints, where g(S) is the payoff to group S in excess of the disagreement point for group S. For a voting game, however, either pS = 1 if S is a minimal winning voting block or pS = 0 otherwise. For this chapter we will assume that for |S| > 1, the disagreement point is expressed simply by a nonnegativity constraint. That is, no group of two or more agents can gain any benefit from working together apart from their participation in a minimal willing coalition.

 
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