A Dual Model of Cooperative Value

Abstract

An expanded model of value in cooperative games is presented in which value has either a linear or a proportional mode, and NTU value has either an input or an output basis. In TU games, the modes correspond to the Shapley (1953) and proportional (Feldman (1999) and Ortmann (2000)) values. In NTU games, the Nash (1950) bargaining solution and the Owen Maschler (1989, 1992) value have a linear mode and an input basis. The egalitarian value (Kalai and Samet (1985)) has a linear mode and an output basis. The output-basis NTU proportional value (Feldman (1999)) and the input-basis variant, identified here, complete the model. The TU proportional value is shown to have a random marginal contribution representation and to be in the core of a positive convex game. The output-basis NTU variant is shown to be the unique efficient Hart and Mas-Colell consistent NTU value based on equal proportional gain in two-player TU games. Both NTU proportional values are shown to be equilibrium payoffs in variations of the bargaining game of Hart and Mas-Colell (1996). In these variations, players' probabilities of participation at any point in the game are a function of their expected payoff at that time. Limit results determine conditions under which players with zero individual worth receive zero value. Further results show the distinctive nature of proportional allocations to players with small individual worths. In an example with a continuum of players bargaining with a monopolist, the monopolist obtains the entire surplus.