Competitive Equilibrium in a Game

Abstract

First Interpretation: The game has a finite number of outcomes, but this set's distributions are also considered as outcomes of the game. These probability distribution outcomes are then points in a linear space. Second Interpretation: The game's outcome is a bundle of goods and, in our case, public goods. We shall give a version of the theory for private goods at the end of this paper. Let us now consider, like von Neumann and Morgenstern, a zero-sum twoperson game between a coalition S and its complement coalition I - S, using an arbitrarily chosen linear function defined on the vector space of outcomes. We denote the outcome by y and the linear function by q. Here S wants to maximize and I - S wants to minimize qy. According to von Neumann and Morgenstern, this game has a value called the characteristic function which is written as r(S, q). Using such a function calls for some explanation and comparison with the other methods employed in economic theory and game theory. The linear forms q