Positively Homogeneous Saddle-Functions and Euler's Theorem in Games

Abstract

Connections are made between solution concepts for games in characteristic function form and Euler's Theorem underlying the neo-classical theory of distribution in which the total output produced is imputed to the marginal products of the inputs producing it. The assumptions for Euler's Theorem are constant returns (positive homogeneity) and differentiability of the production function. Representing characteristic functions in a vector space setting, marginal products of commodity inputs are translated as marginal products of individuals. Marginal products for discrete (resp. infinitesimal) individuals are defined by discrete (resp. infinitesimal) directional derivates. Additivity of directional derivatives underlies the definition of differentiability in both discrete and infinitesimal settings. A key distinction is between characteristic functions defined by von Neumann and Morgenstern (vNM) which do not necessarily exhibit concavity and characteristic function that do. A modification of the definition, interpreted as introducing ``property rights,'' implies concavity. The Shapley value is a redefinition of an individual's marginal product for a (vNM) characteristic function. Concave characteristic functions do not require such redefinition. Concave characteristic functions imply the existence of positively homogeneous saddle-function functions whose saddle-points represent equilbria of the game. The saddle-point property applies to games with populations consisting of any number of individuals each type. When there is a small integer number of each type, the saddle-point property is often, but not always, inconsistent with the Euler condition, that each individual receives its marginal contribution. Conversely, the saddle-point condition is typically, but not always, consistent with the Euler condition when there are a large number of each type.