(Average-) Convexity of Common Pool and Oligopoly Tu-Games

Abstract

The paper studies both the convexity and average-convexity properties for a particular class of cooperative TU-games called common pool games. The common pool situation involves a cost function as well as a (weakly decreasing) average joint production function. Firstly, it is shown that, if the relevant cost function is a linear function, then the common pool games are convex games. The convexity, however, fails whenever cost functions are arbitrary. We present sufficient conditions involving the cost functions (like weakly decreasing marginal costs as well as weakly decreasing average costs) and the average joint production function in order to guarantee the convexity of the common pool game. A similar approach is effective to investigate a relaxation of the convexity property known as the average-convexity property for a cooperative game. An example illustrates that oligopoly games are a special case of common pool games whenever the average joint production function represents an inverse demand function.

Non-binding agreements and fairness in commons dilemma games

“…In addition, concentrating on stylized cooperative bargaining arguments offers some insight into how agreement points can be stabilized. Theoretical results from cooperative game theory that foster the hypothesis -obtained from field studies -that the commons can be successfully managed by selfish individuals have been provided by [7,8,9], and by [22,23].…”

“…[23,7,8,9]). Empirical results are reexamined only for two representative games of the Bloomington type.…”

Convexity of production, common pool and oligopoly games: a survey

“…Definition 1 (cf. [2,8,4]). Let N be a finite set of individuals (players) and let w i ≥ 0 denote the endowment of player i.…”

“…In the context of the study on the convexity property for the common pool game N, v of (8), it turns out that the general framework needs additional assumptions about the cost function c as well as the average joint production function h (cf. [2]). By Theorem 2, for the special case with a linear cost function, the convexity of the common pool game is valid without any further assumptions on the average joint production function h.…”

“…(1) The main purpose of this survey paper is to highlight the uniform approach to establish a new proof of the convexity property for the so-called oligopoly game (to be done in Section 4) similar to the already existing proof of the convexity property for the socalled common pool game (cf. [2], to be recalled in Section 3). The essential part of the forthcoming proof technique is based on the interchangeability of both players i and j concerning the convexity condition (1).…”