Convexity of Asymmetrical TU-CPR Games

As we have shown by Theorem 5.1 on page 107 in Chapter 5 a symmetrical CPR-game with constant marginal costs and a strictly concave Joint production function is convex. We have mentioned that convexity can be interpreted (a) as an incentive for large-scale Cooperation and (b) that the core for a convex game remains nonempty given small perturbation in the parameter space. Large-scale Cooperation and core stability are indications (a) for strong incentives for Cooperation into the grand coalition and (b) that Cooperation does not break down after an exogenous shock has occurred. But now the crucial question arises whether the convexity property can be extended to the case of asymmetrical endowments, more general concave production functions and more arbitrary cost functions. Otherwise, our interpretation that we can expect for common pool situations strong incentives for Cooperation into the grand coalition and, in addition, stability of Cooperation despite observed shocks, would be too strong. In such a case our interpretation would be only true for a small subclass of TU-CPR games. In other words, convexity of TU-CPR games would be a seldom event; and therefore we could not explain mutual Cooperation and stability of Cooperation for a large dass of common pool problems. ... mehrThus, we would have to search for another explanation why the phenomena of Cooperation in contrast to the noncooperative predictions can be observed in so many Situation where appropriators have access to a resource which can be jointly managed. Fortunately, we can extend the convexity result to a large subclass of common pool situations and the Interpretation above is applicable. Moreover, in cases where convexity fails to hold we can derive a weaker convexity result, the so-called average-convexity. For the study of average-convexity in common pool games we refer the reader to Driessen and Meinhardt (2001).