Convexity of Symmetrical TU-CPR Games

Having explicitly worked out general α- and β-core existence results in Chapter 4 the purpose of this chapter is devoted to the study of the convexity property in symmetrical TU-CPR games. As we have already mentioned whenever the core is nonempty, we know that there exists an incentive for mutual Cooperation in the grand coalition in order to realize the gains that are feasible through Cooperation. Core existence results can only explain that incentives for Cooperation exist but neither how strong these incentives are nor whether these incentives are also stable against small perturbation in the underlying economic structure. Especially, the last point is important in common pool situations where we observe subjects carrying on with Cooperation after a small exogenous shock. And indeed, Ostrom (1990) has reported Cooperation in more extreme events. For instance, one can observe despite a heavy dryness that Cooperation does not break down by using jointly a ground-water basin for irrigation purpose (cf. (Ostrom, 1990, pp. 69–82)). More formally spoken, can we expect that the core remains nonempty after small perturbations? This is in general true for convex games, since it is well known that the core of convex transferable Utility games is always nonempty and, further, that the core is relatively large with respect to the imputation set (Shapley (1971)). Therefore we can in general expect that the core remains nonempty against small perturbations in the parameter Space. Due to the generically large size of the core it is pertinent to establish convexity in an economical context.