Abstract:

So far Nash implementation theory has mainly focused on establishing static properties of the equilibria. However, experimental evidence suggests that the fundamental question concerning any actual implementation of a specific mechanism is whether decentralized dynamic learning processes will actually converge to one of the equilibria promised by theory. Based on its attractive theoretical properties and the supporting evidence for these properties in the experimental literature, we focus on supermodularity as a robust stability criterion for Nash-efficient public goods mechanisms.

Five experiments have been conducted with mechanisms having Pareto-optimal Nash equilibria in public goods environments (Smith [1979], Harstad and Marrese [1981, 1982], Chen and Plott [1996] and Chen and Tang [1998]). Sometimes the data converged quickly to the Nash equilibria; other times it did not. Chen and Plott (1996) assessed the performance of the Groves-Ledyard mechanism under different punishment parameters. They found that by varying the punishment parameter the dynamics and stability changed dramatically. This finding was replicated by Chen and Tang (1998) with twenty-one independent sessions and a longer time series in an experiment designed to study the learning dynamics. Chen and Tang (1998) also studied the Walker mechanisms in the same economic environment. They found that all sessions of the Groves-Ledyard mechanism under a high punishment parameter converged very quickly to its stage game Nash equilibrium and remained stable, while the same mechanism did not converge under a low punishment parameter; the Walker mechanism did not converge to its stage game Nash equilibrium either. Because of its good dynamic properties, the Groves-Ledyard mechanism under a high punishment parameter had far better performance than the Groves-Ledyard mechanism under a low punishment parameter and the Walker mechanism, evaluated in terms of system efficiency, close to Pareto optimal level of public goods provision, less violations of individual rationality constraints and convergence to its stage game equilibrium. All these results are statistically highly significant. These results illustrate the importance to design mechanisms which not only have good static properties, but also good dynamic stability properties. Only when the dynamics lead to the convergence to the static equilibrium, can all the nice static properties be realized.

This paper demonstrates that given a quasilinear utility function the Groves-Ledyard mechanism is a supermodular game if and only if the punishment parameter is above a certain threshold while none of the Hurwicz, Walker and Kim mechanisms are supermodular games. These results generalize a previous convergence result on the Groves-Ledyard mechanism by Muench and Walker (1983), and provide a theoretical explanation for the experimental findings of Smith (1979), Harstad and Marrese (1982), Chen and Plott (1996), and Chen and Tang (1998).

The above results also raise the question of whether it is possible to find a Nash mechanism that implements the Lindahl allocations and also possesses a robust stability property, at least given a quasilinear utility function. We provide a positive answer to this question by presenting a new family of Nash mechanisms which implement Lindahl allocations in a general environment; with quasilinear utility functions the new family of mechanisms is supermodular games give a suitable choice of parameters, which implies that they converge under a wide class of learning dynamics, including Bayesian learning, adaptive learning, fictitious play and many others. Thus theoretically the new mechanisms have similar stability properties as the Groves-Ledyard mechanism and are also individually rational.