Symmetric vs Asymmetric Equilibria and Stochastic Stability in a Dynamic Game of Legislative Lobbying

We study a 2-players stochastic dynamic symmetric lobbying differential game. Players have opposite interests; at any date, each player invests in lobbying activities to alter the legislation in her own benefit. The payoffs are quadratic and uncertainty is driven by a Wiener process. We prove that while a symmetric Markov Perfect Equilibrium (MPE) always exists, an asymmetric MPE only emerges when uncertainty is large enough. In the latter case, the legislative state converges to a stationary invariant distribution. Interestingly enough, the implications for the rent dissipation problem are much more involved than in the deterministic counterpart: the symmetric MPE still yields a limited social cost while the asymmetric may yield significant losses. We also characterize the most likely asymptotic state, in particular regarding the level of uncertainty.