Uncertainty-driven symmetry-breaking and stochastic stability in a generic differential game of lobbying

We study a 2-players stochastic differential game of lobbying. Players have opposite
interests; at any date, each player invests in lobbying activities to alter the legislation, the
continuous state variable of the game, 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, (two) asymmetric MPE only emerge when uncertainty
is large enough. In the latter case, the legislative state converges to a stationary invariant
distribution. We fully characterize existence and stochastic stability of the legislative state for
both types of MPE. We finally study the implications for rent dissipation asymptotically. We
show in particular that while the average rent dissipation is lower with asymmetric equilibria
relative to the symmetric, the former yield larger losses at the most likely asymptotic states
for large enough but moderate uncertainty.