In "Lobbying and Policy Extremism in Repeated Elections", Gleason Judd studies a model of repeated elections that features privately informed politicians and ideologically extreme lobby groups. He shows that if office incentives are high, then all equilibria feature strong parties: liberal politicians all choose the same policy, as do all conservative politicians. When the effectiveness of money approaches zero, these equilibrium policies converge to the median, providing a dynamic version of the median voter theorem. When the effectiveness of money becomes large, however, the most polarized strongly partisan equilibria become arbitrarily extremist, and thus highly effective lobbying creates the possibility of arbitrarily extreme policy outcomes. In case the effectiveness of money is not large, lobbying incentives can push politicians to choose more moderate policies than they otherwise would, and an increase in the eectiveness of money can increase the welfare of the median voter.
Bils, Peter, John Duggan, and Gleason Judd. "Lobbying and policy extremism in repeated elections." Journal of Economic Theory 193 (2021).