We study opinion dynamics models where agents evolve via repeated pairwise interactions. In the compromise model, agents with sufficiently close real-valued opinions average their opinions. A steady state is reached with a finite number of isolated, noninteracting opinion clusters (“parties”). As the initial opinion range increases, the number of such parties undergoes a periodic bifurcation sequence, with alternating major and minor parties. In the constrained voter model, there are leftists, centrists, and rightists. A centrist and an extremist can both become centrists or extremists in an interaction, while leftists and rightists do not affect each other. The final state is either consensus or a frozen population of leftists and rightists. The evolution in one dimension is mapped onto a constrained spin-1 Ising chain with zero-temperature Glauber kinetics. The approach to the final state exhibits a nonuniversal long-time tail.
