We model political parties as adaptive decision-makers who compete in a sequence of elections. The key assumptions are that winners satisfice (the winning party in period t keeps its platform in t + 1) while losers search. Under fairly mild assumptions about losers' search rules, we show that the sequence of winning platforms is absorbed into the top cycle of the (finite) set of feasible platforms with probability one. This implies that if there is a majority rule winner then ultimately the incumbent party will espouse it. However, our model, unlike Downs–Hotelling or Kollman–Miller–Page, does not predict full convergence: we show, under weak assumptions about the out-party's search, that losing parties do not stabilize at the majority rule winner (should it exist). We also establish, both analytically and computationally, that the adaptive process is robust: if a majority rule winner nearly exists then the trajectory of winning platforms tends to be close to the trajectory of a process which actually has such a winner.