An infinite sequence of elections with no term limits is modelled. In each period a challenger with privately known preferences is randomly drawn from the electorate to run against the incumbent, and the winner chooses a policy outcome in a one-dimensional issue space. One theorem is that there exists an equilibrium in which the median voter is decisive: an incumbent wins re-election if and only if his most recent policy choice gives the median voter a payoff at least as high as he would expect from a challenger. The equilibrium is symmetric, stationary, and the behavior of voters is consistent with both retrospective and prospective voting. A second theorem is that, in fact, it is the only equilibrium possessing the latter four conditions — decisiveness of the median voter is implied by them.