Gradualism in dynamic influence games

We analyze a dynamic model in which players compete in each period in an all-pay competition to have their ideal action implemented. The winning policy at each competition is implemented for that period, but only if it is ranked higher than the status quo, according to some exogenous order. We show that in any subgame perfect equilibrium of this game, the dynamic process is gradual, i.e., in each period: (i) there is a substantial probability that a higher ranked action is implemented, but,(ii) the probability that the highest ranked action is implemented is bounded away from one. "Progress" is thus inevitable but relatively slow. In an application to a one-dimensional policy space, we show the existence of a fully gradual equilibrium in which all feasible actions are implemented at some period, before converging to the preferred action of the median voter. Such full gradualism arises when the polity is sufficiently polarized and interested parties have policy -as opposed to winning- motives.