Caller Number Five: Timing Games that Morph From One Form to Another

There are two well-studied timing games in economics: In a War of Attrition, having more predecessors helps; in a Pre-emption Game, more predecessors hurts. This paper introduces and explores a rich new spanning class of timing games with _rank-order payoffs_ that subsumes both timing games as special cases. This explains a wide array of economic and social timing phenomena. Indeed, assuming unobserved actions and complete information, we identify equilibria that display both smooth and explosive entry. Anticipation of later timing games influences current play --- from swelling the magnitude of pre-emptive explosions to truncating wars of attrition. We provide conditions for existence, characterize equilibria, and compare outcomes under unobservable and observable actions. Both settings' equilibria show similarities, and the unobservable actions case provides intuitive insights as to likely equilibria with observable actions. The analysis turns in part on Karlin's 1968 theory of total positivity.