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.
There are two varieties of timing games in economics: In a war of attrition, more predecessors helps; in a pre-emption game, more predecessors hurts. In this paper, we introduce and explore a spanning class with rank-order payoffs that subsumes both as special cases. In this environment with unobserved actions and complete information, there are endogenously-timed phase transition moments. We identify equilibria with a rich enough structure to capture a wide array of economic and social timing phenomena -- shifting between phases of smooth and explosive entry. We introduce a tractable general theory of this class of timing games based on potential functions. This not only yields existence by construction, but also affords rapid characterization results. We then flesh out the simple economics of phase transitions: Anticipation of later timing games influences current play -- swelling pre-emptive atoms and truncating wars of attrition. We also bound the number of phase transitions as well as the number of symmetric Nash equilibria. Finally, we compute the payoff and duration of each equilibrium, which we uniformly bound. We contrast all results with those of the standard war of attrition.
timing game, war of attrition, pre-emption game, potential function, Nash equilibrium
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