Timing Games: Probabilistic backrunning and spam

There are $n$ players who compete by timing their actions. An opportunity appears randomly on a time interval. Whoever takes an action the fastest after the opportunity has arisen wins. The occurrence of the opportunity is observed only with a delay. Taking actions is costly. We characterize the unique symmetric equilibrium of this game and study worst-case inefficiency of equilibria. Our main motivation is the study of probabilistic backrunning" on blockchains, where arbitrageurs want to place an order immediately after a trade that impacts the price on an exchange or after an oracle update. In this context, the number of actions taken can be interpreted as a measure of costly spam" generated to compete for the opportunity.