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.

翻译：有$n$名参与者通过选择行动时机展开竞争。某个机会在时间区间上随机出现。在机会出现后最快采取行动者获胜。机会的发生仅存在延迟观测。采取行动需要成本。我们刻画了该博弈的唯一对称均衡，并研究了均衡的最坏情形效率损失。本研究的主要动机在于分析区块链中的“概率性后置交易”现象——套利者期望在交易所发生影响价格的交易后，或在预言机更新后立即提交订单。在此背景下，采取行动的次数可解释为争夺机会所产生的高成本“垃圾信息”度量。