We introduce a model of competing agents in a prophet setting, where rewards arrive online, and decisions are made immediately and irrevocably. The rewards are unknown from the outset, but they are drawn from a known probability distribution. In the standard prophet setting, a single agent makes selection decisions in an attempt to maximize her expected reward. The novelty of our model is the introduction of a competition setting, where multiple agents compete over the arriving rewards, and make online selection decisions simultaneously, as rewards arrive. If a given reward is selected by more than a single agent, ties are broken either randomly or by a fixed ranking of the agents. The consideration of competition turns the prophet setting from an online decision making scenario to a multi-agent game. For both random and ranked tie-breaking rules, we present simple threshold strategies for the agents that give them high guarantees, independent of the strategies taken by others. In particular, for random tie-breaking, every agent can guarantee herself at least $\frac{1}{k+1}$ of the highest reward, and at least $\frac{1}{2k}$ of the optimal social welfare. For ranked tie-breaking, the $i$th ranked agent can guarantee herself at least a half of the $i$th highest reward. We complement these results by matching upper bounds, even with respect to equilibrium profiles. For ranked tie-breaking rule, we also show a correspondence between the equilibrium of the $k$-agent game and the optimal strategy of a single decision maker who can select up to $k$ rewards.
翻译:在预言中,我们引入了竞争代理人的模式,在预言中,奖赏到达网上,并且可以立即和不可逆转地作出决定。奖赏从一开始就不为人知,但从已知的概率分布中抽取。在标准的预言中,一个单一代理人做出选择决定,以尽量扩大预期的奖赏。我们模式的新颖之处是引入竞争环境,让多个代理人争夺即将到来的奖赏,同时做出在线选择决定,当奖赏到达时,每个代理人都可以保证自己至少$\frac{1 ⁇ k+1美元。如果一个以上的代理人选择某一奖赏,关系就会随机地或由代理人固定的排名打破。对竞争的考虑将预言设置从网上决策情景转向多代理人的游戏。对于随机和排名分级的断线规则,我们为那些给予他们高度保证的代理人提出简单的门槛战略,独立于其他人采取的战略。特别是,对于随机断线的,每个代理人可以保证自己至少得到最高奖赏的${1 ⁇ +1美元。如果由某个代理人选择,或者由代理人固定的等级分。对于最优的社会福利的金额,至少可以将预估的奖赏。对于平级的奖赏,让我们的平级平级平级平的奖赏,甚至平级的奖赏。