Competition and Recall in Selection Problems

We extend the prophet inequality problem to a competitive setting. At every period, a new realization of a random variable with a known distribution arrives, which is publicly observed. Then two players simultaneously decide whether to pick an available value or to pass and wait until the next period (ties are broken uniformly at random). As soon as a player gets a value, he leaves the market and his payoff is the value of this realization. In the first variant, namely the "no recall" case, the agents can only bid at each period for the current value. In a second variant, the "full recall" case, the agents can also bid for any of the previous realizations which has not been already selected. For each variant, we study the subgame-perfect Nash equilibrium payoffs of the corresponding game. More specifically, we give a full characterization in the full recall case and show in particular that the expected payoffs of the players at any equilibrium are always equal, whereas in the no recall case the set of equilibrium payoffs typically has full dimension. Regarding the welfare at equilibrium, surprisingly the best equilibrium payoff a player can have may be strictly higher in the no recall case. However, the sum of equilibrium payoffs is weakly larger when the players have full recall. Finally, we show that in the case of 2 arrivals and arbitrary distributions, the prices of Anarchy and Stability in the no recall case are at most 4/3, and this bound is tight.