On a Competitive Selection Problem

We consider the problem in which n items arrive to a market sequentially over time, where two agents compete to choose the best possible item. When an agent selects an item, he leaves the market and obtains a payoff given by the value of the item, which is represented by a random variable following a known distribution with support contained in [0, 1]. We consider two different settings for this problem. In the first one, namely competitive selection problem with no recall, agents observe the value of each item upon its arrival and decide whether to accept or reject it, in which case they will not select it in future. In the second setting, called competitive selection problem with recall, agents are allowed to select any of the available items arrived so far. For each of these problems, we describe the game induced by the selection problem as a sequential game with imperfect information and study the set of subgame-perfect Nash equilibrium payoffs. We also study the efficiency of the game equilibria. More specifically, we address the question of how much better is to have the power of getting any available item against the take-it-or-leave-it fashion. To this end, we define and study the price of anarchy and price of stability of a game instance as the ratio between the maximal sum of payoffs obtained by players under any feasible strategy and the sum of payoffs for the worst and best subgame-perfect Nash equilibrium, respectively. For the no recall case, we prove that if there are two agents and two items arriving sequentially over time, both the price of anarchy and price of stability are upper bounded by the constant 4/3 for any value distribution. Even more, we show that this bound is tight.