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Posted price mechanisms (PPM) constitute one of the predominant practices to price goods in online marketplaces and their revenue guarantees have been a central object of study in the last decade. We consider a basic setting where the buyers' valuations are independent and identically distributed and there is a single unit on sale. It is well-known that this setting is equivalent to the so-called i.i.d. prophet inequality, for which optimal guarantees are known and evaluate to 0.745 in general (equivalent to a PPM with dynamic prices) and 1 − 1/e ≈ 0.632 in the fixed threshold case (equivalent to a fixed price PPM). In this paper we consider an additional assumption, namely, that the underlying market is very large. This is modeled by first fixing a valuation distribution F and then making the number of buyers grow large, rather than considering the worst distribution for each possible market size. In this setting Kennedy and Kertz [Ann. Probab. 1991] breaks the 0.745 fraction achievable in general with a dynamic threshold policy. We prove that this large market benefit continue to hold when using fixed price PPMs, and show that the guarantee of 0.632 actually improves to 0.712. We then move to study the case of selling k identical units and we prove that the revenue gap of the fixed price PPM approaches 1−1/ √ 2kπ. As this bound is achievable without the large market assumption, we obtain the somewhat surprising result that the large market advantage vanishes as k grows.

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.

Dynamic resource allocation problems arise under a variety of settings. In “Survey of Dynamic Resource-Constrained Reward Collection Problems: Unified Model and Analysis,” Balseiro, Besbes, and Pizarro introduce a unifying model for a large class of dynamic optimization problems dubbed dynamic resource-constrained reward collection (DRC2) problems. Surveying the literature, they show that this class encompasses a variety of disparate and classical problems typically studied separately, such as dynamic pricing with capacity constraints, dynamic bidding with budgets, network revenue management, online matching, or order fulfillment. Furthermore, they establish that the DRC2 class is amenable to analysis by characterizing the performance of a central, certainty-equivalent heuristic. Notably, they provide a novel unifying analysis that isolates the drivers of performance, recovers as corollaries some existing specialized results, generalizes other existing results by weakening the assumptions required, and yields new results in specialized settings for which no such characterization was available.