Generalized Prophet Inequalities for Online Auctions

Abstract: Prophet Inequalities for online auctions, as many classical auction settings, traditionally consider agents who strive to maximize quasi-linear utility, i.e. their obtained value minus the price paid. This simplified view of the agents' objectives does not capture real-world settings where agents are almost always constrained by budgets, and can have different objectives altogether (e.g., very prominently, value maximizers in autobidding). We generalize classic prophet inequality auction settings to accommodate unknown, varying types of agent objectives and budgets, modeling them as stochastic input similar to the traditional valuation distributions. In this new model, we analyze (anonymous) posted-price mechanisms, one main category of which are the balanced prices central to many auction results in the prophet inequality model. We demonstrate both the resilience and limits of balanced price mechanisms for generalized prophet inequalities with stochastic agent objectives, budgets and valuations: our generalization of the balanced prices framework incorporates budget constraints and mixed objectives. We show constant competitive ratios for some the most central settings, spanning all the way up to agents with subadditive valuation functions. On the way, we establish trade-offs between competitive ratio and agents' price-sensitivity, and give a general reduction from the budgeted setting to budget-free value maximizers.