Abstract
We give a framework for designing prophet inequalities for combinatorial welfare maximization. Instantiated with different parameters, our framework implies (1) an O(logm/loglogm)-competitive prophet inequality for subadditive agents, (2) an O(Dlogm/loglogm)-competitive prophet inequality for D-approximately subadditive agents, where D∈{1,…,m−1} measures the maximum number of items that complement each other, and (3) as a byproduct, an O(1)-competitive prophet inequality for submodular or fractionally subadditive (a.k.a. XOS) agents, matching the optimal ratio asymptotically. Our framework is computationally efficient given sample access to the prior, value queries and demand queries.
Highlights
Prophet inequalities are a classical topic in stopping theory
Upon seeing the inside of each box, the agent gets to make a choice: she can either (1) take the cash in the box and leave, or (2) let the current box expire, in which case the game proceeds with the remaining boxes
We give a framework for designing prophet inequalities for combinatorial welfare maximization, which implies an O(log m/ log log m)-approximate prophet inequality for subadditive agents, breaking the foregoing logarithmic barrier
Summary
Prophet inequalities are a classical topic in stopping theory. The problem is neat and natural: an agent plays a game, where there are n boxes, each containing a reward (e.g., some amount of cash). The agent can achieve this by executing a simple threshold-based protocol: accept the first box containing a reward exceeding a pre-calculated amount The existence of such 2-approximate protocols lead to the name “prophet inequalities.”. A threshold-based protocol can be translated directly into a take-it-or-leave-it offer – a buyer receives the item (i.e., she buys) iff her value exceeds the pre-calculated price, and so buying is preferred to not buying. Given this connection, various forms of auctions have been considered in the prophet inequality context (see, e.g., the recent survey by Lucier [23]). We generalize our results to accommodate valuations that are approximately subadditive, which have remained largely unexplored even in offline environments
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