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Budget Feasible Procurement Auctions

Nima Anari, Gagan Goel, Afshin Nikzad

2018
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Operations Research
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We consider a simple and well-studied model for procurement problems and solve it to optimality. A buyer with a fixed budget wants to procure, from a set of available workers, a budget feasible subset that maximizes her utility: Any worker has a private reservation price and provides a publicly known utility to the buyer in case of being procured. The buyer's utility function is additive over items. The goal is designing a direct revelation mechanism that solicits workers' reservation prices
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... decides which workers to recruit and how much to pay them. Moreover, the mechanism has to maximize the buyer's utility without violating her budget constraint. We study this problem in the prior-free setting; our main contribution is finding the optimal mechanism in this setting, under the "Small Bidders" assumption. This assumption, also known as the "small bid to budget ratio assumption", states that the bid of each seller is small compared to the buyer's budget. We also study a more general class of utility functions: submodular utility functions. For this class, we improve the existing mechanisms significantly under our assumption. J Hoeffding Bounds 90 We require any mechanism M = (A, P ) to satisfy the following properties: 1. Budget Feasibility: The sum of the payments made to the sellers should not exceed B, i.e., 2. Individual rationality: A winner i ∈ S is paid at least c i . 3. Truthfulness/Incentive-Compatibility: Reporting the true cost should be a dominant strategy for sellers, i.e. for all non-truthful reports c i from seller i, it holds that Defining a Benchmark. Among all mechanisms that satisfy the above properties, we are interested to the one that maximizes the utility of the buyer with respect to the following benchmark. Let U * (c, u) denote the utility of the omniscient mechanism, i.e. the utility of the knapsack optimization problem assuming that costs of the sellers are known to the buyer. 2 When there is no risk of confusion, we also denote U * (c, u) by U * for brevity. Definition 1. A mechanism M is α-competitive when α ∈ [0, 1] is the largest scalar for which the mechanism derives utility at least α · U * (c, u) for all c and u. Our main contribution is finding the mechanism that attains the highest possible competitive ratio in the class of truthful mechanisms. The Small Bidders Assumption Our small bidders assumption states that the cost of a single item is very small compared to the buyer's budget B. The Small Bidders Assumption. Assume that c max B, where c max = max i∈S {c i }. An alternative way to write the assumption is c max = o(B); in other words, we define bid-budget ratio of the market to be θ = cmax B and analyze our mechanisms for when θ → 0. Our mechanisms, however, do not need "very small" θ to perform well; this is elaborated during the discussion of our results in Section 3, where we note that even for θ as large as 1/20 our mechanisms have a very close performance to the optimum performance.

doi:10.1287/opre.2017.1693
fatcat:2jze4wzncnbnbndyssbsl3xkfa