An approximately truthful-in-expectation mechanism for combinatorial auctions using value queries

This manuscript presents an alternative implementation of the truthful-in-expectation (TIE) mechanism of Dughmi, Roughgarden and Yan [2] for combinatorial auctions. Recall that in a combinatorial auction, m goods get allocated to n bidders. Each bidder i has a private valuation vi that describes its value vi(S) for each subset S of goods. The social welfare of an allocation is the sum of the bidders’ values for the goods received. The mechanism of [2] is presented in a “lottery-value” oracle model, where each bidder can be queried about his valuation by means of the following query: given a vector of probabilities over items x ∈ [0, 1], what is the expected value E[vi(x̂)], where x̂ is obtained by independently rounding each coordinate of x to 0 or 1 with probability xi. Such queries can be answered efficiently for certain valuation functions (in particular coverage functions), and this oracle model is a convenient framework for the presentation of the mechanism of [2]. On the other hand, lottery-value queries are #P-hard to answer for the class of matroid rank functions (see Section 4), and hence one can ask how realistic this model is in general. The purpose of this manuscript is to show that the model is “approximately realistic” in the sense that the mechanism of [2] can be implemented in the (weaker) value oracle model at the cost of relaxing the concept of truthfulness in expectation to approximate truthfulness in expectation. (Here, we mean appoximation within an arbitrarily small error, in the sense of an FPTAS.) In particular, we show that the maximal in distributional range (MIDR) allocation rule of [2] can be implemented as an approximately MIDR allocation rule in the value oracle model, and then we present a blackbox transformation from approximately MIDR allocation rules to approximately TIE mechanisms. First, let us define the approximate variants of MIDR and truthfulness in expectation. The exact variants are obtained by setting ǫ = 0.