The Competition Complexity of an auction setting refers to the number of additional bidders necessary in order for the (deterministic, prior-independent, dominant strategy truthful) Vickrey-Clarke-Groves mechanism to achieve greater revenue than the (randomized, priordependent, Bayesian-truthful) optimal mechanism without the additional bidders.We prove that the competition complexity of n bidders with additive valuations over m independent items is at most n(ln(1 + m/n) + 2), and also at most 9 √ nm. When n ≤ m, the first bound is optimal up to constant factors, even when the items are i.i.d. and regular. When n ≥ m, the second bound is optimal for the benchmark introduced in [EFF + 17a] up to constant factors, even when the items are i.i.d. and regular. We further show that, while the Eden et al. benchmark is not necessarily tight in the n ≥ m regime, the competition complexity of n bidders with additive valuations over even 2 i.i.d. regular items is indeed ω(1).Our main technical contribution is a reduction from analyzing the Eden et al. benchmark to proving stochastic dominance of certain random variables.
Martin Weitzman's "Pandora's problem" furnishes the mathematical basis for optimal search theory in economics. Nearly 40 years later, Laura Doval introduced a version of the problem in which the searcher is not obligated to pay the cost of inspecting an alternative's value before selecting it. Unlike the original Pandora's problem, the version with nonobligatory inspection cannot be solved optimally by any simple ranking-based policy, and it is unknown whether there exists any polynomial-time algorithm to compute the optimal policy. is motivates the study of approximately optimal policies that are simple and computationally efficient. In this work we provide the first non-trivial approximation guarantees for this problem. We introduce a family of "commi ing policies" such that it is computationally easy to find and implement the optimal commi ing policy. We prove that the optimal commi ing policy is guaranteed to approximate the fully optimal policy within a 1 − 1 e = 0.63 . . . factor, and for the special case of two boxes we improve this factor to 4/5 and show that this approximation is tight for the class of commi ing policies.
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