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On the approximation ratio of $k$-lookahead auction. (English)
Chen, Ning (ed.) et al., Internet and network economics. 7th international workshop, WINE 2011, Singapore, December 11‒14, 2011. Proceedings. Berlin: Springer (ISBN 978-3-642-25509-0/pbk). Lecture Notes in Computer Science 7090, 61-71 (2011).
Summary: We consider the problem of designing a profit-maximizing single-item auction, where the valuations of bidders are correlated. We revisit the $k$-lookahead auction introduced by Ronen [6] and recently further developed by Dobzinski, Fu and Kleinberg [2]. By a more delicate analysis, we show that the $k$-lookahead auction can guarantee at least $\frac{e^{1-1/k}}{e^{1-1/k}+1}$ of the optimal revenue, improving the previous best results of $\frac{2k-1}{3k-1}$ in [2]. The 2-lookahead auction is of particular interest since it can be derandomized [2, 5]. Therefore, our result implies a polynomial time deterministic truthful mechanism with a ratio of $\frac{\sqrt{e}}{\sqrt{e}+1}\approx 0.622$ for any single-item correlated-bids auction, improving the previous best ratio of 0.6. Interestingly, we can show that our analysis for 2-lookahead is tight. As a byproduct, a theoretical implication of our result is that the gap between the revenues of the optimal deterministically truthful and truthful-in-expectation mechanisms is at most a factor of $\frac{1+\sqrt{e}}{\sqrt{e}}$. This improves the previous best factor of $\frac{5}{3}$ in [2].