Showing papers by "Guido Schäfer published in 2021"
TL;DR: The main results for polytopal congestion games satisfying IDP and box-TDI are shown to show that pure Nash equilibria can be computed in polynomial time and to bound the inefficiency of theseEquilibria and show that this provides a tight bound on the price of stability.
Abstract: We study the computation and efficiency of pure Nash equilibria in combinatorial congestion games, where the strategies of each player i are given by the binary vectors of a polytope Pi. Our main goal is to understand which structural properties of such polytopal congestion games enable us to derive an efficient equilibrium selection procedure to compute pure Nash equilibria with attractive social cost approximation guarantees. To this aim, we identify two general properties of the underlying aggregation polytopePN= ∑ iPi which are sufficient for our results to go through, namely the integer decomposition property (IDP) and the box-totally dual integrality property (box-TDI). Our main results for polytopal congestion games satisfying IDP and box-TDI are as follows: (i) we show that pure Nash equilibria can be computed in polynomial time. In fact, we obtain this result through a general framework for separable convex function minimization, which might be of independent interest. (ii) We bound the inefficiency of these equilibria and show that this provides a tight bound on the price of stability. (iii) We also prove that these results extend to strong equilibria for the “bottleneck variant” of polytopal congestion games. Examples of polytopal congestion games satisfying IDP and box-TDI include common source network congestion games, symmetric totally unimodular congestion games, non-symmetric matroid congestion games and symmetric matroid intersection congestion games (in particular, r-arborescences and strongly base-orderable matroids).
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TL;DR: In this paper, the authors study the social welfare loss caused by a corrupt auctioneer, both in the single-item and the multi-unit auction setting, and derive tight bounds for single item auctions up to the correlated price of anarchy.
Abstract: Corruption in auctions is a phenomenon that is theoretically still poorly understood, despite the fact that it occurs rather frequently in practice. In this paper, we initiate the study of the social welfare loss caused by a corrupt auctioneer, both in the single-item and the multi-unit auction setting. In our model, the auctioneer may collude with the winners of the auction by letting them lower their bids in exchange for a fixed fraction $\gamma$ of the surplus. As it turns out, this setting is equivalent to a $\gamma$-hybrid auction in which the payments are a convex combination (parameterized by $\gamma$) of the first-price and the second-price payments. Our goal is thus to obtain a precise understanding of the (robust) price of anarchy of $\gamma$-hybrid auctions. If no further restrictions are imposed on the bids, we prove a bound on the robust POA which is tight (over the entire range of $\gamma$) for the single-item and the multi-unit auction setting. On the other hand, if the bids satisfy the no-overbidding assumption a more fine-grained landscape of the price of anarchy emerges, depending on the auction setting and the equilibrium notion. We derive tight bounds for single-item auctions up to the correlated price of anarchy and for the pure price of anarchy in multi-unit auctions. These results are complemented by nearly tight bounds on the coarse correlated price of anarchy in both settings.