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 γ of the surplus. As it turns out, this setting is equivalent to a γ-hybrid auction in which the payments are a convex combination (parameterized by γ) 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 γ-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 γ) for the singleitem 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.

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We define and axiomatically characterize a new proportional influence measure for sequential projects with imperfect reliability. We consider a model in which a finite set of players aims to complete a project, consisting of a finite number of tasks, which can only be carried out by certain specific players. Moreover, we assume the players to be imperfectly reliable, i.e., players are not guaranteed to carry out a task successfully. To determine which players are most important for the completion of a project, we use a proportional influence measure. This paper provides two characterizations of this influence measure. The most prominent property in the first characterization is task decomposability. This property describes the relationship between the influence measure of a project and the measures of influence one would obtain if one divides the tasks of the project over multiple independent smaller projects. Invariance under replacement is the most prominent property of the second characterization. If, in a certain task group, a specific player is replaced by a new player who was not in the original player set, this property states that this should have no effect on the allocated measure of influence of any other original player.

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We analyze applications of biform games to linear production (LP) and sequencing processes. Biform games, as introduced by Brandenburger and Stuart ( 2007), apply to problems in which strategic decisions are followed by some cooperative game, where the specific environment of the cooperative game that is played, is in turn determined by these strategic decisions. We extend the work on LP-processes by allowing firms to compete for resources that are scarce or hard to produce, rather than assuming these resource bundles are simply given. With strategy dependent resource bundles that can be obtained from two locations, we show that the induced strategic game has a (pure) Nash equilibrium, using the Owen set or any game-theoretic solution concept that satisfies anonymity to solve the cooperative LP-game. To analyze competition in sequencing processes, we no longer assume an initial processing order is given. Instead, this initial order is strategically determined. Solving the second-stage cooperative sequencing game using a gain splitting rule, we fully determine the set of Nash equilibria of the induced strategic game.