First-price path auctions

We study first-price auction mechanisms for auctioning flow between given nodes in a graph. A first-price auction is any auction in which links on winning paths are paid their bid amount; the designer has flexibility in specifying remaining details. We assume edges are independent agents with fixed capacities and costs, and their objective is to maximize their profit. We characterize all strong ε-Nash equilibria of a first-price auction, and show that the total payment is never significantly more than, and often less than, the well known dominant strategy Vickrey-Clark-Groves mechanism. We then present a randomized version of the first-price auction for which the equilibrium condition can be relaxed to ε-Nash equilibrium. We next consider a model in which the amount of demand is uncertain, but its probability distribution is known. For this model, we show that a simple ex ante first-price auction may not have any ε-Nash equilibria. We then present a modified mechanism with 2-parameter bids which does have an ε-Nash equilibrium. For a randomized version of this 2-parameter mechanism we characterize the set of all eNEs and prove a bound on the total payment in any eNE.