We consider prior-free auctions for revenue and welfare maximization when agents have a common budget. The abstract environments we consider are ones where there is a downward-closed and symmetric feasibility constraint on the probabilities of service of the agents. These environments include position auctions where slots with decreasing click-through rates are auctioned to advertisers. We generalize and characterize the envy-free benchmark from Hartline and Yan [2011] to settings with budgets and characterize the optimal envy-free outcomes for both welfare and revenue. We give prior-free mechanisms that approximate these benchmarks. A building block in our mechanism is a clinching auction for position auction environments. This auction is a generalization of the multi-unit clinching auction of Dobzinski et al. [2008] and a special case of the polyhedral clinching auction of Goel et al. [2012]. For welfare maximization, we show that this clinching auction is a good approximation to the envy-free optimal welfare for position auction environments. For profit maximization, we generalize the random sampling profit extraction auction from Fiat et al. [2002] for digital goods to give a 10.0-approximation to the envy-free optimal revenue in symmetric, downward-closed environments. Even without budgets this revenue maximization question is of interest and we obtain an improved approximation bound of 7.5 (from 30.4 by Ha and Hartline [2012]).

There is only one technique for prior-free optimal mechanism design that generalizes beyond the structurally benevolent setting of digital goods. This technique uses random sampling to estimate the distribution of agent values and then employs the Bayesian optimal mechanism for this estimated distribution on the remaining players. Though quite general, even for digital goods, this random sampling auction has a complicated analysis and is known to be suboptimal. To overcome these issues we generalize the profit extraction and consensus techniques from [5] to structurally rich environments that include, e.g., single-minded combinatorial auctions.