  Robust mechanism design is a rising alternative to Bayesian mechanism design,
which yields designs that do not rely on assumptions like full distributional
knowledge. We apply this approach to mechanisms for selling a single item,
assuming that only the mean of the value distribution and an upper bound on the
bidder values are known. We seek the mechanism that maximizes revenue over the
worst-case distribution compatible with the known parameters. Such a mechanism
arises as an equilibrium of a zero-sum game between the seller and an adversary
who chooses the distribution, and so can be referred to as the max-min
mechanism.
  Carrasco et al. [2018] derive the max-min pricing when the seller faces a
single bidder for the item. We go from max-min pricing to max-min auctions by
studying the canonical setting of two i.i.d. bidders, and show the max-min
mechanism is the second-price auction with a randomized reserve. We derive a
closed-form solution for the distribution over reserve prices, as well as the
worst-case value distribution, for which there is simple economic intuition. In
fact we derive a closed-form solution for the reserve price distribution for
any number of bidders.
  Our technique for solving the zero-sum game is quite different than that of
Carrasco et al.- it involves analyzing a discretized version of the setting,
then refining the discretization grid and deriving a closed-form solution for
the non-discretized, original setting. Our results establish a difference
between the case of two bidders and that of $n \ge 3$ bidders.
