  We study the optimal auction design problem when bidders' preferences follow
the maxmin expected utility model. We suppose that each bidder's set of priors
consists of beliefs close to the seller's belief, where "closeness" is defined
by a divergence. For a given allocation rule, we identify a class of optimal
transfer candidates, named the win-lose dependent transfers, with the following
property: each type of bidder's transfer conditional on winning or losing is
independent of the competitor's type report. Our result reduces the
infinite-dimensional optimal transfer problem to a two-dimensional optimization
problem. By solving the reduced problem, we find that: (i) among efficient
mechanisms with no premiums for losers, the first-price auction is optimal;
and, (ii) among efficient winner-favored mechanisms where each bidder pays
smaller amounts when she wins than loses: the all-pay auction is optimal. Under
a simplifying assumption, these two auctions remain optimal under the
endogenous allocation rule.
