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 infinitedimensional 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.

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