Practical or legal constraints often restrict auctions to being symmetric (anonymous and nondiscriminatory). We examine when this restriction prevents a seller from achieving his objectives. In an independent private value setting with heterogenous buyers, we characterize the set of incentive compatible and individually rational outcomes that can be implemented via a symmetric auction. We show that symmetric auctions can yield many discriminatory outcomes such as revenue maximization and affirmative action. We also characterize the set of implementable outcomes when individual rationality holds ex-post rather than in expectation. This additional requirement may prevent the seller from maximizing revenue.
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