On the Inefficiency of the Uniform Price Auction

We present our results on Uniform Price Auctions, one of the standard sealed-bid multiunit auction formats, for selling multiple identical units of a single good to multi-demand bidders. Contrary to the truthful and economically efficient multi-unit Vickrey auction, the Uniform Price Auction encourages strategic bidding and is socially inefficient in general, partly because of a ”Demand Reduction” effect; bidders tend to bid for fewer (identical) units, so as to receive them at a lower uniform price. Despite its inefficiency, the uniform pricing rule is widely popular by its appeal to the natural anticipation, that identical items should be identically priced. Application domains of its variants include sales of U.S. Treasury notes to investors, trade exchanges over the internet facilitated by popular online brokers, allocation of radio spectrum licenses etc. In this work we study equilibria of the Uniform Price Auction for bidders with (symmetric) submodular valuation functions, over the number of units that they win. We investigate pure Nash equilibria of the auction in undominated strategies; we produce a characterization of these equilibria that allows us to prove that a fraction 1 − e−1 of the optimum social welfare is always recovered in undominated pure Nash equilibrium – and this bound is essentially tight. Subsequently, we study the auction under the incomplete information setting and prove a bound of 4 − 2 k on the economic inefficiency of (mixed) Bayes Nash equilibria that are supported by undominated strategies. ∗Work supported by the research project “DDCOD” (PE6-213). The research project is implemented within the framework of the Action “Supporting Postdoctoral Researchers” of the Operational Program “Education and Lifelong Learning” (Actions Beneficiary: General Secretariat for Research and Technology), and is co-financed by the European Social Fund (ESF) and the Greek State. 1 ar X iv :1 21 1. 18 60 v4 [ cs .G T ] 1 9 Ju n 20 13