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Strategic Ironing in Pay-as-Bid Auctions: Equilibrium Existence with Private Information

It is known that pay-as-bid auctions admit pure-strategy equilibria when bidders have private information, as long as the space of available strategies is discrete; except in parameterized settings, there is no tractable method for computing these equilibria. Continuous approximations frequently prove useful when equilibria are difficult to compute in underlying discretized models, however the presence of payoff discontinuities in pay-as-bid auctions implies that current equilibrium existence results cannot be directly applied to the natural continuous approximation of the pay-as-bid auction. I establish the existence of pure-strategy Bayesian-Nash equilibria in divisible-good pay-as-bid auctions with private information. Existence in the divisible-good model follows from examining a particular limit of strategies in discretized models, taken as the discretization becomes infinitesimal. The techniques employed in demonstrating that properly-defined limiting strategies are mutual best responses may be useful to establishing the existence of pure-strategy equilibria in other models which do not meet the assumptions of the current equilibrium existence literature. That the proof of existence relies on a semi-constructive argument from equilibria in discretized auctions implies that equilibrium in the divisible-good model can approximate equilibrium in nearby multi-unit auctions, hence the divisible-good model may be a reasonable approximation of real-world multi-unit auctions. I further show that equilibrium strategies must exhibit strategic ironing, a phenomenon in which the bidding incentives for an infinitesimal quantity are linked to the bidding incentives for nearby quantities. Strategic ironing implies that incentives in the discriminatory divisible-good auction differ from those in the classical single-unit first-price auction in ways which have not previously been recognized. While equilibrium first-order conditions give the impression that divisible-good auctions may be solved by a system of differential equations, strategic ironing implies that the optimal bid for any particular quantity may depend on non-local constraints, and has negative implications for the tractability of equilibrium strategies.