Auctions often involve the sale of many related goods: Treasury, spectrum and electricity auctions are examples. In multi-unit auctions, a bid for one unit may affect payments for other units won, giving rise to an incentive to shade bids differently across units. We establish that such differential bid shading results generically in ex post inefficient allocations in the uniform-price and pay-as-bid auctions. We also show that, in general, the efficiency and revenue rankings for the two formats are ambiguous. However, in settings with symmetric bidders, the pay-as-bid auction often outperforms. In particular, with diminishing marginal utility, symmetric information and linearity, it yields greater expected revenues. We explain the rankings through multi-unit effects, which have no counterparts in auctions with unit demands. We attribute the new incentives separately to multi-unit but constant marginal utility and diminishing marginal utility.JEL classification: D44, D82, D47, L13, L94

This paper introduces a model of preferences, in which, given beliefs about uncertain outcomes, an individual evaluates an action by a quantile of the induced distribution. The choice rule of Quantile Maximization unifies maxmin and maxmax as maximizing the lowest and the highest quantiles of beliefs distributions, respectively, and offers a family of less extreme preferences. Taking preferences over acts as a primitive, we axiomatize Quantile Maximization in a Savage setting. Our axiomatization also provides a novel derivation of subjective beliefs, which demonstrates that neither the monotonicity nor the continuity conditions assumed in the literature are essential for probabilistic sophistication. We characterize preferences of quantile maximizers towards downside risk. We discuss how the distinct properties of the model, robustness and ordinality, can be useful in studying choice behaviour for categorical variables and in economic policy design. We also offer applications to poll design and insurance problems. Copyright © 2009 The Review of Economic Studies Limited.

Most assets are traded in multiple interconnected trading venues. This paper develops an equilibrium model of decentralized markets that accommodates general market structures with coexisting exchanges. Decentralized markets can allocate risk among traders with different risk preferences more efficiently, thus realizing gains from trade that cannot be reproduced in centralized markets. Market decentralization always increases price impact. Yet, markets in which assets are traded in multiple exchanges, whether they are disjoint or intermediated, can give higher welfare than the centralized market with the same traders and assets. In decentralized markets, demand substitutability across assets is endogenous and heterogeneous among traders. (JEL D43, D44, D85, G11, G12)

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Extensive empirical research has shown that in many markets institutional investors have a significant impact on prices and mitigate its adverse effects through their trading strategies. This paper develops a dynamic model of such thin markets, in which the market structure is one of bilateral oligopoly. The paper demonstrates that market thinness qualitatively changes equilibrium properties of prices and dynamic trading strategies, compared to the existing theories of asset pricing. The predictions match a number of empirical facts that are hard to reconcile with the competitive or Cournot-based models. The paper further establishes that the nonstrategic general-equilibrium approach and the strategic approach to trade via Nash in demands are dual representations of a model with endogenous price impact. The proposed approach yields an analytical framework that can be used to study dynamic markets with bilateral market power.