We consider discriminatory and uniform price auctions for multiple identical units of a good. Players have private values, possibly asymmetrically distributed and for multiple units. Our setting allows for aggregate uncertainty about demand and supply. In this setting, equilibria generally will be inefficient. Despite this, we show that such auctions become arbitrarily close to efficient if they are "large, " and use this to derive an asymptotic characterization of revenue and bidding behavior.

We consider an auction in which k identical objects of unknown value are auctioned off to n bidders. The k highest bidders get an object and pay the k + 1st bid. Bidders receive a signal that provides information about the value of the object. We characterize the unique symmetric equilibrium of this auction. We then consider a sequence of auctions A r with n r bidders and k r objects. We show that price converges in probability to the true value of the object if and only if both k r → ∞ and n r - k r → ∞, i.e., both the number of objects and the number of bidders who do not receive an object go to infinity.

We consider large double auctions with private values. Values need be neither symmetric nor independent. Multiple units may be owned or desired. Participation may be stochastic. We introduce a very mild notion of “a little independence.” We prove that all nontrivial equilibria of auctions that satisfy this notion are asymptotically efficient. For any α>0, inefficiency disappears at rate 1/n2-α.

We show existence of equilibria in distributional strategies for a wide class of private value auctions, including the first general existence result for double auctions. The set of equilibria is invariant to the tie-breaking rule. The model incorporates multiple unit demands, all standard pricing rules, reserve prices, entry costs, and stochastic demand and supply. Valuations can be correlated and asymmetrically distributed. For double auctions, we show further that at least one equilibrium involves a positive volume of trade. The existence proof establishes new connections among existence techniques for discontinuous Bayesian games.

Different extensive form games with the same reduced normal form can have different information sets and subgames. This generates a tension between a belief in the strategic relevance of information sets and subgames and a belief in the sufficiency of the reduced normal form. We identify a property of extensive form information sets and subgames which we term strategic independence. Strategic independence is captured by the reduced normal form, and can be used to define normal form information sets and subgames. We prove a close relationship between these normal form structures and their extensive form namesakes. Using these structures, we are able to motivate and implement solution concepts corresponding to subgame perfection, sequential equilibrium, and forward induction entirely in the reduced normal form, and show close relations between their implications in the normal and extensive form.

We provide general conditions under which principal-agent problems admit mechanisms that are optimal for the principal. Our result covers as special cases those in which the agent has no private information – i.e., pure moral hazard – as well as those in which the agent’s only action is a participation decision – i.e., pure adverse selection. We allow multi-dimensional actions and signals, as well as both …nancial and non-financial rewards. Beyond measurability, we require no a priori restrictions on the space of mechanisms. Consequently, our optimal mechanisms are optimal among all measurable mechanisms. A key to obtaining our result is to permit randomized mechanisms. We also provide conditions under which randomization is unnecessary.

This paper offers a new approach to the study of economic problems usually modeled as games of incomplete information with discontinuous payoffs. Typically, the discontinuities arise from indeterminacies (ties) in the underlying problem. The point of view taken here is that the tie-breaking rules that resolve these indeterminacies should be viewed as part of the solution rather than part of the description of the model. A solution is therefore a tie-breaking rule together with strategies satisfying the usual best-response criterion. When information is incomplete, solutions need not exist; that is, there may be no tie-breaking rule that is compatible with the existence of strategy profiles satisfying the usual best-response criteria. It is shown that the introduction of incentive compatible communication (cheap talk) restores existence. Copyright The Econometric Society 2002.