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We study the sale of two units of a good through simultaneous sealed bid first-price auctions to bidders who have private types and multi-unit demand. Bidders are differentiated based on their valuations and budgets. Under a parametric restriction, we show that this auction game is better-reply secure and thus possesses a Nash equilibrium. In equilibrium, bidders expend their budgets. The equilibrium strategies are essentially pure in the sense that each bidder-type has a unique split (up to a permutation) of her budget between the two auctions. In equilibrium, there always exist bidder-types who submit unequal bids in equilibrium even though they have the same valuations for the units. In addition, the equilibrium is monotone in valuations and budgets: bidders with higher valuations (lower budgets) prefer more unequal splits of their budgets than bidders with lower valuations (higher budgets) and the same budget (valuation).

We study an auction game in which two units of a good are sold via two second-price auctions sequentially to three or more bidders who have private budgets that limit their spending in the auctions. We characterize the unique symmetric equilibrium. In equilibrium, while bidders bid the minimum of their valuations and budgets in the second auction, their aggression in the first round depends on two factors: the competition from other bidders, as measured by the probability of facing high budget bidders, and, the option value from participating the second auction, as measured by the budget difference between various bidders. The former factor makes bidders more aggressive and the latter makes them less aggressive. In particular, when the competition factor dominates, the sequential auctions generate higher revenue in expectation than the uniform price auction.

We study a sequential elimination contest where contestants (e.g., campaigns) spend resources that are provided by strategic players called backers (e.g., donors). In the unique symmetric equilibrium, backers initially provide small budgets, increasing their contributions only if their contestant wins the preliminary round. If backers are only allowed to provide budgets at the start of the game, as opposed to before each round, spending is higher. When unspent resources are refunded to the backer, total spending is higher than when all resources are sunk costs. We also study an extension of the model with asymmetric players.

This paper studies sequential auctions in the presence of ambiguity about the distribution of values. Using the multi-self approach to accommodate the possible time inconsistency that arises with dynamic bidding for maxmin bidders, we characterize the unique symmetric and monotone equilibrium in closed form. We show that the equilibrium prices are a supermartingale if the worst-case beliefs reverse hazard rate dominate the true distribution of values, providing an explanation for the well-documented puzzle “declining price anomaly” in sequential auctions. Importantly, the sufficient condition for declining prices allows for dynamic inconsistency. We also propose a non-parametric procedure to identify bidders' values and beliefs from auction data. The model delivers rich testable implications: on the practical side, declining prices imply that bidders' worst-case beliefs first-order stochastically dominate the true distribution of values; on the theoretical side, dynamic inconsistency generates history dependence in bidding strategies and possibly higher revenues for the seller.

This note studies information disclosure by a planner to firms engaging in a Cournot competition with uncertain demand. The planner can send verifiable, either public or private, messages to firms about the states of the world before they choose quantities but cannot ex ante commit to an information disclosure policy. We show that all equilibria induce the same outcome: firms fully learn the state. Using an example, we also show that without public messages, the commitment solution can be an equilibrium outcome.

This note studies perfect correlated equilibrium introduced by Dhillon and Mertens (1996). We show that the standard revelation principle for perfect correlated equilibrium distributions holds in two-by-two games. We then show that in N -player two-action games, although the revelation principle fails, a canonical correlated device for any perfect correlated equilibrium distribution involves two copies of action sets as the message space.

This paper studies sequential auctions in the presence of Knightian uncertainty about the distribution of values. We propose equilibrium notions based on the multiple selves approach to deal with the possible time inconsistency that arises with dynamic bidding for max-min bidders. We characterize the unique symmetric equilibrium and show that it is robust to different specifications of preferences. We demonstrate that equilibrium prices are a supermartingale under dynamic consistency, providing an explanation for the well-documented puzzle ``declining price anomaly'' in sequential auctions. The model delivers rich testable implications: on the practical side, declining prices imply that bidders' worst-case belief first-order stochastically dominates the true distribution of values; on the theoretical side, dynamic inconsistency, which can arise when bidders have multiple priors, generates history dependence in bidding strategies.

I analyze an auction environment where two units of an object are sold at two simultaneous, sealed-bid, first-price auctions to bidders who have a one-dimensional type space, where a type represents the value a bidder places on each of the two units. All bidders have an identical budget constraint that binds their ability to spend in the auctions. I show that if the valuation distribution is atom-less then there does not exist any equilibrium in behavioral strategies in this auction game.