A geometric approach to mechanism design

We develop a novel geometric approach to mechanism design using an important result in convex analysis: the duality between a closed convex set and its support function. By deriving the support function for the set of feasible interim values we extend the wellknown Maskin-Riley-Matthews-Border conditions for reduced-form auctions to social choice environments. We next refine the support function to include incentive constraints using a geometric characterization of incentive compatibility. Borrowing results from majorization theory that date back to the work of Hardy, Littlewood, and Polya (1929) we elucidate the "ironing" procedure introduced by Myerson (1981) and Mussa and Rosen (1978). The inclusion of Bayesian and dominant strategy incentive constraints result in the same support function, which establishes equivalence between these implementation concepts. Using Hotelling's lemma we next derive the optimal mechanism for any social choice problem and any linear objective, including revenue and surplus maximization. We extend the approach to include general concave objectives by providing a fixed-point condition characterizing the optimal mechanism. We generalize reduced-form implementation to environments with multi-dimensional, correlated types, non-linear utilities, and interdependent values. When value interdependencies are linear we are able to include incentive constraints into the support function and provide a condition when the second-best allocation is ex post incentive compatible.Mechanism design, convex set, support function, duality, majorization, ironing, Hotelling's lemma, reduced-from implementation, BIC-DIC equivalence, concave objectives, interdependent values, second-best mechanisms

Competitive Bidding in Auctions with Private and Common Values

The objects for sale in most auctions display both private and common value characteristics. This salient feature of many real-world auctions has not yet been incorporated into a strategic analysis of equilibrium bidding behavior. This paper reports such an analysis in the context of a stylized model in which bidders receive a private value signal and an independent common value signal. We show that more uncertainty about the common value results in lower efficiency and higher profits for the winning bidder. Information provided by the auctioneer decreases uncertainty, which improves efficiency and increases the seller's revenue. These positive effects of public information disclosure are stronger the more precise the information. Efficiency and revenues are also higher when more bidders enter the auction. Since our model nests both the private and common value case it may lead to an improved specification of empirical models of auctions.Auctions, inefficiencies, information disclosure, competition.

The Amsterdam Auction

Auctions used to sell houses often attract a diverse group of bidders, with realtors and speculators out for a bargain competing against buyers with a real interest in the house. Value asymmetries such as these necessitate careful consideration of the auction format as revenue equivalence cannot be expected to hold. From a theoretical viewpoint, Myerson's (1981) mechanism design approach has identified the seller's optimal choice. The proposed mechanism entails assigning credits to weaker bidders to promote competition and setting bidder-specific reserve prices. In practice, however, sellers often lack the detailed information needed to choose credits and reserve prices optimally, nor can they always discriminate among bidders. A more practical solution to the seller's problem is suggested by the "Amsterdam auction," where a premium is offered to encourage weak bidders to compete aggressively. This auction format, which has been used to sell houses in Amsterdam for centuries, treats all bidders the same and does not rely on detailed information about their value-distributions. In this paper, we consider premium auctions like the one in Amsterdam and demonstrate their revenue-generating virtues in asymmetric situations. We report the results of an experiment, which compares the standard first-price and English formats with two premium auctions in symmetric and asymmetric settings. The introduction of a premium leads weak bidders to set an endogenous reserve price for stronger rivals, with a dramatic effect on the sales price. Awarding a premium raises revenues, especially since Bertrand competition between weaker bidders virtually dissipates the premium to be paid.Auctions, experiments, asymmetries, premium

On the equivalence of Bayesian and dominant strategy implementation in a general class of social choice problems

We consider a standard social choice environment with linear utilities and independent, one-dimensional, private values. We provide a short and constructive proof that for any Bayesian incentive compatible mechanism there exists an equivalent dominant strategy incentive compatible mechanism that delivers the same interim expected utilities for all agents. We demonstrate the usefulness and applicability of our approach with several examples. Finally, we show that the equivalence between Bayesian and dominant strategy implementation breaks down when utilities are non-linear or when values are interdependent, multi-dimensional, or correlated.

An experimental study of costly coordination

This paper reports data for coordination game experiments with random matching. The experimental design is based on changes in an effort-cost parameter, which do not alter the set of Nash equilibria nor do they alter the predictions of adjustment theories based on imitation or best response dynamics. As expected, however, increasing the effort cost lowers effort levels. Maximization of a stochastic potential function, a concept that generalizes risk dominance to continuous games, predicts this reduction in efforts. An error parameter estimated from initial two-person, minimum-effort games is used to predict behavior in other three-person coordination games

An explanation of anomalous behavior in models of political participation

This paper characterizes behavior with “noisy” decision making for models of political interaction characterized by simultaneous binary decisions. Applications include: voting participation games, candidate entry, the volunteer's dilemma, and collective action problems with a contribution threshold. A simple graphical device is used to derive comparative statics and other theoretical properties of a “quantal response” equilibrium, and the resulting predictions are compared with Nash equilibria that arise in the limiting case of no noise. Many anomalous data patterns in laboratory experiments based on these games can be explained in this manner

An Experimental Study of Costly Coordination

This paper reports data for coordination game experiments with random matching. The experimental design is based on changes in an effort-cost parameter, which do not alter the set of Nash equilibria, nor do they alter the predictions of dynamic adjustment theories based on imitation or best responses to others' decisions. As would be expected, however, increases in effort cost result in reduced effort levels. Average behavior in the final periods is consistent with a one-parameter stochastic generalization of the Nash equilibrium that is calculated by maximizing a "stochastic potential function." The noise parameter estimated from the initial two-person, minimum-effort games is used to predict behavior in subsequent experiments with three-person games, using both minimum and medium-effort payoff structures.coordination games, laboratory experiments, stochastic potential, logit equilibrium, bounded rationality, minimum effort game, median effort game

An Explanation of Anomalous Behavior in Binary-Choice Games: Entry, Voting, Public Goods, and the Volunteers' Dilemma

This paper characterizes behavior with "noisy" decision making for a general class of N-person, binary-choice games. Applications include: participation games, voting, market entry, binary step-level public goods games, the volunteer's dilemma, the El Farol problem, etc. A simple graphical device is used to derive comparative statics and other theoretical properties of a "quantal response" equilibrium, and the resulting predictions are compared with Nash equilibria that arise in the limiting case of no noise. Many anomalous data patterns in laboratory experiments based on these games can be explained in this manner.participation games, entry, voting, step-level public goods games, volunteers' dilemma, quantal response equilibrium, El Farol problem, bounded rationality.

Ten Little Treasures of Game Theory and Ten Intuitive Contradictions

This paper reports laboratory data for a series of two-person games that are played only once. These games span the standard categories: static and dynamic games with complete and incomplete information. For each game, the treasure is a treatment for which behavior conforms quite nicely to the predictions of the Nash equilibrium or relevant refinement. In each case we change a key payoff parameter in a manner that does not alter the equilibrium predictions, but this theoretically neutral payoff change has a major (often dramatic) effect on observed behavior. These contradictions are generally consistent with simple economic intuition and with a model of iterated noisy introspection for one-shot games.Nash equilibrium, noncooperative games, experiments, bounded rationality, introspection