Search

We consider a nonlinear pricing problem faced by a dominant firm which competes with a capacity-constrained minor firm for a downstream buyer who may purchase the product from the firms under complete information. Specifically, we analyze a three-stage game in which the dominant firm offers a general tariff first and then the minor firm responds with a per-unit price, followed by the buyer choosing her purchases. By establishing an equivalence between the subgame perfect equilibrium of our asymmetric competition game and the optimal mechanism in a “virtual” principal-agent model, we characterize the dominant firm's optimal nonlinear tariff, which exhibits convexity and yet can display quantity discounts. Our analysis provides a rationale for nonlinear pricing under competition in the absence of private information: The dominant firm can use unchosen offers to constrain its rival’s possible deviations and extract more surplus from the buyer. Antitrust implications are also discussed.

This study considers a mechanism design problem in which service slots are allocated over time to buyers arriving at different periods. Some buyers can accept delayed service, whereas others cannot. Buyers have a multidimensional type representing their valuation and patience level. The seller fully commits to a mechanism with or without complete contingent contracts. We show that a deterministic mechanism is periodic ex-post incentive compatible if and only if the allocation rule is monotone in valuation and the "price equivalence" holds. In cases both with and without contingent contracts, the dynamic pivot mechanism with a reserve price maximizes the seller's expected revenue under a regularity condition, which is characterized by the linearity of the virtual valuation function. When the regularity holds, the fully optimal mechanism with contingent contracts is multi-contracting-proof, whereas the constrained optimal mechanism without contingent contracts is not.

A durable goods monopolist proposes selling mechanisms in two periods, being unable to commit in the first period on the mechanism to propose in the second. Trade is anonymous and resale is not possible. Although buyers have a continuum of possible valuations, the optimal first-period mechanism is a menu with at most two possibilities: a high price guaranteeing delivery and a low price subject to rationing. This characterization is robust to the arrival of additional buyers in the second period. The optimal mechanism is fully characterized for linear demand, with priority pricing being optimal if agents are sufficiently patient.

We derive the revenue maximizing allocation of m units among n symmetric agents who have unit demand, and who take costly actions that in uence their values before participating in the mechanism. The allocation problem with costly actions can be represented by a reduced form model where agents have convex, non-expected utility preferences over the interim probability of receiving an object. Both the uniform m+1 price auction and the discriminatory pay-your-bid auction with reserve price constitute symmetric revenue maximizing mechanisms. Contrasting the case with exogenous valuations, the optimal reserve price reacts to both demand and supply. We also identify a condition under which the optimal mechanism is indeed symmetric, and illustrate the structure of the optimal asymmetric mechanism when the condition fails. The main tool in our analysis is an integral inequality, due to Fan and Lorentz (1954), involving majorization, super-modularity and convexity.

The existing literature on optimal auctions focuses on optimizing the expected revenue of the seller, and is appropriate for risk-neutral sellers. In this paper, we identify good mechanisms for risk-averse sellers. As is standard in the economics literature, we model the risk-aversion of a seller by endowing the seller with a monotone concave utility function. We then seek robust mechanisms that are approximately optimal for all sellers, no matter what their levels of risk-aversion are. We have two main results for multi-unit auctions with unit-demand bidders whose valuations are drawn i.i.d. from a regular distribution. First, we identify a posted-price mechanism called the Hedge mechanism, which gives a universal constant factor approximation; we also show for the unlimited supply case that this mechanism is in a sense the best possible. Second, we show that the VCG mechanism gives a universal constant factor approximation when the number of bidders is even only a small multiple of the number of items. Along the way we point out that Myerson's characterization of the optimal mechanisms fails to extend to utility-maximization for risk-averse sellers, and establish interesting properties of regular distributions and monotone hazard rate distributions.

This paper studies the use of posted prices and auctions in a large dynamic market with many short-lived sellers and long-lived buyers. Although a reserve-price auction maximizes the expected revenue, the optimal revenue decreases when the market becomes more buyer-friendly; namely, when buyers survive longer, face fewer competitors, and become more patient. As the market becomes more buyer-friendly, the revenue advantage of a reserve-price auction over posting a price also decreases, but using posted prices would lead to sale and allocative inefficiencies.

Diversification involves ongoing decisions about firm boundaries and relatedness. We develop a theoretical model that uses a real-option framework combined with optimal mechanism design to analyze how choices of boundaries and relatedness affect firm performance in the face of tradeoffs between governance and flexibility. We find that: (1) optimal boundaries and relatedness are substitutes in determining firm performance; (2) the association between relatedness, the most commonly studied aspect of diversification, and firm performance is indeterminate; (3) the substitution between relatedness and boundaries declines as noise in internal communication increases; (4) variation in relatedness has greater impact than boundary size when headquarters can pick multiple winners; and (5) as the internal market for information becomes more efficient, the lower the value of relatedness in combination with small boundaries. The general conceptual implication of these points is that corporate governance interacts with firm relatedness and boundaries in generating diversification performance.

In the standard herding model, privately informed individuals sequentially see prior actions and then act. An identical action herd eventually starts and public beliefs tend to “cascade sets” where social learning stops. What behaviour is socially efficient when actions ignore informational externalities? We characterize the outcome that maximizes the discounted sum of utilities. Our four key findings are: (1) cascade sets shrink but do not vanish, and herding should occur but less readily as greater weight is attached to posterity. (2) An optimal mechanism rewards individuals mimicked by their successor. (3) Cascades cannot start after period one under a signal log-concavity condition. (4) Given this condition, efficient behaviour is contrarian, leaning against the myopically more popular actions in every period. We make two technical contributions: as value functions with learning are not smooth, we use monotone comparative statics under uncertainty to deduce optimal dynamic behaviour. We also adapt dynamic pivot mechanisms to Bayesian learning.

This paper studies a principal-agent model in which the principal and agent are risk-neutral, there are two actions, adverse selection, moral hazard and limited liability. When the two actions are subject to moral hazard, there is no distortion at the top, the optimal action profile is downward distorted for everyone else and the optimal menu of contract exhibits the one-size-fits-all property; that is, each ability type receives the same contract. The optimal contract pays a bonus when the outcome with the highest likelihood ratio is observed and the limited liability otherwise. When one of the actions is contractible and the other is subject to moral hazard, there is no distortion at the top, the non-contractible action is downward distorted for everyone else, the contractible action can be either upward or downward distorted. The optimal contract no longer exhibits the one-size-fits-all property. The one-size-fits-all property sheds light why we rarely observe menus of contracts in market that use franchising, credit and labor markets, and in regulated industries.

I empirically measure the welfare gains from optimal incentive regulation in the context of electric utilities facing both emissions and rate of return regulation (RORR). I provide evidence that RORR induces lower fuel efficiency, leading to greater coal consumption and higher emissions abatement costs. Replacing RORR with the optimal mechanism of Laffont and Tirole (1986) yields annual welfare gains of $686 million or a 11% reduction in electricity prices. I construct a much simpler two-contract menu that can achieve more than 65% of these welfare gains.

In this paper, I examine the conditions under which rewarding honest taxpayers is an optimal mechanism to reduce tax evasion and improve collection. My theoretical model suggests that when it is expensive to audit contributors, and moral (or pecuniary) evasion cost is high enough, rewarding tax compliance is an effective strategy to increase collection. I provide empirical evidence from Argentina’s province variation of rewards policies and tax collection. Where the data suggests that theoretical conditions hold, I find a positive effect in real estate per capita tax collection between 0.4 and more than 2 standard deviations.

This paper incorporates quasi–hyperbolic discounting into a Mirrlees taxation model to study the design of retirement policies for present-biased agents. I show that the government can improve the screening of productivity by exploiting time inconsistency. This is done by providing commitment to sophisticated agents and taking advantage of the incorrect beliefs of naïve agents. This can be achieved even if the degrees of present bias and sophistication are private information. I also demonstrate how the government can implement the optimal mechanism using retirement savings accounts and social security benefits.

We show that in a class of I‐agent mechanism design problems with evidence, commitment is unnecessary, randomization has no value, and robust incentive compatibility has no cost. In particular, for each agent i, we construct a simple disclosure game between the principal and agent i where the equilibrium strategies of the agents in these disclosure games give their equilibrium strategies in the game corresponding to the mechanism but where the principal is not committed to his response. In this equilibrium, the principal obtains the same payoff as in the optimal mechanism with commitment. As an application, we show that certain costly verification models can be characterized using equilibrium analysis of an associated model of evidence.

This paper revisits the problem of adverse selection in the insurance market of Rothschild and Stiglitz (1976) . We extend the three-stage game in Hellwig (1987) by allowing firms to endogenously choose whether or not to pre-commit on their contractual offers (menus). We show how this mechanism can deliver the Miyazaki–Wilson–Spence allocation as the unique perfect-Bayesian equilibrium. This allocation is the unique incentive-efficient and individually-rational maximizer of the utility of the most profitable type. In fact, given that the informed player has only two types, it is the unique core allocation and thus the unique neutral optimum in the sense of Myerson (1983) .

I study revenue maximizing mechanisms in the presence of agents who are loss averse in the sense of Koszegi and Rabin (Quart J Econ 121:1133–1166, 2006, Am Econ Rev 97:1047–1073, 2007). Since the reference point is formed endogenously as an equilibrium object, the principal can influence the agents’ reference points by announcing a particular mechanism. Two specifications of reference-dependence are considered. If the agents narrowly bracket gains and losses in the good and in the money dimension separately, any optimal mechanism is all pay, and an optimal auction is an all pay auction with minimum bid; with wide bracketing of gains and losses over the entire risk neutral payoff, an optimal auction is a first price auction with minimum bid. Compared to the same environment with risk neutral agents, the minimum bid is always weakly higher with narrow bracketing and always weakly lower with wide bracketing of gains and losses. In case loss aversion is very pronounced, differentiability of the value function fails, and no familiar characterization of incentive compatibility involving the envelope theorem is available.

In this article, we propose an optimal mechanism to reduce congestion when information is asymmetric. Each car driver receives a quantity of traffic rights such that his adjusted marginal benefit is equal to the marginal cost of congestion and payments are based on willingness to pay. We show that the level of congestion achieved is lower and each car user can receive more or fewer rights than under complete information. With symmetric beliefs, the payment rule results from a seconddegree price discrimination. When beliefs are asymmetric, it results simultaneously from a second-degree price discrimination and from a third-degree price discrimination and high willingness-to-pay car users are discriminated against. The revenue raised can be used to reduce distortionary taxes, thereby gaining public acceptability.

In this paper, we analyze the role of negative prizes in contest design with a fixed budget, risk-neutral contestants, and independent private abilities. The effort-maximizing prize allocation rule features a threshold. When the highest effort is above the threshold, all contestants with lower efforts receive negative prizes. These negative prizes are used to augment the prize to the contestant with the highest effort, which better incentivizes contestants with higher abilities. When no contestant's effort exceeds the threshold, all contestants equally split the initial budget (or a portion of it) to ensure their participation. We find that allowing negative prizes can increase the expected total effort dramatically. In particular, if no bound is imposed on negative prizes, the expected total effort can be arbitrarily close to the highest possible effort inducible when all contestants have the maximum ability with certainty. The above contest is shown to be the optimal mechanism for a more general class of mechanisms.

We study the optimal design of mechanisms for the private provision of public goods in a setting in which donors compete for a prize of commonly known value. We discuss equilibrium bidding in mechanisms that promote both conditional cooperation and competition (i.e., the lottery and the all-pay auction with the lowest-bid payment rule) and rank their fund-raising performance vis-a-vis their standard (pay-your-own-bid) counterparts. The theoretically optimal mechanism in this model is the lowest-price all-pay auction—an auction in which the highest bidder wins the prize and all bidders pay the lowest bid. The highest amount for the public good is generated in the unique, symmetric, mixed-strategy equilibrium of this auction. In the laboratory, the theoretically optimal mechanism generates the highest level of donations with three bidders but not with two bidders. (JEL D44, D64)

We model the optimal regulation of continuous, irreversible, capacity expansion, where a regulated firm has private information about capacity costs, investments are financed from the firm's cash flows, and demand is stochastic. The optimal mechanism can be implemented as a revenue tax that increases with the level of the price cap. If the asymmetric information has large support, then the optimal mechanism consists of a laissez-faire regime for low-cost firms. That is, the firm's price cap corresponds to that of an unregulated monopolist, and it is not taxed. This `maximal distortion at the top' is necessary to provide information rents, as direct subsidies are not feasible.

A buyer makes an offer to a privately informed seller for a good of uncertain quality. Quality determines both the seller's valuation and the buyer's valuation and the buyer evaluates each contract according to its worst-case performance over a set of probability distributions. The paper demonstrates that the contract that maximizes the minimum payoff over all possible probability distributions of quality is a screening menu that separates all types, whereas the optimal contract for any given probability distribution is a posted price, which induces bunching. Using the ϵ-contamination model, according to which the buyer's utility is a weighted average of his single prior expected utility and the worst-case scenario, the analysis further shows that for intermediate degrees of confidence the optimal mechanism combines features of both of these contracts.

I consider a broad class of economic environments where a principal contracts with an agent under adverse selection and the agent can credibly disclose information to the principal. I show that there is an equilibrium that interim Pareto dominates the equilibrium without evidence if and only if the optimal mechanism without evidence assigns the outside option to a set of types satisfying a ‘gains from trade’ property. The results apply to a range of economic environments including insurance markets, financial markets and goods markets with quality-based price discrimination.

One of the most topical issues in the sphere of state regulation of environmental security in the region in conditions of economic instability is the improvement of the system of interbudgetary equalization and the search for effective models for redistributing budget funds. The article analyzes the essence and peculiarities of interbudgetary relations in Ukraine. In order to determine the most optimal mechanism for interbudgetary equalization, a centralized, decentralized and mixed model for regulating intergovernmental fiscal relations is studied. The analysis of changes in the structure of tax revenues in state and local budgets is carried out, the main advantages and threats for business entities due to the increase in the level of autonomy of local authorities are determined. Despite the low level of state regulation of environmental safety, as a means of maintaining the balance of social and natural structures, the prerequisite for the formation of regional stability it is proposed measures to improve the process of reforming the system of intergovernmental fiscal relations.

Several agents with privately known social values compete for a prize. The prize is allocated based on the claims of the agents, and the winner is subject to a limited penalty if he makes a false claim. If the number of agents is large, the optimal mechanism places all agents above a threshold onto a shortlist along with a fraction of agents below the threshold, and then allocates the prize to a random agent on the shortlist. When the number of agents is small, the optimal mechanism allocates the prize to the agent who makes the highest claim, but restricts the range of claims above and below. (JEL D63, D82)

We study the optimal mechanism in a dynamic sales relationship where the buyer's arrival date is uncertain, and where his value changes stochastically over time. The buyer's arrival date is the first date at which contracting is feasible and is his private information. To induce immediate participation, the buyer is granted positive expected rents even if his value at arrival is the lowest possible. The buyer is punished for arriving late; i.e., he expects to earn less of the surplus. Optimal allocations for a late arriver are also further distorted below first-best levels. Conditions are provided under which allocations converge to the efficient ones long enough after contracting, and this convergence occurs irrespective of the time the contract is initially agreed (put differently, the so-called “principle of vanishing distortions” introduced by Battaglini (2005) continues to apply irrespective of the buyer's arrival date).

A revenue-maximizing monopolist is selling a single indivisible good to buyers who face a loss if any of its rival buyers obtain it. The rivalry is modeled through a network, an arc between a pair of buyers indicates that a buyer considers another buyer its rival, and the magnitude of the loss is the private information of each buyer. This loss-exposure due to competitive considerations can be viewed as a negative externality. First, using a Myersonian approach we derive the monopolist’s optimal mechanism for any given network. Second, we show that revenues depend on the network structure. Thus, in applications where it is possible, the monopolist might consider designing not only the mechanism but also the network (if not fully, at least partially). Third, we provide and fully describe the solution to the monopolist’s market design problem; that is, the joint network and mechanism design problem. Specifically, despite the nonmonotone impact of additional competition on the monopolist’s revenues, we determine revenue-maximizing rivalry networks (which in turn induce optimal mechanisms) and show that they are independent of distributional assumptions on buyers’ independent private loss values, provided virtual values are bounded from zero. We achieve these results under different restrictions on the network structure and formation. When rivalry is symmetric, matchings are optimal (with at most one path on three vertices). Thus, a market with a fragmented network structure yields higher revenues for the monopolist than a market with a completely connected network structure. However, asymmetric competitive relationships among buyers generate higher revenues than symmetric ones. The optimal asymmetric networks are characterized by (i) every buyer having at least one rival and (ii) the existence of a buyer not considered a rival by anyone. The electronic companion is available at https://doi.org/10.1287/opre.2016.1585 .

We use an economic experiment to identify how people design a mechanism. Our experimental framework is adapted from the market framework utilised by Maskin and Riley, who considered a monopolist seller of a homogeneous good that uses non-linear pricing under incomplete information. Approximately 30 per cent of participants’ offers in each treatment were incentive-compatible (separating) double offers, a requirement of the optimal mechanism. A similar percentage of offers in each treatment were single (nonseparating) offers, which do not require participants to address incentive compatibility. After sufficient rounds, the design of double offers converged to the schedule identified by Maskin and Riley, while the design of single offers converged to the optimal nonseparating (single) offer.

We derive the revenue maximizing allocation of m units among n symmetric agents who have unit demand, and who take costly actions that influence their values before participating in the mechanism. The allocation problem with costly actions can be represented by a reduced form model where agents have convex, non-expected utility preferences over the interim probability of receiving an object. Both the uniform m+1 price auction and the discriminatory pay-your-bid auction with reserve price constitute symmetric revenue maximizing mechanisms. Contrasting the case with exogenous valuations, the optimal reserve price reacts to both demand and supply. We also identify a condition under which the optimal mechanism is indeed symmetric, and illustrate the structure of the optimal asymmetric mechanism when the condition fails. The main tool in our analysis is an integral inequality, due to Fan and Lorentz (1954), involving majorization, super-modularity and convexity.

Auctions with endogenous rationing have been introduced to stimulate competition. Such (procurement) auctions reduce the volume put out to tender when competition is low. This paper finds a strong negative effect of endogenous rationing on participation when bid-preparation is costly, counteracting the aim to stimulate competition. For multiple auctioneer's objectives mentioned in directives, we derive optimal mechanisms, which differ due to different evaluation of the tradeoff between participation and bid-preparation costs. Thus, the auctioneer needs to decide on an objective. However, reducing bid-preparation costs improves the optimal values of multiple objective functions.

We study auction design for bidders equipped with non-expected utility preferences that exhibit constant risk aversion (CRA). The CRA class is large and includes loss-averse, disappointment-averse, mean-dispersion and Yaari's dual preferences as well as coherent and convex risk measures. The optimal mechanism offers "full-insurance" in the sense that each agent's utility is independent of other agents' reports. The seller excludes less types than under risk neutrality, and awards the object randomly to intermediate types. Subjecting intermediate types to a risky allocation while compensating them when losing allows the seller to collect larger payments from higher types. Relatively high types are anyway willing to pay more, and their allocation is efficient. (A previous version has been circulated under the title "Optimal Auctions for Dual Risk Averse Bidders: Myerson meets Yaari".)

We study optimal mechanisms for a utilitarian designer who seeks to assign multiple units of an indivisible good to a group of agents with unit demand. The agents have heterogeneous marginal utilities of money. Heterogeneous marginal utilities of money may naturally arise in environments where agents have different wealth endowments. We show that the ex post efficient allocation rule is not utilitarian optimal in our setting. Firstly, a high willingness to pay may stem from a low marginal utility of money. Secondly, the transfer rule does not only facilitate implementation of the desired social choice function in our setting, but also directly affects social welfare. In the optimal mechanism, rationing may occur, which entails a conflict between ex ante and ex post efficiency. In an extension, we show that it is still not utilitarian optimal to allocate the good solely based on willingness to pay even when redistribution is not possible. Finally, we highlight how our mechanism can be implemented as an auction with minimum bids and bidding subsidies.

Many scarce public resources are allocated at below-market-clearing prices, and sometimes for free. Such "non-market" mechanisms necessarily sacrifice some surplus, yet they can potentially improve equity. In this paper, we develop a model of mechanism design with redistributive concerns. Agents are characterized by a privately observed social welfare weight and willingness to pay for quality, as well as a publicly observed label. A market designer controls allocation and pricing of a set of objects of heterogeneous quality, and maximizes the expectation of a welfare function defined by the social welfare weights. We derive structural insights about the form of the optimal mechanism, leading to a framework for determining how and when to use non-market mechanisms. The key determinant is the strength of the statistical correlation of the unobserved social welfare weights with the label and the willingness to pay that the designer can, respectively, directly observe or elicit through the mechanism.

I consider the welfare and profit maximization problems in markets with externalities. I show that when externalities depend generally on allocation, a Pigouvian tax is often suboptimal. Instead, the optimal mechanism has a simple form: a finite menu of rationing options with corresponding prices. I derive sufficient conditions for a single price to be optimal. I show that a monopolist may ration less relative to a social planner when externalities are present, in contrast to the standard intuition that non-competitive pricing is indicative of market power. My characterization of optimal mechanisms uses a new methodological tool—the constrained maximum principle—which leverages the combined mathematical theorems of Bauer (1958) and Szapiel (1975). This tool generalizes the concavification technique of Aumann and Maschler (1995) and Kamenica and Gentzkow (2011), and has broad applications in economics.

The emergence and ubiquitous presence in everyday life of digital goods such as songs, movies, and e-books give renewed salience to the problem of providing public goods with exclusion. Because digital goods are typically traded via intermediaries like iTunes, Amazon, and Netflix, the question arises as to the optimal pricing mechanism for such club good intermediaries. We derive the direct Bayesian optimal mechanism for allocating club goods when the mechanism designer is an intermediary that neither produces nor consumes the goods, and we develop an indirect mechanism that implements this mechanism. We also derive sufficient conditions for the intermediary-optimal mechanism to be implementable with revenue sharing contracts, which are widely used in e-business.

I study a dynamic relationship where a principal delegates experimentation to an agent. Experimentation is modeled as a one-armed bandit that yields successes following a Poisson process. Its unknown intensity is high or low. The agent has private information, his type being his prior belief that the intensity is high. The agent values successes more than the principal does, so prefers more experimentation. The optimal mechanism is a cutoff rule in the belief space: the cutoff gives pessimistic types total freedom but curtails optimistic types’ behavior. Pessimistic types overexperiment while the most optimistic ones underexperiment. This delegation rule is time consistent. (JEL D23, D82, D83, O30)

A seller wishes to sell multiple goods by a deadline, for example, the end of a season. Potential buyers enter over time and can strategically time their purchases. Each period, the profit-maximizing mechanism awards units to the buyers with the highest valuations exceeding a sequence of cutoffs. We show that these cutoffs are deterministic, depending only on the inventory and time remaining; in the continuous-time limit, the optimal mechanism can be implemented by posting anonymous prices. When incoming demand decreases over time, the optimal cutoffs satisfy a one-period-look-ahead property and prices are defined by an intuitive differential equation.

We analyze optimal procurement mechanisms when firms are specialized. The procurement agency has incomplete information concerning the firms' cost functions and values high quality as well as low price. Lower type firms are cheaper (more expensive) than higher type firms when providing low (high) quality. With specialized firms, distortion is limited and a mass of types earns zero profits. The optimal mechanism can be inefficient: types providing lower second-best welfare win against types providing higher second-best welfare. As standard scoring rule auctions cannot always implement the optimal mechanism, we introduce a new auction format implementing the optimal mechanism.

This paper applies a mechanism design approach to construct a lower bound to the welfare cost of inflation that does not depend on quasi-linear preferences or details of how agents trade. An incentive-feasible trading protocol is derived to minimize the welfare loss subject to frictions rendering money essential. The welfare cost of inflation under this optimal protocol is the lower bound over all pairwise trading protocols of monetary exchange. In general, the first-best is not implementable, even under the Friedman's rule, patient agents and the optimal mechanism. Thus, the lower bound depends on fundamentals like preferences and technology. Finally, I estimate the irreducible cost of 10% inflation with the U.S. data from 1900 to 2000.

We study the optimal allocation of heterogeneous services without using monetary transfers. Agents have private, multi-dimensional utilities over the services, and a social planner has arbitrary priors on the utilities, which may depend on the agents' observable characteristics. The social planner's goal is to maximize a public objective, which may be complex, taking into account diverse considerations such as social welfare, equity, and system costs. Potential applications include the allocation of seats to public schools, spaces in college dorms or courses, and spots in subsidized housing. We propose an "engineering" approach to such problems, in which we simplify the analysis by considering a large-market model with a continuum of agents, and show using real data that a mechanism defined based on this approximation still yields good results in the finite market. In the large-market model, there are finitely many "agent types," and a continuum of agents of each type. (Here, "type" corresponds to an agent's observable information, which is distinct from the private utilities.) The large market assumption implies that a mechanism can be decomposed into a collection of allocation rules, one for each type. We show that any allocation rule that satisfies incentive compatibility and Pareto optimality within type is representable as "Competitive Equilibrium with Equal Incomes" (CEEI). In other words, for each type of agents and for each service, there exists a "virtual price" for a unit of probability of getting that service, and the allocation is induced by giving each agent one unit of "virtual money' and having them purchase their utility-maximizing probabilities to services. In addition, if the mechanism is restricted to being 'ordinal,' which means that it only uses preference rankings but not preference intensities, then an allocation rule that satisfies incentive compatibility and ordinal efficiency within type is 'lottery-plus-cutoff': each agent receives a uniformly random lottery number between zero and one, and for each service and each type, there is a 'lottery cutoff'; an agent is 'admitted' to a service if her lottery number is below the cutoff; each agent is allocated her most preferred service for which she is admitted. Such characterization results reduce the search of the optimal mechanism to a well-defined optimization over the prices and cutoffs for each type. We show how this large market approximation can be applied to an empirically relevant finite setting and yield good results. We consider the allocation of seats in public schools in Boston, which was a real problem faced by a city committee in the 2012-2013 Boston school assignment reform. Students are classified into 868 types by home location and there is a utility model for each type that comes from fitting a multinomial logit discrete choice model on previous years' data. There are 77 elementary schools with given capacities. Depending on their home location, a student is given a menu of options and can submit a preference ranking over the schools in the menu. Given preference submissions, a centralized algorithm computes the assignment using a system of priorities, possibly randomizing to break ties. The goal is to optimize the menus and priorities to maximize a weighted combination of social welfare and max-min welfare, while staying within a budget for expected amount of busing needed. Although this is a finite market problem, we can define a large market approximation. Using the characterization results, we show that the optimal mechanism can be encoded as an exponential-sized linear program, which due to the logit utility structure can be efficiently solved using duality theory. Using the cutoffs that arise from this optimization, we define a feasible finite market mechanism that is based on the Deferred Acceptance algorithm. We evaluate the resultant mechanism by simulating in the finite market setting with real data, and show that compared to the mechanism chosen by the city, this new mechanism significantly improves simultaneously in social welfare, max-min welfare and predictability, while staying within the same expected busing budget. In sum, we incorporate prior information in the allocation problem without monetary transfers, and exhibit computational results in empirically relevant special cases.

This paper analyzes mechanisms for selling advertising inventory in a position auction in which displaying less than the maximal number of ads means the ads that are shown can be dynamically resized and displayed more prominently. I characterize the optimal mechanism with and without dynamic resizing, and illustrate how the optimal reserve prices in a Vickrey–Clarke–Groves mechanism vary with the amount of dynamic resizing and the number of bidders.

We study decentralized mechanisms for allocating firms into search pools. The pools are created in response to noisy preference signals provided by consumers, who then browse the pools via costly random sequential search. Surplus-maximizing search pools are implementable in symmetric Nash equilibrium. Full extraction of the maximal surplus is implementable if and only if the distribution of consumer types satisfies a set of simple inequalities, which involve the relative fractions of consumers who like different products and the Bhattacharyya coefficient of similarity between their conditional signal distributions. The optimal mechanism can be simulated by a keyword auction with broad matching. (JEL C78, D44, D82)

In this paper, we study the optimal provision of a costly public good using an average efficiency criterion. For every fixed cost, we identify a quota mechanism as the optimal mechanism among those that are dominant-incentive-compatible, deficit-free and kind. Moreover, we also consider the asymmetric case and demonstrate that a committee mechanism is optimal for a large class of mechanisms. In particular, this mechanism dominates all VCG (pivotal) mechanisms.

When adjudication is not perfectly accurate, litigants with unmeritorious cases may benefit from court errors, which in turn may result in a dilution of incentives for primary behavior and frivolous litigation. We study how shifting the court fees to the loser may improve the accuracy of adjudication. If litigation costs are high, the American rule performs better than the English rule, and vice versa if litigation costs are low. Yet, the optimal rule typically lies in between the American and the English rule, allowing the court to modulate fee shifting based on the accuracy of the evidence submitted by the parties. Our results rationalize observed patterns of use of fee shifting. In the United States, fee shifting is less common than in Europe, where litigation costs are lower. When used, fee shifting commonly depends on the accuracy of the evidence. Moreover, fee shifting filters extreme cases out of litigation, resulting in a more representative set of cases—including also more extreme ones—being adjudicated under the American rule as compared to the English rule. Finally, we identify a characteristic of fee-shifting rules, their “flatness”, which guarantees that fee shifting does not affect the settlement rate and show that flatness is a feature of an optimal mechanism.

We consider a nonlinear pricing problem faced by a dominant firm which competes with a capacity-constrained minor firm for a downstream buyer who may purchase the product from the firms under complete information. Specifically, we analyze a three-stage game in which the dominant firm offers a general tariff first and then the minor firm responds with a per-unit price, followed by the buyer choosing her purchases. By establishing an equivalence between the subgame perfect equilibrium of our asymmetric competition game and the optimal mechanism in a “virtual” principal-agent model, we characterize the dominant firm's optimal nonlinear tariff, which exhibits convexity and yet can display quantity discounts. Our analysis provides a rationale for nonlinear pricing under competition in the absence of private information: The dominant firm can use unchosen offers to constrain its rival’s possible deviations and extract more surplus from the buyer. Antitrust implications are also discussed.

We consider the problem of a revenue-maximizing seller with m items for sale to n additive bidders with hard budget constraints, assuming that the seller has some prior distribution over bidder values and budgets. The prior may be correlated across items and budgets of the same bidder, but is assumed independent across bidders. We target mechanisms that are Bayesian incentive compatible, but that are ex-post individually rational and ex-post budget respecting. Virtually no such mechanisms are known that satisfy all these conditions and guarantee any revenue approximation, even with just a single item. We provide a computationally efficient mechanism that is a 3-approximation with respect to all BIC, ex-post IR, and ex-post budget respecting mechanisms. Note that the problem is NP-hard to approximate better than a factor of 16/15, even in the case where the prior is a point mass. We further characterize the optimal mechanism in this setting, showing that it can be interpreted as a distribution over virtual welfare maximizers. We prove our results by making use of a black-box reduction from mechanism to algorithm design developed by Cai et al. Our main technical contribution is a computationally efficient 3-approximation algorithm for the algorithmic problem that results from an application of their framework to this problem. The algorithmic problem has a mixed-sign objective and is NP-hard to optimize exactly, so it is surprising that a computationally efficient approximation is possible at all. In the case of a single item (m=1), the algorithmic problem can be solved exactly via exhaustive search, leading to a computationally efficient exact algorithm and a stronger characterization of the optimal mechanism as a distribution over virtual value maximizers., NSF (Awards CCF-0953960 and CCF-1101491)

We study a mechanism design problem in which an indivisible good is auctioned to multiple bidders for each of whom it has a private value that is unknown to the seller and the other bidders. The agents perceive the ensemble of all bidder values as a random vector governed by an ambiguous probability distribution, which belongs to a commonly known ambiguity set. The seller aims to design a revenue-maximizing mechanism that is not only immunized against the ambiguity of the bidder values, but also against the uncertainty about the bidders’ attitude toward ambiguity. We argue that the seller achieves this goal by maximizing the worst-case expected revenue across all value distributions in the ambiguity set and by positing that the bidders have Knightian preferences. For ambiguity sets containing all distributions supported on a hypercube, we show that the Vickrey auction is the unique mechanism that is optimal, efficient, and Pareto robustly optimal. If the bidders’ values are additionally known to be independent, then the revenue of the (unknown) optimal mechanism does not exceed that of a second-price auction with only one additional bidder. For ambiguity sets under which the bidders’ values are dependent and characterized through moment bounds, on the other hand, we provide a new class of randomized mechanisms, the highest-bidder lotteries, whose revenues cannot be matched by any second-price auction with a constant number of additional bidders. Moreover, we show that the optimal highest-bidder lottery is a 2-approximation of the (unknown) optimal mechanism, whereas the best second-price auction fails to provide any constant-factor approximation guarantee. This paper was accepted by Yinyu Ye, optimization.

This paper proves that the welfare of the first price auction in Bayes-Nash equilibrium is at least a .743-fraction of the welfare of the optimal mechanism assuming agents’ values are independently distributed. The previous best bound was 1−1/e≈.63, derived using smoothness, the standard technique for reasoning about welfare of games in equilibrium. In the worst known example, the first price auction achieves a ≈.869-fraction of the optimal welfare, far better than the theoretical guarantee. Despite this large gap, it was unclear whether the 1−1/e bound was tight. We prove that it is not. Our analysis eschews smoothness, and instead uses the independence assumption on agents’ value distributions to give a more careful accounting of the welfare contribution of agents who win despite not having the highest value.

This article determines an optimal forestry contract when firms’ harvesting costs are interdependent. The value of the optimal allocation depends on the private signals of all firms. We show that the optimal rotation of the winning firm must satisfy a modified version of the usual Faustmann rule, which holds under perfect information. This modification is necessary in order to induce the revelation of private signals on the part of all participating firms. We find conditions under which the optimal mechanism can be implemented as a second-price auction. The optimal rotation period and the reservation price are derived. Theoretically and numerically, we show that the predicted forest owner surplus is considerably misestimated under the independent private value paradigm and the predicted forest owner profit is more affected when the interdependence is negative.

A seller maximizes revenue from selling an object in a dynamic environment, with buyers that differ in their patience: Each buyer has a privately known deadline for buying and a privately known valuation. First, we derive the optimal mechanism, neglecting the incentive constraint for the deadline. The deadline of the winner determines the time of the allocation and therefore also the amount of information available to the seller when he decides whether to allocate to a buyer. Depending on the shape of the markup that the seller uses, this can lead to a violation of the neglected incentive constraint. We give sufficient conditions on the type distribution under which the neglected constraint is fulfilled or violated. Second, for the case that the constraint cannot be neglected, we consider a model with two periods and two buyers. Here, the optimal mechanism is implemented by a fixed price in period one and an asymmetric auction in period two. The asymmetry, which is introduced to prevent the patient type of the first buyer from buying in period one leads to pooling of deadlines at the top of the type space.

We take a mechanism design perspective to investigate how environmental agreements should account for multilateral externalities, incentive compatibility, and voluntary participation by sovereign countries. The optimal mechanism involves a tradeoff between a free rider problem in the effort provision of participating countries and the necessity of inducing countries to ratify the agreement. This mechanism can be approximated by a simple menu with attractive implementation and robustness properties. Limits on enforcement and commitment might nevertheless hinder the performance of this menu, making the “business as usual” scenario more likely.