An Efficient Dynamic Mechanism

This paper constructs an efficient, budget‐balanced, Bayesian incentive‐compatible mechanism for a general dynamic environment with quasilinear payoffs in which agents observe private information and decisions are made over countably many periods. First, under the assumption of “private values” (other agents' private information does not directly affect an agent's payoffs), we construct an efficient, ex post incentive‐compatible mechanism, which is not budget‐balanced. Second, under the assumption of “independent types” (the distribution of each agent's private information is not directly affected by other agents' private information), we show how the budget can be balanced without compromising agents' incentives. Finally, we show that the mechanism can be made self‐enforcing when agents are sufficiently patient and the induced stochastic process over types is an ergodic finite Markov chain.     

Common Learning

Consider two agents who learn the value of an unknown parameter by observing a sequence of private signals. The signals are independent and identically distributed across time but not necessarily across agents. We show that when each agent's signal space is finite, the agents will commonly learn the value of the parameter, that is, that the true value of the parameter will become approximate common knowledge. The essential step in this argument is to express the expectation of one agent's signals, conditional on those of the other agent, in terms of a Markov chain. This allows us to invoke a contraction mapping principle ensuring that if one agent's signals are close to those expected under a particular value of the parameter, then that agent expects the other agent's signals to be even closer to those expected under the parameter value. In contrast, if the agents' observations come from a countably infinite signal space, then this contraction mapping property fails. We show by example that common learning can fail in this case.     

Robustly Collusion‐Proof Implementation

A contract with multiple agents may be susceptible to collusion. We show that agents' collusion imposes no cost in a large class of circumstances with risk neutral agents, including both uncorrelated and correlated types. In those circumstances, any payoff the principal can attain in the absence of collusion, including the second‐best level, can be attained in the presence of collusion in a way robust to many aspects of collusion behavior. The collusion‐proof implementation generalizes to a setting in which only a subset of agents may collude, provided that noncollusive agents' incentives can be protected via an ex post incentive compatible and ex post individually rational mechanism. Our collusion‐proof implementation also sheds light on the extent to which hierarchical delegation of contracts can optimally respond to collusion.     

The Limits of ex post Implementation

The sensitivity of Bayesian implementation to agents' beliefs about others suggests the use of more robust notions of implementation such as ex post implementation, which requires that each agent's strategy be optimal for every possible realization of the types of other agents. We show that the only deterministic social choice functions that are ex post implementable in generic mechanism design frameworks with multidimensional signals, interdependent valuations, and transferable utilities are constant functions. In other words, deterministic ex post implementation requires that the same alternative must be chosen irrespective of agents' signals. The proof shows that ex post implementability of a nontrivial deterministic social choice function implies that certain rates of information substitution coincide for all agents. This condition amounts to a system of differential equations that are not satisfied by generic valuation functions.     

Stability and Preference Alignment in Matching and Coalition Formation

We study matching and coalition formation environments allowing complementarities and peer effects. Agents have preferences over coalitions, and these preferences vary with an underlying, and commonly known, state of nature. Assuming that there is substantial variability of preferences across states of nature, we show that there exists a core stable coalition structure in every state if and only if agents' preferences are pairwise‐aligned in every state. This implies that there is a stable coalition structure if agents' preferences are generated by Nash bargaining over coalitional outputs. We further show that all stability‐inducing rules for sharing outputs can be represented by a profile of agents' bargaining functions and that agents match assortatively with respect to these bargaining functions. This framework allows us to show how complementarities and peer effects overturn well known comparative statics of many‐to‐one matching.     

Isotone Equilibrium in Games of Incomplete Information

An isotone pure strategy equilibrium exists in any game of incomplete information in which each player's action set is a finite sublattice of multidimensional Euclidean space, types are multidimensional and atomless, and each player's interim expected payoff function satisfies two “nonprimitive conditions” whenever others adopt isotone pure strategies: (i) single‐crossing in own action and type and (ii) quasi‐supermodularity in own action. Conditions (i), (ii) are satisfied in supermodular and log‐supermodular games given affiliated types,and in games with independent types in which each player's ex post payoff satisfies supermodularity in own action and nondecreasing differences in own action and type. This result is applied to provide the first proof of pure strategy equilibrium existence in the uniform price auction when bidders have multi‐unit demand, nonprivate values, and independent types.     

Sign Restrictions,Structural Vector Autoregressions,and Useful Prior Information

This paper makes the following original contributions to the literature. (i) We develop a simpler analytical characterization and numerical algorithm for Bayesian inference in structural vector autoregressions (VARs) that can be used for models that are overidentified, just‐identified, or underidentified. (ii) We analyze the asymptotic properties of Bayesian inference and show that in the underidentified case, the asymptotic posterior distribution of contemporaneous coefficients in an n‐variable VAR is confined to the set of values that orthogonalize the population variance–covariance matrix of ordinary least squares residuals, with the height of the posterior proportional to the height of the prior at any point within that set. For example, in a bivariate VAR for supply and demand identified solely by sign restrictions, if the population correlation between the VAR residuals is positive, then even if one has available an infinite sample of data, any inference about the demand elasticity is coming exclusively from the prior distribution. (iii) We provide analytical characterizations of the informative prior distributions for impulse‐response functions that are implicit in the traditional sign‐restriction approach to VARs, and we note, as a special case of result (ii), that the influence of these priors does not vanish asymptotically. (iv) We illustrate how Bayesian inference with informative priors can be both a strict generalization and an unambiguous improvement over frequentist inference in just‐identified models. (v) We propose that researchers need to explicitly acknowledge and defend the role of prior beliefs in influencing structural conclusions and we illustrate how this could be done using a simple model of the U.S. labor market.     

Overcoming Incentive Constraints by Linking Decisions1

Consider a Bayesian collective decision problem in which the preferences of agents are private information. We provide a general demonstration that the utility costs associated with incentive constraints become negligible when the decision problem is linked with a large number of independent copies of itself. This is established by defining a mechanism in which agents must budget their representations of preferences so that the frequency of preferences across problems mirrors the underlying distribution of preferences, and then arguing that agents' incentives are to satisfy their budget by being as truthful as possible. We also show that all equilibria of the linking mechanisms converge to the target utility levels. The mechanisms do not require transferable utility or interpersonal comparisons of utility, and are immune to manipulations by coalitions.     

Reputational Bargaining and Deadlines

I highlight how reputational concerns provide a natural explanation for “deadline effects,” the high frequency of deals prior to a deadline in bargaining. Rational agents imitate the demands of obstinate behavioral types and engage in brinkmanship in the face of uncertainty about the deadline's arrival. I also identify how surplus is divided when the prior probability of behavioral types is vanishingly small. If behavioral types are committed to fixed demands, outcomes converge to the Nash bargaining solution regardless of agents' respective impatience. If behavioral types can adopt more complex demand strategies, outcomes converge to the solution of an alternating offers game without behavioral types for the deadline environment.     

A Cooperative Savings Game Approach to a Time Sensitive Capacity Allocation and Scheduling Problem

We consider a competitive scheduling setting with arbitrary number of agents each having the option to utilize two parallel resources to satisfy its demand: (i) an in‐house resource dedicated to process only the tasks of each specific agent, and (ii) a flexible resource capable of processing all agents' workloads. In a noncooperative setting, each agent would determine how much of its demand it will subcontract to the flexible resource with the objective to deliver its entire demand as quickly as possible subject to the priority rules set by the owner of the flexible resource (i.e., third‐party). In this study, we also allow for agents to coalesce with other agents and update their initial subcontracting decisions to attain rescheduling savings. Evidently, a grand coalition of all agents can coordinate to achieve the maximum savings possible, but the resulting schedule may yield individual losses for a subset of agents (which we refer to as “losers”), thus necessitating a transfer payment scheme to distribute the rescheduling savings among the agents in an equitable way. We model the rescheduling interactions among the agents as a cooperative savings game, and propose savings distribution schemes that invoke the core allocation concept.     

INFORMATION GATHERING IN ORGANIZATIONS: EQUILIBRIUM,WELFARE, AND OPTIMAL NETWORK STRUCTURE

We model an organization as a game in which all agents share a common decision problem and some level of coordination is necessary between individual actions. Agents have individual private information concerning the task they have to perform, and they share this private information through pairwise channels of communication. We analyze how this communication pattern, modeled by means of a network structure, affects individual behavior and aggregate welfare. In the unique equilibrium of this Bayesian game, each agent's optimal action depends on a properly defined knowledge index that measures how the aggregation of information helps him to infer higher‐order beliefs about other's information after communication. Adding communication channels is not always beneficial for the organization because it can lead to mis‐coordination. We single out the geometry of interagent communication links that the manager could implement in order to improve the organization's performance.     

How Independent,Competent and Incentivized Should Non‐executive Directors Be? An Empirical Investigation of Good Governance Codes

There is a commonly held conviction among governance scholars and practitioners that increasing the number of non‐executive directors may have beneficial effects on board practices. This view has gained momentum after each wave of scandals. Given the relevance of the issue in governance studies and practices, the aim of this paper is to investigate how independent, competent and incentivized non‐executive directors should be according to governance scholars and board best practices. To answer this question, we conducted a review of the literature on non‐executive directors. We then collected corporate governance codes developed worldwide at the end of 2005, and made a comparative analysis of their recommendations about the independence, the competencies and the incentives of non‐executive directors. Our results show that (i) non‐executive directors' independence is a commonly recommended governance practice, the meaning of which differs widely among countries; (ii) non‐executive directors' competencies and incentives are not considered a governance issue to be regulated in detail; (iii) agency theory and the search for appropriate board demography tend to dominate the recommendations of governance literature and codes. Our findings have implications for both research and practice.     

Approximate versus Exact Equilibria in Dynamic Economies

This paper develops theoretical foundations for an error analysis of approximate equilibria in dynamic stochastic general equilibrium models with heterogeneous agents and incomplete financial markets. While there are several algorithms that compute prices and allocations for which agents' first‐order conditions are approximately satisfied (“approximate equilibria”), there are few results on how to interpret the errors in these candidate solutions and how to relate the computed allocations and prices to exact equilibrium allocations and prices. We give a simple example to illustrate that approximate equilibria might be very far from exact equilibria. We then interpret approximate equilibria as equilibria for close‐by economies; that is, for economies with close‐by individual endowments and preferences. We present an error analysis for two models that are commonly used in applications, an overlapping generations (OLG) model with stochastic production and an asset pricing model with infinitely lived agents. We provide sufficient conditions that ensure that approximate equilibria are close to exact equilibria of close‐by economies. Numerical examples illustrate the analysis.     

On the Equivalence of Bayesian and Dominant Strategy Implementation

We consider a standard social choice environment with linear utilities and independent, one‐dimensional, private types. We prove 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 and the same ex ante expected social surplus. The short proof is based on an extension of an elegant result due to Gutmann, Kemperman, Reeds, and Shepp (1991). We also show that the equivalence between Bayesian and dominant strategy implementation generally breaks down when the main assumptions underlying the social choice model are relaxed or when the equivalence concept is strengthened to apply to interim expected allocations.     

A gradient Markov chain Monte Carlo algorithm for computing multivariate maximum likelihood estimates and posterior distributions: mixture dose-response assessment

Multivariate probability distributions, such as may be used for mixture dose‐response assessment, are typically highly parameterized and difficult to fit to available data. However, such distributions may be useful in analyzing the large electronic data sets becoming available, such as dose‐response biomarker and genetic information. In this article, a new two‐stage computational approach is introduced for estimating multivariate distributions and addressing parameter uncertainty. The proposed first stage comprises a gradient Markov chain Monte Carlo (GMCMC) technique to find Bayesian posterior mode estimates (PMEs) of parameters, equivalent to maximum likelihood estimates (MLEs) in the absence of subjective information. In the second stage, these estimates are used to initialize a Markov chain Monte Carlo (MCMC) simulation, replacing the conventional burn‐in period to allow convergent simulation of the full joint Bayesian posterior distribution and the corresponding unconditional multivariate distribution (not conditional on uncertain parameter values). When the distribution of parameter uncertainty is such a Bayesian posterior, the unconditional distribution is termed predictive. The method is demonstrated by finding conditional and unconditional versions of the recently proposed emergent dose‐response function (DRF). Results are shown for the five‐parameter common‐mode and seven‐parameter dissimilar‐mode models, based on published data for eight benzene–toluene dose pairs. The common mode conditional DRF is obtained with a 21‐fold reduction in data requirement versus MCMC. Example common‐mode unconditional DRFs are then found using synthetic data, showing a 71% reduction in required data. The approach is further demonstrated for a PCB 126‐PCB 153 mixture. Applicability is analyzed and discussed. Matlab^® computer programs are provided.     

Robust Predictions for Bilateral Contracting with Externalities

The paper studies bilateral contracting between one principal and N agents when each agent's utility depends on the principal's unobservable contracts with other agents. We show that allowing deviations to menu contracts from which the principal chooses bounds equilibrium outcomes in a wide class of bilateral contracting games without imposing ad hoc restrictions on the agents' beliefs. This bound yields, for example, competitive convergence as N →∞ in environments in which an appropriately‐defined notion of competitive equilibrium exists. We also examine the additional restrictions arising in two common bilateral contracting games: the “offer game” in which the principal makes simultaneous offers to the agents, and the “bidding game” in which the agents make simultaneous offers to the principal.     

Regression Versus Causation,Revisited

I recently discussed pitfalls in attempted causal inference based on reduced‐form regression models. I used as motivation a real‐world example from a paper by Dr. Sneeringer, which interpreted a reduced‐form regression analysis as implying the startling causal conclusion that “doubling of [livestock] production leads to a 7.4% increase in infant mortality.” This conclusion is based on: (A) fitting a reduced‐form regression model to aggregate (e.g., county‐level) data; and (B) (mis)interpreting a regression coefficient in this model as a causal coefficient, without performing any formal statistical tests for potential causation (such as conditional independence, Granger‐Sims, or path analysis tests). Dr. Sneeringer now adds comments that confirm and augment these deficiencies, while advocating methodological errors that, I believe, risk analysts should avoid if they want to reach logically sound, empirically valid, conclusions about cause and effect. She explains that, in addition to (A) and (B) above, she also performed other steps such as (C) manually selecting specific models and variables and (D) assuming (again, without testing) that hand‐picked surrogate variables are valid (e.g., that log‐transformed income is an adequate surrogate for poverty). In her view, these added steps imply that “critiques of A and B are not applicable” to her analysis and that therefore “a causal argument can be made” for “such a strong, robust correlation” as she believes her regression coefficient indicates. However, multiple wrongs do not create a right. Steps (C) and (D) exacerbate the problem of unjustified causal interpretation of regression coefficients, without rendering irrelevant the fact that (A) and (B) do not provide evidence of causality. This reply focuses on whether any statistical techniques can produce the silk purse of a valid causal inference from the sow's ear of a reduced‐form regression analysis of ecological data. We conclude that Dr. Sneeringer's analysis provides no valid indication that air pollution from livestock operations causes any increase in infant mortality rates. More generally, reduced‐form regression modeling of aggregate population data—no matter how it is augmented by fitting multiple models and hand‐selecting variables and transformations—is not adequate for valid causal inference about health effects caused by specific, but unmeasured, exposures.     

Intelligent Procedures for Intra‐Day Updating of Call Center Agent Schedules

For nearly all call centers, agent schedules are typically created several days or weeks before the time that agents report to work. After schedules are created, call center resource managers receive additional information that can affect forecasted workload and resource availability. In particular, there is significant evidence, both among practitioners and in the research literature, suggesting that actual call arrival volumes early in a scheduling period (typically an individual day or week) can provide valuable information about the call arrival pattern later in the same scheduling period. In this paper, we develop a flexible and powerful heuristic framework for managers to make intra‐day resource adjustment decisions that take into account updated call forecasts, updated agent requirements, existing agent schedules, agents' schedule flexibility, and associated incremental labor costs. We demonstrate the value of this methodology in managing the trade‐off between labor costs and service levels to best meet variable rates of demand for service, using data from an actual call center.     

Stable and Coordinating Contracts for a Supply Chain with Multiple Risk‐Averse Suppliers

We analyze a decentralized supply chain with a single risk‐averse retailer and multiple risk‐averse suppliers under a Conditional Value at Risk objective. We define coordinating contracts and show that the supply chain is coordinated only when the least risk‐averse agent bears the entire risk and the lowest‐cost supplier handles all production. However, due to competition, not all coordinating contracts are stable. Thus, we introduce the notion of contract core, which reflects the agents' “bargaining power” and restricts the set of coordinating contracts to a subset which is “credible.” We also study the concept of contract equilibrium, which helps to characterize contracts that are immune to opportunistic renegotiation. We show that, the concept of contract core imposes conditions on the share of profit among different agents, while the concept of contract equilibrium provide conditions on how the payment changes with the order quantity.     

Fear of Miscoordination and the Robustness of Cooperation in Dynamic Global Games With Exit

This paper develops a framework to assess how fear of miscoordination affects the sustainability of cooperation. Building on theoretical insights from Carlsson and van Damme (1993), it explores the effect of small amounts of private information on a class of dynamic cooperation games with exit. Lack of common knowledge leads players to second guess each other's behavior and makes coordination difficult. This restricts the range of equilibria and highlights the role of miscoordination payoffs in determining whether cooperation is sustainable or not. The paper characterizes the range of perfect Bayesian equilibria as the players' information becomes arbitrarily precise. Unlike in one‐shot two‐by‐two games, the global games information structure does not yield equilibrium uniqueness.