Efficient Random Assignment under a Combination of Ordinal and Cardinal Information on Preferences

Consider a collection of m indivisible objects to be allocated to n agents, where m = n. Each agent falls in one of two distinct categories: either he (a) has a complete ordinal ranking over the set of individual objects, or (b) has a set of “plausible” benchmark von Neumann-Morgenstern (vNM) utility functions in whose non-negative span his “true” utility is known to lie. An allocation is undominated if there does not exist a preference-compatible profile of vNM utilities at which it is Pareto dominated by another feasible allocation. Given an undominated allocation, we use the tools of linear duality theory to construct a profile of vNM utilities at which it is ex-ante welfare maximizing. A finite set of preference-compatible vNM utility profiles is exhibited such that every undominated allocation is ex-ante welfare maximizing with respect to at least one of them. Given an arbitrary allocation, we provide an interpretation of the constructed vNM utilities as subgradients of a function which measures worst-case domination.Random Assignment, Efficiency, Duality, Linear Programming

House allocation with fractional endowments

This paper studies a generalization of the well known house allocation problem in which agents may own fractions of different houses summing to an arbitrary quantity, but have use for only the equivalent of one unit of a house. It departs from the classical model by assuming that arbitrary quantities of each house may be available to the market. Justified envy considerations arise when two agents have the same initial endowment, or when an agent is in some sense disproportionately rewarded in comparison to her peers. For this general model, an algorithm is designed to find a fractional allocation of houses to agents that satisfies ordinal efficiency, individual rationality, and no justified envy. The analysis extend to the full preference domain. Individual rationality, ordinal efficiency, and no justified envy conflict with weak strategyproofness. Moreover, individual rationality, ordinal efficiency and strategyproofness are shown to be incompatible. Finally, two reasonable notions of envy-freeness, no justified envy and equal-endowment no envy, conflict in the presence of ordinal efficiency and individual rationality. All of the impossibility results hold in the strict preference domain.house allocation, fractional endowments, fairness, individual rationality

Pollution Control: When, and How, to be Precautious

The precautionary principle (PP) applied to environmental policy stipulates that, in the presence of physical uncertainty, society must take robust preventive action to guard against worst-case outcomes. It follows that the higher the degree of uncertainty, the more aggressive this preventive action should be. This normative maxim is explored in the case of a stylized dynamic model of pollution control under Knightian uncertainty. At time 0 a decision-maker makes a one-time investment in damage-control technology and subsequently decides on a desirable dynamic emissions policy. Adopting the robust control framework of Hansen and Sargent [10], we investigate optimal damage-control and mitigation policies. We show that optimal investment in damage control is always increasing in the degree of uncertainty, thus confirming the conventional PP wisdom. Optimal mitigation decisions, however, need not always comport with the PP and we provide analytical conditions that sway the relationship one way or the other. This result is interesting when contrasted to a model with fixed damage-control technology, in which it can be easily shown that a PP vis-a-vis mitigation unambiguously holds. We conduct a set of numerical experiments to determine the sensitivity of our results to specific functional forms of damage-control cost. We find that when the cost of damage-control technology is low enough, damage-control investment and mitigation may act as substitutes and a PP with respect to the latter can be unambiguously irrational.Risk, Ambiguity, Robust Control, Precautionary Principle, Pollution Control

Minimizing regret when dissolving a partnership

We study the problem of dissolving an equal-entitlement partnership when the objective is to minimize maximum regret. We initially focus on the family of linear-pricing mechanisms and derive regret-optimizing strategies. We also demonstrate that there exist linear-pricing mechanisms satisfying ex-post efficiency. Next, we analyze a binary-search mechanism which is ex-post individually rational. We discuss connections with the standard Bayesian-Nash framework for both linear and binary-search mechanisms. On a more general level, we show that if entitlements are unequal, ex-post efficiency and ex-post individual rationality impose significant restrictions on permissible mechanisms. In particular, they rule out both linear and binary-search mechanisms.Partnership dissolution; minimax regret; fair division; allocative efficiency

Dynamic nonpoint-source pollution control policy: ambient transfers and uncertainty

When a regulator cannot observe or infer individual emissions, corrective policy must rely on ambient pollution data. Assuming this kind of environment, we study a class of differential games of pollution control with profit functions that are polynomial in the global pollution stock. Given an open-loop emissions strategy satisfying mild regularity conditions, an ambient transfer scheme is exhibited that induces it in Markov-perfect equilibrium (MPE). Proposed transfers are a polynomial function of the difference between actual and desired pollution levels; moreover, they are designed so that in MPE no tax or subsidy is ever levied. Their applicability under stochastic pollution dynamics is studied for a symmetric game of polluting oligopolists with linear demand. We discuss a quadratic scheme that induces agents to adopt Markovian emissions strategies that are stationary and linearly decreasing in total pollution. Total expected ambient transfers are non-positive and their magnitude is linearly increasing in physical volatility, the size of the economy, and the absolute value of the slope of the inverse demand function. However, if the regulator is interested in inducing a constant emissions strategy then, in expectation, transfers vanish. The total expected ambient transfer is compared to its point-source equivalent

The Effect of Uncertainty on Pollution Control Policy

I study a class of differential games of pollution control with profit functions that are polynomial in the global pollution stock. Given an emissions path satisfying mild regularity conditions, a simple polynomial ambient transfer scheme is exhibited that induces it in Markov-perfect equilibrium (MPE). Proposed transfers are a polynomial function of the difference between actual and desired pollution levels; moreover, they are designed so that in MPE no tax or subsidy is ever levied. Their applicability under stochastic pollution dynamics is studied for a symmetric game of polluting oligopolists with linear demand. I discuss a quadratic scheme that induces agents to adopt Markovian emissions strategies that are stationary and linear-decreasing in total pollution. Total expected ambient transfers are always non-positive and increase linearly in volatility and the absolute value of the slope of the inverse demand function. However, if the regulator is interested in inducing a constant emissions strategy then, in expectation, transfers vanish

Ordinal efficiency under the lens of duality theory

An allocation's ordinal efficiency deficit (OED) is defined as the greatest ordinal efficiency loss that can result from its application. More precisely, an allocation's OED is the negative of the greatest total amount by which it may be stochastically dominated by another feasible allocation. Thus, an allocation is ordinally efficient if and only if its OED is zero. Using this insight, we set up a linear program whose optimal objective value corresponds to a given allocation's OED. Furthermore, we show that the OED is a piecewise-linear convex function on the set of allocations. We use the optimal dual variables of the linear program to construct a profile of von Neumann-Morgenstern (vNM) utilities that is compatible with the underlying ordinal preferences, and which is a subgradient of the OED at the given allocation. When the given allocation is ordinally efficient, our analysis implies that it is ex-ante welfare maximizing at the constructed vNM profile, and we recover the ordinal efficiency theorem due to McLennan (2002

The Effect of Uncertainty on Pollution Control Policy

I study a class of differential games of pollution control with profit functions that are polynomial in the global pollution stock. Given an emissions path satisfying mild regularity conditions, a simple polynomial ambient transfer scheme is exhibited that induces it in Markov-perfect equilibrium (MPE). Proposed transfers are a polynomial function of the difference between actual and desired pollution levels; moreover, they are designed so that in MPE no tax or subsidy is ever levied. Their applicability under stochastic pollution dynamics is studied for a symmetric game of polluting oligopolists with linear demand. I discuss a quadratic scheme that induces agents to adopt Markovian emissions strategies that are stationary and linear-decreasing in total pollution. Total expected ambient transfers are always non-positive and increase linearly in volatility and the absolute value of the slope of the inverse demand function. However, if the regulator is interested in inducing a constant emissions strategy then, in expectation, transfers vanish

Multidimensional welfare rankings

Social well-being is intrinsically multidimensional. Welfare indices attempting to reduce this complexity to a unique measure abound in many areas of economics and public policy. Ranking alternatives based on such measures depends, sometimes critically, on how the different dimensions of welfare are weighted. In this paper, a theoretical framework is presented that yields a set of consensus rankings in the presence of such weight imprecision. The main idea is to consider a vector of weights as an imaginary voter submitting preferences over alternatives in the form of an ordered list. With this voting construct in mind, a rule for aggregating the preferences of many plausible choices of weights, suitably weighted by the importance attached to them, is proposed. An axiomatic characterization of the rule is provided, and its computational implementation is developed. An analytic solution is derived for an interesting special case of the model corresponding to generalized weighted means and the $\epsilon$-contamination framework of Bayesian statistics. The model is applied to the Academic Ranking of World Universities index of Shanghai University, a popular composite index measuring academic excellence

Ambiguous Aggregation of Expert Opinions: The Case of Optimal R&D Investment

How should a decision-maker allocate R&D funds when a group of experts provides divergent estimates on a technology's potential effectiveness? To address this question, we propose a simple decision-theoretic framework that takes into account ambiguity over the aggregation of expert opinion and a decision-maker's attitude towards it. In line with the paper's focus on R&D investment, decision variables in our model may affect experts' subjective probability distributions of the future potential of a technology. Using results from convex optimization, we are able to establish a number of analytical results including a closed-form expression of our model's value function, as well as a thorough investigation of its differentiability properties. We apply our framework to original data from a recent expert elicitation survey on solar technology. The analysis suggests that more aggressive investment in solar technology R&D is likely to yield significant dividends even, or rather especially, after taking ambiguous aggregation into account.Aggregation, Ambiguity, R&D, Expert Opinions, Convex/Conic Optimization