Public Decisions: Solidarity and the Status Quo

A public decision model specifies a fixed set of alternatives A, a variable population, and a fixed set of admissible preferences over A, common to all agents. We study the implications, for any social choice function, of the principle of solidarity, in the class of all such models. The principle says that when the environment changes, all agents not responsible for the change should all be affected in the same direction: either all weakly win, or all weakly lose. We consider two formulations of this principle: population-monotonicity (Thomson, 1983); and replacement-domination (Moulin, 1987). Under weak additional requirements, but regardless of the domain of preferences considered, each of the two conditions implies (i) coalition-strategy-proofness; (ii) that the choice only depends on the set of preferences that are present in the society and not on the labels of agents, nor on the number of agents having a particular preference; (iii) that there exists a status quo point, i.e. an alternative always weakly Pareto-dominated by the alternative selected by the rule. We also prove that replacement-domination is generally at least as strong as population-monotonicity.

Priorities in the Location of Multiple Public Facilities

A collective decision problem is described by a set of agents, a profile of single-peaked preferences over the real line and a number k of public facilities to be located. We consider public facilities that do not su¤er from congestion and are non-excludable. We provide a characterization of the class of rules satisfying Pareto-efficiency, object-population monotonicity and sovereignty. Each rule in the class is a priority rule that selects locations according to a predetermined priority ordering among interest groups. We characterize each of the subclasses of priority rules that respectively satisfy anonymity, hiding-proofness and strategy-proofness. In particular, we prove that a priority rule is strategy-proof if and only if it partitions the set of agents into a fixed hierarchy. Alternatively, any such rule can be viewed as a collection of fixed-populations generalized peak-selection median rules (Moulin, 1980), that are linked across populations, in a way that we describe.

Allocation via Deferred-Acceptance under Responsive Priorities

In many economic environments - such as college admissions, student placements at public schools, and university housing allocation - indivisible objects with capacity constraints are assigned to a set of agents when each agent receives at most one object and monetary compensations are not allowed. In these important applications the agent-proposing deferred-acceptance algorithm with responsive priorities (called responsive DA-rule) performs well and economists have successfully implemented responsive DA-rules or slight variants thereof. First, for house allocation problems we characterize the class of responsive DA-rules by a set of basic and intuitive properties, namely, unavailable type invariance, individual rationality, weak non-wastefulness, resource-monotonicity, truncation invariance, and strategy-proofness. We extend this characterization to the full class of allocation problems with capacity constraints by replacing resource- monotonicity with two-agent consistent con ict resolution. An alternative characterization of responsive DA-rules is obtained using unassigned objects invariance, individual rationality, weak non-wastefulness, weak consistency, and strategy-proofness. Various characterizations of the class of "acyclic" responsive DA-rules are obtained by using the properties efficiency, group strategy-proofness, and consistency.

Top trading with fixed tie-breaking in markets with indivisible goods

We study markets with indivisible goods where monetary compensations are not possible. Each individual is endowed with an object and a preference relation over all objects. When preferences are strict, Gale's top trading cycle algorithm finds the unique core allocation. When preferences are not necessarily strict, we use an exogenous profile of tie-breakers to resolve any ties in individuals' preferences and apply Gale's top trading cycle algorithm for the resulting profile of strict preferences. We provide a foundation of these simple extensions of Gale's top trading cycle algorithm from strict preferences to weak preferences. We show that Gale's top trading cycle algorithm with fixed tie-breaking is characterized by individual rationality, strategy-proofness, weak efficiency, non-bossiness, and consistency. Our result supports the common practice in applications to break ties in weak preferences using some fixed exogenous criteria and then to use a 'good and simple' rule for the resulting strict preferences. This reinforces the market-based approach even in the presence of indifferences because always competitive allocations are chosen.

Single-Plateaued Choice

Single-plateaued preferences generalize single-peaked preferences by allowing for multiple best elements. These preferences have played an important role in areas such as voting, strategy-proofness and matching problems. We examine the notion of singleplateauedness in a choice-theoretic setting. Single-plateaued choice is characterized by means of a collinear interval continuity property in the presence of independence of irrelevant alternatives. Further results establish that our notion of single-plateauedness conforms to the motivation underlying the term and we analyze the consequences of alternative continuity properties. The importance of basic assumptions such as closedness and convexity is discussed. Journal of Economic Literature Classification Nos.: D11, D71.

Single-plateaued choice

Single-plateaued preferences generalize single-peaked preferences by allowing for multiple best elements. These preferences have played an important role in areas such as voting, strategy-proofness and matching problems. We examine the notion of single-plateauedness in a choice-theoretic setting. Single-plateaued choice is characterized by means of a collinear interval continuity property in the presence of independence of irrelevant alternatives. Further results establish that our notion of single-plateauedness conforms to the motivation underlying the term and we analyze the consequences of alternative continuity properties. The importance of basic assumptions such as closedness and convexity is discussed. Journal of Economic Literature Classification Nos.: D11, D71.

Single-basined choice

Single-basined preferences generalize single-dipped preferences by allowing for multiple worst elements. These preferences have played an important role in areas such as voting, strategy-proofness and matching problems. We examine the notion of single-basinedness in a choice-theoretic setting. In conjunction with independence of irrelevant alternatives, single-basined choice implies a structure that conforms to the motivation underlying our definition. We also establish the consequenes of requiring single-basined choice correspondences to be upper semicontinuous, and of the revealed preference relation to be Suzumura consistent. Journal of Economic Literature.

House allocation via deferred-acceptance

We study the simple model of assigning indivisible and heterogenous objects (e.g., houses, jobs, offi ces, etc.) to agents. Each agent receives at most one object and monetary compensations are not possible. For this model, known as the house allocation model, we characterize the class of rules satisfying unavailable object invariance, individual rationality, weak non-wastefulness, resource-monotonicity, truncation invariance, and strategy-proofness: any rule with these properties must allocate objects based on (implicitly induced) objects' priorities over agents and the agent-proposing deferred-acceptance-algorithm.

Asymmetrically fair rules for an indivisible good problem with a budget constraint

We study a particular restitution problem where there is an indivisible good (land or property) over which two agents have rights: the dispossessed agent and the owner. A third party, possibly the government, seeks to resolve the situation by assigning rights to one and compensate the other. There is also a maximum amount of money available for the compensation. We characterize a family of asymmetrically fair rules that are immune to strategic behavior, guarantee minimal welfare levels for the agents, and satisfy the budget constraint.

Centralized allocation in multiple markets

We study the problem of centralized allocation of indivisible objects in multiple markets. We show that the set of allocation rules that are group strategy-proof and Pareto-efficient are sequential dictatorships. Therefore, the solution of the joint al-location in multiple markets is significantly narrower than in the single-market case. Our result also applies to dynamic allocation problems. Finally, we provide conditions under which the solution of the single-market allocation coincides with the multiple-market case, and we apply this result to the study of the school choice problem with sibling priorities.

Essays on robust mechanism design

Chapter 1: Under the average common value function, we select almost uniquely the mechanism that gives the seller the largest portion of the true value in the worst situation among all the direct mechanisms that are feasible, ex-post implementable and individually rational. Chapter 2: Strategy-proof, budget balanced, anonymous, envy-free linear mechanisms assign p identical objects to n agents. The efficiency loss is the largest ratio of surplus loss to efficient surplus, over all profiles of non-negative valuations. The smallest efficiency loss is uniquely achieved by the following simple allocation rule: assigns one object to each of the p−1 agents with the highest valuation, a large probability to the agent with the pth highest valuation, and the remaining probability to the agent with the (p+1)th highest valuation. When “envy freeness” is replaced by the weaker condition “voluntary participation”, the optimal mechanism differs only when p is much less than n. Chapter 3: One group is to be selected among a set of agents. Agents have preferences over the size of the group if they are selected; and preferences over size as well as the “stand-outside” option are single-peaked. We take a mechanism design approach and search for group selection mechanisms that are efficient, strategy-proof and individually rational. Two classes of such mechanisms are presented. The proposing mechanism allows agents to either maintain or shrink the group size following a fixed priority, and is characterized by group strategy-proofness. The voting mechanism enlarges the group size in each voting round, and achieves at least half of the maximum group size compatible with individual rationality.

A second order control-volume finite-element least-squares strategy for simulating diffusion in strongly anisotropic media

An unstructured mesh �nite volume discretisation method for simulating di�usion in anisotropic media in two-dimensional space is discussed. This technique is considered as an extension of the fully implicit hybrid control-volume �nite-element method and it retains the local continuity of the ux at the control volume faces. A least squares function recon- struction technique together with a new ux decomposition strategy is used to obtain an accurate ux approximation at the control volume face, ensuring that the overall accuracy of the spatial discretisation maintains second order. This paper highlights that the new technique coincides with the traditional shape function technique when the correction term is neglected and that it signi�cantly increases the accuracy of the previous linear scheme on coarse meshes when applied to media that exhibit very strong to extreme anisotropy ratios. It is concluded that the method can be used on both regular and irregular meshes, and appears independent of the mesh quality.