Strategy-proofness makes the difference: deferred-acceptance with responsive priorities

In college admissions and student placements at public schools, the admission decision can be thought of as assigning indivisible objects with capacity constraints to a set of students such that each student receives at most one object and monetary compensations are not allowed. In these important market design problems, the agent-proposing deferred-acceptance (DA-)mechanism with responsive strict priorities performs well and economists have successfully implemented DA-mechanisms or slight variants thereof. We show that almost all real-life mechanisms used in such environments - including the large classes of priority mechanisms and linear programming mechanisms - satisfy a set of simple and intuitive properties. Once we add strategy-proofness to these properties, DA-mechanisms are the only ones surviving. In market design problems that are based on weak priorities (like school choice), generally multiple tie-breaking (MTB)procedures are used and then a mechanism is implemented with the obtained strict priorities. By adding stability with respect to the weak priorities, we establish the first normative foundation for MTB-DA-mechanisms that are used in NYC.

Strategy-proofness and essentially single-valued cores revisited

We consider general allocation problems with indivisibilities where agents' preferences possibly exhibit externalities. In such contexts many different core notions were proposed. One is the gamma-core whereby blocking is only allowed via allocations where the non-blocking agents receive their endowment. We show that if there exists an allocation rule satisfying ‘individual rationality’, ‘efficiency’, and ‘strategy-proofness’, then for any problem for which the gamma-core is non-empty, the allocation rule must choose a gamma-core allocation and all agents are indifferent between all allocations in the gamma-core. We apply our result to housing markets, coalition formation and networks.

Object allocation via deferred-acceptance: strategy-proofness and comparative statics

We study the problem of assigning indivisible and heterogenous objects (e.g., houses, jobs, offices, school or university admissions etc.) to agents. Each agent receives at most one object and monetary compensations are not possible. We consider mechanisms satisfying a set of basic properties (unavailable-type-invariance, individual-rationality, weak non-wastefulness, or truncation-invariance). In the house allocation problem, where at most one copy of each object is available, deferred-acceptance (DA)-mechanisms allocate objects based on exogenously fixed objects' priorities over agents and the agent-proposing deferred-acceptance-algorithm. For house allocation we show that DA-mechanisms are characterized by our basic properties and (i) strategy-proofness and population-monotonicity or (ii) strategy-proofness and resource-monotonicity. Once we allow for multiple identical copies of objects, on the one hand the first characterization breaks down and there are unstable mechanisms satisfying our basic properties and (i) strategy-proofness and population-monotonicity. On the other hand, our basic properties and (ii) strategy-proofness and resource-monotonicity characterize (the most general) class of DA-mechanisms based on objects' fixed choice functions that are acceptant, monotonic, substitutable, and consistent. These choice functions are used by objects to reject agents in the agent-proposing deferred-acceptance-algorithm. Therefore, in the general model resource-monotonicity is the «stronger» comparative statics requirement because it characterizes (together with our basic requirements and strategy-proofness) choice-based DA-mechanisms whereas population-monotonicity (together with our basic properties and strategy-proofness) does not.

Efficient Strategy-Proof Allocation Functions in Linear Production Economies

In a linear production model, we characterize the class of efficient and strategy-proof allocation functions, and the class of efficient and coalition strategy-proof allocation functions. In the former class, requiring equal treatment of equals allows us to identify a unique allocation function. This function is also the unique member of the latter class which satisfies uniform treatment of uniforms.

Strategy-proof tie-breaking

We study a general class of priority-based allocation problems with weak priority orders and identify conditions under which there exists a strategy-proof mechanism which always chooses an agent-optimal stable, or constrained efficient, matching. A priority structure for which these two requirements are compatible is called solvable. For the general class of priority-based allocation problems with weak priority orders,we introduce three simple necessary conditions on the priority structure. We show that these conditions completely characterize solvable environments within the class of indifferences at the bottom (IB) environments, where ties occur only at the bottom of the priority structure. This generalizes and unifies previously known results on solvable and unsolvable environments established in school choice, housing markets and house allocation with existing tenants. We show how the previously known solvable cases can be viewed as extreme cases of solvable environments. For sufficiency of our conditions we introduce a version of the agent-proposing deferred acceptance algorithm with exogenous and preference-based tie-breaking.

Strategy-proof choice of acts : a preliminary study

We model social choices as acts mapping states of the world to (social) outcomes. A (social choice) rule assigns an act to every profile of subjective expected utility preferences over acts. A rule is strategy-proof if no agent ever has an incentive to misrepresent her beliefs about the world or her valuation of the outcomes; it is ex-post efficient if the act selected at any given preference profile picks a Pareto-efficient outcome in every state of the world. We show that every two-agent ex-post efficient and strategy-proof rule is a top selection: the chosen act picks the most preferred outcome of some (possibly different) agent in every state of the world. The states in which an agent’s top outcome is selected cannot vary with the reported valuations of the outcomes but may change with the reported beliefs. We give a complete characterization of the ex-post efficient and strategy-proof rules in the two-agent, two-state case, and we identify a rich class of such rules in the two-agent case.

On Fixed-Path Rationing Methods.

Moulin (1999) characterizes the fixed-path rationing methods by efficiency, strategy-proofness, consistency, and resource-monotonicity. In this note, we give a straightforward proof of his result.

Probabilistic Assignments of Identical Indivisible Objects and Uniform Probabilistic Rules.

We consider a probabilistic approach to the problem of assigning k indivisible identical objects to a set of agents with single-peaked preferences. Using the ordinal extension of preferences, we characterize the class of uniform probabilistic rules by Pareto efficiency, strategy-proofness, and no-envy. We also show that in this characterization no-envy cannot be replaced by anonymity. When agents are strictly risk averse von-Neumann-Morgenstern utility maximizers, then we reduce the problem of assigning k identical objects to a problem of allocating the amount k of an infinitely divisible commodity.

Efficient Priority Rules

We study the assignment of indivisible objects with quotas (houses, jobs, or offices) to a set of agents (students, job applicants, or professors). Each agent receives at most one object and monetary compensations are not possible. We characterize efficient priority rules by efficiency, strategy-proofness, and reallocation-consistency. Such a rule respects an acyclical priority structure and the allocations can be determined using the deferred acceptance algorithm.

Consistent House Allocation

In practice we often face the problem of assigning indivisible objects (e.g., schools, housing, jobs, offices) to agents (e.g., students, homeless, workers, professors) when monetary compensations are not possible. We show that a rule that satisfies consistency, strategy-proofness, and efficiency must be an efficient generalized priority rule; i.e. it must adapt to an acyclic priority structure, except -maybe- for up to three agents in each object's priority ordering.

Social Choice: Recent Developments

In the past quarter century, there has been a dramatic shift of focus in social choice theory, with structured sets of alternatives and restricted domains of the sort encountered in economic problems coming to the fore. This article provides an overview of some of the recent contributions to four topics in normative social choice theory in which economic modelling has played a prominent role: Arrovian social choice theory on economic domains, variable-population social choice, strategy-proof social choice, and axiomatic models of resource allocation.

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.