Probabilistic Serial Mechanism Research Articles

Random assignment under ordinal preferences: a separation characterisation

Assignment problems include allocating a set of objects among agents; here, only ordinal preferences are revealed. In this paper, we establish a condition of feasible solutions for deterministic assignments. Related to it, we show then a separation characterisation for probabilistic serial (PS) mechanism, based on sd-efficiency, sd-envy-freeness and the definition of PS (where 'sd' stands for first-order stochastic dominance). An application to recent result about PS is also described. Models here are suitable for assignment problems in various fields, such as fair sharing of resources in industry. The separation structure proposed here provides a possibility to divide a large-scale problem into several sub-problems.

Random assignment under ordinal preferences: a separation characterisation

Assignment problems include allocating a set of objects among agents; here, only ordinal preferences are revealed. In this paper, we establish a condition of feasible solutions for deterministic assignments. Related to it, we show then a separation characterisation for probabilistic serial (PS) mechanism, based on sd-efficiency, sd-envy-freeness and the definition of PS (where 'sd' stands for first-order stochastic dominance). An application to recent result about PS is also described. Models here are suitable for assignment problems in various fields, such as fair sharing of resources in industry. The separation structure proposed here provides a possibility to divide a large-scale problem into several sub-problems.

On characterizations of the probabilistic serial mechanism involving incentive and invariance properties

This paper studies the problem of assigning n indivisible objects to n agents when each agent consumes one object and monetary transfers are not allowed. Bogomolnaia and Moulin (2001) proved that for n=3, the probabilistic serial mechanism is characterized by the three axioms of ordinal efficiency, envy-freeness, and weak strategy-proofness. We show that this characterization does not extend to problems of arbitrary size; in particular, it does not hold for any n≥5. A number of general characterizations of the probabilistic serial mechanism have been obtained in the recent literature by replacing weak strategy-proofness with various invariance axioms while retaining ordinal efficiency and envy-freeness. We show that weak strategy-proofness is in fact logically independent of all invariance axioms used in these characterizations.

Efficient lottery design

There has been a surge of interest in stochastic assignment mechanisms that have proven to be theoretically compelling thanks to their prominent welfare properties. Contrary to stochastic mechanisms, however, lottery mechanisms are commonly used in real life for indivisible goods allocation. To help facilitate the design of practical lottery mechanisms, we provide new tools for obtaining stochastic improvements in lotteries. As applications, we propose lottery mechanisms that improve upon the widely used random serial dictatorship mechanism and a lottery representation of its competitor, the probabilistic serial mechanism. The tools we provide here can be useful in developing welfare-enhanced new lottery mechanisms for practical applications such as school choice.

A New Ex-Ante Efficiency Criterion and Implications for the Probabilistic Serial Mechanism

A New Ex-Ante Efficiency Criterion and Implications for the Probabilistic Serial Mechanism

Convex strategyproofness with an application to the probabilistic serial mechanism

We consider two natural notions of strategyproofness in random object-assignment mechanisms based on ordinal preferences. The two notions are stronger than weak strategyproofness but weaker than strategyproofness. We demonstrate that the two notions are equivalent, provide a geometric characterization of the new intermediate property which we call convex strategyproofness, and then show that the (generalized) probabilistic serial mechanism is, in fact, convexly strategyproof. We finish by showing that the property of weak envy-freeness of the random serial dictatorship can be strengthened in an analogous manner.

An equilibrium analysis of the probabilistic serial mechanism

Two prominent mechanisms in the random assignment problem are the random priority (RP) and the probabilistic serial (PS). When agents are truthful, the outcomes obtained under PS have superior efficiency and fairness properties, but unlike RP, PS is vulnerable to strategizing. We study incentives of agents under PS. We find that when agents strategize, in equilibrium an outcome may be obtained under PS which is not efficient or fair and which is worse in some respects than the RP outcome. The results of our equilibrium analysis of PS call for caution when implementing it in “small” assignment problems.

A note on the assignment problem with uniform preferences

Motivated by a problem of scheduling unit-length jobs with weak preferences over time-slots, the random assignment problem is considered on a uniform preference domain. It is shown that the natural extension of the probabilistic serial mechanism to the domain of weak, but uniform, preferences fails strategy-proofness, but so does every other mechanism that is ordinally efficient and treats equals equally. If envy-free assignments are required, any ex-post efficient (probabilistic or deterministic) mechanism must fail even a weak form of strategy-proofness.

Size versus fairness in the assignment problem

When not all objects are acceptable to all agents, maximizing the number of objects actually assigned is an important design concern. We compute the guaranteed size ratio of the Probabilistic Serial mechanism, i.e., the worst ratio of the actual expected size to the maximal feasible size. It converges decreasingly to 1−1e≃63.2% as the maximal size increases. It is the best ratio of any Envy-Free assignment mechanism.

An experimental study on the incentives of the probabilistic serial mechanism

We report an experiment on the Probabilistic Serial (PS) mechanism for allocating indivisible goods. The PS mechanism, a recently discovered alternative to the widely used Random Serial Dictatorship mechanism, has attractive fairness and efficiency properties if people report their preferences truthfully. However, the mechanism is not strategy-proof, so participants may not truthfully report their preferences. We investigate misreporting in a set of simple applications of the PS mechanism. We confront subjects with situations in which the theory suggests that there is an incentive or no incentive to misreport. We find little misreporting in situations where misreporting is a Nash equilibrium. However, we also find a significant degree of misreporting in situations where there is actually no benefit to doing so. These findings suggest that the PS mechanism may have problems in terms of truthful elicitation.

Two axiomatic approaches to the probabilistic serial mechanism

This paper studies the problem of assigning a set of indivisible objects to a set of agents when monetary transfers are not allowed and agents reveal only ordinal preferences, but random assignments are possible. We offer two characterizations of the probabilistic serial mechanism, which assigns lotteries over objects. We show that it is the only mechanism that satisfies non-wastefulness and ordinal fairness, and the only mechanism that satisfies sd-efficiency, sd-envy-freeness, and weak invariance or weak truncation robustness (where “sd” stands for first-order stochastic dominance).

A note on object allocation under lexicographic preferences

We consider the problem of allocating m objects to n agents. Each agent has unit demand, and has strict preferences over the objects. There are qj units of object j available and the problem is balanced in the sense that ∑jqj=n. An allocation specifies the amount of each object j that is assigned to each agent i, when the objects are divisible; when the objects are indivisible and exactly one unit of each object is available, an allocation is interpreted as the probability that agent i is assigned one unit of object j. In our setting, agent preferences over objects are extended to preferences over allocations using the natural lexicographic order. The goal is to design mechanisms that are efficient, envy-free, and strategy-proof. Schulman and Vazirani show that an adaptation of the probabilistic serial mechanism satisfies all these properties when qj≥1 for all objects j. Our first main result is a characterization of problems for which efficiency, envy-freeness, and strategy-proofness are compatible. Furthermore, we show that these three properties do not characterize the serial mechanism. Finally, we show that when indifferences between objects are permitted in agent preferences, it is impossible to satisfy all three properties even in the standard setting of “house” allocation in which all object supplies are 1.

An Experimental Study on the Incentives of the Probabilistic Serial Mechanism

We report an experiment on the Probabilistic Serial (PS) mechanism for allocating indivisible goods. The PS mechanism, a recently discovered alternative to the widely used Random Serial Dictatorship mechanism, has attractive fairness and efficiency properties if people report their preferences truthfully. However, the mechanism is not strategy-proof, so participants may not truthfully report their preferences. We investigate misreporting in a set of simple applications of the PS mechanism. We confront subjects with situations in which theory suggests that there is an incentive or no incentive to misreport. We find little misreporting in situations where misreporting is a Nash equilibrium. However, we also find a significant degree of misreporting in sitations where there is actually no benefit to doing so. These findings suggest that the PS mechanism may have problems in terms of truthful elicitation.

Characterizations of the Probabilistic Serial Mechanism

This paper provides two axiomatizations of the probabilistic serial mechanism. First, the mechanism is characterized by ordinal efficiency, envy-freeness, and truncation robustness. Truncation robustness restricts changes in assignments when agents truncate their preference lists. In this characterization, ordinal efficiency and envy-freeness can be replaced with weaker axioms, 2-ordinal efficiency and weak envy-freeness. Second, the mechanism is characterized by ordinal efficiency, independence of unassigned objects (IUO), and Rawlsian criterion. IUO requires assignments to be independent of whether an agent reports an object as acceptable or not, when she has no chance to obtain it even if she does so. Rawlsian criterion is a fairness condition that specifies how a mechanism should treat the worst-off agent.

The probabilistic serial mechanism with private endowments

A random assignment is ordinally efficient if it is not stochastically dominated with respect to individual preferences over sure objects. When there are no private endowments, the set of ordinally efficient random assignments is characterized by the eating algorithm (Bogomolnaia and Moulin, 2001). When there are private endowments, the main requirement is individual rationality; however, the eating algorithm fails to deliver this property. Our contribution is the natural generalization of the eating algorithm for this general class of problems. The family of this generalized eating algorithm characterizes the set of individually rational and ordinally efficient random assignments. A special solution in this family, the individually rational probabilistic serial ( PS IR ), also achieves a new fairness axiom, no justified-envy. However, it is not immune to strategic manipulation. We show that individual rationality, no justified-envy and strategy-proofness are incompatible.

Incentives in the probabilistic serial mechanism

The probabilistic serial mechanism (Bogomolnaia and Moulin, 2001 [9]) is ordinally efficient but not strategy-proof. We study incentives in the probabilistic serial mechanism for large assignment problems. We establish that for a fixed set of object types and an agent with a given expected utility function, if there are sufficiently many copies of each object type, then reporting ordinal preferences truthfully is a weakly dominant strategy for the agent (regardless of the number of other agents and their preferences). The non-manipulability and the ordinal efficiency of the probabilistic serial mechanism support its implementation instead of random serial dictatorship in large assignment problems.

Random assignment of multiple indivisible objects

We consider random assignment of multiple indivisible objects. When each agent receives one object, [Bogomolnaia, A., Moulin, H., 2001. A new solution to the random assignment problem. Journal of Economic Theory 100, 295–328] show that the probabilistic serial mechanism is ordinally efficient, envy-free and weakly strategy-proof. When each agent receives more than one object, we propose a generalized probabilistic serial mechanism that is ordinally efficient and envy-free but not weakly strategy-proof. Our main result shows that, if each agent receives more than one object, there exists no mechanism that is ordinally efficient, envy-free and weakly strategy-proof.

A solution to the random assignment problem on the full preference domain

We consider the problem of allocating a set of indivisible objects to agents in a fair and efficient manner. In a recent paper, Bogomolnaia and Moulin consider the case in which all agents have strict preferences, and propose the probabilistic serial (PS) mechanism; they define a new notion of efficiency, called ordinal efficiency, and prove that the probabilistic serial mechanism finds an envy-free ordinally efficient assignment. However, the restrictive assumption of strict preferences is critical to their algorithm. Our main contribution is an analogous algorithm for the full preference domain in which agents are allowed to be indifferent between objects. Our algorithm is based on a reinterpretation of the PS mechanism as an iterative algorithm to compute a “flow” in an associated network. In addition we show that on the full preference domain it is impossible for even a weak strategyproof mechanism to find a random assignment that is both ordinally efficient and envy-free.