Indivisible Objects Research Articles

Dynamic Mechanisms with Verification

We consider a principal who allocates an indivisible object among a finite number of agents who arrive on-line, each of whom prefers to have the object than not. Each agent has access to private information about the principal's payoff if he receives the object. The decision to allocate the object to an agent must be made upon arrival of an agent and is irreversible. There are no monetary transfers but he principal can inspect agents' reports at a cost and punish them. A novelty of this paper is a reformulation of this dynamic problem as a compact linear program. Using the formulation we characterize the form of the optimal mechanism and reduce the dynamic version of the inspection problem with identical distributions to an instance of the secretary problem with one fewer secretary and a modified value distribution. This reduction also allows us to derive a prophet inequality for the dynamic version of the inspection problem.

Control of Movement of a Loosely Coupled Self-Propelled Modular Transporter

Arguments are made in favor of the use of transporters intended for transporting large-sized indivisible heavy objects, which are composed of separate self-propelled modules. The basic principles of operating such a loosely coupled vehicle using modern information technologies are given in the context of steering control. Some versions of transporters are presented as are schemes of taxi patterns they can perform. The operation of a three-level movement control system of a loosely coupled transporter is considered including its functional arrangement.

Efficient and fair assignment mechanisms are strongly group manipulable

We study the assignment of indivisible objects to agents without using monetary transfers. We prove that, regardless of whether the number of objects is sufficient, a mechanism that satisfies ex-post Pareto efficiency, equal treatment of equals, equal total assignment (ETA), and uniform-head fairness (UHF) must be strongly manipulable by a group of agents. In other words, by misreporting their preferences, all group members can obtain lotteries that strictly first-order stochastically dominate the lotteries they would have obtained by reporting their true preferences. ETA requires that agents obtain equal total probability shares of objects. UHF requires that if all agents have equal preferences over a subset of objects and prefer the objects in that subset to the remaining objects, then they obtain equal probability shares of the objects in that subset. The random priority and probabilistic serial mechanisms satisfy our axioms.

Robust bilateral trade with discrete types

Bilateral trade problem is the most common market interaction in which a seller and a buyer bargain over an indivisible object, and the valuation of each agent about the object is private information. We investigate the cases where mechanisms satisfying Dominant Strategy Incentive Compatibility (DIC) and Ex-post Individual Rationality (EIR) properties can exhibit robust performance in the face of imprecision in prior structure. We start with the general mathematical formulation for the bilateral trade problem with DIC, EIR properties. We derive necessary and sufficient conditions for DIC, EIR mechanisms to be Ex-post efficient at the same time. Then, we define a new property—Allocation Maximality—and prove that the Posted Price mechanisms are the only mechanisms that satisfy DIC, EIR and Allocation Maximal properties. We also show that Posted Price mechanism is not the only mechanism that satisfies DIC and EIR properties. The last part of the paper introduces different sets of priors for agents’ types and consequently allows ambiguity in the problem framework. We derive robust counterparts and solve them numerically for the proposed objective function under box and ϕ -divergence ambiguity specifications. Results suggest that restricting the feasible set to Posted Price mechanisms can decrease the objective value to different extents depending on the uncertainty set.

A Pseudo-Market Approach to Allocation with Priorities

We propose a pseudo-market mechanism for no-monetary-transfer allocation of indivisible objects based on priorities such as those in school choice. Agents are given token money, face priority-specific prices, and buy utility-maximizing random assignments. The mechanism is asymptotically incentive compatible, and the resulting assignments are fair and constrained Pareto efficient. Hylland and Zeckhauser’s (1979) position-allocation problem is a special case of our framework, and our results on incentives and fairness are also new in their classical setting. (JEL D63, D82, H75, I21, I28)

Efficient and incentive compatible exchange of real-time information

We consider the problem of coordinating the exchange of real-time information. For example, a US Department of Transportation pilot program seeks to reduce traffic accidents by allowing each vehicle to transfer crash-relevant information (e.g. position, speed, braking status) to and from neighboring vehicles. Time is of the essence: vehicle information becomes stale quickly. Electronic files are the medium of information and corrupt/useless when partial; hence, we model them as discrete, indivisible objects. Furthermore, each agent may perfectly replicate an object in his possession and transfer it to another agent. However, replication and transfer takes time (e.g. due to bandwidth constraints), and scarcity arises due to fact that information quickly becomes valueless. How should agents transfer such objects? We study efficiency, strategy-proofness, withholding-proofness, and introduce a new axiom based on the concept of reciprocity. Our results are as follows: when each agent owns one object and consumes only one object, we identify a family of rules satisfying all four axioms. If each agent owns a bundle of objects and consumes a bundle, then the four axioms are incompatible. If agents live in a network, then for a large class of incomplete networks, efficiency is only compatible with strategy-proofness.

Investigating the characteristics of one-sided matching mechanisms under various preferences and risk attitudes

One-sided matching mechanisms are fundamental for assigning a set of indivisible objects to a set of self-interested agents when monetary transfers are not allowed. Two widely-studied randomized mechanisms in multiagent settings are the Random Serial Dictatorship (RSD) and the Probabilistic Serial Rule (PS). Both mechanisms require only that agents specify ordinal preferences and have a number of desirable economic and computational properties. However, the induced outcomes of the mechanisms are often incomparable and thus there are challenges when it comes to deciding which mechanism to adopt in practice. In this paper, we first consider the space of general ordinal preferences and provide empirical results on the (in)comparability of RSD and PS. We analyze their respective economic properties under general and lexicographic preferences. We then instantiate utility functions with the goal of gaining insights on the manipulability, efficiency, and envyfreeness of the mechanisms under different risk-attitude models. Our results hold under various preference distribution models, which further confirm the broad use of RSD in most practical applications.

Redistribution through Markets

Policymakers frequently use price regulations as a response to inequality in the markets they control. In this paper, we examine the optimal structure of such policies from the perspective of mechanism design. We study a buyer-seller market in which agents have private information about both their valuations for an indivisible object and their marginal utilities for money. The planner seeks a mechanism that maximizes agents' total utilities, subject to incentive and market-clearing constraints. We uncover the constrained Pareto frontier by identifying the optimal trade-off between allocative efficiency and redistribution. We find that competitive-equilibrium allocation is not always optimal. Instead, when there is substantial inequality across sides of the market, the optimal design uses a tax-like mechanism, introducing a wedge between the buyer and seller prices, and redistributing the resulting surplus to the poorer side of the market via lump-sum payments. When there is significant same-side inequality, meanwhile, it may be optimal to impose price controls even though doing so induces rationing.

Strategy-proof tie-breaking in matching with priorities

A set of indivisible objects is allocated among agents with strict preferences. Each object has a weak priority ranking of the agents. A collection of priority rankings, a priority structure, is solvable if there is a strategy-proof mechanism that is constrained efficient, i.e. that always produces a stable matching that is not Pareto-dominated by another stable matching. We characterize all solvable priority structures satisfying the following two restrictions: (A) Either there are no ties, or there is at least one four-way tie. (B) For any two agents i and j, if there is an object that assigns higher priority to i than j, there is also an object that assigns higher priority to j than i. We show that there are at most three types of solvable priority structures: The strict type, the house allocation with existing tenants (HET) type, where, for each object, there is at most one agent who has strictly higher priority than another agent, and the task allocation with unqualified agents (TAU) type, where, for each object, there is at most one agent who has strictly lower priority than another agent. Out of these three, only HET priority structures are shown to admit a strongly group strategy-proof and constrained efficient mechanism.

Stable and efficient resource allocation under weak priorities

We study the indivisible object allocation problem without monetary transfer, in which each object is endowed with a weak priority ordering over agents. It is well known that stability is generally not compatible with efficiency. We characterize the priority structures for which a stable and efficient assignment always exists, as well as the priority structures that admit a stable, efficient and (group) strategy-proof rule. While house allocation problems and housing markets are two classic families of allocation problems that admit a stable, efficient and group strategy-proof rule, any priority-augmented allocation problem with more than three objects admits such a rule if and only if it is decomposable into a sequence of subproblems, each of which has the structure of a house allocation problem or a housing market.

On the maximal domain theorem: A corrigendum to “Walrasian equilibrium with gross substitutes”

The maximal domain theorem by Gul and Stacchetti (1999) shows that for markets with indivisible objects, the set of gross substitutable preferences is a largest set for which the existence of a competitive equilibrium is guaranteed. In this paper, we give an example to show that a claim in their proof is false, and provide an alternative proof based on a new characterization of gross substitutability.

Probabilistic assignment of indivisible objects when agents have the same preferences except the ordinal ranking of one object

Bogomolnaia and Moulin (2001) show that there is no rule satisfying equal treatment of equals, stochastic dominance efficiency, and stochastic dominance strategyproofness for a probabilistic assignment problem of indivisible objects. Kasajima (2013) shows that the incompatibility result still holds when agents are restricted to have single-peaked preferences. In this paper, we further restrict the domain and investigate the existence of rules satisfying the three properties. As it turns out, the three properties are still incompatible even if all agents have the same preferences except the ordinal ranking of one object.

A Two Stage Model of Assignment and Market

Centralized matching mechanisms and decentralized markets have been widely studied to allocate indivisible objects. However, they have been analyzed separately. The present paper proposes a new framework, by explicitly formulating a two-stage model where objects are allocated through a matching mechanism in the first stage and traded in the second stage market. In addition, one divisible good called money may or may not be available in the market. Every player demands at most one unit of object besides money. The players may face different priorities at each object type in the first stage. Each object type has a limited amount of capacity, called quota. Each player has a quasi-linear utility function. The present analysis sets forth the equivalence conditions under which stability and efficiency are attained in equilibrium.

Comparing Direct Algorithms for Two-Player Fair Division of Indivisible Items - A Computational Study

We present an exhaustive computational study of algorithms for two-person allocation of indivisible objects. We identify eight algorithms that generate balanced allocations using only players' ordinal rankings and test them over all possible rankings of up to N = 12 items. Algorithms are evaluated according to the fairness and efficiency properties of the allocations they produce, including the ordinal properties of envy-freeness, Pareto-optimality, maximinality, and three similar properties based on Borda count. Our results indicate that, overall, the algorithms manage to identify a large majority of allocations that exhibit all three of the ordinal properties. Algorithms that combine bottom-up and top-down approaches find at least one good allocation whenever one exists, while other algorithms fail sometimes. Performance is not as good for Borda properties: In about 6.6% of the problems in which such an allocation exists, no algorithm can find it. We provide examples that illustrate how algorithms behave, and how they fail. We also discuss the overlap between allocations found by different algorithms, and find that algorithms combining both directions are able to find most of the allocations found by unidirectional algorithms, although there are many instances of allocations that can be found only by an algorithm in the latter group.

Ex-Ante Fair Random Allocation

When allocating indivisible objects, agents might have equal priority rights for some objects. A common practice is to break the ties using a lottery and randomize over deterministic mechanisms. Such randomizations usually lead to unfairness and inefficiency ex-ante. We propose a concept of ex-ante fairness and show the existence of an agent-optimal ex-ante fair solution. Ex- ante fair random allocations are generated using allocation by division, a new method of generating random allocations from deterministic mechanisms. Insights from the two-sided matching theory and the recent random assignment literature are unified and extended. The set of ex-ante fair random allocations forms a complete lattice under first-order stochastic dominance relations. The agent-optimal ex-ante fair mechanism includes both the deferred acceptance algorithm and the probabilistic serial mechanism as special cases.

Fairness and efficiency in strategy-proof object allocation mechanisms

I consider the problem of allocating N indivisible objects among N agents according to their preferences when transfers are absent and an outside option may exist. I study the tradeoff between fairness and efficiency in the class of strategy-proof mechanisms. The main finding is that for strategy-proof mechanisms the following efficiency and fairness criteria are mutually incompatible: (1) ex-post efficiency and envy-freeness, (2) ordinal efficiency and weak envy-freeness, and (3) ordinal efficiency and equal division lower bound. Result 1 is the first impossibility result for this setting that uses ex-post efficiency; results 2 and 3 are more practical than similar results in the literature. In addition, for N=3, I give two characterizations of the celebrated random serial dictatorship mechanism: it is the unique strategy-proof, ex-post efficient mechanism that (4) provides agents that have the same ordinal preferences with assignments not dominated by each other (weak envy-freeness among equals), or (5) provides agents that have the same cardinal preferences with assignments of equal expected utility (symmetry). These results strengthen the characterization by Bogomolnaia and Moulin (2001); result 5 implies the impossibility result by Zhou (1990).

Allocation of an indivisible object on the full preference domain: axiomatic characterizations

We study the problem of allocating an indivisible object to one of several agents on the full preference domain when monetary transfers are not allowed. Our main requirement is strategy-proofness. The other properties we seek are Pareto optimality, non-dictatorship, and non-bossiness. We provide characterizations of strategy-proof rules that satisfy Pareto optimality and non-bossiness, non-dictatorship and non-bossiness, and Pareto optimality and non-dictatorship. As a consequence of these characterizations, we show that a strategy-proof rule cannot satisfy Pareto optimality, non-dictatorship, and non-bossiness simultaneously.

Strategy-proofness and identical preferences lower bound in allocation problem of indivisible objects

We study an allocation problem of heterogeneous indivisible objects among agents without money. Each agent receives at most one object and prefers any object to nothing. We identify the class of rules satisfying strategy-proofness, Pareto-efficiency, and the identical preferences lower bound. Each rule of this class is included in Papai’s (Econometrica 68:1403–1433, 2000) rules and can be described by a top trading cycle rule associated with an inheritance structure that satisfies a symmetry condition called U-symmetry.

Incentive compatible and stable trade mechanisms on networks

We study a network of buyers and sellers where each seller owns an indivisible object and has no incentive to keep it, while each buyer has a downward sloping demand curve which is private information. Only the connected buyer-seller pairs can engage in trade. We search for trade mechanisms that are efficient, strategy-proof, bilateral trade stable and individually rational. In general, there does not exist a trade mechanism simultaneously satisfying these properties. The tension between strategy-proofness and bilateral trade stability is generated by the intersection between sets of competitors of a buyer at different sellers. Such intersections often allow the buyer to manipulate (via demand reductions) the prices paid in the network. The observed tension can be resolved if and only if the underlying network is cycle-free. In such a case, there is a unique trade mechanism which satisfies our four properties, a generalized Vickrey auction.

Efficient Ex Post Implementable Auctions and English Auctions for Bidders with Non-Quasilinear Preferences

We study efficient auction design for a single indivisible object when bidders have interdependent values and non-quasilinear preferences. Instead of quasilinearity, we assume only that bidders have positive wealth effects. Our setting nests cases where bidders are ex ante asymmetric, face financial constrains, are risk averse, and/or face ensuing risk. We give necessary and sufficient conditions for when there exists an ex post implementable and (ex post Pareto) efficient mechanism. Necessary and sufficient conditions for the existence of efficient ex post implementable mechanisms differ between the case where the object is a good and when the object is a bad. In the good setting, there is an efficient ex post implementable mechanism if there is an efficient ex post implementable mechanism in a corresponding quasilinear setting. This result extends established results on efficient ex post equilibria of English auctions with quasilinearity to our non-quasilinear setting. Yet, in the bad setting (i.e. a procurement auction) there is no mechanism that has an ex post efficient equilibrium if the level of interdependence between bidders is sufficiently strong. This result holds even if bidder costs satisfy standard crossing conditions that are sufficient for efficient ex post implementation in the quasilinear setting.