Indivisible Objects Research Articles

Packaging for allocation

The allocation of indivisible objects is studied where individuals claim packages of them and a rule determines the number of each package to be allocated. There exists a rule satisfying feasibility, efficiency, and composition down (or composition up) if and only if the set of institutionally feasible packages is a hierarchy, i.e., for any pair of packages, either that they are disjoint or that one contains the other. Moreover, on a hierarchy, the unique rules satisfying the aforementioned properties are priority rules where a package has higher priority than another if it contains the other.

On the efficiency and fairness of deferred acceptance with single tie-breaking

As a random allocation rule for indivisible object allocation under weak priorities, deferred acceptance with single tie-breaking (DA-STB) is not ex-post constrained efficient. We first observe that it also fails to satisfy equal-top fairness, which requires that two agents be assigned their common top choice with equal probability if they have equal priority for it. Then, it is shown that DA-STB is ex-post constrained efficient, if and only if it is equal-top fair, if and only if the priority structure satisfies a certain acyclic condition. We further characterize the priority structures under which DA-STB is ex-post stable-and-efficient. Based on the characterized priority domains, and using a weak fairness notion called local envy-freeness, new theoretical support is provided for the use of this rule: for any priority structure, among the class of strategy-proof, ex-post stable, symmetric, and locally envy-free rules, each of the above desiderata—ex-post constrained efficiency, ex-post stability-and-efficiency, and equal-top fairness—can be achieved if and only if it can be achieved by DA-STB.

Maximal domains for strategy-proof pairwise exchange

We analyze centralized markets for indivisible objects without money through pairwise exchange when each agent initially owns a single object. We consider rules that for each profile of agents preferences select an assignment of the objects to the agents. We present a family of domains of preferences (minimal reversal domains) that are maximal rich domains for the existence of rules that satisfy individual rationality, efficiency, and strategy-proofness. Each minimal reversal domain is defined by a common ranking of the set of objects, and agents’ preferences over admissible objects coincide with such common ranking but for a specific pair of objects.

Shapley–Scarf Housing Markets: Respecting Improvement, Integer Programming, and Kidney Exchange

In a housing market of Shapley and Scarf, each agent is endowed with one indivisible object and has preferences over all objects. An allocation of the objects is in the (strong) core if there exists no (weakly) blocking coalition. We show that, for strict preferences, the unique strong core allocation “respects improvement”—if an agent’s object becomes more desirable for some other agents, then the agent’s allotment in the unique strong core allocation weakly improves. We extend this result to weak preferences for both the strong core (conditional on nonemptiness) and the set of competitive allocations (using probabilistic allocations and stochastic dominance). There are no counterparts of the latter two results in the two-sided matching literature. We provide examples to show how our results break down when there is a bound on the length of exchange cycles. Respecting improvements is an important property for applications of the housing markets model, such as kidney exchange: it incentivizes each patient to bring the best possible set of donors to the market. We conduct computer simulations using markets that resemble the pools of kidney exchange programs. We compare the game-theoretical solutions with current techniques (maximum size and maximum weight allocations) in terms of violations of the respecting improvement property. We find that game-theoretical solutions fare much better at respecting improvements even when exchange cycles are bounded, and they do so at a low efficiency cost. As a stepping stone for our simulations, we provide novel integer programming formulations for computing core, competitive, and strong core allocations. Funding: P. Biró gratefully acknowledges financial support from the Hungarian Scientific Research Fund, OTKA [Grant K143858] and the Hungarian Academy of Sciences [Grant LP2021-2]. F. Klijn gratefully acknowledges financial support from AGAUR–Generalitat de Catalunya [Grants 2017-SGR-1359 and 2021-SGR-00416] and the Spanish Agencia Estatal de Investigación [Grants ECO2017-88130-P and PID2020-114251GB-I00] (funded by MCIN/AEI/10.13039/501100011033) and the Severo Ochoa Programme for Centres of Excellence in R&D (Barcelona School of Economics) [Grant CEX2019-000915-S]. Research visits related to this work were financed by COST Action [Grant CA15210 ENCKEP], supported by COST (European Cooperation in Science and Technology), http://www.cost.eu/ .

Strategy-proof and envy-free mechanisms for house allocation

We consider the problem of allocating indivisible objects to agents when agents have strict preferences over objects. There are inherent trade-offs between competing notions of efficiency, fairness and incentives in such assignment mechanisms. It is, therefore, natural to consider mechanisms that satisfy two of these three properties in their strongest notions, while trying to improve on the third dimension. In this paper, we are motivated by the following question: Is there a strategy-proof and envy-free random assignment mechanism more efficient than equal division?Our contributions in this paper are twofold. First, we further explore the incompatibility between efficiency, fairness, and strategy-proofness within random assignment mechanisms. We define a new notion of efficiency that is weaker than ex-post efficiency and prove that any strategy-proof and envy-free mechanism must sacrifice efficiency even in this very weak sense. Next, we introduce a new family of mechanisms called Rank Exchange mechanisms that are strategy-proof and envy-free and stochastically dominate equal division. Further, we show that Rank Exchange mechanisms characterize the set of neutral, strategy-proof, and envy-free mechanisms that also satisfy a natural separability axiom that may be of independent interest.

Auction design by an informed seller: A foundation of reserve price signalling

AbstractThis paper studies mechanism design by a seller privately informed of the (continuous) quality of an indivisible object. An important solution to this informed‐principal problem, the Rothschild–Stiglitz–Wilson mechanisms, which correspond to the least‐cost separating allocations in the signalling literature, is characterized. The characterization reveals that reserve prices are the least costly device to separate sellers of different qualities: In the RSW mechanisms, the lowest‐quality seller adopts her public‐information optimal selling procedure, and each higher‐quality seller adopts a selling procedure that differs from her public‐information optimal one only in that the reserve prices are higher. This finding provides a mechanism‐design foundation for reserve‐price signalling studied in the literature.

Legal regulation of forward contracts in construction investment

A brief overview of the legal regulation of forward contracts in construction investment is presented. The relevance of their application in modern conditions, taking into account international standards and the EU acquis, to stimulate the attraction of investments in the national economy, create a competitive environment and sustainable growth in the conditions of financial globalization and the development of communication technologies is substantiated.
 It was found that in order to implement Article 127 of the Association Agreement between Ukraine and the EU regarding the improvement of the financial sector regulation model, new editions of the Laws of Ukraine "On Capital Markets and Organized Commodity Markets" and "On Commodity Exchanges" were adopted, according to which international standards are gradually being implemented functioning of the infrastructure of capital markets, however, the analysis of regulatory and legal regulation and the practice of its application shows a number of problems that negatively affect the development processes of derivative financial instruments in the economy of Ukraine.
 The problems of taxation of forward contracts of joint investment institutes were considered, the analysis of judicial practice was carried out in the context of the research topic. It was concluded that the forward contract, if all the essential conditions and requisites inherent to it are present, is recognized by the courts as a derivative, which is a separate type of product. Fundamental to the determination of the possibility of applying a tax benefit when using forward contracts is the question of the acquisition of such assets at the expense of joint investment and whether they constitute assets of the institution of joint investment. The success of the legislative amendment on detailing the assets of the joint investment institute in relation to real estate objects, including in the form of an indivisible object and (or) divisible object of unfinished construction, future real estate object, is theoretically substantiated.
 It was concluded that the development of financial instruments in Ukraine based on leading international practices is the key to successful improvement of the instrumental configuration of capital markets and organized commodity markets for the integration of the national economy into the world economy. The prospects for the development of legal regulation of forward contracts in construction investment on the Ukrainian market are outlined.

Guaranteeing property rights to real estate objects that will be built in the future

The article analyzes various milestones of legislative regulation of the problem of guaranteeing property rights to real estate objects that will be built in the future. Ukraine has gone a long way towards strengthening institutionalization and guaranteeing property rights to real estate objects that will be built in the future. The current situation in the researched area is characterized by the ratio of the relatively consistent consolidation of the main guarantees of the property rights realization with the obviously unsatisfactory state of their practical implementation, which is expressed in the impossibility for the majority of the population to fully enjoy their rights, in their systematic violations and in the absence of reliable mechanisms for their restoration and protection. The same with regard to further scientifi c research, as author confi dently conclude, that in the current conditions, the epicenter of doctrinal and legal eff orts should shift from the formal recognition and ascertainment of real property rights to real estate objects that will be built in the future, to the implementation of their real guarantee. At the same time, only an indivisible object of unfi nished construction can be the object of ownership rights arising from the moment of state registration of such a right in accordance with the requirements of the Law of Ukraine «On State Registration of Property Rights to Real Property and Their Encumbrances». As a result of the conducted research, the author proposed her own periodization of the main stages of legislative regulation of the issue of guaranteeing property rights to real estate objects that will be built in the future, namely: 1) the judicial stage, 2) the contractual stage, 3) notarial stage. According to the scientist, such an understanding will allow to optimize theoretical research in this area and form a vector for the further development of the investigated challenges. Key words: real estate, rights guarantee, real estate objects, real estate, legal regulation.

Strategy-proof and envy-free random assignment

We study the random assignment of indivisible objects among a set of agents with strict preferences. We show that there exists no mechanism which is unanimous, strategy-proof and envy-free. Weakening the first requirement to q-unanimity – i.e., when every agent ranks a different object at the top, then each agent shall receive his most-preferred object with probability of at least q – we show that a mechanism satisfying strategy-proofness, envy-freeness and ex-post weak non-wastefulness can be q-unanimous only for q≤2n (where n is the number of agents). To demonstrate that this bound is tight, we introduce a new mechanism, Random-Dictatorship-cum-Equal-Division (RDcED), and show that it achieves this maximal bound when all objects are acceptable. In addition, for three agents, RDcED is characterized by the first three properties and ex-post weak efficiency. If objects may be unacceptable, strategy-proofness and envy-freeness are jointly incompatible even with ex-post weak non-wastefulness.

The parameterized complexity of manipulating Top Trading Cycles

We study the problem of exchange when agents are endowed with heterogeneous indivisible objects, and there is no money. In this setting, no rule satisfies Pareto-efficiency, individual rationality, and strategy-proofness; there is no consensus in the literature on satisfactory second-best mechanisms. A natural generalization of the ubiquitous Top Trading Cycles (TTC) satisfies the first two properties on the lexicographic domain, rendering it manipulable. We characterize the computational complexity of manipulating this mechanism; we show that it is \(\mathbf {W[P]}\)-hard by reduction from MONOTONE WEIGHTED CIRCUIT SATISFIABILITY. We provide a matching upper bound for a wide range of preference domains. We further show that manipulation by groups (when parameterized by group size) is \(\mathbf {W[P]}\)-hard. This provides support for TTC as a second-best mechanism. Lastly, our results are of independent interest to complexity theorists: there are few natural \(\mathbf {W[P]}\)-complete problems and, as far as we are aware, this is the first such problem arising from the social sciences.

Strategy-proof popular mechanisms

We consider the allocation of indivisible objects when agents have preferences over their own allocations, but share the ownership of the resources to be distributed. Examples might include seats in public schools, faculty offices, and time slots in public tennis courts. Given an allocation, groups of agents who would prefer an alternative allocation might challenge it. An assignment is popular if it is not challenged by another one. By assuming that agents’ ability to challenge allocations can be represented by weighted votes, we characterize the conditions under which popular allocations might exist and when these can be implemented via strategy-proof mechanisms. Serial dictatorships that use orderings consistent with the agents’ weights are not only strategy-proof and Pareto efficient, but also popular, whenever these assignments exist. We also provide a new characterization for serial dictatorships as the only mechanisms that are popular, strategy-proof, non-wasteful, and satisfy a consistency condition.

Efficient bilateral trade via two-stage mechanisms: Comparison between one-sided and two-sided asymmetric information environments

This paper first considers a bilateral-trade model with one-sided asymmetric information in which one agent (seller) initially owns an indivisible object and is fully informed of its value, while the other agent (buyer) intends to obtain the object whose value is unknown to himself. As no mechanisms can generally result in efficient, voluntary bilateral trades (Jehiel and Pauzner, 2006), we aim to overturn this impossibility result by employing two-stage mechanisms (Mezzetti, 2004) in which first, the outcome (e.g., allocation of the goods) is determined, then the agents observe their own outcome-decision payoffs, and finally, transfers are made. We show that the generalized two-stage Groves mechanism and the shoot-the-liar mechanism both induce efficient, voluntary bilateral trades. We next consider a two-sided asymmetric information setup in which both parties have private information. We show by means of a stylized example that the shoot-the-liar mechanism “sometimes” induces an efficient, voluntary trade, while the generalized two-stage Grove mechanism never induces it.

Efficient Fair Division with Minimal Sharing

When assets are to be divided among several partners, for example, a partnership split, fair division theory can be used to determine a fair allocation. The applicability of existing approaches is limited as they either treat assets as divisible resources that end up being shared among participants or deal with indivisible objects providing only approximate fairness. In practice, sharing is often possible but undesirable, and approximate fairness is not adequate, particularly for highly valuable assets. In “Efficient Fair Division with Minimal Sharing,” Sandomirskiy and Segal-Halevi introduce a novel approach offering a middle ground: the number of shared objects is minimized while maintaining exact fairness and economic efficiency. This minimization can be conducted in polynomial time for generic instances if the number of agents or objects is fixed. Experiments on real data demonstrate a substantial improvement over current methods.

Trading probabilities along cycles

Consider the problem of allocating indivisible objects when agents are endowed with fractional amounts and rules can assign lotteries. We study a natural generalization (to the probabilistic domain) of Gale’s Top Trading Cycles. The latter features an algorithm wherein agents trade objects along a cycle—in our new family of rules, agents now trade probabilities of objects along a cycle. We ask if the attractive properties, namely efficiency, individual rationality, and strategy-proofness extended in the stochastic dominance sense, carry over to the Trading-Probabilities-Along-Cycles (TPAC) rules. All of these rules are sd-efficient. We characterize separately the subclass of TPAC rules satisfying the sd-endowment lower bound and sd-strategy-proofness. Regarding fairness, we follow in spirit to the no-envy in net trade condition of Schmeidler and Vind (1972), where the set of allocations satisfying the property essentially coincides with the set of competitive equilibria, and augment the notion appropriately for our environment. We further generalize the TPAC family while extending results on sd-efficiency and the sd-endowment lower bound, and provide sufficient conditions on parameters for the rules to arbitrarily closely satisfy the sd-no-envy in net trade.

When to Sell an Indivisible Object: Optimal Timing with Markovian Buyers

When to Sell an Indivisible Object: Optimal Timing with Markovian Buyers

On obviously strategy-proof implementation of fixed priority top trading cycles with outside options

We study the implementation of a fixed priority top trading cycles (FPTTC) rule via an obviously strategy-proof (OSP) mechanism (Li, 2017) in the context of assignment problems with outside options, where agents are to be assigned at most one indivisible object and monetary transfers are not allowed. In a model without outside options, Troyan (2019) gives a sufficient (but not necessary) and Mandal and Roy (2020) give a necessary and sufficient condition for an FPTTC rule to be OSP-implementable. This paper shows that in a model with outside options, the two conditions (in Troyan (2019) and Mandal and Roy (2020)) are equivalent for an FPTTC rule, and each of them is necessary and sufficient for an FPTTC rule to be OSP-implementable.

Selling two identical objects

It is well-known that optimal (i.e., revenue-maximizing) selling mechanisms in multidimensional type spaces may involve randomization. We obtain conditions under which deterministic mechanisms are optimal for selling two identical, indivisible objects to a single buyer. We analyze two settings: (i) decreasing marginal values (DMV) and (ii) increasing marginal values (IMV). Thus, the values of the buyer for the two units are not independent.We show that under a well-known condition on distributions (due to McAfee and McMillan (1988)), (a) it is optimal to sell the first unit deterministically in the DMV model and (b) it is optimal to bundle (which is a deterministic mechanism) in the IMV model. Under a stronger sufficient condition on distributions, a deterministic mechanism is optimal in the DMV model.Our results apply to heterogeneous objects when there is a specified sequence in which the two objects must be sold.

OBVIOUSLY STRATEGY‐PROOF IMPLEMENTATION OF ASSIGNMENT RULES: A NEW CHARACTERIZATION

AbstractWe consider assignment problems where individuals are to be assigned at most one indivisible object and monetary transfers are not allowed. We provide a characterization of assignment rules that are Pareto efficient, nonbossy, and implementable in obviously strategy‐proof (OSP) mechanisms. As corollaries of our result, we obtain a characterization of OSP‐implementable fixed priority top trading cycles (FPTTC) rules, hierarchical exchange rules, and trading cycles rules. Troyan (International Economic Review 60 (2019), 1249–61) provides a characterization of OSP‐implementable FPTTC rules when there are equal number of individuals and objects. Our result generalizes this for arbitrary values of those.

Сравнение различных представлений знаний для сложных структурированных объектов при решении задач ИИ

The problem of knowledge representation for a complex structured object is one of the actual problems of AI. This is due to the fact that many of the objects under study are not a single indivisible object characterized by its properties, but complex structures whose elements have some known properties and are in some, often multiplace, relations with each other. An approach to the representation of such knowledge based on first-order logic (predicate calculus formulas) is compared in this paper with two currently widespread approaches based on the representation of data information with the use of finite-valued strings or graphs. It is shown that the use of predicate calculus formulas for description of a complex structured object, despite the NP-difficulty of the solved problems arising after formalization, actually have no greater computational complexity than the other two approaches, what is usually not mentioned by their supporters. An algorithm for constructing an ontology is proposed that does not depend on the methodof desc ribing an object, and is based on the selection of the maximum common property of objects from a given set.

Volumetric procedural models for shape representation

This article describes a volumetric approach for procedural shape modeling and a new Procedural Shape Modeling Language (PSML) that facilitates the specification of these models. PSML provides programmers the ability to describe shapes in terms of their 3D elements where each element may be a semantic group of 3D objects, e.g., a brick wall, or an indivisible object, e.g., an individual brick. Modeling shapes in this manner facilitates the creation of models that more closely approximate the organization and structure of their real-world counterparts. As such, users may query these models for volumetric information such as the number, position, orientation and volume of 3D elements which cannot be provided using surface based model-building techniques. PSML also provides a number of new language-specific capabilities that allow for a rich variety of context-sensitive behaviors and post-processing functions. These capabilities include an object-oriented approach for model design, methods for querying the model for component-based information and the ability to access model elements and components to perform Boolean operations on the model parts. PSML is open-source and includes freely available tutorial videos, demonstration code and an integrated development environment to support writing PSML programs.