Indivisible Objects Research Articles

A tractable multiple agents protocol and algorithm for resource allocation under price rigidities

In many resource allocation problems, economy efficiency must be taken into consideration together with social equality, and price rigidities are often made according to some economic and social needs. We investigate the computational issues of dynamic mechanisms for selling multiple indivisible objects under price rigidities. We propose a multiple agents protocol and algorithm with polynomial time complexity that can achieve the over-demanded sets of items, and then introduce a dynamic mechanism with rationing to discover constrainedWalrasian equilibria under price rigidities in polynomial time. We also address the computation of buyers' expected profits and items' expected prices, and discuss strategical issues in the sense of expected profits.

Hierarchical Exchange Rules and the Core in Indivisible Objects Allocation

Hierarchical Exchange Rules and the Core in Indivisible Objects Allocation

Relaxing strategyproofness in one-sided matching

We consider the one-sided matching problem, where agents must be matched to indivisible objects. Our first result is a novel characterization of strategyproof mechanisms by three intuitive axioms. Furthermore, we introduce partial strategyproofness , a new relaxation of strategyproofness that bridges the gap between full and weak strategyproofness. Partial strategyproofness finds application in the incentive analysis of non-strategyproof mechanisms, such as Probabilistic Serial, different variants of the Boston mechanism, and hybrid mechanisms. In this letter, we summarize the main results from [Mennle and Seuken 2014a] and demonstrate how they can be used for the design and analysis of matching mechanisms.

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 New York City.

Priority-Augmented House Allocation

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 in this problem. 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. One corollary of this result is that there exists a stable hierarchical exchange rule (Papai, 2000) if and only if there exists a stable, efficient and group strategy-proof rule.

Prices and Efficient Assignments Without Transfers

We study the assignment of indivisible objects in environments, such as school choice, in which transfer payments are not used. Our main result shows that every efficient assignments can be decentralized through prices. We thus establish the Second Welfare Theorem for the no-transfer environments with possibly satiated agents. The result builds on a new Separating Hyperplane Theorem, which is of independent interest.

Budget balance, fairness, and minimal manipulability

A common real-life problem is to fairly allocate a number of indivisible objects and a fixed amount of money among a group of agents. Fairness requires that each agent weakly prefers his consumption bundle to any other agent's bundle. In this context, fairness is incompatible with budget balance and nonmanipulability (Green and Laffont 1979). Our approach here is to weaken or abandon nonmanipulability. We search for the rules that are minimally manipulable among all fair and budget-balanced rules. First, we show for a given preference profile, all fair and budget-balanced rules are either (all) manipulable or (all) nonmanipulable. Hence, measures based on counting profiles where a rule is manipulable or considering a possible inclusion of profiles where rules are manipulable do not distinguish fair and budget-balanced rules. Thus, a “finer” measure is needed. Our new concept compares two rules with respect to their degree of manipulability by counting for each profile the number of agents who can manipulate the rule. Second, we show that maximally preferred fair allocation rules are the minimally (individually and coalitionally) manipulable fair and budget-balanced allocation rules according to our new concept. Such rules choose allocations with the maximal number of agents for whom the utility is maximized among all fair and budget-balanced allocations.

Архітектура природно-мовної бази знань

This paper is focused on architecture of software implementation of natural-language knowledge base within the context of modeling of human individual speech system, which is appears as a basis for information natural-language technologies. The architecture is designed taking in the account basic language structure acquired as a result of researches on system level of organization of human speech activity. In context of knowledge base this structure is a primitive element of knowledge. Structural and functional description of core components of knowledge base, process of their interaction and possible extensions are described in detail. The transition from indivisible text objects to structured objects is presented and described, the BSSS implementation in context of OOP paradigm is presented. Also the necessity of implementation or emulation of basic version of linguistic processor as prerequisite for subsequent development of KB is stated. Aside of this, topics of creating of markers net for refinement of search semantics and accessory parts of KB which can be used by linguistic processor in process of synthesis and analysis of text fragments are raised.

Least manipulable Envy-free rules in economies with indivisibilities

We consider envy-free and budget-balanced allocation rules for problems where a number of indivisible objects and a fixed amount of money is allocated among a group of agents. In finite economies, we identify under classical preferences each agent’s maximal gain from manipulation. Using this result we find the envy-free and budget-balanced allocation rules which are least manipulable for each preference profile in terms of any agent’s maximal gain. If preferences are quasi-linear, then we can find an envy-free and budget-balanced allocation rule such that for any problem, the maximal utility gain from manipulation is equalized among all agents.

A general equivalence theorem for allocation of indivisible objects

We consider situations in which n indivisible objects are to be allocated to n agents. A number of recent papers studying such allocation problems have shown various interesting equivalences between randomized mechanisms based on trading and randomized mechanisms based on serial dictatorship. We prove a very general equivalence theorem from which many previous equivalence results immediately follow, and we give several new applications. Our general result sheds some light on why these equivalences hold by presenting the existing serial-dictatorship-based mechanisms as randomizations of a general mechanism which we call serial dictatorship in groups. Our proof technique also streamlines the bijective methods used in previous proofs, showing that it is enough to assemble a bijection from smaller pieces, without needing to construct the pieces explicitly.

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).

One-Sided Matching with Limited Complementarities

The problem of allocating bundles of indivisible objects without transfers arises in the assignment of courses to students, of computing resources like CPU time, memory and disk space to computing tasks and the truck loads of food to food banks. In these settings the complementarities in preferences are small compared with the size of the market. We exploit this to design mechanisms satisfying efficiency, envy-freeness and asymptotic strategy-proofness.Informally, we assume that agents do not want bundles that are too large. There will be a parameter k such that the marginal utility of any item relative to a bundle of size k or larger is zero. We call such preferences k-demand preferences. Given this parameter we show how to represent probability shares over bundles as lotteries over approximately (deterministic) feasible integer allocations. The degree of infeasibility in these integer allocations will be controlled by the parameter k. In particular, ex-post, no good is over allocated by at most k-1 units.

Revenue Maximizing Mechanisms with Strategic Customers and Unknown Demand: Name-Your-Own-Price

A designer allocates several indivisible objects to a stream of randomly arriving agents. The long-lived agents are privately informed about their value for an object, and about their arrival time to the market. The designer learns about future arrivals from past arrivals, while agents strategically choose when to make themselves available for trade. We characterize revenue maximizing direct mechanism and offer a simple indirect mechanism that captures a substantial part of the revenues of the revenue maximizing mechanism.

Optimal mechanism design with resale via bargaining

Abstract In this paper, we examine the optimal mechanism design of selling an indivisible object to one regular buyer and one publicly known buyer, where inter-buyer resale cannot be prohibited. The resale market is modeled as a stochastic ultimatum bargaining game between the two buyers. We fully characterize an optimal mechanism under general conditions. Surprisingly, in this optimal mechanism, the seller never allocates the object to the regular buyer regardless of his bargaining power in the resale market. The seller sells only to the publicly known buyer, and reveals no additional information to the resale market. The possibility of resale causes the seller to sometimes hold back the object, which under our setup is never optimal if resale is prohibited. We find that the sellerʼs revenue is increasing in the publicly known buyerʼs bargaining power in the resale market. When the publicly known buyer has full bargaining power, Myersonʼs optimal revenue is achieved; when the publicly known buyer has no bargaining power, a conditionally efficient mechanism prevails.

Reclaim-proof allocation of indivisible objects

We study desirability axioms imposed on allocations in indivisible object allocation problems. The existing axioms in the literature are various conditions of robustness to blocking coalitions with respect to agentsʼ ex ante (individual rationality and group rationality) and ex post (Pareto efficiency) endowments. We introduce a stringent axiom that encompasses and strengthens the existing ones. An allocation is reclaim-proof if it is robust to blocking coalitions with respect to any conceivable interim endowments of agents. This is an appealing property in dynamic settings, where the assignments prescribed by an allocation to be implemented need to be made in multiple rounds rather than all in one shot. We show that an allocation is reclaim-proof if and only if it is induced by a YRMH–IGYT mechanism (introduced by Abdulkadiroğlu and Sönmez, 1999) and if and only if it is a competitive allocation.

Two New Perspectives on Multi-Stage Group Testing

The group testing problem asks to find d≪n defective elements out of n elements, by testing subsets (pools) for the presence of defectives. In the strict model of group testing, the goal is to identify all defectives if at most d defectives exist, and otherwise to report that more than d defectives are present. If tests are time-consuming, they should be performed in a small constant number s of stages of parallel tests. It is known that a test number O(dlogn), which is optimal up to a constant factor, can be achieved already in s=2 stages. Here we study two aspects of group testing that have not found major attention before. (1) Asymptotic bounds on the test number do not yet lead to optimal strategies for specific n,d,s. Especially for small n we show that randomized strategies significantly save tests on average, compared to worst-case deterministic results. Moreover, the only type of randomness needed is a random permutation of the elements. We solve the problem of constructing optimal randomized strategies for strict group testing completely for the case when d=1 and s≤2. A byproduct of our analysis is that optimal deterministic strategies for strict group testing for d=1 need at most 2 stages. (2) Usually, an element may participate in several pools within a stage. However, when the elements are indivisible objects, every element can belong to at most one pool at the same time. For group testing with disjoint simultaneous pools we show that Θ(sd(n/d)1/s) tests are sufficient and necessary. While the strategy is simple, the challenge is to derive tight lower bounds for different s and different ranges of d versus n.

Maximal Manipulation in Fair Allocation

We herein consider the problems of fairly allocating indivisible objects and money. The objective of the present study is to examine strategic manipulation under envy-free solutions. We therefore investigate the consequence of individual manipulation. Each individual is observed to obtain the welfare level of his “optimal” envy-free allocation by maximally manipulating the solutions. This maximal manipulation theorem leads to several fruitful results on the manipulability of envy-free solutions: (i) we present a characterization of non-manipulable preference profiles under a given envy-free solution; (ii) we analyze the set of Nash equilibrium allocations in the direct revelation games associated with a given envy-free solution; (iii) we formulate envy-free solutions that are strategy-proof for at least one individual; and (iv) we identify the functional form of the least manipulable envy-free solutions on the quasi-linear domain.

Fraction auctions: The tradeoff between efficiency and running time

This paper studies the sales of a single indivisible object where bidders have continuous valuations. In Grigorieva et al. [14] it was shown that, in this setting, query auctions necessarily allocate inefficiently in equilibrium. In this paper we propose a new sequential auction, called the c-fraction auction. We show the existence of an ex-post equilibrium, called bluff equilibrium, in which bidders behave truthfully except for particular constellations of observed bids at which it is optimal to pretend a slightly higher valuation. We show c-fraction auctions guarantee approximate efficiency at any desired level of accuracy, independent of the number of bidders, when bidders choose to play the bluff equilibrium. We discuss the running time and the efficiency in the bluff equilibrium. We show that by changing the parameter c of the auction we can trade off efficiency against running time.

Secure Implementation in Queueing Problems

This paper studies secure implementability (T. Saijo, T. Sjostrom and T. Yamato, “Secure Implementation,” Theoretical Economics, Vol. 2, No. 3, 2007, pp. 203-229) in queueing problems. Our main result shows that the social choice function satisfies strategy-proofness and strong non-bossiness (Z. Ritz, “Restricted Domains, Arrow-Social Welfare Functions and Noncorruptible and Non-Manipulable Social Choice Correspondences: The Case of Private Alternatives,” Mathematical Social Science, Vol. 4, No. 2, 1983, pp. 155-179), both of which are necessary for secure imple- mentation, if and only if it is constant on the domains that satisfy weak indifference introduced in this paper. Weak in- difference is weaker than minimal richness (Y. Fujinaka and T. Wakayama, “Secure Implementation in Economies with Indivisible Objects and Money,” Economics Letters, Vol. 100, No. 1, 2008, pp. 91-95). Our main result illustrates that secure implementation is too difficult in queueing problems since many reasonable domains satisfy weak indifference, for example, convex domains.

A Cooperative Approach to Queue Allocation of Indivisible Objects

We consider the allocation of a finite number of indivisible objects to the same number of agents according to an exogenously given queue. We assume that the agents collaborate in order to achieve an efficient outcome for society. We allow for side-payments and provide a method for obtaining stable outcomes.