Indivisible Goods Research Articles

Santa claus meets hypergraph matchings

We consider the restricted assignment version of the problem of max-min fair allocation of indivisible goods, also known as the Santa Claus problem . There are m items and n players. Every item has some nonnegative value, and every player is interested in only some of the items. The goal is to distribute the items to the players in a way that maximizes the minimum of the sum of the values of the items given to any player. It was previously shown via a nonconstructive proof that uses the Lovász local lemma that the integrality gap of a certain configuration LP for the problem is no worse than some (unspecified) constant. This gives a polynomial-time algorithm to estimate the optimum value of the problem within a constant factor, but does not provide a polynomial-time algorithm for finding a corresponding allocation. We use a different approach to analyze the integrality gap. Our approach is based upon local search techniques for finding perfect matchings in certain classes of hypergraphs. As a result, we prove that the integrality gap of the configuration LP is no worse than 1/4. Our proof provides a local search algorithm which finds the corresponding allocation, but is nonconstructive in the sense that this algorithm is not known to converge to a local optimum in a polynomial number of steps.

Fair allocation of indivisible goods: the two-agent case

One must allocate a finite set of indivisible goods among two agents without monetary compensation. We impose Pareto-efficiency, anonymity, a weak notion of no-envy, a welfare lower bound based on each agent's ranking of the subsets of goods, and a monotonicity property w.r.t. changes in preferences. We prove that there is a rule satisfying these axioms. If there are three goods, it is the only rule, together with one of its subcorrespondences, satisfying each fairness axiom and not discriminating between goods.

Probabilistic assignment of indivisible goods with single-peaked preferences

We consider the problem of assigning indivisible goods among a group of agents with lotteries when the preference profile is single-peaked. Unfortunately, even on this restricted domain of preferences, equal treatment of equals, stochastic dominance efficiency, and stochastic dominance strategy-proofness are incompatible.

Probabilistic assignment of objects: Characterizing the serial rule

We study the problem of assigning a set of objects (i.e., indivisible goods) to a set of agents, when each agent is supposed to receive only one object and has strict preferences over the objects. In the absence of monetary transfers, we focus on the probabilistic rules, which only take the ordinal preferences as input (the ordering over objects by each agent is submitted, but the relative cardinal intensities of their preferences are not). We present a characterization of the serial rule, proposed by Bogomolnaia and Moulin (2001) in this model. The serial rule is the only rule satisfying sd e¢ ciency, sd no-envy, and bounded invariance (where “sd”stands for “stochastic dominance”). A special representation of feasible assignment matrices by means of consumption processes is the key to the simple and intuitive proof of our main result. This technique also allows us to present a simple unifying argument for a number of related earlier and concurrent results. JEL classi…cation: C70, D61, D63

Characterizing Welfare-egalitarian Mechanisms with Solidarity When Valuations are Private Information

In the problem of assigning indivisible goods and monetary transfers, we characterize welfare-egalitarian mechanisms (that are decision-efficient and incentive compatible) with an axiom of solidarity under preference changes and a fair ranking axiom of order preservation. This result is in line with characterizations of egalitarian rules with solidarity in other economic models. We also show that we can replace order-preservation with egalitarian-equivalence or no-envy (on the subadditive domain) and still characterize the welfare-egalitarian class. However, if we weaken order preservation to symmetry, mechanisms that are not welfare-egalitarian exist. We also study upper bounds on deficit and welfare lower bounds that characterize subclasses of the welfare-egalitarian class.

Asymmetrically Fair Rules for an Indivisible Good Problem with a Budget Constraint

We study a particular restitution problem where there is an indivisible good (land or property) over which two agents have rights: the dispossessed agent and the owner. A third party, possibly the government, seeks to resolve the situation by assigning rights to one and compensate the other. There is also a maximum amount of money available for the compensation. We characterize a family of asymmetrically fair rules that are immune to strategic behavior, guarantee minimal welfare levels for the agents, and satisfy the budget constraint.

The Efficiency of Fair Division

In this paper we study the impact of fairness on the efficiency of allocations. We consider three different notions of fairness, namely proportionality, envy-freeness, and equitability for allocations of divisible and indivisible goods and chores. We present a series of results on the price of fairness under the three different notions that quantify the efficiency loss in fair allocations compared to optimal ones. Most of our bounds are either exact or tight within constant factors. Our study is of an optimistic nature and aims to identify the potential of fairness in allocations.

A Convex Optimization Framework for Almost Budget Balanced Allocation of a Divisible Good

We address the problem of allocating a single divisible good to a number of agents. The agents have concave valuation functions parameterized by a scalar type. The agents report only the type. The goal is to find allocatively efficient, strategy proof, nearly budget balanced mechanisms within the Groves class. Near budget balance is attained by returning as much of the received payments as rebates to agents. Two performance criteria are of interest: the maximum ratio of budget surplus to efficient surplus, and the expected budget surplus, within the class of linear rebate functions. The goal is to minimize them. Assuming that the valuation functions are known, we show that both problems reduce to convex optimization problems, where the convex constraint sets are characterized by a continuum of half-plane constraints parameterized by the vector of reported types. We then propose a randomized relaxation of these problems by sampling constraints. The relaxed problem is a linear programming problem (LP). We then identify the number of samples needed for “near-feasibility” of the relaxed constraint set. Under some conditions on the valuation function, we show that value of the approximate LP is close to the optimal value. Simulation results show significant improvements of our proposed method over the Vickrey-Clarke-Groves (VCG) mechanism without rebates. In the special case of indivisible goods, the mechanisms in this paper fall back to those proposed by Moulin, by Guo and Conitzer, and by Gujar and Narahari, without any need for randomization. Extension of the proposed mechanisms to situations when the valuation functions are not known to the central planner are also discussed.

Walrasian Equilibrium in Large, Quasilinear Markets

In an economy with indivisible goods, a continuum of agents and quasilinear utility, we show that equilibrium exists regardless of the nature of agents' preferences over bundles. This contrasts with results for economies with a finite number of agents, which require restrictions on preferences (such as substitutability) to guarantee existence. When the distribution of preferences has full support, equilibrium prices are unique.Available at http://econtheory.org. DOI: 10.3982/TE1060

An Ascending Vickrey Auction for Selling Bases of a Matroid

Consider selling bundles of indivisible goods to buyers with concave utilities that are additively separable in money and goods. We propose an ascending auction for the case when the seller is constrained to sell bundles whose elements form a basis of a matroid. It extends easily to polymatroids. Applications include scheduling, allocation of homogeneous goods, and spatially distributed markets, among others. Our ascending auction induces buyers to bid truthfully and returns the economically efficient basis. Unlike other ascending auctions for this environment, ours runs in pseudopolynomial or polynomial time. Furthermore, we prove the impossibility of an ascending auction for nonmatroidal independence set-systems.

Allocation problems with indivisibilities when preferences are single-peaked

We consider allocation problems with indivisible goods when preferences are single-peaked. In this paper we identify the family of efficient, non-manipulable, consistent, and balanced solutions. We refer those solutions as Temporary Satisfaction Methods,that can be viewed as extensions to the indivisible case of the so-called uniform rule.

Pareto-Optimal Assignments by Hierarchical Exchange

A version of the Second Fundamental Theorem of Welfare Economics that applies to a money-free environment, in which a set of indivisible goods needs to be matched to some set of agents, is established. In such environments, trade can be identified with the set of hierarchical exchange mechanisms defined by Papai (2000). Papai (2000)'s result – that any such mechanism yields Pareto-optimal allocations – can be interpreted as a version of the First Fundamental Theorem of Welfare Economics for the given environment. In this note, I show that for any Pareto-optimal allocation and any hierarchical exchange mechanism one can find an initial allocation of ownership rights, such that the given Pareto-optimal allocation arises as a result of trade.

Exchange of indivisible goods and indifferences: The Top Trading Absorbing Sets mechanisms

There is a wide range of economic problems that involve the exchange of indivisible goods with no monetary transfers, starting from the housing market model of the seminal paper by Shapley and Scarf (1974) to problems such as the kidney exchange or the school choice problem. The classical solution to many of these models is to apply a mechanism called Top Trading Cycles, attributed to David Gale, which satisfies good properties for the case of strict preferences. In this paper, we propose a family of mechanisms, called Top Trading Absorbing Sets mechanisms, which generalize the Top Trading Cycles to the general case in which individuals are allowed to report indifferences, while preserving a maximal possible set of its desirable properties.

The Core of Shapley–Scarf markets with couples

Abstract We extend the Shapley–Scarf model of markets for indivisible goods without money to the case where couples of agents have joint preferences over the set of allocations. We show that the domain of (weakly) lexicographic preferences is maximal (for inclusion) for the existence of Core allocations. This result also holds in the case where the set of agents is partitioned into non-empty coalitions with any size, as well as for the class of markets studied in Konishi et al. (2001), where individuals exchange several types of purely indivisible goods.

The Pareto-dominant strategy-proof and fair rule for problems with indivisible goods

We study the problem of allocating a set of indivisible goods among a set of agents when monetary transfers are not allowed. We consider two interesting cases of this problem: (1) the supply of each object is exactly one; and (2) the supply of an object may be greater than one. Our central requirements are strategy-proofness and ex post fairness. We propose a particular rule satisfying strategy-proofness and no-envy (as well as equal treatment of equals). For the first case, it Pareto dominates any other rule satisfying strategy-proofness and equal treatment of equals. For the second case, it Pareto dominates any other rule satisfying strategy-proofness and no-envy.

Are incentives against economic justice?

We consider the problem of fairly allocating a social endowment of indivisible goods and money when the domain of admissible preferences contains, but is not restricted to, quasi-linear preferences. We analyze the manipulability of the Generalized Money Rawlsian Fair (GMRF) solutions. (i) We show that the Nash and strong Nash equilibrium correspondences of the “preference revelation game form” associated with each GMRF solution coincide with the no-envy solution (in equilibrium, efficiency is preserved according to agents' true preferences). (ii) A corollary is that the GMRF solutions “naturally implement” the no-envy solution in Nash and strong Nash equilibria.

Inequality Aversion and Efficiency with Ordinal and Cardinal Social Preferences – An Experimental Study

In this paper, we report on a series of free-form bargaining experiments in which two players have to distribute four indivisible goods among themselves. In one treatment, players are informed about the monetary payoffs associated with each bundle of goods; in a second treatment only the ordinal ranking of the bundles is given. We find that in both cases, inequality aversion plays a prominent role. In the ordinal treatment, individuals apparently use the ranks in the respective preference orderings over bundles of goods as a substitute for the unknown monetary value. Allocations that distribute the value (money or ranks, respectively) most equally serve as natural “reference points” for the bargaining processes. Frequently, such “equal split” allocations are chosen by our subjects even though they are Pareto dominated; but also if they are rejected for that reason they matter in a specific way: whether a Pareto optimal allocation is chosen or not depends on whether or not it is a Pareto improvement relative to the equal split. Interestingly, we find much less Pareto-damaging behavior due to inequality aversion in the ordinal treatment.

Inequality aversion and efficiency with ordinal and cardinal social preferences—An experimental study

In this paper, we report on a series of free-form bargaining experiments in which two players have to distribute four indivisible goods among themselves. In one treatment, players are informed about the monetary payoffs associated with each bundle of goods; in a second treatment only the ordinal ranking of the bundles is given. We find that in both cases, inequality aversion plays a prominent role. In the ordinal treatment, individuals apparently use the ranks in the respective preference orderings over bundles of goods as a substitute for the unknown monetary value. Allocations that distribute the value (money or ranks, respectively) most equally serve as natural “reference points” for the bargaining processes. Frequently, such “equal split” allocations are chosen by our subjects even though they are Pareto dominated. Whether a Pareto optimal allocation is chosen or not depends on whether or not it is a Pareto improvement relative to the “equal split” reference allocation. We find less Pareto-damaging behavior due to inequality aversion in the ordinal than in the cardinal treatment.

STRATEGY-PROOF ALLOCATION OF INDIVISIBLE GOODS AMONG COUPLES

We consider the problem of allocating indivisible goods among couples. Agents in a couple share the indivisible good assigned to them. The main result is that an allocation rule is strategy-proof, neutral and non-bossy if and only if it is serially dictatorial. An allocation rule is serially dictatorial if there is a priority order of couples and a function that identifies who chooses in each couple, such that for all preference profiles, a good assigned to couple i is the best element according to the preference of the identified agent in couple i among the remaining goods when the couples with higher priorities have made their choice.

Evolution of cooperation in rotating indivisible goods game

Collective behavior is theoretically and experimentally studied through a public goods game in which players contribute resources or efforts to produce goods (or pool), which are then divided equally among all players regardless of the amount of their contribution. However, if goods are indivisible, only one player can receive the goods. In this case, the problem is how to distribute indivisible goods, and here therefore we propose a new game, namely the "rotating indivisible goods game." In this game, the goods are not divided but distributed by regular rotation. An example is rotating savings and credit associations (ROSCAs), which exist all over the world and serve as efficient and informal institutions for collecting savings for small investments. In a ROSCA, members regularly contribute money to produce goods and to distribute them to each member on a regular rotation. It has been pointed out that ROSCA members are selected based on their reliability or reputation, and that defectors who stop contributing are excluded. We elucidate mechanisms that sustain cooperation in rotating indivisible goods games by means of evolutionary simulations. First, we investigate the effect of the peer selection rule by which the group chooses members based on the players reputation, also by which players choose groups based on their reputation. Regardless of the peer selection rule, cooperation is not sustainable in a rotating indivisible goods game. Second, we introduce the forfeiture rule that forbids a member who has not contributed earlier from receiving goods. These analyses show that employing these two rules can sustain cooperation in the rotating indivisible goods game, although employing either of the two cannot. Finally, we prove that evolutionary simulation can be a tool for investigating institutional designs that promote cooperation.