Indivisible Items Research Articles

Dissolving a partnership dynamically

In financial disputes arising from divorce, inheritance, or the dissolution of a partnership, frequently the need arises to assign ownership of an indivisible item to one member of a group. This paper introduces and analyzes a dynamic auction for simply and efficiently allocating an item when participants are privately informed of their values. In the auction, the price rises continuously. A bidder who drops out of the auction, in return for surrendering his claim to the item, obtains compensation equal to the difference between the price at which he drops and the preceding drop price. When only one bidder remains, that bidder wins the item and pays the compensations of his rivals. We characterize the unique equilibrium with risk-neutral and CARA risk averse bidders. We show that dropout prices are decreasing as bidders become more risk averse. Each bidder's equilibrium payoff is at least 1/N-th of his value for the item.

Proportional Borda allocations.

In this paper we study the allocation of indivisible items among a group of agents, a problem which has received increased attention in recent years, especially in areas such as computer science and economics. A major fairness property in the fair division literature is proportionality, which is satisfied whenever each of the n agents receives at least frac{1}{n} of the value attached to the whole set of items. To simplify the determination of values of (sets of) items from ordinal rankings of the items, we use the Borda rule, a concept used extensively and well-known in voting theory. Although, in general, proportionality cannot be guaranteed, we show that, under certain assumptions, proportional allocations of indivisible items are possible and finding such allocations is computationally easy.

Maximin Envy-Fee Division of Indivisible Items

Assume that two players have strict rankings over an even number of indivisible items. We propose two algorithms to find balanced allocations of these items that are maximin—maximize the minimum rank of the items that the players receive—and are envy-free and Pareto-optimal, if such allocations exist. To determine whether an envy-free allocation exists, we introduce a simple condition on preference profiles; in fact, our condition guarantees the existence of a maximin, envy-free, and Pareto-optimal allocation. Although not strategy-proof, our algorithms would be difficult to manipulate unless a player has complete information about its opponent’s ranking. We assess the applicability of the algorithms to real-world problems, such as allocating marital property in a divorce or assigning people to committees or projects.

Maximin Envy-Free Division of Indivisible Items

Assume that two players have strict rankings over an even number of indivisible items. We propose two algorithms to find balanced allocations of these items that are maximin—maximize the minimum rank of the items that the players receive—and are envy-free and Pareto-optimal, if such allocations exist. To determine whether an envy-free allocation exists, we introduce a simple condition on preference profiles; in fact, our condition guarantees the existence of a maximin, envy-free, and Pareto-optimal allocation. Although not strategy-proof, our algorithms would be difficult to manipulate unless a player has complete information about its opponent’s ranking. We assess the applicability of the algorithms to real-world problems, such as allocating marital property in a divorce or assigning people to committees or projects.

Lower bounds and algorithms for the minimum cardinality bin covering problem

This paper introduces the minimum cardinality bin covering problem where we are given m identical bins with capacity C and n indivisible items with integer weights wj(j=1,…,n). The objective is to minimize the number of items packed into the m bins so that the total weight of each bin is at least equal to C. We discuss reduction criteria, derive several lower bound arguments and propose construction heuristics as well as a powerful subset sum-based improvement algorithm that is even optimal when m=2. Moreover, we present a tailored branch-and-bound method which is able to solve instances with up to 20 bins and several hundreds of items within a reasonable amount of time. In a comprehensive computational study on a wide range of randomly generated instances, our algorithmic approach proved to be much more effective than a commercial solver.

Optimal robust bilateral trade: Risk neutrality

A risk neutral seller and buyer with private information bargain over an indivisible item. We prove that optimal robust bilateral trade mechanisms are payoff equivalent to non-wasteful randomized posted prices.

How to Divide Things Fairly

SummaryWe propose an intuitively simple sequential algorithm (SA) for the fair division of indivisible items that are strictly ranked by two or more players. We analyze several properties of the allocations that it yields and discuss SA's application to real-life problems, such as dividing the marital property in a divorce or assigning people to committees or projects.

Approximating the nash social welfare with indivisible items

In this letter we briefly discuss our main result from [Cole and Gkatzelis 2015]. Given a set of indivisible items and a set of agents having additive valuations, our goal is to allocate the items to the agents in order to maximize the geometric mean of the agents' valuations, i.e., the Nash social welfare. This problem is known to be NP-hard, and our main result is the first efficient constant-factor approximation algorithm for this objective.

QoS and Energy Efficient Resource Allocation in Uplink SC-FDMA Systems

In this paper, we present and evaluate the performance of a resource allocation algorithm to enhance the Quality of Service (QoS) provision and energy efficiency of uplink Long Term Evolution (LTE) systems. The proposed algorithm considers the main constraints in uplink LTE resource allocation, i.e., the allocation of contiguous sets of resource blocks of the localized Single Carrier—Frequency Division Multiple Access (SC-FDMA) physical layer to each user, and the imperfect knowledge of the users' uplink buffer status and packet waiting time. The optimal resource allocation is formulated as a discrete connected cake-cutting problem, where different agents are allocated consecutive subsequences of a sequence of indivisible items. This problem is NP-hard, therefore a suboptimal algorithm is introduced, which performs resource allocation using information on the estimated uplink packet delay, the average delay and data rate of past allocations, as well as the required uplink power per resource block. Based on simulation results, the proposed algorithm achieves significant performance improvement in terms of packet timeout rate, goodput, and fairness. Moreover, the effect of poor QoS provision on energy efficiency is demonstrated through the evaluation of the performance in terms of energy consumption per successfully received bit.

Truthful unit-demand auctions with budgets revisited

We consider auctions of indivisible items to unit-demand bidders with budgets. This setting was suggested as an expressive model for single sponsored search auctions. Prior work presented mechanisms that compute bidder-optimal outcomes and are truthful for a restricted set of inputs, i.e., inputs in so-called general position. This condition is easily violated. We provide the first mechanism that is truthful in expectation for all inputs and achieves for each bidder no worse utility than the bidder-optimal outcome. Additionally we give a complete characterization for which inputs mechanisms that compute bidder-optimal outcomes are truthful.

On revenue maximization with sharp multi-unit demands

We consider markets consisting of a set of indivisible items, and buyers that have sharp multi-unit demand. This means that each buyer $$i$$ wants a specific number $$d_i$$ of items; a bundle of size less than $$d_i$$ has no value. We consider the objective of setting prices and allocations in order to maximize the total revenue of the market maker. The pricing problem with sharp multi-unit demand buyers has a number of properties that the unit-demand model does not possess, and is an important question in algorithmic pricing. We consider the problem of computing a revenue maximizing solution for two solution concepts: competitive equilibrium and envy-free pricing. For unrestricted valuations, these problems are NP-complete; we focus on a realistic special case of “correlated values” where each buyer $$i$$ has a valuation $$v_iq_j$$ for item $$j$$ , where $$v_i$$ and $$q_j$$ are positive quantities associated with buyer $$i$$ and item $$j$$ respectively. We present a polynomial time algorithm to solve the revenue-maximizing competitive equilibrium problem. For envy-free pricing, if the demand of each buyer is bounded by a constant, a revenue maximizing solution can be found efficiently; the general demand case is shown to be NP-hard.

An efficient multi-item dynamic auction with budget constrained bidders

An auctioneer wishes to sell several heterogeneous indivisible items to a group of potential bidders. Each bidder has valuations over the items but may face a budget constraint and therefore be unable to pay up to his values. In such markets, a Walrasian equilibrium may fail to exist. We develop a novel dynamic auction and prove that the auction always finds a core allocation. In the auction prices that have been increased can be later decreased if they have become too high. The core allocation consists of an assignment of the items and its associated supporting price vector, achieves Pareto efficiency, and is robust against the threat of deviation by any coalition of market participants.

How to Divide Things Fairly

We analyze a simple sequential algorithm (SA) for allocating indivisible items that are strictly ranked by n ≥ 2 players. It yields at least one Pareto-optimal allocation which, when n = 2, is envy-free unless no envy-free allocation exists. However, an SA allocation may not be maximin or Borda maximin — maximize the minimum rank, or the Borda score — of the items received by a player. Although SA is potentially vulnerable to manipulation, it would be difficult to manipulate in the absence of one player’s having complete information about the other players’ preferences. We discuss the applicability of SA, such as in assigning people to committees or allocating marital property in a divorce.

An Algorithm for the Proportional Division of Indivisible Items

An allocation of indivisible items among n ≥ 2 players is proportional if and only if each player receives a proportional subset—one that it thinks is worth at least 1/n of the total value of all the items. We show that a proportional allocation exists if and only if there is an allocation in which each player receives one of its minimal bundles, from which the subtraction of any item would make the bundle worth less than 1/n. We give a practicable algorithm, based on players’ rankings of minimal bundles, that finds a proportional allocation if one exists; if not, it gives as many players as possible minimal bundles. The resulting allocation is maximin, but it may be neither envy-free nor Pareto-optimal. However, there always exists a Pareto-optimal maximin allocation which, when n = 2, is also envy-free. We compare our algorithm with two other 2-person algorithms, and we discuss its applicability to real-world disputes among two or more players.

Optimality versus practicality in market design: A comparison of two double auctions

We consider a market for indivisible items with m buyers and m sellers. Traders privately know their values/costs, which are statistically dependent. Two mechanisms are considered. The buyer's bid double auction collects bids and asks from traders and determines the allocation by selecting a market-clearing price. It fails to achieve all possible gains from trade because of strategic bidding. The designed mechanism is a revelation mechanism in which honest reporting of values/costs is incentive compatible and all gains from trade are achieved. This optimality, however, comes at the expense of plausibility: (i) the monetary transfers among the traders are defined in terms of the traders' beliefs about each other's value/cost; (ii) a trader may suffer a loss ex post; (iii) the mechanism may run a surplus/deficit ex post. We compare the virtues of the simple yet mildly inefficient buyer's bid double auction to the flawed yet perfectly efficient designed mechanism.

Approximate Random Allocation Mechanisms

We extend the scope of random allocation mechanisms, in which the mechanism first identifies a feasible "expected allocation" and then implements it by randomizing over nearby feasible integer allocations. Previous literature had shown that the cases in which this is possible are sharply limited. We show that if some of the feasibility constraints can be treated as goals rather than hard constraints then, subject to weak conditions that we identify, any expected allocation that satisfies all the constraints and goals can be implemented by randomizing among nearby integer allocations that satisfy all the hard constraints exactly and the goals approximately. By defining ex post utilities as goals, we are able to improve the ex post properties of several classic assignment mechanisms, such as the random serial dictatorship. We use the same approach to prove the existence of e-equilibrium in large markets with indivisible items and feasibility constraints.

Two-Person Fair Division of Indivisible Items: An Efficient, Envy-Free Algorithm

Many procedures have been suggested for the venerable problem of dividing a set of indivisible items between two players. We propose a new algorithm (AL), related to one proposed by Brams and Taylor (BT), which requires only that the players strictly rank items from best to worst. Unlike BT, in which any item named by both players in the same round goes into a “contested pile,” AL may reduce, or even eliminate, the contested pile, allocating additional or more preferred items to the players. The allocation(s) that AL yields are Pareto-optimal, envy-free, and maximal; as the number of items (assumed even) increases, the probability that AL allocates all the items appears to approach infinity if all possible rankings are equiprobable. Although AL is potentially manipulable, strategizing under it would be difficult in practice.

Fair division of indivisible items between two players: design parameters for Contested Pile methods

Contested Pile methods are two-phase procedures for the fair allocation of indivisible items to two players. In the Generation Phase, items over which the players’ preferences differ widely enough are allocated. “Contested” items are placed in the Contested Pile, which is then allocated in the Splitting Phase. Each phase can be carried out using several different techniques; we perform a comprehensive analysis of the resulting design variants using a computational model. The properties of fairness and efficiency, generally achieved in the Generation Phase, must be traded off against robustness to manipulation. We find that the recently developed Undercut procedure for the Splitting Phase outperforms alternative methods in both fairness and efficiency. In general, procedures that keep the Contested Pile relatively small and incorporate the Undercut procedure score well in both fairness and efficiency, but are prone to manipulation.

Two-Person Fair Division of Indivisible Items: An Efficient, Envy-Free Algorithm

Many procedures have been suggested for the venerable problem of dividing a set of indivisible items between two players. We propose a new algorithm (AL), related to one proposed by Brams and Taylor (BT), which requires only that the players strictly rank items from best to worst. Unlike BT, in which any item named by both players in the same round goes into a “contested AL may reduce, or even eliminate the contested pile, allocating additional or more preferred items to the players. The allocation(s) that AL yields are Pareto-optimal, envy-free, and maximal; as the number of items (assumed even) increases, the probability that AL allocates all the items appears to approach infinity if all possible rankings are equi-probable. Although AL is potentially manipulable, strategizing under it would be difficult in practice.

An Equilibrium Analysis of Knaster’s Fair Division Procedure

In an incomplete information setting, we analyze the sealed bid auction proposed by Knaster (cf. Steinhaus (1948)). This procedure was designed to efficiently and fairly allocate multiple indivisible items when participants report their valuations truthfully. In equilibrium, players do not follow truthful bidding strategies. We find that, ex-post, the equilibrium allocation is still efficient but may not be fair. However, on average, participants receive the same outcome they would have received if everyone had reported truthfully—i.e., the mechanism is ex-ante fair.