Allocation Of Indivisible Goods: A General Model And Some Complexity Results Sylvain Bouveret and Michel Lemaˆıtre

Hel ´ ene ` Fargier and Jer ´ ome ˆ Lang

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ABSTRACT Many industrial or research activities are so expensive that it is often benefitable for the involved agents to cofund the construction or the purchase of a common required resource. This resource will then be exploited in common, therefore in a shared way. The rules for resource sharing should take account of the possibly antagonistic preferences: each agent wants to maximize its own satisfaction, whereas, from the collective point of view, decisions which both are equitable and exploit the resource in an optimal way are looked for. We give in this article a formal model for indivisible goods resource sharing without monetary compensations and with arbitrary feasability constraints. We also give some complexity results about this model. The model is applied to a real-world case, namely satellite resource sharing.

Categories and Subject Descriptors I.2.4 [Computing Methodologies]: Artificial Intelligence— Knowledge Representation Formalisms and Methods

General Terms Theory

Keywords fair allocation, resource sharing, preference aggregation, logic, computational compexity.

1.

INTRODUCTION

Allocation of indivisible goods has been considered both in social choice theory, where fairness is a key-point but where computational complexity and algorithmics are often neglected, and in artificial intelligence, where research has focused on combinatorial auctions, where fairness is not relevant (see [7]).

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Whereas fair division has been widely studied by the economists (see e.g. [5]), it seems to lack complexity results and languages for compact expression of preferences. This is the main goal of this paper: after having introduced a general framework for representing the allocation of indivisible goods problem, it presents some complexity results for criteria that can vary from utilitarianism to egalitarianism. Then, a real-life problem is presented, and lastly some connections to the literature are suggested.

2. A GENERAL MODEL AND A REPRESENTATION LANGUAGE We shortly describe in this section our general model for representing the problem of sharing a set of indivisible goods or objects (denoted by O), between some agents (denoted by N = {1, . . . , n}). In this model, an allocation is a n-uple x = hx1 , . . . , xn i where ∀i, xi ⊆ O is the share of agent i. Let us describe how the agents can express their preferences about the objects they want to receive. In our model, they are allowed to express dependencies between objects using a compact representation language (see e.g. [2] for more on compact representation). Using logic is particulary appropriate here, thus our language L+ O is based on the set of propositional symbols O, and the usual connectives ∧ and ∨. Each agent can express a set of demands. Each demand consists of a bundle, described by a formula of L+ O , and its associated weight w ∈ R. If some constraints restrict the set of feasible allocations, they can be expressed in the full propositional language generated by the set of propositional symbols alloc(o, i), where o ∈ O and i ∈ N . alloc(o, i) is true iff o ∈ xi . The last issue is to compute the collective utility of a resource allocation. This is achieved by a two-steps aggregation of the agents’ preferences. The first step computes the individual utility ui of each agent by aggregating the weights of her satisfied demands, and the second step computes the collective utility uc by aggregating the individual ones. The two aggregation functions are respectively denoted by f and g: 9 f w11 , . . . , wp11 7→ u1 > > = g .. 7→ uc . . > > ; f w1n , . . . , wpnn 7→ un Most appropriate functions are +, max and leximax for f and +, min and leximin for g.

Constraints Demands f g

NPC h Phh ((( hhhhh any lPP none volume hhhonly l PP

(( preemption( only ((((

( NPC  X  XXXatomic only any  XX  NPC NPC aa max HH max +  + aa  H NPC NPC NPC P b + %min/lex + %min/lex + min @ e  lex +/min b % @ % e   b LLlex

NPCa NPC

NPC NPC

Pa NPC1 NPC

P* NPC*

exclusion only

NPC any

NPC any

NPC @max   @

+

+

NPC*

NPC + min/lex @

@

P ?

NPC @max

@

NPC* NPC +/min/lex NPC

NPC* NPC*

a Already known from combinatorial auctions. * Immediate. 1 P if equal weights.

Figure 1: The complexity classes of different instances of the AIG problem. To summarize, an instance of the allocation problem consists of a tuple hN, O, C, (f, g)i: N = {1, . . . , n}, a finite set of agents, O, a finite set of objects, C, a finite set of constraints over possible allocations, a pair (f, g) of aggregation functions. The output is a feasible allocation maximizing the collective utility.

3.

COMPUTATIONAL COMPLEXITY

In this section, we give some results concerning the complexity of the allocation of indivisible goods (aig) problem: Given an instance of an allocation problem and an integer K, is there a feasible allocation x such that uc (x) ≥ K ? The complexity classes of different instances of the aig problem are shown in figure 1. These instances are classified using their constraint types, the syntax of demands, and the aggregation functions f and g. Preemption means that each object can be attributed to at most one agent, exclusion means that a given set of objects cannot be simultaneously allocated, and volume means that there is a maximal number (or weight if the objects have weights) of objects that can be simultaneously allocated.

4.

This application can be formally stated using the previous model. In this framework, the objects are the images, and the constraints and the agents’ demands can be expressed using the appropriate language.

5. RELATED WORK AND CONCLUSION Allocation of indivisible goods has widely been considered, both in social choice theory and in artificial intelligence. Most works concerning fair resource allocation in social choice theory deal with divisible goods. There are a few recent works about fair division of indivisible goods ; they generally do not consider compact representation issues. For example, the work [4] studies approximation schemes for another fairness criterion: envy-freeness (see also [1]). On the contrary, several “bidding languages” have been developped in the community of combinatorial auctions (see e.g. [6] for a recent overview). Most of these languages allow for expressing complex preferences about bundles. Complexity issues have been studied, for the winner determination problem (that is known to be NP-hard), and some restrictions that make it tractable. However, fairness is irrelevant in combinatorial auctions, which corresponds to a purely utilitarianistic resource allocation. 1

APPLICATION

The latter model can be applied to a real-world problem, that we describe here (see also [3]). The aim of Earth Observing Satellites (EOS) is to aquire images of specified aeras on the Earth. Like other space projects, EOS are often coexploited by several agents, leading to a sharing problem. Moreover, the exploitation of the satellite must obey a set of physical constraints (e.g. memory and energy management). Due to these constraints, the agents’ demands are often conflicting, and the processing center must choose between the requests to be satisfied. The resulting choice has to be both efficient (the satellite cannot be underexploited) and equitable (each agent wants to get a return on investments proportional to its financial contribution). Each demand of each agent is associated to a weight, that measures the importance of this demand for the involved agent. The requests can express dependencies between images. The satisfaction of an agent can be measured by the sum of the weights of the allocated images. The general quality of an allocation can be ensured by maximizing the collective utility function, whose role must be to guarantee the overall fairness of the sharing. Note that intercomparability of the individual utilities (and thus their normalization) is required.

6. REFERENCES [1] S. Bouveret and J. Lang. Efficiency and envy-freeness in fair division of indivisible goods: logical representation and complexity. In Proc. of IJCAI05. [2] J. Lang. Logical preference representation and combinatorial vote. Annals of Mathematics and Artificial Intelligence, 42:37–71, 2004. [3] M. Lemaˆıtre, G. Verfaillie, and N. Bataille. Exploiting a Common Property Resource under a Fairness Constraint: a Case Study. In Proc. of IJCAI-99, pages 206–211, Stockholm, Sweden, 1999. [4] R. Lipton, E. Markakis, E. Mossel, and A. Saberi. On approximately fair allocations of indivisible goods. In Proceedings of EC’04, 2004. [5] H. Moulin. Fair division and collective welfare. MIT Press, 2003. [6] N. Nisan. Combinatorial auctions, chapter Bidding languages. 2005. [7] Y. S. Peter Cramton and R. Steinberg, editors. Combinatorial Auctions. MIT Press, 2006. forthcoming. 1

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