Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

07/28/2018 ∙ by Aurélie Beynier, et al. ∙ 0

In fair division of indivisible goods, using sequences of sincere choices (or picking sequences) is a natural way to allocate the objects. The idea is as follows: at each stage, a designated agent picks one object among those that remain. Another intuitive way to obtain an allocation is to give objects to agents in the first place, and to let agents exchange them as long as such "deals" are beneficial. This paper investigates these notions, when agents have additive preferences over objects, and unveils surprising connections between them, and with other efficiency and fairness notions. In particular, we show that an allocation is sequenceable iff it is optimal for a certain type of deals, namely cycle deals involving a single object. Furthermore, any Pareto-optimal allocation is sequenceable, but not the converse. Regarding fairness, we show that an allocation can be envy-free and non-sequenceable, but that every competitive equilibrium with equal incomes is sequenceable. To complete the picture, we show how some domain restrictions may affect the relations between these notions. Finally, we experimentally explore the links between the scales of efficiency and fairness.

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1 Introduction

In this paper, we investigate fair division of indivisible goods. In this problem, a set of indivisible objects or goods has to be allocated to a set of agents, taking into account the agents’ preferences about the objects. This classical collective decision making problem has plenty of practical applications, among which the allocation of space resources (Lemaître et al., 1999; Bianchessi et al., 2007), of tasks to workers in crowdsourcing market systems, papers to reviewers (Goldsmith and Sloan, 2007) or courses to students (Budish, 2011).

This problem can be tackled from two different perspectives. The first possibility is to resort to a benevolent entity in charge of collecting in a centralized way the preferences of all the agents. This entity then computes an allocation that takes into account these preferences and satisfies some fairness (e.g. envy-freeness) and efficiency (e.g. Pareto-optimality) criteria, or optimizes a well-chosen social welfare ordering. The second possibility is to have a distributed point of view, e.g. by starting from an initial allocation and letting the agents negotiate to swap their objects (Sandholm, 1998; Chevaleyre et al., 2005). A somewhat intermediate approach consists in allocating the objects to the agents using a protocol, which allows to build an allocation interactively by asking the agents a sequence of questions. Protocols are at the heart of works mainly concerning the allocation of divisible resources (cake-cutting) (Brams and Taylor, 1996), but have also been studied in the context of indivisible goods (Brams and Taylor, 1996; Brams et al., 2012).

In this paper, we focus on a particular allocation protocol: sequences of sincere choices (also known as picking sequences). This very simple protocol works as follows. A central authority chooses a sequence of agents before the protocol starts, having as many agents as the number of objects (some agents may appear several times in the sequence). Then, each agent appearing in the sequence is asked to choose in turn one object among those that remain. For instance, according to the sequence , agent is going to choose first, then agent will pick two consecutive objects, and agent will take the last object. This protocol, actually used in a lot of everyday situations, has been studied for the first time by Kohler and Chandrasekaran (1971). Later, Brams and Taylor (2000) have studied a particular version of this protocol, namely alternating sequences, in which the sequence of agents is restricted to a balanced () or strict () alternation of agents. Bouveret and Lang (2011) have further formalized this protocol, whose properties (especially related to game theoretic aspects) have been characterized (Kalinowski et al., 2013a, b). Finally, Aziz et al. (2015b) have studied the complexity of problems related to finding whether a particular assignment (or bundle) is achievable by a particular class of picking sequences. In their work (not specifically dedicated to picking sequences) focusing on a situation where the agents have ordinal preferences, Brams and King (2005) make an interesting link between this protocol and Pareto-optimality, showing, among others, that picking sequences always result in a Pareto-optimal allocation, but also that every Pareto-optimal allocation can be obtained in this way.

In this paper, we elaborate on these ideas and analyze the links between sequences, certain types of deals among agents, and some efficiency and fairness properties, in a more general model in which the agents have numerical additive preferences on the objects. Our main contributions are the following. We give a formalization of the link between allocations and sequences of sincere choices, highlighting a simple characterization of the sequenceability of an allocation (Section 3). Then, we show that in this slightly more general framework than the one by Brams et al., surprisingly, Pareto-optimality and sequenceability are not equivalent anymore (Section 4). By unveiling the connection between sequenceability and cycle deals among agents (Section 5), we obtain a rich “scale of efficiency” that allows us to characterize the degree of efficiency of a given allocation. Interestingly, some domain restrictions have significant effects on this hierarchy (Section 6). We also highlight a link between sequenceability and another important economical concept: the competitive equilibrium from equal income (CEEI). Another contribution is the experimental exploration of the links between the scale of efficiency and fairness properties.

2 Model and Definitions

The aim of the fair division of indivisible goods, also called MultiAgent Resource Allocation (MARA), is to allocate a finite set of objects to a finite set of agents . A sub-allocation on

is a vector

of bundles of objects, such that with (a given object cannot be allocated to more than one agent) and (all the objects from are allocated). is called agent ’s share on . is a sub-allocation of when for each agent . Any sub-allocation on the entire set of objects will be denoted and just called allocation.

Any satisfactory allocation must take into account the agents’ preferences on the objects. Here, we will make the classical assumption that these preferences are numerically additive. Each agent has a utility function measuring her satisfaction when she obtains share , which is defined as follows:

where is the weight given by agent to object . This assumption, as restrictive as it may seem, is made by a lot of authors (Lipton et al., 2004; Bansal and Sviridenko, 2006, for instance) and is considered as a good compromise between expressivity and conciseness.

Definition 1.

An instance of the additive multiagent resource allocation problem (add-MARA instance for short) is a tuple with and as defined above and is a mapping with being the weight given by agent to object .

We say that the agents’ preferences are strict on objects if, for each agent and each pair of objects , we have . Similarly, we say that the agents’ preferences are strict on shares if, for each agent and each pair of shares , we have . Note that preferences strict on shares entails preferences strict on objects; the converse is false.

We will denote by the set of allocations for .

The following definition will play a prominent role.

Definition 2.

Given an agent and a set of objects , let be the subset of objects in having the highest weight for agent (such objects will be called top objects of ).

A (sub-)allocation is said frustrating if no agent receives one of her top objects in (formally: for each agent ), and non-frustrating otherwise.

In the following, we will consider a particular way of allocating objects to agents: allocation by sequences of sincere choices. Informally the agents are asked in turn, according to a predefined sequence, to choose and pick a top object among the remaining ones.

Definition 3.

Let be an add-MARA instance. A sequence of sincere choices (or simply sequence when the context is clear) is a vector of . We will denote by the set of possible sequences for the instance .

Let . is said to generate allocation if and only if can be obtained as a possible result of the non-deterministic222The algorithm contains an instruction choose splitting the control flow into several branches, building all the allocations generated by . Algorithm 1 on input and .

Input: an instance and a sequence
Output: an allocation
1 empty allocation (such that );
2 ;
3 for  from 1 to  do
4       ;
5       choose object ;
6       ;
7      
Algorithm 1 Execution of a sequence
Definition 4.

An allocation is said to be sequenceable if there exists a sequence that generates , and non-sequenceable otherwise. For a given instance I, we will denote by the binary relation defined by if and only if can be generated by .

Example 1.

Let be the instance represented by the following weight matrix:333In this example and the following ones, we represent instances by a matrix where the value at row and column represents the weight . We also use as a shorthand for .

The binary relation between and can be graphically represented as follows:

For instance, sequence generates two possible allocations: and , depending on whether agent 2 chooses object 1 or 3 that she both prefers. Allocation can be generated by three sequences. Allocations and are non-sequenceable.

For any instance , . Note also that the number of objects allocated to an agent by a sequence is the number of times the agent appears in the sequence.

The notion of frustrating allocation and sequenceability were already implicitly present in the work by Brams and King (2005), and sequenceability has been extensively studied by Aziz et al. (2015b) with a focus on sub-classes of sequences (e.g. alternating sequences). However, a fundamental difference is that in our setting, the preferences might be non strict on objects, which entails that the same sequence can yield different allocations (in the worst case, an exponential number), as Example 1 shows.

3 Sequenceable allocations

We have seen in Example 1 that some allocations are non-sequenceable. We will now formalize this and give a precise characterization of sequenceable allocations that is, we will try to identify under which conditions an allocation is achievable by the execution of a sequence of sincere choices. We first start by noticing that in every sequenceable allocation, the first agent of the sequence gets a top object, so every frustrating allocation is non-sequenceable. It is even possible to find a non-sequenceable allocation that gives her top object to one agent (as allocation in Example 1) or even to all:

Example 2.

Consider this instance:

In the circled allocation , every agent receives her top object. However, after objects 1 and 2 have been allocated (they must be allocated first by all sequences generating ), the dotted sub-allocation remains. This sub-allocation is obviously non-sequenceable because it is frustrating. Hence is not sequenceable either.

This property of containing a frustrating sub-allocation exactly characterizes the set of non-sequenceable allocations:

Proposition 0.

Let be an instance and be an allocation of this instance. The two following statements are equivalent:

  • is sequenceable.

  • No sub-allocation of is frustrating (in every sub-allocation, at least one agent receives a top object).

Proof.

(B) (A). Let us suppose that for all subsets of objects there is at least one agent obtaining one of her top objects in . We will show that is sequenceable. Let be a sequence of agents and be a sequence of sets of objects jointly defined as follows:

  • and is an agent that receives one of her top objects in ;

  • , where and is an agent that receives one of her top objects in , for .

From the assumption on , we can check that the sequence is perfectly defined. Moreover, is one of the allocations generated by .

(A) (B) by contraposition. Let be an allocation containing a frustrating sub-allocation . Suppose that there exists a sequence generating . We can notice that in a sequence of sincere choices, when an object is allocated to an agent, all the objects that are strictly better for her have already been allocated at a previous step. Let , and let be the agent that receives in . Since is frustrating, there is another object such that . From the previous remark, is necessarily allocated before in the execution of . We can deduce, from the same line of reasoning on and agent that receives it, that there is another object allocated before in the execution of the sequence. The set being finite, using the same argument iteratively, we will necessarily find an object which has already been encountered before. This leads to a cycle in the precedence relation of the objects in the execution of the sequence. Contradiction: no sequence can thus generate . ∎

Beyond the fact that it characterizes a sequenceable allocation, the proof of Proposition 1 gives a practical way of checking if an allocation is sequenceable, and, if it is the case, of computing a sequence that generates this allocation.

Proposition 0.

Let be an instance and be an allocation of this instance. We can decide in time if is sequenceable.

The proof is based on the execution of Algorithm 2. This algorithm is similar in spirit to the one proposed by Brams and King (2005) but Algorithm 2 is more general because (i) it can involve non-strict preferences on objects, and (ii) it can conclude with non-sequenceability.

Proof.

We show that Algorithm 2 returns a sequence generating the input allocation if and only if there is one. Suppose that the algorithm returns a sequence . Then, by definition of the sequence (in the loop from line 2 to line 2), at each step , can choose an object in , that is one of her top objects. Conversely, suppose the algorithm returns NonSeq. Then, at a given step , , . By definition, is therefore, at this step, a frustrating sub-allocation of . By Proposition 1, is thus non-sequenceable. The loop from line 2 to line 2 runs in time , because searching for the top objects in the preferences of each agent is in . This loop being executed times, the algorithm runs in .

Input: and
Output: a sequence generating or NonSeq
1 ;
2 for  from 1 to  do
3       if  such that  then
4             Append to ;
5             let ;
6             ;
7            
8      else  return NonSeq ;
9      
10return ;
Algorithm 2 Sequencing an allocation

4 Pareto-optimality

An allocation is Pareto-optimal if there is no other allocation dominating it. In our context, allocation dominates allocation if for all agent , and for at least one agent . When an allocation is generated from a sequence, in some sense, a weak form of efficiency is applied to build the allocation: each successive (picking) choice is “locally” optimal. This raises a natural question: is every sequenceable allocation Pareto-optimal?

This question has already been extensively discussed independently by Aziz et al. (2016b) and in a previous version of this work (Bouveret and Lemaître, 2016b). We complete the discussion here to give more insights about the implications of the previous results in our framework.

Brams and King (2005, Proposition 1) prove the equivalence between sequenceability and Pareto-optimality. However, they have a different notion of Pareto-optimality, because the agents’ preferences are given as linear orders over objects. To be able to compare bundles, these preferences are lifted on subsets using the responsive set extension . This extension leaves many bundles incomparable and leads to define possible and necessary Pareto-optimality. Brams and King’s notion is possible Pareto-optimality. Aziz et al. (2015a) show that, given a linear order on objects and two bundles and , if and only if for all additive utility functions compatible with (that is, such that if and only if ). This characterization of responsive dominance yields the following reinterpretation of Brams and King’s result: an allocation is sequenceable if and only if for each other allocation , there is a set of additive utility functions, respectively compatible with such that for at least one agent .

The latter notion of Pareto-optimality is very weak, because (unlike in our context) the set of additive utility functions is not fixed — we just have to find one that works. Under our stronger notion, the equivalence between sequenceability and Pareto-optimality no longer holds.444Actually, since it is known (de Keijzer et al., 2009; Aziz et al., 2016a) that testing Pareto-optimality with additive preferences in coNP-complete, and that testing sequenceability is in →(Proposition 2), they cannot be equivalent unless →= coNP.

Example 3.

Let us consider the following instance:

The sequence generates allocation giving utilities . is then sequenceable but it is dominated by , giving utilities (and generated by ). Observe that, under ordinal linear preferences, would not dominate , but they would be incomparable.

The last example shows that a sequence of sincere choices does not necessarily generate a Pareto-optimal allocation. What about the converse? We can see, as a trivial corollary of the reinterpretation of Brams and King’s result in our terminology, that the answer is positive if the preferences are strict on shares. The following result is more general, because it holds even without this assumption:

Proposition 0 (Aziz et al., 2016b; Bouveret and Lemaître, 2016b).

Every Pareto-optimal allocation is sequenceable.

Before giving the formal proof, we illustrate it on a concrete example (Bouveret and Lemaître, 2016a, Example 5).

Example 4.

Let us consider the following instance:

The circled allocation is not sequenceable: indeed, every sequence that could generate it should start with , leaving the frustrating sub-allocation in a dotted box.

Let us now choose an arbitrary agent who does not receive a top object in , for instance agent . Let be her top object (of weight 11 in this case). The agent receiving in is . This agent prefers object (of weight 15), held by , already encountered. We have built a cycle , in other words , that tells us exactly how to build another sub-allocation dominating . This sub-allocation can be built by replacing in the attributions by the attributions . Hence, each agent involved in the cycle obtains a strictly better object than the previous one. Doing the same substitutions in the initial allocation yields an allocation that dominates (marked with in the matrix above).

Now we will give the formal proof.555This proof is similar to the one by Brams and King (2005, Proposition 1, necessity). However we give it entirely because it is more general and will be reused in Proposition 10.

Proof.

As stated in the example, we will now prove the contraposition of the proposition: every non-sequenceable allocation is dominated. Let be a non-sequenceable allocation. From Proposition 1, in a non-sequenceable allocation, there is at least one frustratring sub-allocation. Let be such a sub-allocation (that can be itself). We will, from , build another sub-allocation dominating it. Let us choose an arbitrary agent involved in , receiving an object not among her top ones in . Let be a top object of in , and let () be the unique agent receiving it in . Let be a top object of . We can notice that (otherwise would obtain one of her top objects and would not be frustrating). Let be the unique agent receiving in , and so on. Using this argument iteratively, we form a path starting from and alternating agents and objects, in which two successive agents and objects are distinct. Since the number of agents and objects is finite, we will eventually encounter an agent which has been encountered at a previous step of the path. Let be the first such agent and be the last object seen before her in the sequence ( is the unique agent receiving ). We have built a cycle in which all the agents and objects are distinct, and that has at least two agents and two objects. From this cycle, we can modify to build a new sub-allocation by giving to each agent in the cycle a top object instead of another less preferred object, all the agents not appearing in the cycle being left unchanged. More formally, the following attributions in (and hence in ): are replaced by: where means that is attributed to . The same substitutions operated in yield an allocation that dominates . ∎

Corollary 1.

No frustrating allocation can be Pareto-optimal (equivalently, in every Pareto-optimal allocation, at least one agent receives a top object).

Proposition 3 implies that there exists, for a given instance, three classes of allocations: (1) non-sequenceable (therefore non Pareto-optimal) allocations, (2) sequenceable but non Pareto-optimal allocations, and (3) Pareto-optimal (hence sequenceable) allocations. These three classes define a “scale of efficiency” that can be used to characterize the allocations. What is interesting and new here is the intermediate level. We will see that this scale can be further detailed.

5 Cycle deals-optimality

Pareto-optimality can be thought as a reallocation of objects among agents using improving deals (Sandholm, 1998), as we have seen, to some extent, in the proof of Proposition 3. Trading cycles or cycle deals constitute a sub-class of deals, which is classical and used, e.g., by Varian (1974, page 79) and Lipton et al. (2004, Lemma 2.2) in the context of envy-freeness. Trying to link efficiency concepts with various notions of deals is thus a natural idea.

Definition 5.

Let be an add-MARA instance and be an allocation of this instance. A -cycle deal of is a sequence of transfers of items , where, for each , ,, and . The allocation resulting from the application of to is defined as follows:

  • for ;

  • ;

  • if .

A cycle deal will be written .

In other word, in a cycle deal (we omit and when they are not necessary to understand the context), each agent gives a subset of at most items from her share to the next agent in the sequence and receives in return a subset from the previous agent. -cycle deals will be denoted by -cycle deals. -cycle deals will be called swap-deals. Among these cycle deals, some are more interesting: those where each agent improves her utility by trading objects. More formally, a deal will be called weakly improving if with at least one of these inequalities being strict, and strictly improving if all these inequalities are strict.

Intuitively, if it is possible to improve an allocation by applying an improving cycle deal, then it means that this allocation is inefficient. Reallocating the items according to the deal will make everyone better-off. It is thus natural to derive a concept of efficiency from this notion of cycle-deal.

Definition 6.

An allocation is said to be --Cycle Optimal (resp. --Cycle Optimal) if it does not admit any strictly (resp. weakly) improving -cycle deal for any .

We begin with easy observations. First, -cycle optimality implies -cycle optimality, and these two notions become equivalent when the preferences are strict on shares. Moreover, restricting the size of the cycles and the size of the bundles exchange yield less possible deals and hence lead to weaker optimality notions. Note that for and --cycle-optimality and --cycle-optimality are incomparable. These observations show that cycle-deal optimality notions form a (non-linear) hierarchy of efficiency concepts of diverse strengths. The natural question is whether they can be related to sequenceability and Pareto-optimality. Obviously, Pareto-optimality implies both -cycle-optimality and -cycle-optimality. An easy adaptation of the proof of Proposition 3 leads to the following stronger result:

Proposition 0.

An allocation is sequenceable if and only if it is --cycle optimal (with ).

Proof.

Let be a non-sequenceable allocation. Then by Proposition 1, there is at least one frustrating sub-allocation in . Using the same line of arguments as in the proof of Proposition 3 we can build a --cycle. Hence is not -cycle optimal. Conversely, suppose that is a --cycle. Then obviously this cycle yields an allocation that Pareto-dominates . ∎

The scale of efficiency introduced in Section 4 can then be refined with an intermediate hierarchy of -cycle optimality notions between sequenceable and non-sequenceable allocations. As for -cycle optimality, it forms a parallel hierarchy between Pareto-optimal and non-sequenceable allocations.666Note that -CO and sequenceability are incomparable notions. There exist allocations which are -swap optimal but not sequenceable and the other way around.

A corollary of Propositions 2 and 4 is that checking whether an allocation is --cycle optimal can be made in polynomial time (by checking whether it is sequenceable).

More generally, we can observe that checking whether an allocation is -cycle optimal can be done by iterating over all -uples of agents, and for each one iterating over all possible transfers involving less than objects.777This is sufficient to also run through all the cycles involving strictly less than agents: such a cycle can be simulated just by appending at the end of the cycle some agents whose role is just to pass the objects they receive to the next agent. In total, there are -uples of agents (which is upper-bounded by ). For each -uple, there are at most possible transfers, which is again upper-bounded by . Hence, in total, checking whether an allocation is -cycle optimal can be done in time . This is polynomial in and if both and are bounded (as for swap deals).

6 Restricted Domains

We now study the impact of several preference restrictions on the hierarchy of efficiency notions introduced in Section 5.

Strict preferences on objects

When the preferences are strict on objects, then obviously every sequence generates exactly one allocation. The following proposition is stronger and shows that the reverse is also true:

Proposition 0.

Preferences are strict on objects iff is a mapping from to .

Proof.

If preferences are strict on objects, then each agent has only one possible choice at her turn in the sequence of sincere choices and hence every sequence generates one and only one allocation.

Conversely, if preferences are not strict on objects, at least one agent (suppose w.l.o.g. agent 1) gives the same weight to two different objects. Suppose that there is at least objects ranked above. Then obviously, the following sequence generates two allocations, depending on agent 1’s choice at step . ∎

Same order preferences

We say that the agents have same order preferences if there is a permutation such that for each agent and each pair of objects and , if then .

Proposition 0.

All the allocations of an instance with same order preferences are sequenceable (and actually cycle-deal optimal). Conversely, if all the allocations of an instance are sequenceable, then this instance has same order preferences.

Proof.

Let be an instance with same order preferences, and let be an arbitrary allocation. In every sub-allocation of at least one agent obtains a top object (because the preference order is the same among agents) and hence cannot be frustrating. By Proposition 1, is sequenceable.

Conversely, let us assume for contradiction that is an instance not having same order preferences. Then there are two distinct objects and and two distinct agents and such that and , one of the two inequalities being strict (assume w.l.o.g. the first one). The sub-allocation such that and is frustrating. By Proposition 1, every allocation containing this frustrating sub-allocation (hence such that and ) is non-sequenceable. ∎

Let us now characterize the instances for which is a one-to-one correspondence.

Proposition 0.

For a given instance, the following two statements are equivalent.

  • Preferences are strict on objects and in the same order.

  • The relation is a one-to-one correspondence.

The proof is a consequence of Propositions 5 and 6.

Single-peaked preferences

An interesting domain restriction are single-peaked preferences (Black, 1948; Elkind et al., 2016), which, beyond voting, is also relevant in resource allocation settings (Bade, 2017; Damamme et al., 2015). Formally, in this context, single-peakness can be defined as follows.

There exists a linear order over the set of objects . Let be the preferred object of . An agent has single-peaked preferences wrt. if, for any two objects such that either or (i.e. lying on the same “side” of the agent’s peak), it is the case that prefers over .

Interestingly, when preferences are single-peaked, the hierarchy of -cycle optimality collapses at the second level:

Proposition 0.

If all the preferences are single-peaked (and additive), then an allocation is --cycle optimal iff it is swap-optimal.

Proof.

(Revisiting Damamme et al. (2015)) First, note that --cycle optimality trivially implies swap-optimality. Let us now show the conserve.

Let us consider for the sake of contradiction an allocation that is swap-optimal and such that there exists a --cycle , with . Without loss of generality, let us suppose that . We show by induction on , the length of , that such a cycle can not exist.

Base case: A 1-cycle of length is a swap-deal but as is swap-optimal, no improving swap-deal exists in hence the contradiction.

Induction step: Let us assume that for each such that , no --cycle exists in and let us show that no cycle of length exists.

To exhibit a contradiction we will need to use the following necessary (Ballester and Haeringer, 2011): to be single-peaked, a profile needs to be worst-restricted, i.e. for any triple of resources there always exists a resource such that there exists an agent with (Sen, 1966).

Because is a --cycle, for all agent involved in we have and . As no --cycle exists, with , for all agents involved in and for all resources in , and , we have . Moreover for all resource in , and , we have . If the preferences do not respect these conditions, a --cycleexists with .

Because the profile is worst-restricted, for all the triple of resources in , at most two resources of can be ranked last among by the agents. Let us call one of these resources ranked last by agent and held by agent . Thanks to the previous paragraph, we know that and so, because her preferences are single-peaked, puts in last position among . The same holds for agent who ranks in last position among (because ). Therefore when we focus only on the three resources , each of them is ranked last among them by one agent which violates the condition of worst-restriction. The contradiction is set, no --cycle exists in . ∎

Together with Proposition 4, Proposition 8 gives another interpretation of sequenceability in this domain:

Corollary 2.

If all the preferences are single-peaked (and additive), then an allocation is sequenceable if and only if it is swap-optimal.

Proposition 1 by Damamme et al. (2015) is much stronger than our Corollary 2, because it shows that swap-optimality is actually equivalent to Pareto-efficiency when each agent receives a single resource. Unfortunately, in our context where each agent can receive several items, this is no longer the case, as the following example shows:

Example 5.

Consider this instance, single-peaked with respect to :

The circled allocation is swap-optimal, but Pareto-dominated by the allocation marked with dags.

7 Envy-Freeness and CEEI

The use of sequence of sincere choices can also be motivated by the search for a fair allocation protocol. We will focus on two fairness properties, envy-freeness and competitive equilibrium from equal income and analyze their link with sequenceability. Envy-freeness (Tinbergen, 1953; Foley, 1967; Varian, 1974)

is probably one of the most prominent fairness properties:

Definition 7.

Let be an add-MARA instance and be an allocation. verifies the envy-freeness property (or is simply envy-free), when , (no agent strictly prefers the share of any other agent).

The notion of competitive equilibrium is an old and well-known concept in economics (Walras, 1874; Fisher, 1892). If equal incomes are imposed among the stakeholders, this concept becomes the competitive equilibrium from equal incomes (Moulin, 2003)

, yielding a very strong fairness concept which has been recently explored in artificial intelligence

(Othman et al., 2010; Budish, 2011; Bouveret and Lemaître, 2016a).

Definition 8.

Let be an add-MARA instance, an allocation, and a vector of prices. A pair is said to form a competitive equilibrium from equal incomes (CEEI) if

In other words, is one of the maximal shares that can buy with a budget of 1, given that the price of each object is .

We will say that allocation satisfies the CEEI test (is a CEEI allocation for short) if there exists a vector such that forms a CEEI.

As Bouveret and Lemaître (2016a) and Brânzei (2015) have shown, every CEEI allocation is envy-free in the model we use. In this section, we investigate the question of whether an envy-free or CEEI allocation is necessarily sequenceable. For envy-freeness, the answer is negative.

Proposition 0.

There exists non-sequenceable envy-free allocations, even if the agents’ preferences are strict on shares.

Proof.

A counterexample with strict preferences on shares is given in Example 4 above, for which we can check that the circled allocation is envy-free and non-sequenceable. ∎

However, for CEEI, the answer is positive:

Proposition 0.

Every CEEI allocation is sequenceable.

It was already known that every CEEI allocation is Pareto-optimal if the preferences are strict on shares (Bouveret and Lemaître, 2016a). From Proposition 3, if the preferences are strict on shares, then every CEEI allocation is sequenceable. But Proposition 10 is more general: no assumption is made on the strict order on shares (nor on objects).

Note that a CEEI allocation can be ordinally necessary Pareto-dominated, as the following example shows.

The circled allocation is CEEI (with prices 0.5, 1, 1, 0.5) but is ordinally necessary (hence also additively) dominated by the allocation marked with .

Proof.

We will show that no allocation can be at the same time non-sequenceable and CEEI. Let be a non-sequenceable allocation. We can use the same terms and notations than in the proof of Proposition 3, especially concerning the dominance cycle.

Let be the set of agents concerned by the cycle. contains the following shares:

whereas the allocation that dominates it, contains the following shares:

the other shares being unchanged from to .

Suppose that is CEEI. This allocation must satisfy two kinds of constraints. First, must satisfy the price constraint. If we write , we have, , (1).

Next, must be optimal: every share having a higher utility for an agent than her share in costs strictly more than 1. Provided that (because substitutes more preferred objects to less preferred objects in ), this constraint can be written as , (2).

By summing equations (1) and (2), provided that all shares are disjoint, we obtain

and

Yet, (because the allocation is obtained from by simply swapping objects between agents in ). The two previous equations are contradictory. ∎

8 Experiments

We have exhibited in Section 4 a “hierarchy of allocation efficiency” made of several steps: Pareto-optimal (PO), sequenceable (Seq), {cycle-deal-optimal}, non-sequenceable (–). A natural question is to know, for a given instance, which proportion of allocations are located at each level of the scale. We give a first answer by experimentally studying the distribution of allocations between the different levels. For cycle-deal optimality, we focus on the simplest type of deals, namely, -swap-deals. We thus have a linear scale of efficiency concepts, from the strongest to the weakest: PO Seq Swap –. We also analyze the relation between efficiency and various notions of fairness by linking this latter scale with the 6-level scale of fairness introduced by Bouveret and Lemaître (2016a): CEEI Envy-Freeness (EF) min-max share (mFS) proportionality (PFS) max-min share (MFS) –. We generate 50 add-MARA instances involving 3 agents-8 objects, using two different models. For both models, a set of weights are uniformly drawn in the interval and the instances are then normalized. For the second model, these weights are reordered afterwards to make the preferences single-peaked. For each instance, we generate all 6561 allocations, and identify for each of them the highest level of fairness and efficiency satisfied. The average number of allocations with min-max interval is plotted as a box for each level on a logarithmic scale in Figure 1. The figure also shows for each fairness criterion the proportion of allocations that satisfy each efficiency criterion, on a linear scale.

Uniform generation

-

MMS

PFS

mMS

EF

CEEI

NS

Swap

Seq

PO

NS (proportion)

Swap (proportion)

Seq (proportion)

PO (proportion)

Single-Peaked uniform generation

-

MMS

PFS

mMS

EF

CEEI

NS

Swap

Seq

PO

NS (proportion)

Swap (proportion)

Seq (proportion)

PO (proportion)
Figure 1: Distribution of the number of allocations by pair of (efficiency, fairness) criteria.

Note that some fairness and efficiency tests require to solve NP-hard or coNP-hard problems (MMS, mMS, and PO tests). These tests are delegated to an external ILP solver. This is especially interesting for the CEEI test which is known to be NP-hard (Brânzei, 2015), and for which, to the best of our knowledge, no practical method had been described before. The implementation is available as a fully documented and tested Free Python library.888Available at: https://gricad-gitlab.univ-grenoble-alpes.fr/bouveres/fairdiv.

We note several interesting facts. First, a majority of allocations do not have any efficiency nor fairness property (first black bar on the left). Second, the distribution of allocations on the efficiency scale seems to be related to the fairness criteria: a higher proportion of swap-optimal or sequenceable allocations are found among envy-free allocations than among allocations that do not satisfy any fairness property, and for CEEI allocations, there are even more Pareto-optimal allocations than just sequenceable ones. Lastly, the absence of vertical bar for swap-optimality in the experiments concerning single-peaked preferences confirms the results of Corollary 2: in this context, no allocation can be swap-optimal but not Sequenceable; hence, all the allocations that are swap-optimal are contained in the bars concerning sequenceable or Pareto-optimal allocations. Similarly, the absence of bars for swap-optimality and – (non-sequenceable) in both graphs confirms the result of Proposition 10.

9 Conclusion

In this paper, we have shown that picking sequences and cycle-deals can be reinterpreted to form a rich hierarchy of efficiency concepts. Many interesting questions remain open, such as the complexity of computing cycle-deals, the precise relation between Pareto-efficiency and -cycle-optimality or the link between efficiency concepts and social welfare. One could also think of further extending the efficiency hierarchy by studying restrictions on possible sequences (e.g. alternating) or extending the types of deals to non-cyclic ones.

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