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. envyfreeness) and efficiency (e.g. Paretooptimality) criteria, or optimizes a wellchosen 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 (cakecutting) (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 Paretooptimality, showing, among others, that picking sequences always result in a Paretooptimal allocation, but also that every Paretooptimal 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, Paretooptimality 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 suballocation 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 suballocation of when for each agent . Any suballocation 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 (addMARA 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 nonfrustrating 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 addMARA 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 nondeterministic^{2}^{2}2The algorithm contains an instruction choose splitting the control flow into several branches, building all the allocations generated by . Algorithm 1 on input and .
Definition 4.
An allocation is said to be sequenceable if there exists a sequence that generates , and nonsequenceable 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:^{3}^{3}3In 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 nonsequenceable.
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 subclasses 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 nonsequenceable. 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 nonsequenceable. It is even possible to find a nonsequenceable 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 suballocation remains. This suballocation is obviously nonsequenceable because it is frustrating. Hence is not sequenceable either.
This property of containing a frustrating suballocation exactly characterizes the set of nonsequenceable allocations:
Proposition 0.
Let be an instance and be an allocation of this instance. The two following statements are equivalent:

is sequenceable.

No suballocation of is frustrating (in every suballocation, 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 suballocation . 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 nonstrict preferences on objects, and (ii) it can conclude with nonsequenceability.
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 suballocation of . By Proposition 1, is thus nonsequenceable. 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 .
∎
4 Paretooptimality
An allocation is Paretooptimal 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 Paretooptimal?
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 Paretooptimality. However, they have a different notion of Paretooptimality, 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 Paretooptimality. Brams and King’s notion is possible Paretooptimality. 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 Paretooptimality 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 Paretooptimality no longer holds.^{4}^{4}4Actually, since it is known (de Keijzer et al., 2009; Aziz et al., 2016a) that testing Paretooptimality with additive preferences in coNPcomplete, 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 Paretooptimal 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 Paretooptimal 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 suballocation 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 suballocation dominating . This suballocation 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.^{5}^{5}5This 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 nonsequenceable allocation is dominated. Let be a nonsequenceable allocation. From Proposition 1, in a nonsequenceable allocation, there is at least one frustratring suballocation. Let be such a suballocation (that can be itself). We will, from , build another suballocation 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 suballocation 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 Paretooptimal (equivalently, in every Paretooptimal allocation, at least one agent receives a top object).
Proposition 3 implies that there exists, for a given instance, three classes of allocations: (1) nonsequenceable (therefore non Paretooptimal) allocations, (2) sequenceable but non Paretooptimal allocations, and (3) Paretooptimal (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 dealsoptimality
Paretooptimality 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 subclass 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 envyfreeness. Trying to link efficiency concepts with various notions of deals is thus a natural idea.
Definition 5.
Let be an addMARA 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 swapdeals. 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 betteroff. It is thus natural to derive a concept of efficiency from this notion of cycledeal.
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 cycleoptimality and cycleoptimality are incomparable. These observations show that cycledeal optimality notions form a (nonlinear) hierarchy of efficiency concepts of diverse strengths. The natural question is whether they can be related to sequenceability and Paretooptimality. Obviously, Paretooptimality implies both cycleoptimality and cycleoptimality. 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 nonsequenceable allocation. Then by Proposition 1, there is at least one frustrating suballocation 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 Paretodominates . ∎
The scale of efficiency introduced in Section 4 can then be refined with an intermediate hierarchy of cycle optimality notions between sequenceable and nonsequenceable allocations. As for cycle optimality, it forms a parallel hierarchy between Paretooptimal and nonsequenceable allocations.^{6}^{6}6Note 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.^{7}^{7}7This 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 upperbounded by ). For each uple, there are at most possible transfers, which is again upperbounded 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 cycledeal 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 suballocation 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 suballocation such that and is frustrating. By Proposition 1, every allocation containing this frustrating suballocation (hence such that and ) is nonsequenceable. ∎
Let us now characterize the instances for which is a onetoone 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 onetoone correspondence.
Singlepeaked preferences
An interesting domain restriction are singlepeaked 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, singlepeakness can be defined as follows.
There exists a linear order over the set of objects . Let be the preferred object of . An agent has singlepeaked 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 singlepeaked, the hierarchy of cycle optimality collapses at the second level:
Proposition 0.
If all the preferences are singlepeaked (and additive), then an allocation is cycle optimal iff it is swapoptimal.
Proof.
(Revisiting Damamme et al. (2015)) First, note that cycle optimality trivially implies swapoptimality. Let us now show the conserve.
Let us consider for the sake of contradiction an allocation that is swapoptimal 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 1cycle of length is a swapdeal but as is swapoptimal,
no improving swapdeal 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 singlepeaked, a profile needs to be worstrestricted, 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 worstrestricted, 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 singlepeaked, 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 worstrestriction. 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 singlepeaked (and additive), then an allocation is sequenceable if and only if it is swapoptimal.
Proposition 1 by Damamme et al. (2015) is much stronger than our Corollary 2, because it shows that swapoptimality is actually equivalent to Paretoefficiency 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, singlepeaked with respect to :
The circled allocation is swapoptimal, but Paretodominated by the allocation marked with dags.
7 EnvyFreeness 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, envyfreeness and competitive equilibrium from equal income and analyze their link with sequenceability. Envyfreeness (Tinbergen, 1953; Foley, 1967; Varian, 1974)
is probably one of the most prominent fairness properties:
Definition 7.
Let be an addMARA instance and be an allocation. verifies the envyfreeness property (or is simply envyfree), when , (no agent strictly prefers the share of any other agent).
The notion of competitive equilibrium is an old and wellknown 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 addMARA 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 envyfree in the model we use. In this section, we investigate the question of whether an envyfree or CEEI allocation is necessarily sequenceable. For envyfreeness, the answer is negative.
Proposition 0.
There exists nonsequenceable envyfree 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 envyfree and nonsequenceable. ∎
However, for CEEI, the answer is positive:
Proposition 0.
Every CEEI allocation is sequenceable.
It was already known that every CEEI allocation is Paretooptimal 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 Paretodominated, 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 nonsequenceable and CEEI. Let be a nonsequenceable 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: Paretooptimal (PO), sequenceable (Seq), {cycledealoptimal}, nonsequenceable (–). 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 cycledeal optimality, we focus on the simplest type of deals, namely, swapdeals. 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 6level scale of fairness introduced by Bouveret and Lemaître (2016a): CEEI EnvyFreeness (EF) minmax share (mFS) proportionality (PFS) maxmin share (MFS) –. We generate 50 addMARA instances involving 3 agents8 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 singlepeaked. 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 minmax 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.
Note that some fairness and efficiency tests require to solve NPhard or coNPhard 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 NPhard (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.^{8}^{8}8Available at: https://gricadgitlab.univgrenoblealpes.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 swapoptimal or sequenceable allocations are found among envyfree allocations than among allocations that do not satisfy any fairness property, and for CEEI allocations, there are even more Paretooptimal allocations than just sequenceable ones. Lastly, the absence of vertical bar for swapoptimality in the experiments concerning singlepeaked preferences confirms the results of Corollary 2: in this context, no allocation can be swapoptimal but not Sequenceable; hence, all the allocations that are swapoptimal are contained in the bars concerning sequenceable or Paretooptimal allocations. Similarly, the absence of bars for swapoptimality and – (nonsequenceable) in both graphs confirms the result of Proposition 10.
9 Conclusion
In this paper, we have shown that picking sequences and cycledeals can be reinterpreted to form a rich hierarchy of efficiency concepts. Many interesting questions remain open, such as the complexity of computing cycledeals, the precise relation between Paretoefficiency and cycleoptimality 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 noncyclic ones.
References
 Aziz et al. (2016a) Aziz, H., Biró, P., Lang, J., Lesca, J., Monnot, J., 2016a. Optimal reallocation under additive and ordinal preferences. In: Proceedings of the 15th International Conference on Autonomous Agents and Multiagent Systems (AAMAS’16). International Foundation for Autonomous Agents and Multiagent Systems, Richland, SC, pp. 402–410.
 Aziz et al. (2015a) Aziz, H., Gaspers, S., Mackenzie, S., Walsh, T., 2015a. Fair assignment of indivisible objects under ordinal preferences. Artificial Intelligence 227, 71–92.
 Aziz et al. (2016b) Aziz, H., Kalinowski, T., Walsh, T., Xia, L., 2016b. Welfare of sequential allocation mechanisms for indivisible goods. In: Kaminka, G. A., Fox, M., Bouquet, P., Hüllermeier, E., Dignum, V., Dignum, F., van Harmelen, F. (Eds.), 22nd European Conference on Artificial Intelligence (ECAI 2016). IOS Press, pp. 787–794.
 Aziz et al. (2015b) Aziz, H., Walsh, T., Xia, L., 2015b. Possible and necessary allocations via sequential mechanisms. In: Proceedings of the 24th International Joint Conference on Artificial Intelligence (IJCAI’15). AAAI Press, pp. 468–474.
 Bade (2017) Bade, S., 2017. Matching with singlepeaked preferences, working Paper.
 Ballester and Haeringer (2011) Ballester, M. A., Haeringer, G., 2011. A characterization of the singlepeaked domain. Social Choice and Welfare 36 (2), 305–322.

Bansal and Sviridenko (2006)
Bansal, N., Sviridenko, M., 2006. The santa claus problem. In: Proceedings of the thirtyeighth annual ACM symposium on Theory of computing. STOC ’06. ACM, New York, NY, USA, pp. 31–40.

Bianchessi et al. (2007)
Bianchessi, N., Cordeau, J.F., Desrosiers, J., Laporte, G., Raymond, V., Mar. 2007. A heuristic for the multisatellite, multiorbit and multiuser management of earth observation satellites. European Journal of Operational Research 177 (2), 750–762.
 Black (1948) Black, D., 1948. On the rationale of group decisionmaking. The journal of political economy, 23–34.
 Bouveret and Lang (2011) Bouveret, S., Lang, J., Jul. 2011. A general elicitationfree protocol for allocating indivisible goods. In: Walsh, T. (Ed.), Proceedings of the 22nd International Joint Conference on Artificial Intelligence (IJCAI11). IJCAI/AAAI, Barcelona, Spain, pp. 73–78.
 Bouveret and Lemaître (2016a) Bouveret, S., Lemaître, M., 2016a. Characterizing conflicts in fair division of indivisible goods using a scale of criteria. Autonomous Agents and MultiAgent Systems 30 (2), 259–290.
 Bouveret and Lemaître (2016b) Bouveret, S., Lemaître, M., 2016b. Efficiency and sequenceability in fair division of indivisible goods with additive preferences. In: Proceedings of the Sixth International Workshop on Computational Social Choice (COMSOC’16). Toulouse, France.
 Brams et al. (2012) Brams, S. J., Kilgour, M. D., Klamler, C., 2012. The undercut procedure: an algorithm for the envyfree division of indivisible items. Social Choice and Welfare 39 (23), 615–631.
 Brams and King (2005) Brams, S. J., King, D., 2005. Efficient fair division—help the worst off or avoid envy? Rationality and Society 17 (4), 387–421.
 Brams and Taylor (1996) Brams, S. J., Taylor, A. D., 1996. Fair Division — From Cakecutting to Dispute Resolution. Cambridge University Press.
 Brams and Taylor (2000) Brams, S. J., Taylor, A. D., 2000. The Winwin Solution. Guaranteeing Fair Shares to Everybody. W. W. Norton & Company.
 Brânzei (2015) Brânzei, S., February 2015. Computational fair division. Ph.D. thesis, Department of Computer Science, Aarhus Universitet, Denmark.
 Budish (2011) Budish, E., dec 2011. The combinatorial assignment problem: Approximate competitive equilibrium from equal incomes. Journal of Political Economy 119 (6), 1061–1103.
 Chevaleyre et al. (2005) Chevaleyre, Y., Endriss, U., Maudet, N., Aug. 2005. On maximal classes of utility functions for efficient onetoone negociation. In: Kaelbling, L. P., Saffiotti, A. (Eds.), Proceedings of the 19th International Joint Conference on Artificial Intelligence (IJCAI05). Professional Book Center, Edinburgh, Scotland, pp. 941–946.
 Damamme et al. (2015) Damamme, A., Beynier, A., Chevaleyre, Y., Maudet, N., May 2015. The Power of Swap Deals in Distributed Resource Allocation. In: The 14th International Conference on Autonomous Agents and Multiagent Systems (AAMAS 2015). Istanbul, Turkey, pp. 625–633.
 de Keijzer et al. (2009) de Keijzer, B., Bouveret, S., Klos, T., Zhang, Y., October 2009. On the complexity of efficiency and envyfreeness in fair division of indivisible goods with additive preferences. In: Proceedings of the 1st International Conference on Algorithmic Decision Theory (ADT’09). Lecture Notes in Artificial Intelligence. Springer Verlag, Venice, Italy, pp. 98–110.
 Elkind et al. (2016) Elkind, E., Lackner, M., Peters, D., 2016. Preference restrictions in computational social choice: Recent progress. In: Proceedings of the TwentyFifth International Joint Conference on Artificial Intelligence, IJCAI 2016, New York, NY, USA, 915 July 2016. pp. 4062–4065.
 Fisher (1892) Fisher, I., 1892. Mathematical Investigations in the Theory of Value and Prices, and Appreciation and Interest. Augustus M. Kelley, Publishers.
 Foley (1967) Foley, D. K., 1967. Resource allocation and the public sector. Yale Economic Essays 7 (1), 45–98.
 Goldsmith and Sloan (2007) Goldsmith, J., Sloan, R. H., 2007. The AI conference paper assignment problem. In: Proc. AAAI Workshop on Preference Handling for Artificial Intelligence, Vancouver. pp. 53–57.
 Kalinowski et al. (2013a) Kalinowski, T., Narodytska, N., Walsh, T., Aug. 2013a. A social welfare optimal sequential allocation procedure. In: Rossi, F. (Ed.), Proceedings of the 23rd International Joint Conference on Artificial Intelligence (IJCAI13). IJCAI/AAAI, Beijing, China, pp. 227–233.
 Kalinowski et al. (2013b) Kalinowski, T., Narodytska, N., Walsh, T., Xia, L., Jul. 2013b. Strategic behavior when allocating indivisible goods sequentially. In: Proceedings of the 26th AAAI Conference on Artificial Intelligence (AAAI12). AAAI Press, Bellevue, WA, pp. 452–458.
 Kohler and Chandrasekaran (1971) Kohler, D. A., Chandrasekaran, R., 1971. A class of sequential games. Operations Research 19 (2), 270–277.
 Lemaître et al. (1999) Lemaître, M., Verfaillie, G., Bataille, N., Jul. 1999. Exploiting a common property resource under a fairness constraint: a case study. In: Proceedings of the 16th International Joint Conference on Artificial Intelligence (IJCAI99). Stockholm, Sweden, pp. 206–211.
 Lipton et al. (2004) Lipton, R., Markakis, E., Mossel, E., Saberi, A., May 2004. On approximately fair allocations of divisible goods. In: Proceedings of the 5th ACM Conference on Electronic Commerce (EC04). ACM, New York, NY, pp. 125–131.
 Moulin (2003) Moulin, H., 2003. Fair Division and Collective Welfare. MIT Press.
 Othman et al. (2010) Othman, A., Sandholm, T., Budish, E., May 2010. Finding approximate competitive equilibria: efficient and fair course allocation. In: van der Hoek, W., Kaminka, G. A., Lespérance, Y., Luck, M., Sen, S. (Eds.), Proceedings of the 9th International Conference on Autonomous Agents and MultiAgent Systems (AAMAS10). IFAAMAS, Toronto, Canada, pp. 873–880.
 Sandholm (1998) Sandholm, T. W., 1998. Contract types for satisficing task allocation: I. theoretical results. In: Sen, S. (Ed.), Proceedings of the AAAI Spring Symposium: Satisficing Models. AAAI Press, Menlo Park, California, pp. 68–75.
 Sen (1966) Sen, A. K., 1966. A possibility theorem on majority decisions. Econometrica, 491–499.
 Tinbergen (1953) Tinbergen, J., 1953. Redeljke Inkomensverdeling. N. V. DeGulden Pers., Haarlem.
 Varian (1974) Varian, H. R., 1974. Equity, Envy and Efficiency. Journal of Economic Theory 9, 63–91.
 Walras (1874) Walras, L., 1874. Éléments d’économie politique pure ou Théorie de la richesse sociale, 1st Edition. L. Corbaz.