The Vigilant Eating Rule: A General Approach for Probabilistic Economic Design with Constraints
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The Vigilant Eating Rule:A General Approach for Probabilistic EconomicDesign with Constraints
Haris Aziz † Florian Brandl ‡ We consider the problem of probabilistic allocation of objects under ordinal prefer-ences. Our main contribution is an allocation mechanism, called the vigilant eatingrule (VER), that applies to nearly arbitrary feasibility constraints. It is constrainedordinally efficient, can be computed efficiently for a large class of constraints, and treatsagents equally if they have the same preferences and are subject to the same constraints.When the set of feasible allocations is convex, we also present a characterization of ourrule based on ordinal egalitarianism. Our results about VER do not just apply to al-location problems but to all collective choice problems in which agents have ordinalpreferences over discrete outcomes. As a case study, we assume objects have prioritiesfor agents and apply VER to sets of probabilistic allocations that are constrained bystability. VER coincides with the (extended) probabilistic serial rule when priorities areflat and the agent proposing deterministic deferred acceptance algorithm when prefer-ences and priorities are strict. While VER always returns a stable and constrainedefficient allocation, it fails to be strategyproof, unconstrained efficient, and envy-free.We show, however, that each of these three properties is incompatible with stabilityand constrained efficiency.
1. Introduction
The theory and application of allocation and matching mechanisms have proved to be one of themajor success stories of algorithmic economics. An ongoing challenge is designing mechanisms thatcan handle complex constraints arising in new applications. In this paper, we present a versatile † UNSW Sydney and Data61 CSIRO, Australia, [email protected] ‡ Princeton University, USA, [email protected] raft – October 27, 2020 and robust allocation rule that achieves fair and efficient outcomes for a plethora of economic designproblems including matching market design. It simultaneously generalizes celebrated rules in theliterature including the probabilistic serial rule and the deferred acceptance algorithm.We study the problem of allocating a set of indivisible objects to a group of agents based onthe agents’ preferences over objects. The objects could be seats at schools, dormitory rooms, jobplacements, or kidney transplants for example. In these applications, it is important, perhapseven mandatory, that allocations are fair in the sense that no agent (justifiably) envies some otheragent. Since objects are indivisible, envy-free allocations may however not exist. One possibleremedy for this dilemma is to consider randomizations over deterministic allocations, which canrestore fairness ex-ante. This approach is deeply rooted in the literature on resource allocation andhas gained popularity in recent years. The seminal works of Hylland and Zeckhauser (1979) andBogomolnaia and Moulin (2001) provide allocation mechanisms for the case when each agent de-mands exactly one object and preferences are given by linear utility functions or ordinal preferencesover objects, respectively. We shall be concerned with ordinal preferences under two generaliza-tions: first, agents may receive more than one object, and, second, there may be constraints onwhich random allocations are feasible.Bogomolnaia and Moulin’s probabilistic serial rule can be pictured as follows. Time runs contin-uously from 0 to 1. At time 0, each agent starts off by eating her most-preferred object. An objectbecomes unavailable once the cumulative time agents spent eating it equals 1. Whenever an objectbecomes unavailable, the agents who have been eating it switch to their next most-preferred objectamong those which are still available. The probability with which an agent receives an object in thefinal random allocation equals the time she spent eating that object. The Birkhoff-von NeumannTheorem ensures that these probabilities can be attained by randomizing over deterministic alloca-tions. The probabilistic serial rule enjoys several appealing properties: it is weakly strategyproof,can be computed efficiently, and always yields an allocation that is efficient and envy-free (whenthe agents’ preferences over probabilistic allocations are based on stochastic dominance).A tacit assumption in the formulation of the probabilistic serial rule is that all allocations arefeasible. But several applications of object allocation require allocation rules to respect variousfeasibility constraints. For example, allocating students to courses may be subject to curricularconstraints that require students to take a minimal number of courses or courses in different sub-jects. Likewise, when allocating donor’s kidneys to patients, logistics and blood type compatibilityimpose constraints on which exchanges are feasible. In these examples, one has constraints ondeterministic allocations and stipulates that a probabilistic allocation is feasible if it can be writtenas a convex combination of deterministic allocations that meet the constraints. We will allow fora more general class of constraints where the primitive is a set of feasible probabilistic allocations.This of course includes the previous example. More generally, it can capture ex-ante constraints In this case, the set of feasible probabilistic allocations is the face of the simplex of probabilistic allocations spanned raft – October 27, 2020 on the probabilistic allocation and possibly different constraints on the ex-post allocation. Forexample, a system designer may impose ex-ante stability constraints in addition to ex-post con-straints. In that case, a probabilistic allocation is feasible if it satisfies the ex-ante constraintsand can be decomposed into deterministic allocations that meet the ex-post constraints (see, forexample, Ashlagi et al., 2019b; Akbarpour and Nikzad, 2020).Our main contribution is a generalization of the probabilistic serial rule, called the vigilant eatingrule (VER) , which can handle multi-object allocation under nearly arbitrary constraints. For-mally, it requires that the set of feasible probabilistic allocations is closed. Previous generalizationsof the probabilistic serial rule required that feasible probabilistic allocations are given by con-straints that form a bi-hierarchical structure (Budish et al., 2013), constitute lower and upper quo-tas (Ashlagi et al., 2019b), or exclude a fixed set of deterministic allocations (Aziz and Stursberg,2014). In each of these three domains, VER coincides with the proposed generalization of theprobabilistic serial rule. Despite being generally applicable, VER retains many of the propertiesof the probabilistic serial rule: it always yields an allocation that is efficient among feasible alloca-tions, gives the ex-ante same allotment to agents who have the same preferences and are subjectto the same constraints, and can be computed efficiently whenever the set of feasible allocations isa union of polytopes (described by polynomially many linear constraints). Clearly, no rule can beunconstrained efficient or envy-free without restrictions on the set of feasible allocations.A class of problems that falls under the umbrella of object allocation is two-sided allocation,where, in addition to preferences, agents have different priorities for objects. The literature ontwo-sided allocation is well-developed (see, for example, Roth and Sotomayor, 1990). Almost uni-versally, allocations are required to be stable: if an agent prefers an object to the one she hasbeen allocated, the former object is given to an agent who has a higher priority for it. We willview stable two-sided allocation as a special case of allocation under constraints, where only stableallocations are feasible. Unlike when constraints come from quotas, stability constraints depend onthe preferences of the agents and are thus harder to handle.In the classical formulation of two-sided allocation as marriage markets, preferences and prioritiesare strict and one considers only deterministic mechanisms. The most prominent representativeis the (agent proposing) deferred acceptance rule of Gale and Shapley (1962). It enjoys severalappealing properties such as strategyproofness and Pareto efficiency (when interpreting prioritiesas preferences). Yet, applications like school choice where priorities are typically coarse, that is,contain large indifference classes, motivate the search for rules that can deal with this more generalproblem domain. There are various reasons for considering probabilistic rules when priorities areweak. Perhaps the most evident one is that it allows for fairness ex-ante, which is not alwaysattainable when deterministically allocating objects. Take the example of two agents preferring by deterministic allocations satisfying the constraints. raft – October 27, 2020 object a over object b . If both agents have the same priority for a and for b , allocating a to eitheragent with a probability of is fair ex-ante, while no deterministic allocation is fair.We propose VER as a promising probabilistic mechanism for stable two-sided allocation whenpreferences and priorities are weak. Stability can be generalized to probabilistic allocations invarious ways. We consider four different notions of stability that have been considered in theliterature and all reduce to pairwise stability for deterministic allocations. For each stability notion,we can apply VER to the set of stable allocations and thereby obtain a mechanism that alwaysyields stable and constrained efficient allocations. Each of these mechanisms provides a naturaltransition between the deferred acceptance algorithm and the probabilistic serial rule. It coincideswith the former if priorities are strict and with the latter if priorities are completely flat (andpreferences are strict in both cases). Contributions
We explore the space of desirable rules and algorithms for probabilistic allocation.In contrast to previous work probabilistic rules for stable allocation, we allow the agents to haveweak preferences. Our main contribution is the
Vigilant Eating Rule (VER) , which applies toany class of distributional constraints, including ones given by stability notions. For example,it applies to non -bi-hierarchical constraints, which are not handled by the generalization of theprobabilistic serial rule due to Budish et al. (2013). It also applies to non-convex constraints arisingfrom integrality requirements or some stability notions. We show that the outcome of VER isefficient within the set of feasible allocations with respect to stochastic dominance (that is, SD -efficient). If the feasible set is convex, the outcome is ordinally egalitarianism in the sense defined byBogomolnaia (2017). Our results about VER are not restricted to allocation problems. They applyto collective decision making in general whenever agents have ordinal preferences over deterministicoutcomes.For object allocation under priorities, we study VER in more detail and argue that it leadsto compelling stable allocation rules. VER returns a stable allocation that is constrained SD -efficient within the set of stable allocations. For several properties that VER does not satisfy, weprove that they are incompatible with stability, constrained SD -efficiency, or their conjunction. Inparticular, we prove that even for the weakest stability notion we consider (claimwise stability),the set of stable probabilistic allocations can be disjoint from the set of SD -efficient allocations andthe set of weakly SD -envy-free allocations. For each of our stability notions, no rule can be weakly SD -strategyproof and always yield a stable and constrained (within the set of stable allocations) SD -efficient allocation.More broadly, we present a generalization of the probabilistic serial rule that can handle a moregeneral class of constraints than previous generalizations. In restricted domains, VER coincideswith well-established rules (see Table 1). We also show that VER can also be useful when norandomization is allowed. 4 raft – October 27, 2020 Constraints Preferences Priorities Resulting Rule– strict – probabilistic serial (PS) (Bogomolnaia and Moulin, 2001)– weak – extended PS (Katta and Sethuraman, 2006) SD -individual rationality strict – controlled consuming (Athanassoglou and Sethuraman, 2011)upper quotas strict – generalized PS (Budish et al., 2013)lower & upper quotas strict – generalized PS (Ashlagi et al., 2019b)feasible ex-post allocations weak – egalitarian simultaneous reservation (Aziz and Stursberg, 2014)feasible ex-post allocations strict – generalized constrained PS (Balbuzanov, 2019)ex-post stability strict strict agent proposing DA (Gale and Shapley, 1962)claimwise stability strict weak constrained PS (Afacan, 2018)– dichotomous dichotomous egalitarian rule (Bogomolnaia and Moulin, 2004)integral allocations; unit demand strict – serial dictatorship– strict over bundles – PS for bundles (Chatterji and Liu, 2020) Table 1.: Overview of generalizations of the probabilistic serial rule to particular constraints andpreference and priority structures. VER coincides with each of these rules in the corre-sponding domain.As we endeavor to apply the theory of fair allocation and market design to facilitate applications,it is critical to develop robust mechanisms that can flexibly handle domain-specific constraints. Ourapproach is especially well-suited for settings with ordinal preferences. We hope that the framework,methodology, and rule we present can be useful for new application domains.
2. Related Work
We explore two-sided allocation while relaxing the assumption that allocations have to be integral ordeterministic. In our paper, an allocation specifies, for each pair of agent and object, the probabilitythat the agent receives that object. This probabilistic approach is important on several accounts.We refer the reader to Aziz (2019) for an in-depth discussion. In short, randomization allows oneto achieve minimal fairness requirements such as equal treatment of equals that are unobtainablethrough deterministic allocations. Secondly, probabilistic allocations can also be interpreted astime-sharing or fractional allocations.
Probabilistic matching market design
Although most of the work on two-sided allocationhas focussed on deterministic allocations, some authors have pointed out connections betweenintegral stable allocations and their linear relaxations (see, for example, Roth et al. (1993); Budish et al. (2013) assume that the constraints can be written as the union of two hierarchies of constraints. Acollection of constraints forms a hierarchy if for any two constraints, the set of agent-object pairs they apply toare either disjoint or one is contained in the other. The SD -individual rationality constraints require that each agent’s allocation stochastically dominates her initialendowment. The endowments are part of the specification of an allocation instance in addition to the agents’preferences. raft – October 27, 2020 Teo and Sethuraman (1998)). Erdil and Ergin (2008, 2017) and Kesten (2010) undertook a studyof school choice when schools have weak priorities. They highlighted that tie-breaking can lead to aloss of efficiency. However, they focus on achieving constrained efficiency of deterministic or integralallocations. Similarly, Ashlagi et al. (2019a) consider the impact of various types of tie-breaking onthe efficiency of the allocations. Kamada and Kojima (2015, 2017a,b) and Kojima et al. (2018) con-sider deterministic two-sided matching under general distributional constraints. These constraintscapture several real-life scenarios, such as restrictions on the number of doctors in a particular re-gion. As noted before, deterministic outcomes cannot guarantee ex-ante equal treatment of equalsor capture time-sharing. Deterministic allocation also has several other aspects that are differentfrom its probabilistic counterpart. For example, the problem of computing deterministic allocationsthat satisfy a set of distributional constraints is NP-hard for many classes of constraints (see, forexample, Biró et al., 2010). Considering a probabilistic allocation and then suitably rounding it canbe an indirect, but computationally more tractable approach to arrive at a deterministic allocation(see, for example, Akbarpour and Nikzad (2020)).Kesten and Unver (2015) initiated a serious study of stability notions and mechanisms for prob-abilistic allocation under weak priorities. They focussed on a strong version of ex-ante stabilityand proposed two mechanisms. First, they present the fractional deferred acceptance algorithm,which returns a strongly ex-ante stable allocation. They then modify it to derive the fractionaldeferred acceptance and trading algorithm that returns a strongly ex-ante stable allocation whichis also Pareto efficient with respect to stochastic dominance among stable allocations. Han (2017)presented a similar algorithm that returns an ex-ante stable allocation.Afacan (2018) considered a more general model in which objects have probabilities for prioritizingone agent over another. He proposed a weak stability notion called claimwise stability and showedthat no weakly strategyproof mechanism always returns an allocation that is claimwise stableand efficient among claimwise stable allocations. Both strategyproofness and efficiency assumethat preferences are based on stochastic dominance. Afacan presented his constrained probabilisticserial algorithm, which returns an allocation that is constrained efficient among claimwise stableallocations and can be computed in polynomial time.All of the algorithms that return constrained efficient and stable probabilistic alloca-tions discussed above assume that agents have strict preferences (Afacan, 2018; Han, 2017;Kesten and Unver, 2015). We will not make this assumption and our main results are valid forweak preferences. Furthermore, our approach is general enough to allow for any type of feasibilityconstraints.When considering probabilistic stable allocations, there are various notions of stability thatcoincide with the classical pairwise stability notion for deterministic allocations. Aziz and Klaus(2019) investigate the taxonomy of stability notions for probabilistic allocations and map out thelogical relationships between them. 6 raft – October 27, 2020
Caragiannis et al. (2019) considered fractional stable allocations under cardinal utilities. A frac-tional allocation is stable if no pair gets higher utility in being integrally matched to each other thanunder the fractional allocation. They explore computational problems of finding stable fractionalallocations with high welfare. He et al. (2018) also considered cardinal utilities and presented apseudo-market mechanism that simultaneously generalizes the deferred acceptance algorithm andthe mechanism of Hylland and Zeckhauser (1979). It returns an allocation that is ex-ante stable.Echenique et al. (2019) have extended the pseudo-market rule to handle constraints on determin-istic allocations.
Extensions of probabilistic serial without priorities
There are several papers on probabilisticallocation when the objects have no priorities. Our model allows for weak priorities and, thus, hasno or flat priorities as a special case. One of the seminal works on random allocation under ordinalpreferences without priorities is by Bogomolnaia and Moulin (2001), who compared the randompriority rule with the probabilistic serial rule. They showed that the probabilistic serial rule is SD -efficient and SD -envy-free, whereas the random priority rule satisfies neither property.The attractive properties of the probabilistic serial rule have led to a whole line of work onextensions to more general settings. Those include weak preferences (Katta and Sethuraman, 2006),multi-unit demands (Kojima, 2009), endowments (Athanassoglou and Sethuraman, 2011; Yilmaz,2010; Yu and Zhang, 2020), bundled allocations (Chatterji and Liu, 2020) and the probabilisticvoting setting (Aziz and Stursberg, 2014). VER coincides with each of these rules on their domain.It captures and generalizes the key insights that underly the probabilistic serial rule. Moreover, ourrule provides a conceptually easier formulation of the previously presented algorithms for specificdomains (Katta and Sethuraman, 2006; Yilmaz, 2010; Athanassoglou and Sethuraman, 2011).Budish et al. (2013) consider probabilistic allocation without priorities. They presented a gener-alization of the probabilistic serial rule which allows for distributional constraints that are given byupper quotas and form a bi-hierarchy. Fujishige et al. (2018) extend the probabilistic serial rule tosubmodular constraints. In their algorithms, agents continue eating a most preferred object untilsome upper quota on a set of agent-object pairs is met. This approach works if there are upperquotas on agent-object groups that form a bi-hierarchical constraint structure; it does howevernot extend to lower quotas on agent-object groups or constraint structures that do not form abi-hierarchy. Ashlagi et al. (2019b) consider probabilistic allocation with lower and upper boundson the types of students for each school without priorities and give a suitable generalization of theprobabilistic serial rule. The class of constraints to which VER applies includes those consideredby Budish et al. (2013), Fujishige et al. (2018), and Ashlagi et al. (2019b).Our model allows for constraints both on the set of feasible ex-post allocations as well as con-straints on the ex-ante allocation. Note that if we are only concerned about constraints on the setof ex-post allocations, we can explicitly list the feasible ex-post allocations as the set of alternatives,7 raft – October 27, 2020 and run the egalitarian simultaneous reservation rule of Aziz and Stursberg (2014), which is con-strained SD -efficient and has been referred to as the appropriate extension of the probabilistic serialrule to the social choice domain (Bogomolnaia, 2017). However, in allocations problems, there areseveral advantages of VER. Firstly, the constraints themselves may be ex-ante. For example, threeof the four stability notions we consider are defined by constraints on the ex-ante allocation matrixand not by enumerating the set of feasible ex-post allocations. Another example is time-sharing where the focus is on the fractional allocations and not on lotteries over deterministic allocations.Secondly, even if certain ex-ante constraints can be captured by enumerating ex-post allocations,the approach may be computationally prohibitive as one may need to enumerate an exponentialnumber of allocations.
3. The Model
Let N be a set of n agents and O be a set of m objects. Each agent i ∈ N has a weak preferencerelation (complete, reflexive, and transitive) % i over objects. The symmetric and asymmetric partsof a relation % are denoted by ∼ and ≻ as usual. A preference relation is strict if it is a linear order;otherwise it is weak. By E i = O/ ∼ i we denote the collection of equivalence classes of objects inducedby % i . Abusing notation, we also write % i for the relation on E i induced by % i . A preference profile % N = ( % i ) i ∈ N consists of preferences for each agent.A probabilistic allocation of objects to agents is a matrix p ∈ R N × O + such that all columns sum toat most 1. We will write p ( i ) for the allotment of agent i , that is, the row of p corresponding to agent i . If O ′ ⊂ O , p ( i, O ′ ) = P o ∈ O ′ p ( i, o ) is the probability with which i receives an object in O ′ . Wesay that p is complete if all columns sum to 1 meaning that all objects are fully allocated; otherwiseit is partial. If p ∈ { , } N × O , it is a deterministic allocation. Every probabilistic allocation canbe written as a lottery over deterministic allocations. Whenever left unspecified, allocations areassumed to be complete and not necessarily deterministic. An allocation p satisfies equal treatmentof equals for a profile % N if % i = % j implies p ( i, E ) = p ( j, E ) for all E ∈ E i = E j .We introduce two extensions of preferences over objects to preferences over probabilistic alloca-tions. Let % be a preference relation over O and x, y ∈ R O + . We say that x (first-order) stochastically More recently, Balbuzanov (2019) proposed an adaptation of the probabilistic serial rule for strict preferences, calledthe generalized constrained probabilistic serial rule , where the set of feasible ex-post allocations are enumerated.Algorithmically, the results of Balbuzanov (2019) can be achieved by enumerating the feasible ex-post allocationsand running the egalitarian simultaneous reservation algorithm of Aziz and Stursberg (2014), which seamlesslyhandles weak preferences. In fact, one can use the Birkhoff-von Neumann Theorem to show the following: if p is a probabilistic allocationwith p ( O ) = α i for all i , then p can be written as a lottery over deterministic allocations in which agent i receiveseither ⌊ α i ⌋ or ⌈ α i ⌉ objects. raft – October 27, 2020 dominates y with respect to % , written x % SD y , if X o ′ % o ( x ( o ′ ) − y ( o ′ )) ≥ for all o ∈ O (stochastic dominance)If at least one inequality is strict, we write x ≻ SD y . Secondly, x lexicographically dominates y with respect to % , written x ≻ LD y , ifthere is E ∈ E with x ( E ) > y ( E ) and x ( E ′ ) = y ( E ′ ) for all E ′ ∈ E with E ′ ≻ E (lexicographic dominance)If x ( E ) = y ( E ) for all E ∈ E , we have x ∼ LD y and x ∼ SD y and simply write x ∼ y . Notice thatlexicographic dominance refines stochastic dominance.We assume that each agent’s preferences over allocations only depend on her own allotment.That is, for allocations p and q , p % SD i q if and only if p ( i ) % SD i q ( i ) . Moreover, p stochasticallydominates q if p ( i ) % SD i q ( i ) for all i ∈ N and at least one agent has a strict preference; p is SD -efficient if it is not stochastically dominated by any allocation. When X ⊂ R N × O + is a set ofallocations, we say that p is constrained SD -efficient for X if it is not stochastically dominatedby any allocation in X . The analogous definitions apply when preferences are extended based onlexicographic dominance. Since preferences based on lexicographic dominance refine preferencesbased on stochastic dominance, LD -efficiency is more restrictive than SD -efficiency. If p ( i ) ∼ i q ( i ) for all i , we say that all agents are indifferent between p and q .An allocation mechanism f maps a preference profile % N and a set of feasible (complete) allo-cations X to an allocation f ( % N , X ) ∈ X . For any property for allocations, we say that f has thisproperty if f ( % N , X ) has the property for all preference profiles % N and sets X . For example, f is SD -efficient if f ( % N , X ) is SD -efficient for all % N and X .
4. Vigilant Eating Rules
The probabilistic serial rule, sometimes also called the eating algorithm, was introduced byBogomolnaia and Moulin (2001) and roughly works as follows. Time runs continuously from 0onward. At time 0, each agent starts off by eating her most-preferred object. An object becomesunavailable if the cumulative time agents spent eating it equals 1. Whenever an object becomesunavailable, the agents who have been eating it switch to their next most-preferred object amongthose which are still available. The algorithm terminates when all objects are unavailable (in whichcase the probabilities for each object sum to 1). The probability with which an agent receives anobject in the final allocation equals the time she spent eating it.The description of the eating rule presumes that the agents always have a unique favorite objectand thus strict preferences. If an agent is indifferent between multiple objects, one can think of9 raft – October 27, 2020 eating from that equivalence class as increasing the probability of getting some object from thatclass. Another assumption is that all allocations are feasible. The challenge in defining eating ruleswhen not all allocations are feasible is to not let agents eat objects if this would make it impossibleto continue eating in a way that results in a feasible allocation. In other words, we need to ensurethat the partial allocation at each time can be extended to a feasible allocation. Even if the set offeasible allocations consists of all complete allocations satisfying upper bounds on the probabilities,this cannot always be achieved by simply letting agents eat objects as long as this does not violateany of the bounds. To see this, consider the following
Example 1.
Assume two agents have the preferences depicted below. ≻ : a b ≻ : a b Suppose we want to get an allocation p in which each agent gets one unit of objects: p (1 , a )+ p (1 , b ) =1 and p (2 , a )+ p (2 , b ) = 1 . Suppose additionally that there is a constraint that p (1 , a )+ p (2 , b ) ≤ / .If agents start eating their most preferred object a , they each get / of a . Up to this point, noconstraint is violated. However, there is no completion of the partial allocation that satisfies theconstraints. The example shows that the Generalized Probabilistic Serial Rule of Budish et al.(2013) cannot handle the above constraints. We thus need a more “vigilant” approach. The idea is to let an agent eat an object only if theresulting partial allocation can be extended to a complete allocation. In general, it could be however,that agents i and j both can individually eat object o , but not simultaneously. Thus, which objectsan agent can eat may depend on which objects other agents eat (and these dependencies may becyclic). Our approach to cope with the above mentioned issues is formalized in Algorithm 1. It isformulated for weak preferences, so instead of eating objects, agents increase their probability forobjects in the equivalence classes induced by their preferences. We keep track of how much of everyequivalence class each agent is guaranteed, while always ensuring that there is a feasible allocationof objects which meets all guarantees.Start with setting all guarantees to 0. At the beginning of each round, we decide for every agentfor which equivalence class she gets to increase her guarantee. An equivalence class E is availableto agent 1 if there exists a feasible allocation that meets all of the previous guarantees and assigns astrictly higher probability for E to i than her current guarantee for E . Agent 1 gets to increase herprobability for her most preferred equivalence class E among those available to her. In general, E is available to agent i if there exists a feasible allocation that meets all guarantees establishedin the previous rounds, assigns a higher probability for E j to j for each j < i , and assigns a higher Budish et al. (2013) defined the Generalized Probabilistic Serial Rule for bi-hierarchical constraint structures. Theconstraints in Example 1 do not form a bi-hierarchy. raft – October 27, 2020 probability for E to i that the current guarantee. Agent i gets to eat her favorite equivalence class E i among those available to her. We let the agents increase their probabilities of the classes E i we have just determined for as long as all the resulting guarantees can be met by some feasibleallocation. More precisely, if π k stores the guarantees at the beginning of round k , find the largest δ so that there exists a feasible allocation which meets all guarantees imposed by π k and assignsa probability of at least π k ( i, E i ) + δ for E i to i for all i . If δ ∗ is this maximal value, we let π k +1 be array that increases i ’s guarantee for E i by δ ∗ compared to π k and is otherwise identical to π k .After at most m · n rounds, no guarantee can be further increased and the process terminates. Byconstruction, X contains an allocation that meets all of the guarantees and all agents are indifferentbetween all such allocations. Hence, it is irrelevant which of those we choose as the final allocation.We call this mechanism the vigilant eating rule (VER). Whenever the set of feasible allocations X is non-empty and closed, VER results in a feasible allocation that is constrained SD -efficientwith respect to X (see Theorems 1 and 2). For feasible sets that are not closed, constrained SD -efficient allocations may not exist, which shows that our requirement is minimal. If X is convex,the outcome of VER is ordinally egalitarian (Theorem 4). Moreover, if X is a union of polytopes,it can be computed in polynomial time in number of inequalities used to describe X (Theorem 5).If X meets certain symmetry conditions, VER satisfies equal treatment of equals (Theorem 6). Example 2 (Illustration of VER) . Let us revisit Example 1, where we had two agents with thefollowing preferences. ≻ : a b ≻ : a b As before, we want to get an allocation p satisfying the row constraints p (1 , a ) + p (1 , b ) = 1 and p (2 , a ) + p (2 , b ) = 1 and the diagonal constraint p (1 , a ) + p (2 , b ) ≤ / . We illustrate how VERworks in this example. The order of consumption of objects over time is illustrated in Figure 1.• Round 1: In the first round, we identify the most preferred object for agent that she canconsume. That object is a . Next, we find the most preferred object for agent that shecan consume given that agent 1 receives a with positive probability. That object is also a .The algorithm computes the maximum amount that the agents can consume from a whileensuring that there exists a feasible allocation extending the current partial allocation. Thisamount is / since if agent were to consume more of object a , the row constraints wouldforce agent 2 to consume more than / of object b . A violation of the diagonal constraintwould thus be inevitable. Hence, agent 1 has move on to consuming object b . At the end ofthe round, we have the following guarantees: π (1 , a ) ≥ / and π (2 , a ) ≥ / .• Round 2: In the second round, agent consumes / of object b whereas agent consumes / more of object a . At this point, object a becomes unavailable and agent has to move on to11 raft – October 27, 2020 Algorithm 1.
The Vigilant Eating Rule
Input:
A preference profile % N and a non-empty and closed set X of allocations Output:
An allocation p ∈ X k ←− (the round of the algorithm) N ′ ←− N (the set of active agents) π ( i, E ) ←− for all i ∈ N and E ∈ E i while N ′ = ∅ do for i ∈ N ′ in increasing order do F i ←− { E ∈ E i : there is p ∈ X so that p ( j, E ′ ) ≥ π k ( j, E ′ ) for all j ∈ N and E ′ ∈E j , p ( i, E ) > π k ( i, E ) , and p ( j, E j ) > π k ( j, E j ) for all j ∈ N ′ with j < i } if F i = ∅ then E i ←− max % i F i else N ′ ←− N ′ \ { i } end if end for Compute δ ∗ as follows δ ∗ = max δ,p δ s.t. p ( i, E i ) ≥ π k ( i, E i ) + δ ∀ i ∈ N ′ (agent i consumes E i ) p ( i, E ) ≥ π k ( i, E ) ∀ i ∈ N and E ∈ E i ( p extends π k ) p ∈ X ( p is feasible) π k +1 ( i, E ) ←− π k ( i, E ) for all i ∈ N and E ∈ E i π k +1 ( i, E i ) ←− π k ( i, E i ) + δ ∗ for all i ∈ N ′ k ←− k + 1 end while return p (as computed in Line 13) 12 raft – October 27, 2020 Time Agent Agent aa ba bb Figure 1.: Illustration of VER for the problem in Example 1.consuming object b . At the end of the round, we have the following guarantees: π (1 , a ) ≥ / , π (1 , b ) ≥ / , and π (2 , b ) ≥ / .• Round 3: Now both agents consume / more of b until b becomes unavailable. At the endof round 3, we have the following guarantees: π (1 , a ) ≥ / , π (1 , b ) ≥ / , π (2 , a ) ≥ / and π (2 , b ) ≥ / . Since both objects are unavailable, the algorithm terminates with thefollowing allocation satisfying all guarantees and the original constraints. p = a b ! / / / / We first verify that each step of VER is well-defined and that it always terminates. It is thenclear from the definition that it results in a feasible allocation. The proof proceeds by showingthat agents consume equivalence classes in decreasing order of their preferences and then observingthat in each round some agent moves to a less preferred class or becomes inactive since she cannotincrease any of her guarantees further. (All proofs are in the appendix.)
Theorem 1.
For every closed set X of allocations, Algorithm 1 terminates after at most m · n rounds with an allocation p ∈ X . Intuitively, it is clear that VER yields an allocation that is LD -efficient (and, thus, SD -efficient)among feasible allocations. To see this, assume that the allocation p of VER were dominated bysome allocation q in X . We can view q as the result of some eating trajectory different from theone which yields p in Algorithm 1. Assuming agents eat objects at unit speed in decreasing orderof their preference pins down a unique trajectory for q . Consider the earliest point in time atwhich both trajectories diverge and let i ∗ be the lexicographically smallest agent who starts eatingdifferent objects (or, more generally, equivalence classes) in both trajectories. Since q dominates p , i ∗ strictly prefers her object in the q -trajectory to that in the p -trajectory. But this contradictsthe choice of o i ∗ at that time in Line 8 of Algorithm 1. Theorem 2.
When VER is applied to any closed set of allocations X , it returns an allocation thatis constrained LD -efficient among the allocations in X . raft – October 27, 2020 One can generalize VER by equipping each agent with an eating speed function which deter-mines how fast she can increase her probability for an equivalence class as a function of time.Bogomolnaia and Moulin (2001, Theorem 1) show that for unit demand object allocation withoutconstraints, an allocation is SD -efficient if and only if it is the outcome of their eating mechanismfor some profile of eating speed functions. The “if part” remains true in our setting, that is, for ev-ery profile of eating speed functions, VER yields a constrained LD -efficient and, thus, SD -efficientallocation. But not every SD -efficient allocation is the outcome of VER for some profile of eatingspeed functions. Consider the following preferences. ≻ : a b c ≻ : b c a ≻ : c a b Let X be the convex hull of the allocations p and q . p = a b c / /
22 1 / / / / q = a b c / / /
32 1 / / /
33 1 / / / Every allocation in X is constrained SD -efficient. However, for any profile of eating speed functions,VER will yield p . This is because at each point during the eating process, every agent’s favoriteobject o will be available to her until she has consumed of o .The reason for the equivalence derived by Bogomolnaia and Moulin is that SD -efficiency and LD -efficiency coincide in their setting as shown by Cho and Dogan (2016). This is no longer truein our case as observed above. We can however show that when the set of feasible allocations isconvex, every constrained LD -efficient allocation is unanimously indifferent to the outcome of VERfor some profile of eating speed functions. To this end, it suffices to consider eating speed functionsbased on indicator functions. Those result in one-at-a-time-VER, a more flexible version of VERwhere at any point only one agent is allowed to increase her probability for some equivalence class.Clearly, the outcome of VER can be achieved by some instance of one-at-a-time-VER. Theorem 3.
Let X be a closed and convex set of allocations and p ∈ X . Then p is constrained LD -efficient with respect to X if and only if all agents are indifferent between p and the outcomeof some instance of one-at-a-time-VER on the set X . An interesting instance of one-at-a-time-VER is the vigilant priority rule . Fix some order of theagents. The first agent in that order increases her probability guarantees for each of her equivalenceclasses in order of her preferences by as much as possible. Once no further increase is possible for thefirst agent, it is the second agent’s turn to increase all of her guarantees in order of her preferences14 raft – October 27, 2020 by as much as possible. We proceed in the same way for the remaining agents. Appendix C gives apseudo-code formulation of this algorithm. The vigilant priority rule is clearly unfair to agents whocome later in the order. It is (strongly) SD -strategyproof whenever the set of feasible allocationsis independent of the agents’ preferences. If we choose the order of agents uniformly at random,then the rule is no longer LD -efficient. The reason is that it coincides with the classic randompriority rule in the assignment domain, which is well-known to violate SD -efficiency (and thus LD -efficiency).Bogomolnaia (2017) proposed a solution notion called ordinal egalitarianism that characterizesthe probabilistic serial rule (which can be argued to be the fairest rule for the allocation of objectsto agents under ordinal preferences). Ordinal egalitarianism is defined as follows. We evaluate anygiven allocation by the list of numbers t ji , the total probability agent i gets of objects from herfirst j equivalence classes, for all i and j . An allocation is ordinally egalitarian if it is a leximinmaximizer of the signature vector t ↑ ( p ) = ( t ji ( p )) i,j over all feasible allocations. For any signaturevector t ↑ ( p ) , we will denote by t ↑ ( p )( j ) as the j -th entry of the non-decreasing ordered entries. Wewill denote by t i ( p ) as the signature vector t ↑ ( p ) in which only the entries corresponding to agent i are considered. If an allocation is ordinally egalitarian, then it is LD -efficient. We show that underconvex constraints, VER returns an allocation that is ordinally egalitarian. Theorem 4.
The VER allocation for a closed and convex set of allocations X is ordinally egali-tarian among the allocations in X . The steps in Algorithm 1 that are potentially computationally taxing are determining the bestobjects an agent can eat at a given time in Line 8 and the amount of time the current eating schemecan be continued in Line 13. Both require maximizing a linear objective over the intersection of X with a polytope. Whenever X is itself a polytope (described by polynomially many linearinequalities), this can be done efficiently by solving a linear program. More generally, if X is aunion of polytopes, we can solve the problem for each polytope individually and then pick the bestsolution (which is the best object o i agent i can eat or the maximal amount of time δ ∗ the currenteating scheme can continue). Theorem 5. If X is a finite union of polytopes, the outcome of VER can be computed in polynomialtime in the number of inequalities used to describe X . One of the desiderata when assigning objects to agents is to treat all agents equally. Roughly, thismeans that if two agents have the same preferences and are subject to the same constraints, thenthey should receive the same allocation. Thus, there are two ways in which asymmetry betweenagents in an allocation can arise: the mechanism that determines it is inherently asymmetric orthe set of feasible allocations is biased toward certain agents. To guarantee that the convex combination of the outcomes of the vigilant priority rule for different orders is feasible,we need to assume that the set of feasible allocations is convex. raft – October 27, 2020 VER introduces some asymmetry of the first kind by letting agents choose their most-preferredavailable object in lexicographic order. In general, this order is relevant since whether or not anobject is available to an agent may depend on the choices of previous agents. It is clear thatthis asymmetry cannot be eliminated entirely. For example, two agents may have the same most-preferred object and only the allocations assigning that object to one of them with probability1 may be feasible. (This is just the special case of deterministic mechanisms.) However, if X isconvex, whether an object is available does not depend on the choices of previous agents and theorder in which agents choose objects becomes irrelevant. A second source of asymmetry is the setof feasible allocations X itself. Clearly, if X is asymmetric, we may not be able to give the sameallocation to agents with the same preferences.These considerations motivate the conditions under which VER treats agents equally. Let i and i ′ be two agents. We say that a set of allocations X is symmetric for { i, j } if p ∈ X implies p ( ij ) ∈ X ,where p ( ij ) is identical to p except that the allocations of i and j are swapped. Moreover, X isconvex for { i, i ′ } if for all λ ∈ [0 , , p ∈ X and p ( ii ′ ) ∈ X implies λp + (1 − λ ) p ( ii ′ ) ∈ X . Theorem 6.
Let X be a closed set of allocations and % N be a preference profile. If X is symmetricand convex for all i, i ′ ∈ N with % i = % ′ i , then the outcome of VER satisfies equal treatment ofequals. We conclude this section by giving concrete classes of constraints that VER can handle.
Distributional constraints on the ex-ante allocations
We have pointed out that VER applies toa wide variety of constraints on the ex-ante allocations. A natural and general family of constraintsarises from imposing lower and upper bounds on the cumulative probability a subset of agentscan obtain from a subset of objects. These constraints appear in applications like school choice ifa certain number of seats is reserved for students who live close to the school. Another class ofconstraints obtains from diversity requirements, which put bounds on the fraction of objects fromwithin some subset that is assigned to a certain subset of agents (instead of bounding the absolutenumber).
Distributional constraints on the ex-post allocations
As mentioned in the introduction, ex-postconstraints, that is, constraints on deterministic allocations, can be expressed as ex-ante constraintson the set of probabilistic allocations by taking only those allocations to be feasible which can bedecomposed into deterministic allocations satisfying the ex-post constraints. These can captureupper and lower bounds on the number of objects a subset of agents can receive from some subsetof objects ex-post, for example. One instance of this are bi-hierarchical constraints on deterministicallocations of the type studied by Budish et al. (2013). Similarly, one can also formulate diversityconstraints ex-post. 16 raft – October 27, 2020
Ex-post constraints can be combined with possibly different ex-ante constraints on probabilisticallocations. For example, one may stipulate that the ex-ante constraints hold exactly, while theex-post constraints only need to be satisfied approximately. Akbarpour and Nikzad (2020) showthat if the ex-ante and ex-post constraints are the same, then any allocation satisfying the ex-ante constraints can be decomposed into deterministic allocations that approximately achieve theconstraints. Hence, in that case, there is little need to impose additional constraints which guaranteethat a decomposition into deterministic allocations satisfying the constraints exists.
Allocation with endowments
VER can be applied to problems in which agents haveendowments to obtain individually rational and possibly more efficient re-allocations.Athanassoglou and Sethuraman (2011) presented the controlled consuming algorithm, which can beviewed as an extension of the probabilistic serial rule. It ensures that each agent gets an allocationshe weakly prefers to her endowment with respect to stochastic dominance. The requirement of SD -individual rationality can be embedded in our framework by using a linear number of inequal-ities for each agent. Hence, we can use linear programming to capture the controlled consumingalgorithm. In principle, we can even combine individual rationality constraints and priorities. Deterministic and integral allocations
Our approach has been framed in the context of random-ization or time-sharing. However, it also applies when no randomization is involved or allowed byrendering all non-deterministic allocations infeasible. Then the δ increment in the probability ofan agent for an object (cf. Algorithm 1, Line 13) always has to be one. This ensures that an agenteither gets an object with probability one or zero. Not allowing probabilistic allocations underfeasibility constraints may however render many problems NP-hard and thus computationally in-tractable. We note that for the house allocation problem with deterministic allocations (Svensson,1999), VER coincides with the serial dictatorship rule. Welfare requirements
If the agents additionally have cardinal preferences over the objects, thenminimum social welfare guarantees can also be treated as feasibility constraints.
5. Stable Probabilistic Allocations
One way in which feasibility constraints can occur is if agents have different priorities for objectsand we require allocations to be stable. How to generalize stability to probabilistic allocations is notat all unambiguous, however. Various notions have been proposed in the literature, of which we willdiscuss four. Then we examine VER as a mechanism for stable probabilistic object allocation underpriorities. Many of its properties follow directly from the general statements we have derived above.On the other hand, VER lacks properties such as strategyproofness, unconstrained efficiency, and17 raft – October 27, 2020 envy-freeness. We show that this is unavoidable if one insists on stability.We augment our formal model by assuming that every object o comes with a (complete, reflexive,and transitive) relation % o over agents, which specifies the agents’ priorities for o . A profile ( % N , % O ) = (( % i ) i ∈ N , ( % o ) o ∈ O ) consists of preferences for each agent and priorities for each object.The priorities give rise to a relaxed notion of equal treatment of equals, which only requires thattwo agents receive the same allotment if they have the same priority for all objects. That is, anallocation p satisfies limited equal treatment of equals for a profile ( % N , % O ) if p ( i ) = p ( j ) for all i, j ∈ N with % i = % j , cap( i ) = cap( j ) , and i ∼ o j for all o ∈ O . We consider four notions of stability, all of which coincide with the standard (pairwise) stabilityfor deterministic allocations if the agents’ preferences over allocations are responsive. Throughoutthis section, we assume that every agent i has a capacity cap( i ) ∈ N of objects she can receive. Westipulate that for every allocation p , p ( i, O ) ≤ cap( i ) . Recall that a deterministic allocation p is stable if for all i ∈ N and o ∈ O , either P o ′ % i o p ( i, o ′ ) = cap( i ) or p ( o ) % o i .Our most restrictive notion is ex-ante stability, which has been introduced by Kesten and Unver(2015). It prescribes that an agent i can only receive a positive probability for an object o ifevery agent j with a higher priority for o can meet the capacity cap( j ) with objects j prefers to o .Formally, p is ex-ante stable if for all i, j ∈ N and o ∈ O , j ≻ o i and p ( i, o ) > implies X o ′ % j o p ( j, o ′ ) = cap( j ) (ex-ante stability)Thus, ex-ante stability requires that j has no justified envy toward i even before knowing therealization of the random allocation p .Analogously, one can ask that no agent should have justified envy ex-post, that is, after adeterministic allocation has been selected according to the random allocation. This leads us todefine p as ex-post stable if p is a convex combination of deterministic stable allocations (ex-post stability)The third stability notion, called fractional stability, requires that for all i ∈ N and o ∈ N , X o ′ % i o, o ′ = o p ( i, o ′ ) + cap( i ) X j % o i p ( j, o ) ≥ cap( i ) (fractional stability)These inequalities originate from the work of Roth et al. (1993), who observed that for determin-istic allocations, their conjunction is equivalent to stability. Baïou and Balinski (2000) showedthat when preferences and priorities are strict, fractional stability is equivalent to ex-post stability.18 raft – October 27, 2020 Aziz and Klaus (2019) give an example, attributed to Battal Doğan, which shows that this equiv-alence breaks down if one omits the strictness assumption for both preferences and priorities. OurExample 3 in the appendix shows that weak priorities alone to break the equivalence.A motivation for fractional stability under unit capacities, adopted from Aziz and Klaus (2019),is that if the inequality for a pair ( i, o ) fails, then i justifiably envies the set of agents with lowerpriority for o for jointly consuming more of o than i consumes of objects i weakly prefers to o .Another reason for considering fractional stability is as a proxy for ex-post stability in situationswhere it is computationally prohibitive to handle the latter. Since the set of fractionally stableallocations is described by m · n linear inequalities, it is typically much more well-behaved in thatrespect.Lastly, we consider claimwise stability, which has been introduced by Afacan (2018). We saythat agent i has a justified claim against j for object o if i has higher priority for o than j and j ’sprobability for o is larger than i ’s probability for objects i weakly prefers to o . An allocation p isclaimwise stable if no agent has a justified claim, that is, if for all i, j ∈ N and o ∈ O , i ≻ o j implies X o ′ % i o, o ′ = o p ( i, o ′ ) ≥ p ( j, o ) (claimwise stability)Aziz and Klaus (2019) showed that each of our four stability notions implies the ones below it inthe list, while none of the converse implications holds. In this section, we study the properties of VER when it is applied to sets of stable allocations. If S is one of our stability notions, then S -VER denotes the mechanism which, for a profile ( % N , % O ) ,runs VER for the preferences % N on the set X of S -stable allocations for ( % N , % O ) , so that S - VER ( % N , % O ) = VER ( % N , X ) . Most of the properties of VER on sets of stable allocationsfollow directly from the results we have proved in Section 4. In particular, for every stability notion S defined in Section 5.1, S -VER yields an allocation that is S -stable, constrained SD -efficient, andsatisfies limited equal treatment of equals. We summarize these results in the following corollary. Corollary 1.
For each of our four stability notions S , S -VER returns an S -stable allocation thatis SD -efficient among S -stable allocations and satisfies limited equal treatment of equals. For the extreme cases of coarseness of priorities and strict preferences, VER coincides with well-known mechanisms. If priorities are flat, that is, if all agents have the same priority for everyobject, every allocation is stable for each of our stability notions. In that case, VER reduces to theprobabilistic serial rule of Bogomolnaia and Moulin (2001), which corresponds to unconstrainedeating. The opposite extreme is that priorities are strict. Then VER returns the agent optimaldeterministic stable allocation, which is also the outcome of the agent-proposing deferred acceptance19 raft – October 27, 2020 algorithm. Intuitively, this checks out since VER is optimal for agents in the sense that it allowsthem to eat their most preferred object available to them.
Corollary 2.
Assume that preferences and priorities are strict. Then, for each of our four stabilitynotions S , S -VER returns the agent optimal deterministic stable allocation. In view of Corollary 1 and Corollary 2, we also recover the well-known result that the agentproposing deferred acceptance algorithm returns an allocation that is Pareto efficient within theset of stable allocations when the preferences and priorities are strict. Corollary 2 shows that S -VER results in the same mechanism for all four stability notions in that case even though the sets ofstable allocations are not the same (except for ex-post stability and fractional stability). For weakpriorities, all four instantiations of VER are in fact distinct. We provide examples in Appendix B.An alternative interpretation of our formal model is that the entities on both sides of the marketare agents who have preferences over the other side (instead of one side being objects with prioritiesover agents). Then, instead of considering SD -efficiency for one side, we could ask for allocationsthat are SD -efficient with respect to the preferences of both sides, called two-sided SD -efficiency.Formally, an allocation p is two-sided SD -efficient for a profile ( % N , % O ) if there is no allocation q such that q ( i ) % SD i p ( i ) for all i ∈ N and q ( o ) % o p ( o ) for all o ∈ O and at least one preferenceis strict. For ex-ante stability and fractional stability, we can show that VER always returns anallocation that is two-sided SD -efficient among all allocations, not only among stable allocations.For ex-post and claimwise stability, this is open. Proposition 1.
Assume that preferences are strict. For ex-ante stability and fractional stability, S -VER returns an allocation that is two-sided SD -efficient. Let us now consider the computational complexity of VER for sets of stable allocations. It is easyto see from the definitions that the set of fractionally stable allocations and the set of claimwisestable allocations are polytopes described by on the order of m · n and m · n linear inequalities,respectively. Thus, Theorem 5 implies that the corresponding vigilant eating rules can be computedin polynomial time. Corollary 3.
FS-VER and CWS-VER can be computed in polynomial time.
The exact complexity of computing VER on the set of ex-post stable and ex-ante stable allocationsis not settled. Since the set of ex-post stable allocations is convex, it would by Theorem 5 sufficeto describe it by a polynomial number of linear inequalities. For the case of strict preferences andpriorities, this has been done by Baïou and Balinski (2000) as discussed earlier. For weak priorities,it is an interesting open problem, as also pointed out by Kesten and Unver (2015). Ex-ante stabilitycan be captured by a set of constraints, each of which is a disjunction of two linear equalities: foreach i ∈ N and o ∈ O , either P o ′ % i o p ( i, o ) = 1 or P j ≺ o i p ( j, o ) = 0 . In general, solving problemsinvolving disjunctions of equalities is NP-hard, however.20 raft – October 27, 2020 For each of our stability notions, VER on the set of stable allocations violates unconstrainedefficiency, weak envy-freeness, and weak strategyproofness when preferences over probabilistic al-locations are based on stochastic dominance. Neither of those shortcomings is specific to VER,but the consequence of an inherent incompatibility of each of these properties with stability andconstrained efficiency. We address them in turn.
Efficiency
Roth (1982) showed that there may be no deterministic allocation that is both stableand Pareto efficient. Since any ex-post stable and SD -efficient allocation has to be a convexcombination of stable and Pareto efficient, it follows that ex-post stability is incompatible withunconstrained SD -efficiency. We prove that this conflict remains even when weakening stability toclaimwise stability. Proposition 2.
There may be no allocation that is claimwise stable and SD -efficient. Hence, for any stability notion that is stronger than claimwise stability, the set of stable alloca-tions can be disjoint from the set of SD -efficient allocations. Envy-Freeness
In the absence of priorities, envy-freeness requires that each agent prefers herallocation to that of any other agent. This definition is no longer compelling for non-trivial prioritiessince an agent may legitimately receive an allocation that some other agent would prefer to her ownbecause of a higher priority for some objects. One definition of fairness in this context is limitedequal treatment of equals, which we discussed above. Another one is that of justified envy. Itapplies only to pairs of agents i, j who have the same priority for all objects and requires that i does not prefer j ’s allotment to her own when comparing them via stochastic dominance. That is,an allocation p is weakly SD -envy-free if for all i, j ∈ N , i ∼ o j for all o ∈ O implies p ( j ) SD i p ( i ) (weak SD -envy-freeness)Even this weak notion of envy-freeness turns out to be incompatible with claimwise stability andhence with all stability notions stronger than that. This can be seen from the example in the proofof Proposition 2. Proposition 3.
There may be no allocation that is claimwise stable and weakly SD -envy-free. Strategyproofness
Comparing allocations via stochastic dominance results in incomplete prefer-ences. Thus, there are two notions of strategyproofness associated with it. The stronger, usuallycalled SD -strategyproofness, requires that the allotment obtained by truth-telling weakly stochas-tically dominates any allotment that can be obtained otherwise. Bogomolnaia and Moulin (2001)21 raft – October 27, 2020 proved that when priorities are flat, there exists no mechanism that is SD -strategyproof, SD -efficient, and satisfies equal treatment of equals. Under flat priorities, SD -efficiency is the sameas constrained efficiency for each of our stability notions since stability has no bite. We can thusnot hope for a mechanism that returns a stable allocation and satisfies SD -strategyproofness, con-strained efficiency, and limited equal treatment of equals, irrespective of which stability notion wechoose.The weak notion of SD -strategyproofness only prescribes that no agent can obtain a strictly SD -dominating allotment by misrepresenting her preferences. Formally, a mechanism f is weakly SD -strategyproof if for all agents i ∈ N and all profiles ( % N , % O ) , ( % ′ N , % O ) with % j = % ′ j for all j ∈ N \ { i } , f ( % ′ N , % O )( i ) SD i f ( % N , % O )( i ) . Now for flat priorities and strict preferences, there is a mechanism that satisfies weak SD -strategyproofness, SD -efficiency, and equal treatment of equals: the probabilistic serial rule. More-over, the generalized probabilistic serial rule for upper quotas forming a bi-hierarchy of Budish et al.(2013) is weakly strategyproof. But imposing stability again results in an impossibility. Afacan(2018) proved that for at least 4 agents, there exists no mechanism that is weakly SD -strategyproofand always returns a claimwise stable and constrained SD -efficient allocation. We extend his resultto the remaining three stability notions. That is, for any stability notion in our list, no mechanism isjointly weakly SD -strategyproof, stable, and constrained efficient (with respect to the set of stableallocations). Note that constrained efficiency becomes weaker as stability becomes stronger, sincethe set of stable allocations becomes smaller. Thus, none of these statements implies the other.Our proof does not rely on weak preferences and requires only 3 agents. Proposition 4.
No mechanism satisfies weak SD -strategyproofness, S-stability, and S-constrainedefficiency for S ∈ { ex-ante, ex-post, FS } even if preferences are strict. All the properties are needed for the conclusion of Proposition 4. VER satisfies S-stability andS-constrained efficiency. Ignoring the priorities and running the probabilistic serial rule gives amechanism that is weakly SD -strategyproof and efficient (and thus S-constrained efficient), but notS-stable. The deferred acceptance algorithm with lexicographic tie-breaking of priorities is SD -strategyproof and S-stable. For ex-post stability, fractional stability, and claimwise stability, onecan simultaneously achieve limited equal treatment of equals by breaking ties uniformly at random. We discuss other mechanisms presented in the literature. Since most of them have only been definedfor agents with strict preferences, we assume that preferences are strict for the comparison. We havealready discussed the probabilistic serial rule in the introduction. Except for the first mechanism22 raft – October 27, 2020 (random priority), all the other mechanism are extensions of the deferred acceptance algorithm,which is typically defined for strict preferences and strict priorities (Gale and Shapley, 1962; Roth,2008) and returns a stable allocation.
Random Priority (or random serial dictatorship) chooses an ordering of the agents uniformlyat random and then lets each agent pick her most preferred object among the ones remainingin that order. (Bogomolnaia and Moulin, 2001; Aziz et al., 2013a). The rule does not take intoaccount the priorities of the objects. For the basic assignment problem, random priority is knownto be strategyproof. It satisfies equal treatment of equals but is not SD -efficient or SD -envy-free (Bogomolnaia and Moulin, 2001). If the priority order is chosen deterministically, then SD -efficiency is regained, but equal treatment of equals is lost. Deferred acceptance with lexicographic tie-breaking
A simple adaptation of the deferred accep-tance algorithm to the case of weak priorities is to break the ties and then run deferred acceptance.If the tie-breaking is pre-determined, for example lexicographically over agents, and thus does notdepend on the agents’ preferences, the resulting mechanism is strategyproof. Moreover, it returnsa deterministic allocation that is stable and hence ex-ante stable. However, it is not SD -efficienteven among deterministic stable allocations. Like any other mechanism that returns deterministicallocations, it also violates limited equal treatment of equals. Deferred acceptance with random tie-breaking
Another natural approach is to break ties in thepriorities uniformly at random and then run deferred acceptance. Afacan (2018) referred to thisextension as probabilistic deferred acceptance . To compute the random allocation, we need to takethe mean of the outcomes over all possible tie-breakings. Under flat priorities, probabilistic deferredacceptance is equivalent to random priority. Since the latter is well-known to violate SD -efficiency,it follows that probabilistic deferred acceptance does not satisfy constrained SD -efficiency. Fractional deferred acceptance and trading
Kesten and Unver (2015) presented the fractionaldeferred acceptance and trading mechanism. It returns an allocation that is strongly ex-ante stableand SD -efficient constrained to the set of strongly ex-ante stable allocations. It is different fromthe probabilistic serial rule under flat priorities and thus also from VER. Constrained probabilistic serial
Afacan (2018) introduced the constrained probabilistic serial rule,which is an adaptation of the probabilistic serial rule that obtains a claimwise stable allocation.Because of the simplicity of the constraints imposed by claimwise stability, this algorithm does notneed to look ahead to check when an agent should stop eating an object. Suppose an agent i startseating object o . At that point, we put an upper bound on all agents j who have a lower priority23 raft – October 27, 2020 S -VER probabilistic random deferred acceptance deferred acceptance fractional deferredserial priority (lexicographic) (uniform) acceptance and trading S -stability + – – + + + S -constrained SD -efficiency + + – – – – SD -efficiency – + – – – – SD -envy freeness – + – – – –weak SD -envy freeness – + + – – –limited equal treatment of equals + + + + – + Table 2.: Summary of the properties satisfied by the allocation mechanisms discussed in Section 5.4. S may stand for each of the four stability notions we define in Section 5.1. In order toenable a comparison of all rules, we assume that preferences are strict.for o than i . If j had been eating o all the time while i was eating more preferred objects, we canstop j from eating more. If j had been eating o only part of the time, we put a limit on how much j can eat o . The upper limit is equal to the amount of time i was eating objects weakly morepreferred objects. The constrained probabilistic serial rule can be viewed as a careful version ofthe probabilistic serial rule that handles stability constraints dynamically. By contrast, VER forclaimwise stability makes look-ahead checks to see how much of an object an agent can eat beforea stability violation becomes unavoidable. One can show that in this case, the look-ahead checksare not necessary and both mechanisms coincide.Table 2 summarizes the properties satisfied by various mechanisms.
6. Extension Beyond Allocation Problems
The VER framework is not restricted to allocation problems. It applies just as well to other settingswhere fractional or probabilistic outcomes are feasible, such as social choice (Bogomolnaia et al.,2005; Brandt, 2017), coalition formation (Bogomolnaia and Jackson, 2002; Aziz et al., 2013b), net-works (Jackson and Wolinsky, 1996), and other models discussed by Sönmez (1999). In this section,we discuss how to extend our model to capture these applications.Instead of a set of objects, we now consider an abstract set of alternatives A and each agent has apreference relation over A . An outcome is an element of R A + . The agents’ preferences over outcomesare again based on stochastic dominance. By a problem we denote a pair ( % N , X ) , where % N is apreference profile and X ⊂ R A + is a non-empty set of feasible outcomes. As before, a mechanism f maps a problem ( % N , X ) to an outcome f ( % N , X ) ∈ X . The deferred acceptance algorithm with lexicographic tie-breaking satisfies ex-post stability and all weaker stabilitynotions. It violates ex-ante stability, however. The fractional deferred acceptance and trading algorithm violates S -constrained SD -efficiency when S is ex-poststability or any weaker stability notion. It satisfies SD -efficiency constrained to the set of ex-ante stable allocations,however. raft – October 27, 2020 A generalization of VER then works as follows: in each round, agents are addressed in lexi-cographic order as in Algorithm 1. When it is agent i ’s turn, we determine i ’s most preferredalternative a i whose probability can still be increased (while also increasing the probability foralternatives a j with j < i ). Having determined a i for every agent i , we find the maximal δ sothat the probability for all a i can be increased by at least δ . Note that the probability of everyalternative a i is increased by the same amount independently of how many agents nominate a i .This version of VER can address several classes of problems. Probabilistic voting
Our abstract model immediately captures probabilistic voting (see, for ex-ample, Brandt, 2019) where voters have preferences over the alternatives A and an outcome is aprobability distribution over the alternatives. That is, X = ∆( A ) ⊂ R A + . In this context, VERcoincides with the ESR rule of Aziz and Stursberg (2014). Participatory budgeting
The probabilistic voting setting can also be interpreted as determiningthe share of the budget allocated to each of the alternatives (Airiau et al., 2019). Since our modelallows for arbitrary constraints, it can capture natural constraints such as enforcing lower bounds(reflecting the minimum funding required) on alternatives that get at least some funding.
Probabilistic allocation with bundles
Let O be a set of objects and assume every agent i has apreference relation ˆ % i over subsets of O (see, for example, Chatterji and Liu, 2020). A deterministicallocation of objects to agents is an ordered N -partition of objects (which may include empty sets).Let A be the set of all such partitions. By a ( i ) we denote the set of objects agent i receives in theallocation a . The preference relation % i over A has a % i b if and only if a ( i ) ˆ % i b ( i ) for all a, b ∈ A .A random allocation is an element of the unit simplex ∆( A ) ⊂ R A + and X ⊂ ∆( A ) specifies a setof feasible probabilistic allocations. Two-sided probabilistic matching
Let N and N be disjoint sets of agents. Each agent i ∈ N has a preference relation ˆ % i over agents in N and likewise for agents in N . A deterministicmatching is a subset µ of N × N such that ( i, j ) , ( i, j ′ ) ∈ µ implies j = j ′ and ( i, j ) , ( i ′ , j ) ∈ µ implies i = i ′ . Let A be the set of all deterministic matchings. The preference relation % i of i ∈ N over A has µ % i µ ′ if and only if ( i, j ) ∈ µ and ( i, j ′ ) ∈ µ ′ with j ˆ % i j ′ or ( i, j ) ∈ µ ′ for no j ∈ N ; preferences for agents in N are defined analogously. A random matching is an elementof the unit simplex ∆( A ) and X specifies a set of feasible random matchings. We can apply VERto the problem with two-sided preferences and let both sides eat simultaneously. This approach ispromising because many standard mechanisms for two-sided matching are asymmetric in that theytreat the two sides differently. If we do not impose any stability constraints and both sides havedichotomous preferences, then VER is equivalent to Bogomolnaia and Moulin’s (2004) egalitarian25 raft – October 27, 2020 rule. We assumed that agents have complete preference orders over alternatives. VER and its prop-erties extend to preferences given by partial orders. The change that is required is the same thatKatta and Sethuraman (2006) suggested for adapting the extended probabilistic serial rule to par-tial orders. Instead of trying to increase the probability for most preferred alternatives, agentstry to increase the probability for those alternatives that are not strictly dominated by any otheralternative.
Acknowledgements
The authors thank Fuhito Kojima, Debasis Mishra, Hervé Moulin, Barton Lee, and Arunava Sen forhelpful comments. Florian Brandl acknowledges support by the Deutsche Forschungsgemeinschaftunder grant BR 5969/1-1.
APPENDIXA. Proofs From Section 4
Theorem 1.
For every closed set X of allocations, Algorithm 1 terminates after at most m · n rounds with an allocation p ∈ X .Proof. First, we show that the optimization problem in Line 13 always has a solution. Second, weshow that from each round to the next, either some some agent is removed from the set of activeagents N ′ or some agent moves to a less preferred equivalence class.Let k be the index of a round in the algorithm. For the first statement, we have to show thatthere exists ( p, δ ) that satisfies the constraints in Line 13 in round k . If k = 0 , this is trivial since π ( i, E ) = 0 for all i ∈ N and E ∈ E i and X is non-empty. If k > , let ( p, δ ∗ ) be an optimal solutionto the optimization problem computed in round k − . It follows from the definition of π k (at theend of round k − ) that ( p, δ ∗ ) satisfies all constraints of the optimization problem in round k .Thus, the set of feasible points in round k is non-empty. Since X is closed, the problem has anoptimal solution.For the second statement, denote by F ki the set of equivalence classes available to agent i inround k (cf. Line 8), by N ′ k the set of agents so that F ki = ∅ , and by E ki = max % i F ki for i ∈ N ′ their most preferred equivalence classes. Define F k +1 i , N ′ k +1 , and, for i ∈ N ′ k +1 , E k +1 i similarly.Note that N ′ k = ∅ since we would not have reached round k + 1 otherwise. Moreover, N ′ k = ∅ implies that δ ∗ as computed in Line 13 in round k is strictly positive by definition of F ki . Bogomolnaia and Moulin (2004) showed that under dichotomous preferences, core stability, ex-ante efficiency, andex-post efficiency are equivalent. raft – October 27, 2020 Now if N ′ k +1 = ∅ , some agent is removed from the set of active agents and there is nothing leftto show. So assume N ′ k +1 = ∅ . We want to show that for i ∈ N ′ , F k +1 i ⊂ F ki . If E ∈ F k +1 i ,there is p ∈ X such that p ( j, E ′ ) ≥ π k +1 ( j, E ′ ) for all j ∈ N and E ′ ∈ E j , p ( i, E ) > π k +1 ( i, E ) ,and p ( j, E j ) > π k +1 ( j, E j ) for all j ∈ N ′ k +1 with j < i . From the definition of π k +1 and the factthat δ ∗ > in round k , it follows that p is a witness that E ∈ F ki . So we get E ki % i E k +1 i for all i ∈ N ′ k +1 . If N ′ k +1 = N ′ k , it follows from the choice of δ ∗ that this preference is strict for at leastone i ∈ N ′ k +1 . Otherwise N ′ k +1 ( N ′ k . It follows that the algorithm terminates after at most m · n rounds. Clearly, the returned allocation p is in X . Theorem 2.
When VER is applied to any closed set of allocations X , it returns an allocation thatis constrained LD -efficient among the allocations in X .Proof. Let % N be a preference profile and p be the allocation returned by VER for X and % N . Forevery allocation q and t ≥ , let q t be the allocation where every agent i receives a prefix (accordingto her preferences) of q ( i ) summing to min { t, q ( i, O ) } . Formally, q t ( i, O ) = min { t, q ( i, O ) } and, for E i ∈ E i , q t ( i, E i ) > implies q t ( i, E ′ i ) = q ( i, E ′ i ) for all E ′ i ≻ i E i .Assume there is an allocation q ∈ X that lexicographically dominates p . Let t ∗ = max { t ≥ p t ( i ) ∼ LD i q t ( i ) for all i ∈ N } Let i ∗ be the lexicographically first agent so that q t + ǫ ( i ) ≻ LD i p t + ǫ ( i ) for all ǫ > .Observe that p t ∗ ( i ∗ , E i ) = p ( i ∗ , E i ) whenever p t ∗ ( i ∗ , E i ) > . To see this, let E ∗ i be i ∗ ’s leastpreferred equivalence class those with p t ∗ ( i ∗ , E i ) > . It certainly holds that p t ∗ ( i ∗ , E i ) = p ( i ∗ , E i ) for all E i with E i ≻ i ∗ E ∗ i by definition of p t ∗ . Now if p ( i ∗ , E ∗ i ) > p t ∗ ( i ∗ , E ∗ i ) , let < ǫ < p ( i ∗ , E ∗ i ) − p t ∗ ( i ∗ , E ∗ i ) . We have p t ∗ + ǫ ( i ∗ , E i ) = q t ∗ + ǫ ( i ∗ , E i ) for all E i ≻ i ∗ E ∗ i by the choice of t ∗ . Moreover, bythe choice of ǫ , p t ∗ + ǫ ( i ∗ , O ) = q t ∗ + ǫ ( i ∗ , O ) and so p t ∗ + ǫ ( i ∗ , E ∗ i ) = q t ∗ + ǫ ( i ∗ , E ∗ i ) . But this contradicts q t ∗ + ǫ ( i ∗ ) ≻ LD i ∗ p t ∗ + ǫ ( i ∗ ) . It follows that agent i ∗ moves to a less preferred equivalence class aftersecuring t ∗ of equivalence classes at least as good as E ∗ i . That is, there is a round k in Algorithm 1such that p t ∗ ( i, E i ) = π ( i, E i ) for all i ∈ N and E i ∈ E i .Let N ′ be the set of active agents at the end of round k , that is, all agents who increase theirguarantee for some equivalence class in round k . For every i ∈ N ′ , let E i be as determined byLine 8 in round k . Since p is the outcome of VER, it follows that p ( i, E i ) > π k ( i, E i ) for all i ∈ N ′ Moreover, q ( i, E i ) > π k ( i, E i ) for all i < i ∗ (by the choice of i ∗ ) and, since q t ∗ + ǫ ≻ LD i ∗ p t ∗ + ǫ , there is E ′ i ∗ ≻ i E i ∗ such that q ( i ∗ , E ′ i ∗ ) > p ( i ∗ , E ′ i ∗ ) ≥ π k ( i ∗ , E ′ i ∗ ) . This contradicts the choice of E i ∗ since q is a witness that i ∗ could have chosen E ′ i ∗ instead of E i ∗ . Theorem 3.
Let X be a closed and convex set of allocations and p ∈ X . Then p is constrained LD -efficient with respect to X if and only if all agents are indifferent between p and the outcomeof some instance of one-at-a-time-VER on the set X . raft – October 27, 2020 Proof.
Our arguments in the proof of Theorem 2 go through to show that the outcome of anyinstance of one-at-a-time-VER is constrained LD -efficient.For the converse, assume that p is constrained LD -efficient with respect to X . Our argumentsare similar to those used in the proof of Theorem 8 by Aziz and Stursberg (2014). We show thatif π is an allocation to equivalence classes so that p ( i, E i ) ≥ π ( i, E i ) for all i ∈ N and E i ∈ E i withat least one strict inequality, then there is an agent i ∗ so that the most-preferred equivalence class E i ∗ for which i ∗ can increase her probability has p ( i ∗ , E i ∗ ) > π ( i ∗ , E i ∗ ) . Then, by allowing i ∗ toeat, we can get closer to our target allocation p and the claim follows.Suppose we have π as above. For every i ∈ N , let F i = { E ∈ E i : there is q ∈ X so that q ( j, E j ) ≥ π ( j, E j ) for all j ∈ N and E j ∈ E j and q ( i, E ) > π ( i, E ) } . Let N ′ = { i ∈ N : F i = ∅} and for i ∈ N ′ , let E i = max % i F i . Note that N ′ is non-empty since we assume that p ( i, E ) > π ( i, E ) forsome i ∈ N and E ∈ E i . Since X is convex, we can find p ∗ ∈ X so that p ∗ ( i, E ) ≥ π ( i, E ) for all i ∈ N and E ∈ E i and p ∗ ( i, E i ) > π ( i, E i ) for all i ∈ N ′ .Now if p ( i, E ′ i ) > π ( i, E ′ i ) for E ′ i ∈ E i , then E i % i E ′ i since p is a witness that E ′ i ∈ F i . Since p is LD -efficient, it is not LD -dominated by p ∗ . So either p ( i ) ∼ LD i p ∗ ( i ) for all i ∈ N or thereis i ∈ N such that p ( i ) ≻ LD i p ∗ ( i ) . In the first case we can choose i ∗ ∈ N ′ arbitrarily. In thesecond case, choose i ∈ N so that p ( i ) ≻ LD i p ∗ ( i ) . So we can find E ′ i with p ( i, E ′ i ) > p ∗ ( i, E ′ i ) and p ( i, E ′′ i ) = p ∗ ( i, E ′′ i ) for all E ′′ i ≻ i E ′ i . It follows that E ′ i ∈ F i and so i ∈ N ′ and E i % i E ′ i . Thelatter implies that p ( i, E i ) ≥ p ∗ ( i, E i ) > π ( i, E i ) . We can thus choose i ∗ = i . Theorem 4.
The VER allocation for a closed and convex set of allocations X is ordinally egali-tarian among the allocations in X .Proof. Consider an allocation q that is OE and an allocation p that is returned by VER. Consider t ↑ ( q ) = ( t ji ( q )) i,j the signature vector of q and t ↑ ( p ) = ( t ji ( p )) i,j the signature vector of p .Consider that during the run of VER, lower bound constraints of the following form are added: p ( E ji ) ≥ λ . The set of such constraints can alternatively be written as lower bounds on the uppercontour set as follows: p ( S jℓ =1 E ℓi ) ≥ λ ′ . Equivalently, they can be written as t ji ≥ λ ′ . At the startof round k , we denote by g ( i, k ) the number of the first equivalence class E g ( i,k ) i of agent i for whichthe lower bound has not been fixed.We prove by induction on the rounds k of the algorithm that VER finds the largest s such that t g ( i,k ) i ≥ s and that the following values are present in t ↑ ( q ) : min i ′ ∈ N p ( S g ( i ′ ,k ) ℓ =1 E ℓi ′ )) as well as p ( S k ′ ℓ =1 E ℓi ′ )) for i ∈ N and k ′ < g ( i ′ , k ) .For k = 0 , in the first round, VER tries to maximizes the lower bound on E g ( i, i for all i ∈ N .Suppose that p ( E g ( i, i ) ≥ λ ′ for all i ∈ N . Then λ ′ is by definition δ ∗ as computed by VER. Italso follows that for all i ∈ N , p ( E ji ) = 0 for all j < g ( i, . Hence, we have established that theminimum non-zero entry in the vector t ↑ ( p ) is the same as the minimum non-zero entry in vector t ↑ ( q ) . By convexity of the feasible region it follows that it is not possible to have an allocation28 raft – October 27, 2020 in which p ( S jℓ =1 E ℓi ) > for any j < g ( i, . Therefore, the corresponding entries t ℓi = 0 are alsoentries in t ↑ ( q ) .Now suppose k rounds have passed. By the induction hypothesis, for each k ′ ≤ k , t g ( i,k ′ ) i ispresent in the vector t ↑ ( q ) . We also note that VER has fixed a weight for each of S jℓ =1 E ℓi where j < g ( i, k + 1) . At this point, VER computes the largest δ that can be additionally guaranteed foreach E g ( i,k +1) i . Equivalently, it computes the largest s such that p ( S g ( i,k +1) ℓ =1 E ℓi ) ≥ s for all i ∈ N .Then min i ′ ∈ N p ( S g ( i ′ ,k +1) ℓ =1 E ℓi ′ )) is the optimised weight of the next heavy upper contour set. Hence min i ′ ∈ N p ( S g ( i ′ ,k +1) ℓ =1 E ℓi ′ )) is present in t ↑ ( q ) . Theorem 5. If X is a finite union of polytopes, the outcome of VER can be computed in polynomialtime in the number of inequalities used to describe X .Proof. Let X = S sr =1 P r so that P r is a polytope for each r . The only computationally non-trivialsteps are determining the set F i in Line 8 and the computation of δ ∗ in Line 13.First, observe that given a round k of the algorithm, an agent i , an equivalence class E ∈ E i , anda polytope P r , the problem ǫ ∗ = max ǫ s.t. q ( j, E j ) ≥ p k ( j, E j ) + ǫ for all j ∈ N ′ with j < iq ( i, E ) ≥ p k ( i, E ) + ǫq ( j, E ′ ) ≥ p k ( j, E ′ ) for all j ∈ N and E ′ ∈ E j q ∈ P r can be solved in polynomial time in the number of linear inequalities used to describe P r . If ǫ ∗ > ,agent i can increase her probability for E . By solving this problem for every r and every equivalenceclass E ∈ E i , we can determine the most-preferred equivalence class E i of which agent i can increaseher share in polynomial time in the number of inequalities used to describe X .Second, the linear program δ ∗ r = max δ s.t. q ( i, E i ) ≥ p k ( i, E i ) + δ for all i ∈ N ′ q ( i, E ) ≥ p k ( i, E ) for all i ∈ N and E ∈ E i q ∈ P r can be solved in polynomial time for every P r . Since δ ∗ = max r δ ∗ r , the value of δ ∗ in Line 13 canbe computed in polynomial time.Theorem 1 ensures that the number of iterations of the while-loop in Line 4 is bounded by m · n . 29 raft – October 27, 2020 Theorem 6.
Let X be a closed set of allocations and % N be a preference profile. If X is symmetricand convex for all i, i ′ ∈ N with % i = % ′ i , then the outcome of VER satisfies equal treatment ofequals.Proof. Let p be the outcome of VER for the set X and the profile % N . We prove by inductionthat for each round k of Algorithm 1, π k ( i, E ) = π k ( i ′ , E ) for all E ∈ E i = E ′ i . It then follows that p ( i, E ) = p ( i ′ , E ) for all E ∈ E i .For k = 0 , this is clear since π ( j, E ) = 0 for all j ∈ N and E ∈ E i . Now let k be arbitrary andassume that π k ( i, E ) = π k ( i ′ , E ) for all E ∈ E i . Let N ′ be the set active agents at the beginning ofround k . We show that F i = F i ′ , where the F j are determined as in Line 8.Assume that i < i ′ . Since X is symmetric for i and i ′ , it follows that F i ′ ⊆ F i . If F i = ∅ , weare done. Otherwise, let j = max { j ′ ∈ N ′ : j ′ < i ′ and F j ′ = ∅} (note that j ≥ i ). By definitionof E j , there is an allocation p ∈ X such that p ( j ′ , E ) ≥ π k ( j ′ , E ) for all j ′ ∈ N and E ∈ E j ′ and p ( j ′ , o ′ j ) > π k ( j ′ , E j ′ ) for all j ′ ∈ N ′ with j ′ ≤ j and F j ′ = ∅ . Since X is symmetric and convex for i and i ′ , the allocation p ′ = p + p ( ii ′ ) is in X . Thus defined, p ′ satisfies the constraints on p above.For j ′ = i, i ′ , this is obvious since p ′ ( j ′ ) = p ( j ′ ) . For i and i ′ , it follows from p ( i, E i ) > π k ( i, E i ) and π k ( i, E ) = π k ( i ′ , E ) . Clearly, p ′ ( i ) = p ′ ( i ′ ) and so p ′ ( i ′ , E i ) > π k ( i ′ , E i ) . It follows that E i ∈ F i ′ .Thus, E i = E i ′ and so π k +1 ( i, E ) = π k +1 ( i ′ , E ) for all E ∈ E i . B. Proofs From Section 5
Corollary 1.
For each of our four stability notions S , S -VER returns an S -stable allocation thatis SD -efficient among S -stable allocations and satisfies limited equal treatment of equals.Proof. Stability and constrained SD -efficiency of the VER allocation follow from Theorems 1 and 2.To show that VER satisfies limited equal treatment of equals, let i, i ′ ∈ N such that % i = % ′ i , cap( i ) = cap( i ′ ) , and i ∼ o i ′ for all o ∈ O . We prove that the set of S-stable allocations is symmetricand convex for { i, i ′ } . Then Theorem 6 yields the desired conclusion. Symmetry is obvious for allstability notions. Also, for ex-post stability, fractional stability, and claimwise stability, the set ofstable allocations is convex and thus convex for { i, i ′ } . The set of ex-ante stable allocations is notin general convex. It is convex for { i, i ′ } , however, as we show now.Let p be ex-ante stable and λ ∈ [0 , ; let q = λp + (1 − λ ) p ( ii ′ ) . Note that q ( j ) = p ( j ) for all j = i, i ′ . Thus, it suffices to check envy-freeness for pairs of agents in { i, i ′ } × N \ { i, i ′ } . Let j ∈ N \ { i, i ′ } . First, if i ≻ o j and q ( j, o ) > , then p ( j, o ) > since p ( j ) = p ( ii ′ ) ( j ) . Thus, P o ′ % i o p ( i, o ′ ) = P o ′ % i ′ o p ( i ′ , o ′ ) = cap( i ) = cap( i ′ ) . It follows that P o ′ % i o q ( i, o ′ ) = cap( i ) andsimilarly for i ′ . Second, if j ≻ o i and q ( i, o ) = q ( i ′ , o ) > , then without loss of generality, p ( i, o ) > . Hence, P o ′ % j o p ( j, o ′ ) = P o ′ % j o p ( ii ′ ) ( j, o ′ ) = cap( j ) . We get P o ′ % j o q ( j, o ′ ) = cap( j ) asdesired. Together with the fact that q ( j ) = p ( j ) for j = i, i ′ , this shows that q is ex-ante stable.30 raft – October 27, 2020 Corollary 2.
Assume that preferences and priorities are strict. Then, for each of our four stabilitynotions S , S -VER returns the agent optimal deterministic stable allocation.Proof. First notice that if preferences and priorities are strict, ex-post stability and fractional sta-bility coincide. We use that the sets of ex-ante stable allocations and of ex-post stable allocationsare lattices when join and meet are defined via stochastic dominance. This has been shown byAlkan and Gale (2003, Theorem 5) for ex-ante stability (and a more general class of preferences)and by Juárez et al. (2020) for ex-post stability. The upper bound (with respect to the join opera-tion) of both lattices is the agent-optimal deterministic stable allocation, which thus stochasticallydominates every other ex-post or ex-ante stable allocation according the the agents’ preferences.Since by Corollary 1 the outcome of S -VER is SD -efficient among S -stable allocations, it followsthat S -VER returns the agent-optimal deterministic stable allocation when S is ex-ante, ex-post,or fractional stability.Now consider VER for claimwise stability. Let p be the outcome of CWS-VER for the profile ( ≻ N , ≻ O ) . We show that p is a deterministic stable allocation. It then follows from the fact that p is SD -efficient among (claimwise) stable allocations that p is the agent-optimal deterministic stableallocation. Let i, j ∈ N . In the first round of Algorithm 1 ( k = 0 ), agents i and j eat objects o i and o j , respectively. (Since preferences are strict, agents eat objects instead of equivalence classes.)We have shown in Theorem 1 that agents eat objects in decreasing order of their preference, so o i % i o for all o ∈ O with p ( i, o ) > and similarly for j . Suppose j ≻ o i i . If o i % j o j , then P o ≻ j o i p ( j, o ) < p ( i, o i ) , which contradicts claimwise stability of p . Thus, either o j ≻ j o i or i ≻ o i j . Since this holds for all i, j , the deterministic allocation q that assigns o i to i for all i isclaimwise stable. Moreover, q weakly stochastically dominates p . Since, by Proposition 2, p is SD -efficient among claimwise stable allocations, it follows that p = q . For deterministic allocations andstrict priorities, claimwise stability reduces to stability and so p is the agent optimal deterministicstable allocation.Alternatively, one can show that CWS-VER is equivalent to the claimwise probabilistic serialrule defined by Afacan (2018) and apply his Proposition 4. Proposition 1.
Assume that preferences are strict. For ex-ante stability and fractional stability, S -VER returns an allocation that is two-sided SD -efficient.Proof. First we consider fractional stability. We show that on the set of fractionally stable al-locations, (one-sided) constrained SD -efficiency implies two-sided SD -efficiency. Then the claimfollows from Corollary 1. Let p an allocation that is constrained SD -efficient among fractionallystable allocations. Assume that q = p two-sided stochastically dominates p . Frist, observe that q is fractionally stable since the inequalities defining fractional stability are preserved under im-provements with respect to stochastic dominance. By assumption, all agents weakly prefer q to p according to stochastic dominance. Since q = p and preferences are strict, this preference is31 raft – October 27, 2020 strict for at least one agent. In summary, q is a fractionally stable allocation that stochasticallydominates p , which is a contradiction.Ex-ante stability requires more work. Let PS be a profile and p be an allocation that is ex-antestable and SD -efficient among ex-ante stable allocations. By Corollary 1, it suffices to show that p is two-sided SD -efficient.If p is not two-sided SD -efficient, there exists a cycle ( i , o ) , ( i , o ) , . . . , ( i k , o k ) such that ( i , o ) = ( i k , o k ) and for all l = 1 , . . . , k , o l ≻ i l o l − , i l − % o l − i l , and p ( i l , o l − ) > (cf.Dogan and Yildiz, 2016, Proposition 1). For ǫ ∈ (0 , min { p ( i, o ) : i ∈ N and o ∈ O } ) , let q beequal to p except that q ( i l , o l ) = p ( i l , o l ) + ǫ and q ( i l , o l − ) = p ( i l , o l − ) − ǫ for all l = 1 , . . . , k .Thus, agent i l passes a fraction of o l − on to agent i l − . Note that q stochastically dominates p forboth sides and at least one agent i ∈ N strictly prefers q to p since preferences are strict.We show that q is ex-ante stable. Let i, j ∈ N and o ∈ O with i ≻ o j and q ( j, o ) > . If p ( j, o ) > ,ex-ante stability of p implies p ( i, o ′ ) = 0 for all o ′ with o ≻ i o ′ . Since q ( i ) % SD i p ( i ) , it follows that q ( i, o ′ ) = 0 for all o ′ with o ≻ i o ′ . Otherwise, j = i l − and o = o l − for some l = 1 , . . . , k . Now p ( i l , o l − ) ≥ ǫ > and i l − % o l − i l . It follows that i ≻ o l − i l . Since p is ex-ante stable, p ( i, o ′ ) = 0 for all o ′ with o l − ≻ o ′ . And again, since q ( i ) % SD i p ( i ) , it follows that q ( i, o ′ ) = 0 for all o ′ with o l − ≻ i o ′ . So q is ex-ante stable. Thus, p is not SD -efficient among ex-ante stable allocations,which contradicts the assumption. Proposition 2.
There may be no allocation that is claimwise stable and SD -efficient.Proof. Consider the following instance. ≻ : b a c ≻ : a b c ≻ : a b c ≻ a : 1 3 2 ≻ b : 2 1 3 ≻ c : 2 1 3 The only deterministic stable allocation is the following one. a b c We show that it is also the only claimwise stable allocation. Consider any claimwise stableallocation p . We first claim that p (2 , a ) = 0 . If p (2 , a ) > , agent 1 will have a justified claimagainst agent 2 for object a . 32 raft – October 27, 2020 p = a b c ? ? ? ? ? ? ? ?Next, we claim that p (1 , b ) = 0 . Since p (2 , a ) = 0 , agent cannot let agent get any part of b orelse will have a justified claim against agent for object b . p = a b c ? ? ? ? ? ? ?We now claim that p (3 , a ) = 0 . If p (3 , a ) > , then we know that p (1 , b ) = 0 so 1 cannot let take any part of a or else it will have a justified claim against for object a . p = a b c ? ? ? ? ? ?Since the matrix is bistochastic, we complete some columns and rows. p = a b c ? ? ? ?Next, we argue that p (3 , b ) = 0 . If p (3 , b ) > , then we know that p (2 , b ) = 0 so agent will notlet get any part of b or else will have a justified claim against for object b . Hence, p = a b c ? ? ?We can now complete the matrix. 33 raft – October 27, 2020 p = a b c We have established that p is the only claimwise stable allocation. However, it is not SD -efficient.In particular, the allocation a b c SD -dominates p . Proposition 4.
No mechanism satisfies weak SD -strategyproofness, S-stability, and S-constrainedefficiency for S ∈ { ex-ante, ex-post, FS } even if preferences are strict.Proof. In all profiles in the proof, the sets of S-stable allocations with be the same for S ∈{ ex-ante, ex-post, FS } . Thus, it will prove the statement for all three stability notions at once.Let f be a mechanism that is weakly SD -strategyproof, S-stable, and S-constrained efficient.Consider the following profile PS . ≻ : b c a ≻ : b c a ≻ : a b c % a : { , } % b : 3 { , } % c : 3 { , } There are two S-stable and S-constrained efficient deterministic allocations. a b c a b c Thus, all S-stable and S-constrained efficient allocations are of the following form. (Here we usethat the set of S-stable allocations in the same for all three stability notions.) p α = a b c − α α α − α raft – October 27, 2020 for some α ∈ [0 , . Suppose f ( PS ) = p α for some α > . Then, agent 1 can misreport by swapping c and a resulting in the preferences ≻ ′ N below. ≻ ′ : b a c ≻ ′ : b c a ≻ ′ : a b c S-stability and S-constrained efficiency imply that f (( ≻ ′ N , % O )) = p .Note that p (1) ≻ SD i p α and so agent 1 can successfully manipulate in the profile PS , whichcontradicts strategyproofness of f . If − α > , we can use a symmetric argument where agent misreports. Example 3.
Fractional stability is strictly weaker than ex-post stability for weak prioritiesand strict preferences. To see this, consider the following example. Let N = { , , , , } , O = { a, b, c, d, e } , and cap( i ) = 1 for all i ∈ N . The preferences and priorities are as follows. ≻ : c d e a b ≻ : c d e a b ≻ : a b d c e ≻ : a b c d e ≻ : a b c d e % a : [1 , , , , % b : [1 , , , , % c : [3 , ,
5] [1 , % d : [3 , ,
5] [1 , % e : [1 , , , , Then the allocation p = a b c d e / / / / / / / / / / / / /
65 1 / / / / is fractionally stable, but not ex-post stable.To see that p is fractionally stable, observe that (fractional stability) holds for each agent-objectpair including agents 1 or 2 (since they have the lowest priority for each object) or objects a, b , or e (since all agents have the same priority for those). For the remaining pairs, we note that agents 3,4,and 5 each receive probability for objects they prefer to c and d and each of c and d is assignedto agents { , , } with probability .Next we argue that p is not ex-post stable. Any decomposition of p into deterministic allocationsmust contain a deterministic allocation p that assigns object e to agent 5. Neither 1 nor 2 canreceive c or d in p , since then (5 , c ) and (5 , d ) would be blocking pairs, respectively. Hence, p assigns c to agent 4 and d to agent 3. On the other hand, p ( i, a ) = p ( i, b ) = 0 and so p cannotassign a or b to agent 1. It follows that 1 remains unmatched in p , which violates stability of p .35 raft – October 27, 2020 Example 4.
For each of our stability notions S , S -VER results in a different mechanism.Consider again Example 3. One can check that the allocation p is the outcome of VER forfractional stability. Hence, not only is fractional stability different from ex-post stability, but italso leads to different instantiation of VER.VER for claimwise stability in Example 3 yields the following allocation. a b c d e / /
10 1 / / /
10 1 / / / /
15 1 /
54 1 / / /
15 1 /
55 1 / / /
15 1 / Thus, VER for fractional stability and claimwise stability are different.The following example shows that EAS-VER is different from ExpS-VER. Let N = { , . . . , } , O = { a, . . . , h } , and cap( i ) = 1 for all i ∈ N . We truncate the preferences to the part that isrelevant for computing ExpS-VER. All agents have the same priority for objects other than e . ≻ : a e g ≻ : a f g ≻ : b e g ≻ : b f g ≻ : c e h ≻ : c f h ≻ : d e h ≻ : d f h % e : 1,3 5,7 2,4,6,8ExpS-VER (as well as the probabilistic serial rule) yield the allocation shown below. Note that p is not ex-ante stable as agent wants to get more of e and has higher priority for e than and . p = a b c d e f g h / / / / / / / / / / / / / / / / / / / / / / / / raft – October 27, 2020 Algorithm 2.
Vigilant Priority
Input:
A preference profile % N , a non-empty and closed set X of allocations, and a permutation σ over N Output:
An allocation p ∈ X π ( i, E ) ←− for all i ∈ N and E ∈ E i for i ∈ N in order of σ do for E i ∈ E i in order of % i do Compute δ ∗ as follows δ ∗ = max δ,p δ s.t. p ( i, E i ) ≥ π ( i, E i ) + δ (agent i eats E i ) p ( j, E ) ≥ π ( j, E ) ∀ j ∈ N and E ∈ E j ( p extends π ) p ∈ X ( p is feasible) π ( i, E i ) ←− π ( i, E i ) + δ ∗ end for end for return p (as computed in Line 4)To see that p is ex-post stable, we observe that it can be written as the uniform convex combinationof the following four deterministic stable allocations.1 2 3 4 5 6 7 8 (cid:16) (cid:17) a f b g e c h d (cid:16) (cid:17) a g b f h c e d (cid:16) (cid:17) g a e b c f d h (cid:16) (cid:17) e a g b c h d f Since p is ex-post stable and coincides with the allocation produced by the probabilistic serial rule,it is also the outcome of ExpS-VER. Since p is not ex-ante stable, it cannot be the outcome ofEAS-VER. C. The Vigilant Priority Rule raft – October 27, 2020 References
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