Axiomatic characterizations of the constrained probabilistic serial mechanism

Abstract

Afacan (Games and Economic Behavior 110: 71-89, 2018) introduces an object allocation with random priorities problem. He proposes the constrained probabilistic serial (CPS) mechanism. This study, for the first time in the literature, provides axiomatic characterizations of CPS. The first result characterizes it via non-wastefulness, claimwise stability, constrained ordinal fairness, and surplus invariance to truncations. The other axiomatizes CPS via constrained stochastic efficiency, claimwise stability, and constrained ordinal fairness. The independence of the axioms is provided.

1 Introduction

Afacan (2018) considers an object allocation problem where the objects’ priorities are random. Besides its richer theoretical scope, it has practical appeal. In many real-life object allocation problems, priorities come with large indifference classes.¹ Ties are first randomly broken, and then the outcome is calculated based on the obtained strict priorities. This, in turn, means that priorities are random ex-ante, thereby falling into Afacan (2018)’s setting.²

Afacan (2018) proposes a fairness notion, so-called “claimwise stability". He then introduces the constrained probabilistic serial (CPS) mechanism. It is mainly built on the well-known “probabilistic serial" (PS) mechanism of Bogomolnaia and Moulin (2001). Informally speaking, CPS lets agents consume objects in decreasing order of their preferences until either relevant claimwise stability constraints start binding or objects are exhausted (whichever occurs first). Afacan (2018) shows that CPS is claimwise stable and constrained stochastically efficient (henceforth, constrained sd-efficient) in the sense that its outcome is never stochastically dominated by another claimwise stable matching.³⁴ No claimwise stable and constrained sd-efficient mechanism is strategy proof; hence, in particular, CPS is not strategy-proof.⁵ Afacan (2018) also shows that CPS satisfies a weaker version of equal treatment of equals.

In this study, we provide two characterizations of CPS: (i) A mechanism is non-wasteful, claimwise stable, constrained ordinally fair, and surplus-invariant to truncations if and only if it is CPS, and (ii) A mechanism is constrained sd-efficient, claimwise stable, and constrained ordinally fair if and only if it is CPS. We also obtain the independence of the axioms.

Non-wastefulness is a standard property that requires all preferred objects to be exhausted. Claimwise stability is a fairness property, motivated by both the usual fairness notion in object allocations and the proportional allocation principle in claim problems. In terms of the latter, claimwise stability advocates that objects’ shares should be distributed in proportion to the probabilistic priorities. On the other hand, if we think of objects as perfectly divisible and interpret the assigned probabilities as the time-shares for which agents consume them, then claimwise stability rules out any time-share in which a pair of agents compete for the same object, but the lower priority one consumes it. In other words, it requires the usual fairness⁶ over the whole consumption process ( Bogomolnaia and Heo (2012)), where agents acquire objects’ shares over the unit-time interval at the speed of one.

Constrained ordinal fairness is an adaptation of the ordinal fairness property of Kesten et al. (2011) and Hashimoto et al. (2014), which is used in their PS characterization, to our framework. Given a matching, let us call the agent i’s total share of the objects that are at least as good as object a “surplus of agent i at object a." If agent i’s surplus at object a is less than that of agent j while the latter obtains a positive share of object a, then constrained ordinal fairness guarantees that some claimwise stability constraint that is imposed on agent i for object a is binding at the matching. Hence, giving more of object a to agent i would violate claimwise stability.⁷ Therefore, constrained ordinal fairness can be deemed a fairness property in terms of surpluses but subject to claimwise stability. The last axiom—surplus invariance to truncations—is a mild invariance axiom. It imposes that after an agent truncates his preferences below an object a, his total share from the objects that are at least as good as object a remains the same. More formal motivations of the axioms are provided in the model.

This study is the first to offer an axiomatization of CPS. The most relevant works are the existing PS characterizations. Bogomolnaia and Moulin (2001) axiomatize PS for three-agent economies.⁸ Then PS has been characterized for any number of agents by Kesten et al. (2011), Hashimoto and Hirata (2011), Bogomolnaia and Heo (2012), and Hashimoto et al. (2014). Besides, a recent related study is Balbuzanov (2022) where the author proposes a “generalized constrained probabilistic serial mechanism" (GCPS) to accommodate constraints over deterministic assignments into (random) matchings. In a priority-free setting, Balbuzanov (2022) characterizes constraints as upper-bound inequalities. GCPS works as PS except agents stop consuming an object whenever the relevant upper-bound constraints start binding. While our mechanisms CPS and GCPS have similar logic, the former addresses the claimwise stability constraints, which are not captured in Balbuzanov (2022).

2 The model

There are finite sets of agents N and objects $\hat{O}$. There is also a null-object, denoted by $\emptyset$. It represents receiving no object. Each agent $i\in N$ has a preference relation $\mathrel {R_i}$, which is a complete, transitive, and antisymmetric binary relation over $\hat{O} \cup \{\emptyset \}$. For its asymmetric part, we write $P_i$, which is defined as $a \mathrel {\mathrel {P_i}} b$ whenever $a \mathrel {\mathrel {R_i}} b$ and $a\ne b$. Let $\mathcal {R}$ be the set of all preference relations. Object $a\in \hat{O}$ is acceptable to agent i if $a \mathrel {P_i} \emptyset$, and otherwise, it is unacceptable. We write $q_a$ for the number object a copies in the problem. In the rest of the paper, for ease of exposition, we assume that $q_a=1$ for each $a\in \hat{O}$. However, as we discuss in Remark 7, the whole analysis carries over to the multi-copy case. The null-object is not scarce, that is, $q_{\emptyset }=|N|$.

For object $a\in \hat{O}$, we write $\succ _a$ for its strict priority ordering over N. Let $\succ =(\succ _a)_{a\in \hat{O}}$ and $\zeta$ be the priority profile and the set of all such profiles, respectively. By going beyond these priorities, we let the objects have a random priority profile $\Delta$, which is a probability distribution over $\zeta$. Let $\Delta (\succ )$ be the probability of $\succ =(\succ _a)_{a\in \hat{O}}$ under $\Delta$. We write $\Delta _a$ for the (marginal) priority distribution of object $a\in \hat{O}$ under $\Delta$.⁹ There is no restriction on $\Delta$; hence, the objects’ priorities may be independent as well as correlated. We write $supp(\Delta )=\{\succ \in \zeta :\ \Delta (\succ ) > 0 \}$ for the support of $\Delta$. We define $Pr_{\Delta } ( i \rhd _a j)=\sum _{\succ \in \zeta :\ i \succ _a j } \Delta (\succ )$. In words, it is the probability that agent i has a higher priority than agent j at object a. In the rest of the paper, we fix the set of agents and objects and write $(R,\Delta )$ for the problem.

For ease of writing, let $O=\hat{O} \cup \{\emptyset \}$. A matching $\sigma =[\sigma _{i,a}]_{i\in N, a\in O}$ is a matrix such that (i) for each $i\in N$ and $a\in O$, $0\le \sigma _{i,a} \le 1$, (ii) $\sum _{a\in O} \sigma _{i,a}=1$, and (iii) for each $c\in \hat{O}$, $\sum _{i\in N} \sigma _{i,c} \le 1$. Here, $\sigma _{i,a}$ represents the probability that agent i is matched with object a. Let $\sigma _i=[\sigma _{i,a}]_{a\in O}$ denote the matching of agent i. A matching $\sigma$ is deterministic if $\sigma _{i,a}\in \{0,1\}$ for each $i\in N$ and $a\in O$. Let $\mathcal {X}$ be the set of all matchings. We write $\mathcal {M}$ for the proper subset of $\mathcal {X}$ that consists of the deterministic matchings.

A probability distribution $\lambda$ over $\mathcal {M}$ is called a lottery. Formally, $\lambda =(\lambda _{\mu })_{\mu \in \mathcal {M}}$ is such that for each $\mu \in \mathcal {M}$, $0\le \lambda _{\mu } \le 1$ and $\sum _{\mu \in \mathcal {M}} \lambda _{\mu }=1$. We write $\sigma ^{\lambda }$ for the matching induced by $\lambda$, that is, for each agent–object pair (i, a), $\sigma ^{\lambda }_{i,a}=\sum _{\mu \in \mathcal {M}:\ \mu _i=a} \lambda _{\mu }$.

Fact 1

(Birkoff-von Neumann and Kojima and Manea (2010)) Every matching is induced by a lottery $\lambda$ over $\mathcal {M}$.

Because of the above well-known fact, in the rest of the paper, we consider matchings instead of lotteries. A mechanism $\psi$ is a systematic procedure that produces a matching for each problem $(R,\Delta )$. We write $\psi (R,\Delta )$ for the outcome of $\psi$ in problem $(R,\Delta )$.

2.1 The axioms

For an agent i and object a, let $SU(\mathrel {R_i},a)=\{c\in O:\ c \mathrel {\mathrel {P_i}} a\}$ (the strict upper contour set of agent i at object a), and $U(\mathrel {R_i},a)=SU(\mathrel {R_i},a) \cup \{a\}$ (the upper contour set of agent i at object a).

A matching $\sigma$ is non-wasteful if, for any objects $a,b\in O$ and agent i, $a \mathrel {P}_i b$ and $\sigma _{i,b} > 0$, then $\sum _{j\in N} \sigma _{j,a}=q_a$. Note that non-wastefulness implies individual rationality.¹⁰ A matching $\sigma$ induces a justified claim if there are a pair of agents i and j and object $a\in \hat{O}$ such that $\sigma _{j,a} > Pr_{\Delta }(j \rhd _a i)+\sum _{c\in SU(\mathrel {R_i},a)} \sigma _{i,c}$. A matching is claimwise stable if it does not induce a justified claim.

To better understand the spirit of claimwise stability, by following Afacan (2018), let us utilize the consumption process representation of matchings by Bogomolnaia and Heo (2012). Each matching can be deemed as the outcome of a consumption process where, over the unit-time interval, agents continuously acquire objects’ shares at the speed of one and in decreasing order of their preferences. Agents’ attained shares are interpreted as time-shares during which they consume them in the consumption process.

Fig. 1

Consumption process motivation of claimwise stability

Accompanied by Fig. 1, consider a pair of agents i, j, object a, and a matching $\sigma$. We can interpret $Pr_{\Delta }(j \rhd _a i)$—orange region R.1—as the time-share in which agent j has a higher priority than agent i for object a. Similarly, $\sum _{c\in SU(\mathrel {R_i},a)} \sigma _{i,c}$—green region R.2—is the time-share in which agent i consumes objects that are better than object a. Therefore, it is unarguably fair to let agent j consume object a over these regions, which amounts to $Pr_{\Delta }(j \rhd _a i)+\sum _{c\in SU(\mathrel {R_i},a)} \sigma _{i,c}$. However, once agent j’s share exceeds it (more than R.1+R.2), he consumes object a during some time-share falling into the red region R.4. This is problematic on the fairness ground because over R.4, both agents i and j compete for object a (note that agent i would prefer consuming object a outside R.2), and agent i has a higher priority than agent j (R.3, the time-share in which agent i has a higher priority than agent j, includes R.4). That is, the standard fairness is violated for some time-share in R.4.

To make the motivation behind claimwise stability more concrete, let us consider a problem with three agents i, j, k, and two objects a, b. Let $Pr_{\Delta }(k \succ _a i)=3/4$ and $Pr_{\Delta }(i \succ _b j)=1$. Let the preferences be such that $P_i:\ a, b, \emptyset$; $P_j:\ b, \emptyset$; and $P_k:\ a, \emptyset$. Let us consider a matching $\sigma$ given in the table below.

	a	b	$\emptyset$
i	1/4	1/2	1/4
j	0	1/2	1/2
k	3/4	0	1/4

Here, $\sigma _{j,b}$ $(=1/2) > Pr_{\Delta } ( j \succ _b i) (=0)+\sum _{c\in SU(R_i, b)} \sigma _{i,c} (=1/4)$, constituting a violation of claimwise stability (there is no other claimwise stability violation). Agent i desires to consume object b after having done with acquiring object a’s share. As $\sigma _{i,a}=1/4$, during a time-share of 3/4, agent i demands object b. On the other hand, $\sigma _{j,b} > 1/4$ means that in some positive time-share during which agent i demands object b, agent j manages to consume object b (note that this time-share has the mass of $\sigma _{j,b}-1/4 > 0$). Moreover, in this time-share, agent i has a higher priority than agent j at object b.¹¹ This goes against the spirit of fairness that the higher priority agents should be treated favorably in receiving objects.

Remark 1

Claimwise stability considers agent i’s acquired shares of the objects that are better than object a. This is because agent i demands object a once he finishes consuming these better objects. That is, his object a demand is in the amount of $1-\sum _{c\in SU(R_i,a)}\sigma _{i,c}$. On the other hand, if we considered the agent i’s object a share as well in calculating his demand, we would have underestimated the agent i’s demand for object a (that is, $1-\sum _{c\in U(R_i,a)}\sigma _{i,c}$ might be less than the actual agent i’s object a demand, which is $1-\sum _{c\in SU(R_i,a)} \sigma _{i,c})$. Thus, agent j would have been allowed to consume more than what is needed to ensure fairness over the whole consumption process.

Remark 2

Our model features randomness both in priorities and assignments. That is, it exhibits double uncertainty. This makes it hard to relate ex-post and ex-ante properties. Neither ex-post fairness nor claimwise stability does not easily imply the other. To see this, let us consider two agents i, j and one object a. Suppose each agent has a higher priority with the probability of 1/2. The only claimwise stable matching assigns 1/2 of object a to each agent. This matching is not ex-post fair irrespective of the priority realization (either of the agents has a higher priority ex-post, yet in the matching’s lottery decomposition, it is always the case that the (ex-post) lower priority agent receives object a with the probability of 1/2). Similarly, any ex-post fair matching assigns the whole share to the (ex-post) higher priority agent. This assignment cannot be claimwise stable as each agent must receive 1/2 of object a under claimwise stability.

For a matching $\sigma$, agent i, and object a, let $F(\mathrel {R_i},a,\sigma _i)=\sum _{c\in U(\mathrel {R_i},a)} \sigma _{i,c}$. Matching $\sigma$ stochastically (sd) dominates $\sigma '$ if, for each agent i and object a, $F(\mathrel {R_i},a,\sigma _i) \ge F(\mathrel {R_i},a,\sigma '_i)$, where this strictly holds for some agent–object pair. Matching $\sigma$ is sd-efficient if it is not sd-dominated. Afacan (2018) shows that there does not always exist a claimwise stable and sd-efficient matching. Thus, matching $\sigma$ is constrained sd-efficient if it is not sd-dominated by a claimwise stable matching.

While constrained sd-efficiency is weaker than sd-efficiency, it still implies non-wastefulness, as formally stated below. All the proofs are relegated to Appendices.

Proposition 1

In any problem $(R,\Delta )$, a constrained sd-efficient matching is non-wasteful.

Remark 3

To see how constrained sd-efficiency implies non-wastefulness, suppose that object a is wasted at a constrained sd-efficient matching $\sigma$. This means that some agent, say j, would rather consume more of object a; yet this would violate claimwise stability (as, otherwise, $\sigma$ could not have been constrained sd-efficient). That is, for some agent $i\ne j$, $\sigma _{j,a}=Pr_{\Delta }(j \succ _a i)+\sum _{c\in SU(R_i,a)} \sigma _{i,c}$. This implies that $\sigma _{i,a} < 1-\sum _{c\in SU(R_i,a)} \sigma _{i,c}$.¹² Therefore, agent i also prefers to consume more of object a. However, for the same reason above, it would go against claimwise stability. This means that $\sigma _{i,a}=Pr_{\Delta }(i \succ _a k)+\sum _{c\in SU(R_k,a)} \sigma _{k,c}$ for some agent $k\ne i$. Note that as $Pr_{\Delta }(i \succ _a j)+Pr_{\Delta }(j \succ _a i)=1$, agent k have to be different from agent j (as, otherwise, $\sigma _{i,a}+\sigma _{j,a}=1$, contradicting our supposition that object a is wasted). Thus, no pair of agents simultaneously prevent each other from consuming more of object a. In the proof of Proposition 1, we also observe that it holds for any collection of agents of size more than two. Thus, whenever an object is wasted at a constrained sd-efficient matching, we must have a chain of agents of the size of infinity, where each agent prevents the previous one from consuming more of a wasted object. This, however, is not possible, as there are finitely many agents.

By following Kesten et al. (2011) and Hashimoto et al. (2014), we refer to $F(\mathrel {R_i},a,\sigma _i)$ as “agent i’s surplus at object a under $\sigma$." We adapt the ordinal fairness property of Kesten et al. (2011) and Hashimoto et al. (2014) to our setting as follows.¹³

Definition 1

In problem $(R,\Delta )$, matching $\sigma$ is constrained ordinally fair if, for any pair of agents i, j, and object a, $F(\mathrel {R_i},a,\sigma _i) < F(R_j,a,\sigma _j)$ and $\sigma _{j,a} > 0$, then there exists an agent k such that $\sigma _{i,a}=Pr_{\Delta }(i \rhd _a k)+\sum _{c\in SU(R_k,a)} \sigma _{k,c}$.

In words, constrained ordinal fairness says that if agent i’s surplus at object a is less than that of agent j while, at the same time, agent j receives a positive amount of object a—an ordinal fairness violation instance—then some claimwise stability constraint has to bind for agent–object pair (i, a). Therefore, giving more of object a to agent i would go against claimwise stability. In the presence of constrained ordinal fairness violation at the expense of agent i, we can increase agent i’s surplus at object a (by a trade with agent j for an arbitrarily small amount of object a) while preserving claimwise stability. Hence, constrained ordinal fairness advocates equality in terms of surpluses to the extent that claimwise stability permits.

Remark 4

Ordinal fairness violation is not solely based on the object a’s allotment, but on the total share of the objects that are at least weakly better than object a. Thus, it does not necessarily imply that agent i, at the expense of whom ordinal fairness is violated, receives a lesser amount of object a than agent j. However, the condition that agent j receives a positive amount of object a ensures that agent i’s surplus can be increased by taking away some positive amount of object a from agent j.¹⁴ However, under a constrained ordinally fair matching, doing that would go against claimwise stability.

Remark 5

Constrained ordinal fairness partly justifies ordinal fairness violations on the basis of allocation. That is, the condition of $\sigma _{i,a}=Pr_{\Delta }(i \rhd _a k)+\sum _{c\in SU(R_k,a)} \sigma _{k,c}$ depends on the agent k’s share from $SU(R_k,a)$ at $\sigma$ (the other term of $Pr_{\Delta }(i \rhd _a k)$ purely comes from the primitives). While perhaps sounds contriving at first glance, it could be supported via a myopic view. Suppose agent i objects the allocation because of the ordinal fairness violation against himself at object a. That is, $F(\mathrel {R_i},a,\sigma _i) < F(R_j,a,\sigma _j)$ and $\sigma _{j,a} > 0$ for some agent j. From a myopic perspective, it is plausible to fix the other objects’ assignments and look for a possible increase in the agent i’s object a allotment, especially given the fact that agent j, for the sake of whom violation occurs, receives a positive amount of object a. Put another way, one can imagine that the assignments of all the objects except object a are taken as granted and search for a room to increase agent i’s object a allotment. Constrained ordinal fairness ensures that any such increase is impossible under claimwise stability. Our approach to justifying a violation based on the allocation is common in the field.¹⁵

A mechanism $\psi$ is <non-wasteful, claimwise stable, constrained ordinally fair, constrained sd-efficient> if, for each problem $(R,\Delta )$, $\psi (R,\Delta )$ is <non-wasteful, claimwise stable, constrained ordinally fair, constrained sd-efficient>

A preference relation $R'_i$ is the truncation of $\mathrel {R_i}$ from some object $a\in \hat{O}$ if (i) $a \mathrel {P_i} \emptyset$, (ii) for each pair of objects $c,c'\in \hat{O}$, $c \mathrel {R'_i} c'$ if and only if $c \mathrel {R_i} c'$; and (iii) $c \mathrel {P'_i} \emptyset$ if and only if $c\in U(R_i,a)$.

Definition 2

A mechanism $\psi$ is surplus-invariant to truncations if, for any problem $(R,\Delta )$ and any agent i, $F(\mathrel {R_i},a,\psi _i(R'_i,R_{-i},\Delta ))=F(\mathrel {R_i},a,\psi _i(R,\Delta ))$ where $R'_i$ is the truncation of $R_i$ from object a.¹⁶¹⁷

The invariance axiom above requires no change in agent i’s surplus at object a after he truncates his preferences from object a.¹⁸ It is easy to see that surplus invariance to truncations is necessary for a mechanism to be strategy-proof; hence, it is desirable for the strategic concerns,¹⁹ It is also of practical interest. In many real-life problems, there is a cap on the number of objects that can be ranked in preference lists.²⁰ Under such caps, it is natural for agents to truncate their preferences and submit a ranking only over their best objects. Our invariance property ensures that agents’ surpluses at their last object under the truncated preferences would remain the same even in the absence of caps; thus, caps do not hurt agents in this sense.

2.2 Constrained probabilistic serial (CPS)

We relegate the formal description of CPS to Appendix A. Here, instead, we informally outline how it works and run it on a simple example. Let us consider the objects as perfectly divisible. In the course of CPS, over the unit-time interval and in decreasing order of their preferences, agents continuously acquire object shares at the speed of one. An agent stops consuming an object whenever the relevant claimwise stability constraints start binding, or the object is exhausted (whichever occurs first). The algorithm moves to the next step whenever an agent stops consuming an object, and it terminates when each agent has consumed a total share of one.

Below, we run CPS on a simple example. Let $N=\{i,j,k\}$ and $\hat{O}=\{a,b,c\}$. The preferences are given below.

$\mathrel {\mathrel {R_i}}:\ a,b,c, \emptyset$; $\mathrel {R_j}:\ b, c, a, \emptyset$; $\mathrel {R_k}:\ a,c,b, \emptyset$.

Let $supp(\Delta )$ contain the following deterministic priorities.

$\succ$			$\succ '$
a	b	c	a	b	c
j	i	j	j	i	j
i	j	k	k	j	i
k	k	i	i	k	k

Let $\Delta (\succ )=3/4$ and $\Delta (\succ ')=1/4$. Below demonstrates the working of CPS on the example.

Step 1. Each agent first attempts to consume his favorite object. The best object of both agents i and k is object a and $Pr_{\Delta }( i \rhd _a k)=3/4$. Hence, because of the claimwise stability constraints that each of these agents imposes on the other one for object a, agents i and k can consume at most 3/4 and 1/4 of object a, respectively. As agent j attempts to consume object b, no constraint is imposed on him in this round. Hence, the agents start consuming their respective objects, and this round terminates at the time of $t^1=1/4$ when agent k stops consuming object a. By the end of this step, each agent has consumed 1/4 of his top object.

Step 2. Agent k now attempts to consume his second-best object, which is object c. The others continue attempting to consume their previous objects. As the agents’ respective objects are all different, no claimwise stability constraint is imposed on the agents in this step. Hence, they start consuming their objects and keep consuming until time $t^2=3/4$ when object a has become exhausted. By the end of this step, each agent has consumed 1/2 of his object.

Step 3. Agent i now attempts to consume his second-best object, which is object b. Hence, both him and agent j desire to consume the same object in this round. We have $Pr_{\Delta }( i \rhd _b j)=1$. However, agent i has already acquired 3/4 of his more preferred object a. Therefore, agent j can consume at most 3/4 of object b. However, as he has already consumed that much, he is not allowed to consume more of object b. Therefore, he stops consuming object b, implying that the step terminates at the time of $t^3=3/4$, and no agent acquires a positive amount of any object.

Step 4. Agent j now attempts to consume his second-best object c. Therefore, both agents j and k attempt to consume object c in this round. We have $Pr_{\Delta }( j \rhd _c k)=1$. However, as agent j has already consumed 3/4 of his more preferred object b, agent k is allowed to consume at most 3/4 of object c. Note that agent k has already consumed 1/2 of object c up to this step. Hence, the agents start consuming their respective objects, and this step terminates at the time of $t^4=1$. In the course of this step, each agent has acquired 1/4 of his object. By the end of this step, all the agents have acquired a total share of one; hence, CPS terminates. Below is the CPS outcome:

	a	b	c	$\emptyset$
i	3/4	1/4	0	0
j	0	3/4	1/4	0
k	1/4	0	3/4	0

2.3 The results

We are now ready to state our results, whose proofs are relegated to Appendix B.

Theorem 1

A mechanism $\psi$ is CPS if and only if it is non-wasteful, claimwise stable, constrained ordinally fair, and surplus-invariant to truncations.

Theorem 2

A mechanism $\psi$ is CPS if and only if it is constrained sd-efficient, claimwise stable, and constrained ordinally fair.

The independence of the axioms is shown in Appendix C.

Remark 6

Constrained ordinal fairness can be seen as very tailored to CPS. However, as we see in the independence analyses (in Appendix C), each property is independent of others. Moreover, for constrained ordinally fair mechanisms different from CPS, we can think of PS; and a mechanism that applies the agent-proposing deferred acceptance ( Gale and Shapley (1962)) whenever the priorities are deterministic, and otherwise CPS. As PS is ordinally fair ( Kesten et al. (2011) and Hashimoto et al. (2014)), it is constrained ordinally fair. One can easily verify that any (deterministic) stable matching is constrained ordinally fair in the deterministic priority domain.²¹ This, as well as the constrained ordinal fairness of CPS, implies that the second mechanism is also constrained ordinally fair.

Remark 7

We present the whole analysis for the case of unit-copy objects. In the case of multi-copy objects, claimwise stability is problematic, as extensively discussed in Afacan (2018). Its caveat is that even though an agent may already satisfy all of his demand with some object a, he may continue preventing someone else from consuming the object through claimwise stability. For instance, consider a problem with two agents i, j and one object a with the capacity of 2. Let $Pr_{\Delta }(i \succ _a j)=1$, and both agents prefer consuming object a. Here, agent j cannot consume object a even if agent i receives 1 unit of object a (this is because $SU(R_i,a)=\emptyset$, hence $\sum _{c\in SU(R_i,a)} \sigma _{i,c}=0$ at any matching $\sigma$). Afacan (2018) overcomes this caveat by transforming multi-quota problems into those with unit-quota. The transformation is such that each copy of the same object is treated as a different object with the same priority as the original. Agents’ preferences over this extended set of objects are such that the ranking over the original objects is preserved while the same object’s copies can be ranked in any matter. We can find a solution in the artificial problem and straightforwardly transform it to the original problem (each agent’s share from an object is the sum of the acquired shares from its copies in the artificial problem). Similarly, any matching in the multi-copy problem can be written as a matching in the unit-copy problem in the same manner. Thus, we can only consider unit-copy problems without losing generality; thereby, our results can be invoked in a multi-copy object problem through its corresponding unit-copy object market.

Appendices

Appendix A: the formal description of CPS

The description is taken from Afacan (2018). Let $|N|=n$ and $|O|=m$. For agent i and $O'\subseteq O$, let $top(i,O')$ denote the favorite object of agent i in $O'$. Given a $n\times m$ matrix $\sigma$, $A=(A_i)_{i\in N}$ where $A_i\subseteq O$, a priority order profile $\Delta$, below we define $\Theta ^a_i$ for each agent $i\in N$ and each object $a\in O$:

$\Theta ^a_i(\sigma ,\Delta ,A)=min \{Pr_{\Delta }(i \rhd _a j)+\sum _{c\in SU(\mathrel {R_j},a)} \sigma _{j,c}-\sigma _{i,a} :\ j\in N\setminus \{i\}\ \ \text {such that}\ a=top(j,A_j)\}$.

($\Theta ^a_i(\sigma ,\Delta ,A)=1$ if the above set is empty or a is the null-object).

$\Theta ^a_i$ will keep track of how much more agent i can consume object a within each step of the algorithm without violating claimwise stability constraints.

We define $M(a,A,N)=\{i\in N:\ a=top(i,A_i)\}$. Let $A^0=(A^0_i)_{i\in N}$ where $A^0_i=O$ for each $i\in N$ and $\sigma ^0=[0]$ (matrix of zeros). Suppose that $A^{s-1}$ and $\sigma ^{s-1}$ are already defined. Then, for each $a\in \bigcup _{i\in N} A^{s-1}_i$:

$$\begin{aligned} y^s(a)=\left\{ \begin{array}{ll} \dfrac{1-\sum _{i\in N} \sigma ^{s-1}_{i,a}}{|M(a,A^{s-1},N)|} &{} \hbox {if } M(a,A^{s-1},N)\ne \emptyset \hbox { and }a\ne \emptyset \\ 1 &{} \hbox {otherwise}\\ \end{array} \right. \end{aligned}$$

Let us define the followings,

$$\begin{aligned} y^s= & {} min \{y^s(a):\ a\in \cup _{i\in N} A^{s-1}_i\}\\ \theta ^s= & {} min \{\Theta ^{top(i,A^{s-1}_i)}_i (\sigma ^{s-1},\Delta ,A^{s-1}):\ i\in N\}\\ c^s= & {} min \{y^s,\theta ^s\} \end{aligned}$$

For each $i\in N$ and object $a\in O$:

$\sigma ^s_{i,a}=\left\{ \begin{array}{ll} \sigma ^{s-1}_{i,a}+c^s &{} \hbox {if} a=top(i,A^{s-1}_i)\\ \sigma ^{s-1}_{i,a} &{} \hbox {otherwise} \end{array} \right.$

$E=\{a\in \bigcup _{i\in N} A^{s-1}_i:$ $a\ne \emptyset$ and $\sum _{i\in N} \sigma ^s_{i,a}=1\}$

$A^s_i=\left\{ \begin{array}{ll} A^{s-1}_i\setminus (E \cup top(i,A^{s-1}_i)) &{} \hbox {if } c^s=\Theta ^{top(i,A^{s-1}_i)}_i (\sigma ^{s-1},\Delta ,A^{s-1})\\ A^{s-1}_i\setminus E &{} \hbox {otherwise} \end{array} \right.$

Note that $A^s_i$ keeps track of the set of objects from which agent i can keep consuming in the next step. Here, E consists of the already exhausted objects. We also exclude the top object of agent i, $top(i,A^{s-1}_i)$, whenever the relevant claimwise stability constraint starts binding.

The algorithm terminates whenever each agent has received a total object share of one. As the algorithm moves to the next step when an agent stops consuming an object and everything (i.e., both agents and objects) is finite, the algorithm terminates in a finite step. If the algorithm finishes by the end of step s, then $\sigma ^s$, becomes the CPS outcome.

Appendix B: the proofs of proposition 1, Theorem 1, and Theorem 2

Proof of Proposition 1

Let $\sigma$ be a constrained sd-efficient matching at problem $(R,\Delta )$. Assume for a contradiction that there exist an agent i and objects a, b such that $a \mathrel {P_i} b$, $\sigma _{i,b} >0$, $a\ne \emptyset$ (we will consider the null-object case later in the proof), and $\sum _{k\in N} \sigma _{k,a} < 1$. Then, by constrained sd-efficiency, we have $\sigma _{i,a}=Pr_{\Delta } ( i \rhd _a j)+\sum _{c\in SU(R_j,a)} \sigma _{j,c}$ for some agent j. This is because, otherwise, we can give an arbitrarily small amount of object a to agent i in return of the same amount of object b. While this trade would benefit agent i, it would not violate claimwise stability, contradicting the constrained sd-efficiency of $\sigma$. Note that if $a=\emptyset$, then as no claimwise stability constraint is imposed for the allocation of null-object, the same kind of trade would directly go against the constrained sd-efficiency of $\sigma$. Hence, in the rest of the proof, we assume that $a\ne \emptyset$.

As $Pr_{\Delta } ( i \rhd _a j) \ge 0$, we have $\sigma _{i,a} \ge \sum _{c\in SU(R_j,a)} \sigma _{j,c}$. This, along with our initial supposition $\sum _{k\in N} \sigma _{k,a} < 1$, shows that $\sigma _{j,a} < 1-\sum _{c\in SU(R_j,a)} \sigma _{j,c}$. This, in turn, implies that there exists an object d such that $a \mathrel {P_j} d$ and $\sigma _{j,d} > 0$. By the same argument as above, it implies that there exists an agent $\ell$ such that $\sigma _{j,a}=Pr_{\Delta } (j \rhd _a \ell )+\sum _{c\in SU(R_{\ell },a)} \sigma _{\ell ,c}$. We now claim that $\ell \ne i$. Assume for a contradiction that $\ell =i$. Then we have $\sigma _{i,a} \ge 1-Pr_{\Delta } ( j \rhd _a i)$ and $\sigma _{j,a}\ge 1-Pr_{\Delta } ( i \rhd _a j)$. These inequalities imply that $\sigma _{i,a}+\sigma _{j,a} \ge 2-Pr_{\Delta } (i \rhd _a j)-Pr_{\Delta } ( j \rhd _a i)$. From here, since $Pr_{\Delta } ( i \rhd _a j)=1-Pr_{\Delta } ( j \rhd _a i)$, we obtain $\sigma _{i,a}+\sigma _{j,a} \ge 1$, which contradicts our initial supposition ($\sum _{k\in N} \sigma _{k,a} < 1$); hence, $\ell \ne i$.

Next, by the same reasoning as before, there exists an object z such that $a \mathrel {P_{\ell }} z$ and $\sigma _{\ell ,z} > 0$. This in turn implies that there exists an agent h such that $\sigma _{\ell ,a}=Pr_{\Delta }(\ell \rhd _a h)+\sum _{c\in SU(R_h,a)} \sigma _{h,c}$. Then, by following the same steps in the previous paragraph, we can easily show that $h\ne j$. Furthermore, in what follows, we show that $h\ne i$ as well.

Assume for a contradiction that $h=i$. We already have $\sigma _{i,a} \ge 1-Pr_{\Delta } ( j\rhd _a i)$ and $\sigma _{j,a} \ge 1- Pr_{\Delta } ( \ell \rhd _a j)$. The last finding also implies that $\sigma _{\ell ,a} \ge 1-Pr_{\Delta } ( i \rhd _a \ell )$. Now, we can decompose $Pr_{\Delta } ( i\rhd _a \ell )$ as follows: $Pr_{\Delta } ( i \rhd _a \ell )=\sum _{\succ \in supp(\Delta ):\ j \succ _a i \succ _a \ell } \Delta (\succ )+\sum _{\succ \in supp(\Delta ):\ i \succ _a \ell \succ _a j} \Delta (\succ )+\sum _{\succ \in supp(\Delta ):\ i \succ _a j \succ _a \ell } \Delta (\succ )$. The sum of the last two terms is less than or equal to $Pr_{\Delta } ( i \rhd _a j)$. On the other hand, we have $\sigma _{i,a} \ge Pr_{\Delta _a} ( i \rhd _a j)$. The first term, moreover, is less than or equal to $Pr_{\Delta } ( j \rhd _a \ell )$. Similar to above, we also have $\sigma _{j,a} \ge Pr_{\Delta } ( j \rhd _a \ell )$. These findings, therefore, show that $Pr_{\Delta } ( i \rhd _a \ell ) \le \sigma _{i,a}+\sigma _{j,a}$. This, together with $\sigma _{\ell ,a} \ge 1- Pr_{\Delta } ( i \rhd _a \ell )$, implies that $\sigma _{\ell ,a} \ge 1-\sigma _{i,a}-\sigma _{j,a}$. This, in turn, means that $\sigma _{i,a}+\sigma _{j,a}+\sigma _{\ell ,a} \ge 1$, which contradicts our initial supposition. Hence, $h\in N{\setminus } \{i,j,\ell \}$.

If we continue in the same manner as above, we obtain an object u such that $a \mathrel {P_h} u$ and $\sigma _{h,u} > 0$. This implies that there exists an agent l such that $\sigma _{h,a}=Pr_{\Delta }( h \rhd _a l)+\sum _{c\in SU(R_l,a)} \sigma _{l,c}$. Then by following the same steps as above, we can easily show that $l\in N{\setminus } \{i,j,\ell ,h\}$.

If we continue applying the same argument to agent l, then we find an agent who is different from all the previously considered agents. Each iteration gives us a different agent. This, however, is impossible because there are finitely many agents, yielding a contradiction. Hence, $\sigma$ is non-wasteful. □

Proof of Theorem 1

"If” Part: From Afacan (2018), CPS is claimwise stable and constrained sd-efficient. Hence, by Proposition 1, it is non-wasteful as well. For constrained ordinal fairness, let us assume that at some problem $(R,\Delta )$, for a pair of agents i, k and object a, $F(\mathrel {R_i},a, CPS_i(R,\Delta )) < F(R_k,a,CPS_k(R,\Delta ))$ and $CPS(R,\Delta )_{k,a} > 0$. As, in the course of CPS, agents consume objects in decreasing order of their preferences, it implies that agent i stops consuming object a before it is totally exhausted. By the definition of CPS, it happens only if some claimwise stability constraint imposed on agent i for object a starts binding. That is, for some agent j, $CPS(R,\Delta )_{i,a}=Pr_{\Delta } (i \rhd _a j)+\sum _{c\in SU(R_j,a)} CPS(R,\Delta )_{j,c}$. Hence, CPS is constrained ordinally fair.

For surplus invariance to truncations, for some agent i, let $R'_i$ be the truncation of $\mathrel {R_i}$ from some object a. By its definition, CPS works the same at the truncated preference profile until the time of $t=F(\mathrel {R_i},a,CPS_i(R,\Delta ))$. Therefore, $F(\mathrel {R_i},a,CPS_i(R,\Delta ))=F(\mathrel {R_i},a,CPS_i(R'_i,R_{-i},\Delta ))$, showing that CPS is surplus-invariant to truncations.

"Only If” Part: Let $\psi$ be a mechanism that is non-wasteful, claimwise stable, constrained ordinally fair, and surplus-invariant to truncations. Let $(R,\Delta )$ be any problem. For ease of notation, let $\sigma =CPS(R,\Delta )$ and $\sigma '=\psi (R,\Delta )$. We now claim that $\sigma =\sigma '$. We prove it through obtaining that $F(\mathrel {R_i},a,\sigma _i)=F(\mathrel {R_i},a,\sigma '_i)$ for every agent–object (i, a) pair. Assume for a contradiction that for some agent–object pair (i, a), $F(\mathrel {R_i},a,\sigma _i)\ne F(\mathrel {R_i},a,\sigma '_i)$. □

Claim 1

There exists an agent–object pair (i, a) such that $a\ne \emptyset$, $F(\mathrel {R_i},a,\sigma '_i) < F(\mathrel {R_i},a,\sigma _i)$, and $\sigma _{i,a} > \sigma '_{i,a}$.

First, it cannot be that for any agent–object pair (j, c), $F(R_j,c,\sigma '_j) \ge F(R_j,c,\sigma _j)$, with strictly holding for some agent–object pair. This is because otherwise $\sigma '$ would sd-dominate $\sigma$. This, along with the claimwise stability of $\sigma '$, would contradict the constrained sd-efficiency of CPS ( Afacan (2018)). Hence, for some agent–object pair (i, a), $F(\mathrel {R_i},a,\sigma '_i) < F(\mathrel {R_i},a,\sigma _i)$ and $\sigma _{i,a} > \sigma '_{i,a}$ (the latter is true because whenever $F(\mathrel {R_i},a,\sigma '_i) < F(\mathrel {R_i},a,\sigma _i)$, there always exists an object $b\in U(\mathrel {R_i},a)$ such that $F(\mathrel {R_i},b,\sigma '_i) < F(\mathrel {R_i},b,\sigma _i)$ and $\sigma _{i,b} > \sigma '_{i,b}$). Moreover, $a\ne \emptyset$ because $F(\mathrel {R_i},\emptyset ,\sigma _i)=F(\mathrel {R_i},\emptyset ,\sigma '_i)=1$ (due to the non-wastefulness of $\psi$ and CPS), finishing the proof of Claim 1.

Let $Z=\{F(\mathrel {R_i},a,\sigma '_i):$ $(i,a)\in N\times \hat{O}$ such that $F(\mathrel {R_i},a,\sigma '_i) < F(\mathrel {R_i},a,\sigma _i)$ and $\sigma '_{i,a} < \sigma _{i,a}\}$. Moreover, define $Z'=\{F(\mathrel {R_i},a,\sigma '_i)\in Z:$ for any $F(R_k,c,\sigma '_k)\in Z$, $F(\mathrel {R_i},a,\sigma '_i) \le F(R_k,c,\sigma '_k)\}$. From Claim 1, $Z\ne \emptyset$, thereby $Z'\ne \emptyset$. Let $F(\mathrel {R_i},a,\sigma '_i)\in Z'$.

Claim 2

For every agent-object pair (k, c) where $c\ne \emptyset$ and $F(R_k,c,\sigma '_k) < F(\mathrel {R_i},a,\sigma '_i)$, $F(R_k,c,\sigma '_k)=F(R_k,c,\sigma _k)$.

For each agent–object pair (k, c) with $F(R_k,c,\sigma '_k) < F(\mathrel {R_i},a,\sigma '_i)$, we have $F(R_k,c,\sigma '_k) \ge F(R_k,c,\sigma _k)$. This is because, otherwise, $F(R_k,c,\sigma '_k) < F(R_k,c,\sigma _k)$. This implies that for some object $d\in U(R_k,c)$, we have $F(R_k,d,\sigma '_k) < F(R_k,d,\sigma _k)$ and $\sigma '_{k,d} < \sigma _{k,d}$ (as $\sigma '$ is non-wasteful, d is not the null-object, that is, $d\in \hat{O}$). Hence, $F(R_k,d,\sigma '_k)\in Z$. As $d\in U(R_k,c)$, $F(R_k,d, \sigma '_k) \le F(R_k,c,\sigma '_k)$; hence, $F(R_k,d, \sigma '_k) < F(\mathrel {R_i},a,\sigma '_i)$. This, as well as $F(R_k,d, \sigma '_k)\in Z$, contradicts the fact that $F(\mathrel {R_i},a,\sigma '_i)\in Z'$.

Let $D=\{(k,c)\in N\times \hat{O}:\ F(R_k,c,\sigma '_k) < F(\mathrel {R_i},a,\sigma '_i)\}$. Note that if D is empty, then there is nothing to prove. Hence, suppose that $D\ne \emptyset$. Let $t=\max \{F(R_k, c, \sigma '_k): (k,c)\in D\}$ (as both the sets of agents and objects are finite, this maximal selection–t– is well-defined).

In the course of CPS, agents consume objects in decreasing order of their preferences until claimwise stability constraints start binding or objects are totally exhausted. Hence, in particular, they do so until the time of t. This, as well as $F(R_k,c,\sigma '_k) \ge F(R_k,c,\sigma _k)$ for each $(k,c)\in D$, implies that $F(R_k,c,\sigma '_k)= F(R_k,c,\sigma _k)$ for each $(k,c)\in D$ (because, otherwise, it would imply that for some $(k,c)\in D$, agent k stops consuming object c in CPS before it gets exhausted or the relevant claimwise stability constraints start binding (the latter is due to fact that $\sigma '$ is claimwise stable)). Therefore, for each $(k,c)\in D$, we have $F(R_k, c, \sigma '_k)=F(R_k, c, \sigma _k)$, proving Claim 2.

Claim 3

There exists an agent j such that $F(\mathrel {R_i},a,\sigma '_i) < F(R_j,a,\sigma '_j)$ and $\sigma '_{j,a} > 0$.

Assume for a contradiction that for each agent j with $\sigma '_{j,a} > 0$, we have $F(R_j,a,\sigma '_j)\le F(\mathrel {R_i},a,\sigma '_i)$. If we define $N(a)=\{j\in N:\ \sigma '_{j,a} > 0\}$, then due to the non-wastefulness of $\psi$ (note that $F(\mathrel {R_i},a,\sigma '_i) < 1$ implies that agent i consumes some positive amount of a less preferred object to object a at matching $\sigma '$), $\sum _{j\in N(a)} \sigma '_{j,a}=1$. On the other hand, $\sigma _{i,a} > \sigma '_{i,a}$ implies that for some agent $k\in N(a)$, we have $\sigma _{k,a} < \sigma '_{k,a}$. From here, if $SU(R_k,a)\ne \emptyset$, then for any $b\in SU(R_k,a)$, we have $F(R_k,b,\sigma '_k) < F(\mathrel {R_i},a,\sigma '_i)$, therefore $F(R_k,b,\sigma _k)=F(R_k,b,\sigma '_k)$ (by Claim 2). This, along with $\sigma _{k,a} < \sigma '_{k,a}$, implies that $F(R_k,a,\sigma _k)< F(R_k,a,\sigma '_k) \le F(\mathrel {R_i},a,\sigma '_i) < F(\mathrel {R_i},a,\sigma _i)$. Note that if $SU(R_k,a)=\emptyset$, then as $\sigma _{k,a} < \sigma '_{k,a}$, we directly obtain the above inequality. Hence, $F(R_k,a,\sigma _k) < F(\mathrel {R_i},a,\sigma _i)$, and moreover, $\sigma _{i,a} > 0$. Then, by constrained ordinal fairness of CPS, $\sigma _{k,a}=Pr_{\Delta } (k \rhd _a s)+ \sum _{c\in SU(R_s,a)} \sigma _{s,c}$ for some agent s. This implies that $\sum _{c\in SU(R_s,a)} \sigma _{s,c} \le F(R_k,a,\sigma _k) < F(\mathrel {R_i},a,\sigma '_i)$, hence $\sum _{c\in SU(R_s,a)} \sigma _{s,c}= \sum _{c\in SU(R_s,a)} \sigma '_{s,c}$ (by Claim 2). As $\sigma _{k,a} < \sigma '_{k,a}$, we have $\sigma '_{k,a} > Pr_{\Delta } (k \rhd _a s)+ \sum _{c\in SU(R_s,a)} \sigma '_{s,c}$, contradicting the claimwise stability of $\psi$. Hence, there exists an agent j such that $F(\mathrel {R_i},a,\sigma '_i) < F(R_j,a,\sigma '_j)$ and $\sigma '_{j,a} > 0$, finishing the proof of Claim 3.

As $F(\mathrel {R_i},a,\sigma '_i) < F(R_j,a,\sigma '_j)$ and $\sigma '_{j,a} > 0$, by the constrained ordinal fairness of $\psi$, we have $\sigma '_{i,a}=Pr_{\Delta }(i \rhd _a h)+\sum _{c\in SU(R_h,a)} \sigma '_{h,c}$ for some agent h. As $\sigma _{i,a} > \sigma '_{i,a}$, due to the claimwise stability of $\sigma$, we have $\sum _{c\in SU(R_h,a)} \sigma _{h,c} > \sum _{c\in SU(R_h,a)} \sigma '_{h,c}$. This also implies that a is not the top object of agent h (note that $a \mathrel {P}_h \emptyset$, as otherwise due to the non-wastefulness of $\psi$ and CPS, we would have $\sum _{c\in SU(R_h,a)} \sigma _{h,c}= \sum _{c\in SU(R_h,a)} \sigma '_{h,c}=1$). Let object b be the least preferred object in $SU(R_h,a)$. Then we have $F(R_h,b,\sigma _h) > F(R_h,b,\sigma '_h)$.

To summarize our findings so far, we start with different matchings $\sigma$ and $\sigma '$ produced by CPS and $\psi$ at $(R,\Delta )$, respectively. We then obtain that for some agent–object pair (i, a), $\sigma _{i,a} > \sigma '_{i,a}$, and moreover, with another agent h, we find the triplet $(i,h,a)\in N\times N\times O$ such that $\sigma '_{i,a}=Pr_{\Delta }(i \rhd _a h)+\sum _{c\in SU(R_h,a)} \sigma '_{h,c}$. Moreover, $F(R_h,b,\sigma _h) > F(R_h,b,\sigma '_h)$ where b is the least preferred object by agent h among the objects in $SU(R_h,a)$.

Let us now consider the truncation $R'_h$ of $R_h$ from object b. Let $R'=(R'_h,R_{-h})$, $CPS(R',\Delta )=\nu$, and $\psi (R',\Delta )=\nu '$. As both CPS and $\psi$ are surplus-invariant to truncations, we have $F(R_h,b,\nu _h)=F(R_h,b,\sigma _h) > F(R_h,b,\nu '_h)=F(R_h,b,\sigma '_h)$. Therefore, in particular, $\nu \ne \nu '$.

If we apply the same whole analysis above to the matchings $\nu$ and $\nu '$, we find another agent–object pair (j, d) such that $\nu _{j,d} > \nu '_{j,d}$, and moreover, with another agent k, we find a triplet $(j,k,d)\in N\times N \times O$ such that $\nu '_{j,d}=Pr_{\Delta }(j \rhd _d k)+\sum _{c\in SU(R_k,d)} \nu '_{k,c}$. In this case, however, as $\sum _{c\in SU(R'_h,a)} \nu '_{h,c}=1$ (due to the non-wastefulness of $\psi$ and $\emptyset \in SU(R'_h,a)$), $(j,k,d)\ne (i,h,a)$.

Then, as the same as above, if we let agent k truncate his preferences from the least preferred object in $SU(R_k,d)$ and write $R'_k$ for this truncated preferences, we obtain different CPS and $\psi$ outcomes at $\tilde{R}=(R'_h,R'_k,R_{N{\setminus } \{h,k\}})$.²²²³

If we keep applying the same steps above to the obtained different outcomes of CPS and $\psi$ at the generated truncated preferences, then each iteration gives us a different triplet for which associated claimwise stability constraint binds. But then, as both agents and objects are finite, we eventually reach the preference profile $R''$ where for each agent i, the only acceptable object is his top choice at the original preference $\mathrel {R_i}$. Moreover, if we let $CPS(R'',\Delta )=\tau$ and $\psi (R'',\Delta )=\tau '$, then we have $\tau \ne \tau '$.

As the same as above, at $\tau$ and $\tau '$, there exists an agent–object pair (s, e) such that $\tau _{s,e} > \tau '_{s,e}$ (where object $e\ne \emptyset$), and moreover, with another agent $s'$, there exists a triplet $(s,s',e)$ such that $\tau '_{s,e}=Pr_{\Delta }(s \rhd _e s')+\sum _{c\in SU(R''_{s'},e)} \tau '_{s',c}$. Due to the definition of $R''$ and the non-wastefulness of $\psi$ and CPS, we have $\sum _{c\in SU(R''_{s'},e)} \tau '_{s',c}=\sum _{c\in SU(R''_{s'},e)} \tau _{s',c}$. This, along with the fact that $\tau _{s,e} > \tau '_{s,e}$, implies $\tau _{s,e} > Pr_{\Delta }(s \rhd _e s')+\sum _{c\in SU(R''_{s'},e)} \tau _{s',c}$, contradicting the claimwise stability of CPS. Hence, for any agent–object pair (i, a), $F(\mathrel {R_i},a,\sigma _i)=F(\mathrel {R_i},a,\sigma '_i)$, implying that $\sigma =\sigma '$, which finishes the proof.

Proof of Theorem 2

"If" Part. Afacan (2018) shows that CPS is constrained sd-efficient. This, as well as Theorem 1, shows this part.

"Only If" Part. Let $\psi$ be a constrained sd-efficient, claimwise stable, and constrained ordinally fair mechanism. Assume for a contradiction that $\psi \ne CPS$. This means that for some problem $(R, \Delta )$, $CPS(R,\Delta )=\sigma$, $\psi (R, \Delta )=\sigma '$, and $\sigma \ne \sigma '$.

For the proof, we invoke the consumption process representation of matchings by Bogomolnaia and Heo (2012): Any matching can be obtained as the outcome of a consumption process where, over the unit-time interval, agents continuously acquire probability shares of objects in decreasing order of their preferences at the speed of one.

We now claim that for some agent–object pair (i, a), $F(R_i, a, \sigma _i) > F(R_i,a, \sigma '_i)$. This holds, as otherwise, $\sigma '$ sd-dominates $\sigma$, implying that $\sigma$ cannot be constrained sd-efficient. This, however, contradicts the fact that CPS is constrained sd-efficient ( Afacan (2018)).

Let $Z=\{F(R_i,a,\sigma _i): (i,a)\in N\times O$ and $F(R_i, a, \sigma _i) > F(R_i,a, \sigma '_i)\}$. From above, $Z\ne \emptyset$. Let (i, a) be such that $F(R_i,a,\sigma _i)\in Z$ and $F(R_i,a,\sigma _i) \le F(R_j,b,\sigma _j)$ for each $F(R_j, b, \sigma _j)\in Z$. For each $(k,c)\in N\times O$ with $F(R_k, c, \sigma _k) < F(R_i,a, \sigma _i)$, we have $F(R_k,c,\sigma '_k) \ge F(R_k,c,\sigma _k)$. (1)

In CPS, each agent continues acquiring probability shares of objects until claimwise stability constraints start binding or the objects are exhausted (whichever occurs first). This, as well as the claimwise stability of $\psi$ and (1), implies that for each (k, c) with $F(R_k,c,\sigma _k) < F(R_i,a,\sigma _i)$, we have $F(R_k,c, \sigma _k)=F(R_k, c, \sigma '_k)$. (2)

As $F(R_i, a, \sigma _i) > F(R_i,a, \sigma '_i)$, for some object $c\in U(R_i,a)$, we have $\sigma _{i,c} > \sigma '_{i,c}$ and $F(R_i, c, \sigma _i) > F(R_i,c, \sigma '_i)$. Without loss of generality, let $\sigma _{i,a} > \sigma '_{i,a}$. By (2), under both $\sigma$ and $\sigma '$, the consumption process works the same until time $t=F(R_i,a, \sigma '_i)$. This, as well as $\sigma _{i,a} > \sigma '_{i,a}$, implies that under $\sigma '$, agent i stops consuming object a before it gets exhausted. Moreover, as $\sigma '$ is constrained sd-efficient (hence, by Proposition 1, it is non-wasteful), for some agent j, $\sigma '_{j,a} > 0$ and $F(R_i,a, \sigma '_i) < F(R_j,a, \sigma '_j)$. Then by (2) and the constrained ordinal fairness of $\psi$, some agent, say k, starts consuming object a in the consumption process by time $t=F(R_i,a,\sigma '_i)$, and the claimwise stability constraint on agent i (for object a) due to agent k starts binding. Therefore, without loss of generality, we can take agent k as agent j, and we have $\sigma '_{i,a}=Pr_{\Delta }(i \rhd _a j)+\sum _{c\in SU(R_j, a)} \sigma '_{j,c}$. Thus, so far we have a group of agents $\{i,j\}$ and object a, where $\sigma '_{j,a} > 0$, and for some object d, $\sigma '_{i,d} > 0$ and $a \mathrel {P_i} d$ (the latter is because $F(R_i,a,\sigma '_i) < F(R_i,a,\sigma _i) \le 1$).

We have $\sigma '_{i,a} \ge \sum _{c\in SU(R_j, a)} \sigma '_{j,c}$. Therefore, for agent j, if object b is the least preferred object to object a, then $F(R_j, b, \sigma '_j) \le F(R_i, a, \sigma '_i) < F(R_i, a, \sigma _i)$ (note that if object a is the best alternative of agent j, then we take $a=b$). If $F(R_j, b, \sigma _j) < F(R_i,a,\sigma _i)$, then by (2), $F(R_j, b, \sigma _j)=F(R_j,b,\sigma '_j)$. This, as well as $\sigma '_{i,a} < \sigma _{i,a}$, implies that $\sigma _{i,a} > Pr_{\Delta }(i \rhd _a j)+\sum _{c\in SU(R_j, a)} \sigma _{j,c}$, contradicting the claimwise stability of $\sigma$. Therefore, we have $F(R_j, b, \sigma _j)\ge F(R_i, a, \sigma _i)$, implying that $F(R_j, b, \sigma '_j) < F(R_j, b, \sigma _j)$.

Let $t'=F(R_j, b, \sigma '_j) < F(R_i, a, \sigma _i)$. As $F(R_j, b, \sigma '_j) < F(R_j, b, \sigma _j)$, for some object $d\in U(R_j,b)$, $\sigma '_{j,d} < \sigma _{j,d}$. Without loss of generality, let $\sigma '_{j,b} < \sigma _{j,b}$. From (2), under both $\sigma$ and $\sigma '$, the consumption process works the same until time $t'$. This, as well as $\sigma '_{j,b} < \sigma _{j,b}$, implies that under $\sigma '$, agent j stops consuming object b before it gets exhausted. Moreover, as $\sigma '$ is constrained sd-efficient, for some agent k, $\sigma '_{k,b} > 0$ and $F(R_j,b, \sigma '_j) < F(R_k,b, \sigma '_k)$. Then as the same as above, by (2) and the constrained ordinal fairness of $\psi$, some agent, say h, starts consuming object b in the consumption process by time $t'$, and the claimwise stability constraint on agent j (for object b) due to agent h starts binding. Therefore, without loss of generality, let $k=h$, and $\sigma '_{j,b}=Pr_{\Delta }(j \rhd _b k)+\sum _{c\in SU(R_k, b)} \sigma '_{k,c}$. Hence, we now have a group of agents $\{i,j,k\}$ and objects a, b, d such that $\sigma '_{j,a} >0$, $\sigma '_{k,b} > 0$, $\sigma '_{i,d} >0$, $a \mathrel {P_i} d$, and $b \mathrel {P_j} a$.

For agent k, let object e be the least preferred object to object b (if object b is the top object, then we take $e=b$). From above, $F(R_k, e, \sigma '_k)< F(R_j, b, \sigma '_j) < F(R_i, a, \sigma _i)$. If $F(R_k,e, \sigma _k) < F(R_i, a, \sigma _i)$, then by (2), we have $F(R_k, e, \sigma '_k)=F(R_k,e, \sigma _k)$. This, as well as $\sigma '_{j,b} < \sigma _{j,b}$, implies that $\sigma _{j,b} > Pr_{\Delta }(j \rhd _b k)+\sum _{c\in SU(R_k, b)} \sigma _{k,c}$, contradicting the claimwise stability of $\sigma$. Therefore, we have $F(R_k,e, \sigma _k) \ge F(R_i, a, \sigma _i)$. This implies that $F(R_k, e, \sigma '_k)< F(R_k,e, \sigma _k)$. We then apply the same arguments above for the agent–object pair (k, e). As both agents and objects are finite, continuing in the same manner gives us a set of agents $\{i_1,..,i_m\}$ and set of objects $\{c_1,..,c_m\}$ such that for each $k=1,..,m$, $\sigma '_{i_k, c_k} > 0$ and $c_{k+1} \mathrel {P_{i_k}} c_k$, where $c_{m+1}=c_1$. For an arbitrarily small amount, each agent $i_k$ can receive $c_{k+1}$ in return of giving the same amount $c_k$ to agent $i_{k-1}$. We keep everything else the same as under $\sigma '$. By this way, we can obtain a matching that sd-dominates $\sigma '$. Moreover, as these agents receive better objects, the claimwise stability constraints imposed on others (because of themselves) relax. This means that for an arbitrarily small amount of cyclic trade, we can preserve claimwise stability. This, however, contradicts constrained sd-efficiency of $\sigma '$. Hence, $\sigma '=\sigma$, finishing the proof. □

Appendix C: the independence of the axioms for Theorem 1, and Theorem 2

We provide four examples below to show the independence of the axioms. The table below summarizes the properties that are simultaneously obtained and those that are not in each example.

	Claim. St	NW	Const. Sd-Eff	Const. OF	Surp. In
Ex 1.	$\times$	$\checkmark$	$\checkmark$	$\checkmark$	$\checkmark$
Ex 2.	$\checkmark$	$\times$	$\times$	$\checkmark$	$\checkmark$
Ex 3.	$\checkmark$	$\checkmark$	$\times$	$\checkmark$	$\times$
Ex 4.	$\checkmark$	$\checkmark$	$\checkmark$	$\times$	$\checkmark$

Example 1

From Hashimoto et al. (2014), we know that PS is non-wasteful, ordinally fair, hence constrained ordinally fair. It is easy to verify that PS is also surplus-invariant to truncations. PS is sd-efficient, hence constrained sd-efficient. Yet, it is not claimwise stable whenever $|N|\ge 2$ (PS is defined for problems without priorities, hence it ignores $\Delta$).

Example 2

The mechanism that always leaves each agent unassigned is claimwise stable, constrained ordinally fair, surplus-invariant to truncations, yet it is not non-wasteful. By Proposition 1, it is not constrained sd-efficient either.

Example 3

Let $N=\{i,j\}$ and $O=\{a,b\} \cup \{\emptyset \}$. Consider the following preference and deterministic priority profile:

$\mathrel {R_i}: a, b, \emptyset$ and $R_j: b, a, \emptyset$.

$\succ _a: j, i$ and $\succ _b: i , j$.

Consider a mechanism $\psi$ that gives the deterministic matching $\mu$ where $\mu _i=b$ and $\mu _j=a$ at the above problem and coincides with CPS at all other problems. Note that the CPS outcome at the above problem is such that both agents receive their top choices. Hence, $\psi \ne CPS$. Notice that CPS sd-dominates $\psi$; hence, $\psi$ is not constrained sd-efficient. As $\mu$ is claimwise stable and non-wasteful, $\psi$ is claimwise stable and non-wasteful. Moreover, it is easy to verify that constrained ordinal fairness is satisfied at $\mu$. This, along with the constrained ordinal fairness of CPS, shows that $\psi$ is constrained ordinally fair as well. However, $\psi$ is not surplus-invariant to truncations. To see this, let $R'_j: b, \emptyset$; and let $\sigma$ be the outcome of $\psi$ at the truncated profile $(R_i, R'_j)$. Note that $\sigma$ is deterministic such that $\sigma _j=b$. Hence, $F(R_j,b,\mu _j)=0$ and $F(R_j,b,\sigma _j)=1$, showing that $\psi$ fails to be surplus-invariant to truncations.

Example 4

Let us consider a problem where $N=\{i,j,k\}$ and $O=\{a,b\} \cup \{\emptyset \}$. Consider the following problem $(R, \Delta )$;

$$\begin{aligned} \mathrel {R_i}=R_j=R_k:\ a, b,\emptyset . \end{aligned}$$

$\succ$		$\succ '$		$\succ ''$
a	b	a	b	a	b
i	i	j	k	j	i
j	j	i	i	i	j
k	k	k	j	k	k

Let $\Delta (\succ )=1/2$, $\Delta (\succ ')=1/10$, and $\Delta (\succ '')=2/5$. Let $\psi$ be the mechanism that gives the following matching $\sigma$ at the above instance and coincides with CPS at every other instance.

	a	b	$\emptyset$
i	1/2	1/6	1/3
j	1/2	7/30	4/15
k	0	3/5	2/5

Note that the CPS outcome at the above problem is different from $\sigma$ as, for at least (j, b) pair, $CPS(P, \Delta )_{j,b}=1/6$. Hence, $CPS\ne \psi$. It is easy to verify that $\psi$ is constrained sd-efficient (hence, non-wasteful), claimwise stable, and surplus-invariant to truncations. However, $F(\mathrel {R_i},b,\sigma _i) < F(R_j,b,\sigma _j)$ and $\sigma _{j,b} > 0$, and for every agent $k\ne i$, $\sigma _{i,b} < Pr_{\Delta }(i \rhd _b k)+\sum _{c\in SU(R_k,b)} \sigma _{k,c}$. Hence, $\psi$ fails to be constrained ordinally fair.