Random matching under priorities: stability and no envy concepts

Abstract

We consider stability concepts for random matchings where agents have preferences over objects and objects have priorities for the agents. When matchings are deterministic, the standard stability concept also captures the fairness property of no (justified) envy. When matchings can be random, there are a number of natural stability and fairness concepts that coincide with stability and no envy whenever matchings are deterministic. We formalize known stability concepts for random matchings for a general setting that allows weak preferences and weak priorities, unacceptability, and an unequal number of agents and objects. We then present a clear taxonomy of the stability concepts and identify logical relations between them. Finally, we present a transformation from the most general setting to the most restricted setting, and show how almost all our stability concepts are preserved by that transformation.

Appendices

Appendix: Omitted proofs, auxiliary results, and examples

Section 2.1

Definition 25

(Aharoni–Fleiner fractional stability) A random matching p is Aharoni–Fleiner fractionally stable if for each pair $(i,o)\in N\times O$,

$$\begin{aligned} \sum _{o':o' \succsim _i o}p(i,o')=1\quad \text{ or }\quad \sum _{j:j\succsim _o i}p(j,o)=1. \end{aligned}$$

Proposition 15

A random matching is Aharoni–Fleiner fractionally stable if and only if it has no ex-ante envy.

Proof of Proposition 15

Suppose random matching p has ex-ante envy. Then, there exist $i,j\in N$ and $o,o'\in O$ such that $p(i,o')>0$, $p(j,o)>0$, $o\succ _i o'$, and $i\succ _o j$. Thus, there exists a pair $(i,o)\in N\times O$ such that $\sum _{o'':o'' \succsim _i o}p(i,o'')<1$ and $\sum _{k:k\succsim _o i}p(k,o)<1$. Hence, p is not Aharoni–Fleiner fractionally stable.

Suppose random matching p is not Aharoni–Fleiner fractionally stable. Then, there exists a pair $(i,o)\in N\times O$ such that $\sum _{o'':o''\succsim _i o}p(i,o'')<1$ and $\sum _{k:k\succsim _o i}p(k,o)<1$. Thus, there exist $i,j\in N$ and $o,o'\in O$ such that $p(i,o')>0$, $p(j,o)>0$, $o\succ _i o'$, and $i\succ _o j$. Hence, p has ex-ante envy. $\square$

Note that under weak preferences and weak priorities, the definition of Aharoni–Fleiner fractional stability (Definition 25) remains the same and its equivalence to no ex-ante envy follows as before.

We say that a stability concept $*$ is convex if the convex combination of $*$-stable matchings is $*$-stable as well ($*$-stability stands for any of our stability concepts for random matchings). Since the stability constraints for fractional and claimwise stability are linear, there are simple (linear) arguments why both stability concepts are convex. We will later show that ex-ante stability and robust ex-post stability are not convex.

Lemma 3

Fractional stability is convex.

Proof of Lemma 3

Let p and q be fractionally stable random matchings. Then, they both satisfy inequalities (5). For each $\lambda \in [0,1]$, by taking the convex combinations of the corresponding inequalities, we can see that $\lambda p + (1-\lambda ) q$ also satisfies inequalities (5). $\square$

Lemma 4

Claimwise stability is convex.

The proof for the convexity of claimwise stability is similar to the Proof of Lemma 3 and we omit it.

Proof that ex-ante stability implies robust ex-post stability

Consider a random matching p that is not robust ex-post stable. This means that p can be decomposed into deterministic matchings such that one of them is not stable. Let q be such an unstable deterministic matching. Since q is unstable, there exist agents $i,j\in N$ and objects $o,o'\in O$ such that $q(i,o')=1$, $q(j,o)=1$, $o\succ _i o'$, and $i\succ _o j$. Since q is part of a decomposition of p (with positive weight), it follows that then $p(i,o')>0$, $p(j,o)>0$, $o\succ _i o'$, and $i\succ _o j$. Hence, p is not ex-ante stable. $\square$

Proof that robust ex-post stability does not necessarily imply ex-ante stability

Let $N=\{1,2,3\}$ and $O=\{x,y,z\}$. Consider the following preferences and priorities; they are the same as in Roth et al. (1993, Example 2) but we use them to prove a different statement:

Then, consider $p^A$, which is the deterministic agent optimal stable matching,⁷

and consider $p^O$, which is the deterministic object optimal stable matching,⁸

Let $q=\frac{1}{2}p^A+\frac{1}{2}p^O$. Thus,

We first show that q’s only decomposition into deterministic matchings is the one with respect to $p^A$ and $p^O$: if the decomposition involves a deterministic matching in which agent 1 gets object x, then the only deterministic matching consistent with q is $p^A$ (because $q(2,z)=0$); if the decomposition involves a deterministic matching in which agent 1 gets object z, then the only deterministic matching consistent with q is $p^O$ (because $q(3,x)=0$); since $q(1,y)=0$, no deterministic matching consistent with q allows for agent 1 to get object y. Hence, we have proven that a convex decomposition of q can only involve deterministic matchings $p^A$ and $p^O$. Since both $p^A$ and $p^O$ are stable, it follows that q is robust ex-post stable.

Second, we show that q is not ex-ante stable. Note that for agents $1,2\in N$ and objects $z,y\in O$ we have that $q(1,z)>0$, $q(2,y)>0$, $1\succ _y 2$, and $y\succ _1 z$, i.e., agent 1 ex-ante envies agent 2 for his probability share of object y. Hence, q is not ex-ante stable.

Thus, q is robust ex-post stable but not ex-ante stable. $\square$

Proof that robust ex-post stability implies ex-post stability

By definition, if all decompositions of the random matching involve deterministic stable matchings, then there exists at least one decomposition that involves only deterministic stable matchings. $\square$

Proof that ex-post stability does not necessarily imply robust ex-post stability

Our example and proof is the same as in Roth et al. (1993, Example 2). Let $N=\{1,2,3\}$ and $O=\{x,y,z\}$. Consider the following preferences and priorities:

Then, consider $p^A$, which is the deterministic agent optimal stable matching,

and consider $p^O$, which is the deterministic object optimal stable matching,

The only other deterministic stable matching is

Let q be the uniform random matching. Thus,

Note that

$$\begin{aligned} q= \frac{1}{3} \begin{pmatrix} 1&{}\quad 0&{}\quad 0\\ 0&{}\quad 1&{}\quad 0\\ 0&{}\quad 0&{}\quad 1 \end{pmatrix} +\frac{1}{3} \begin{pmatrix} 0&{}\quad 0&{}\quad 1\\ 1&{}\quad 0&{}\quad 0\\ 0&{}\quad 1&{}\quad 0 \end{pmatrix} +\frac{1}{3} \begin{pmatrix} 0&{}\quad 1&{}\quad 0\\ 0&{}\quad 0&{}\quad 1\\ 1&{}\quad 0&{}\quad 0 \end{pmatrix}=\frac{1}{3}p^A+\frac{1}{3}p^O+\frac{1}{3}p. \end{aligned}$$

Since q can be decomposed into deterministic stable matchings, it is ex-post stable.

We now show that the uniform random matching q is not robust ex-post stable. Note that

$$\begin{aligned} q= \frac{1}{3} \begin{pmatrix} 0&{}\quad 1&{}\quad 0\\ 1&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 1 \end{pmatrix} +\frac{1}{3} \begin{pmatrix} 0&{}\quad 0&{}\quad 1\\ 0&{}\quad 1&{}\quad 0\\ 1&{}\quad 0&{}\quad 0 \end{pmatrix} +\frac{1}{3} \begin{pmatrix} 1&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 1\\ 0&{}\quad 1&{}\quad 0 \end{pmatrix} \end{aligned}$$

where all the deterministic matchings in the decomposition are unstable. Hence, q is not robust ex-post stable.

Thus, q is ex-post stable but not robust ex-post stable. $\square$

Proof that ex-post stability implies fractional stability

If a random matching is ex-post stable then by definition it can be written as a convex combination of deterministic stable matchings. All of these deterministic stable matchings are fractionally stable. Since the set of fractionally stable matchings is convex (Lemma 3), a convex combination of deterministic stable matchings is fractionally stable. $\square$

Proof that fractional stability implies ex-post stability

As already mentioned when introducing fractional stability, for strict priorities, the extreme points of the polytope defined by the linear inequalities (1), (2), (3), and (4) are exactly the (incidence vectors of the) deterministically stable matchings (Vande Vate 1989; Rothblum 1992; Roth et al. 1993). Since, by definition, fractionally stable random matchings are solutions to the linear inequalities, a fractionally stable random matching can be decomposed into deterministic stable matchings, which implies that a fractionally stable random matching is ex-post stable. $\square$

Proof that fractional stability implies claimwise stability

Consider a random matching p that is not claimwise stable. Then, for some pair $(i,o)\in N\times O$ and some $j\in N$ such that $i\succ _oj$, strict inequality (7) applies:

$$\begin{aligned} p(j,o)>\sum _{o':o'\succ _i o}p(i,o'), \end{aligned}$$

i.e., agent i has a claim against agent j with respect to object o. But this implies that

$$\begin{aligned} \sum _{k:k\prec _o i}p(k,o)> \sum _{o':o'\succ _i o}p(i,o'). \end{aligned}$$

Hence, p is not fractionally stable. Thus, fractional stability implies claimwise stability. $\square$

Proof that claimwise stability does not necessarily imply ex-post/fractional stability

Let $N=\{1,2,3\}$ and $O=\{x,y,z\}$. Consider the following preferences and priorities:

Then, consider $p^A$, which is the deterministic agent optimal stable matching,

and consider $p^O$, which is the deterministic object optimal stable matching,

Let q be the uniform random matching. Thus,

First, since $p^O$ is the deterministic object optimal stable matching, agent 1 does not get y in any deterministic stable matching. Hence, random matching q is not ex-post stable. Alternatively, we can check that fractional stability is violated and inequality (6) holds for agent 2 and object x:

$$\begin{aligned} \frac{2}{3}=\sum _{j:j\prec _x 2}q(j,x)>\sum _{o':o'\succ _2 x}q(2,o') =\frac{1}{3} . \end{aligned}$$

Second, we show that random matching q is claimwise stable by checking if there are claims of an agent i against an agent j, i.e., are there $(i,o)\in N\times O$ and $j\in N$ such that $i\succ _oj$ and $q(j,o)>\sum _{o':o'\succ _i o}q(i,o')$? We show that there are no claims.

• For an agent $i\in N$, a claim for a higher probability for his most preferred object against any of the other agents is not justified because all other agents have higher priority for that object.

• For an agent $i\in N$, a claim for a higher probability for his second preferred object against any of the other agents is not justified because he gets an object in the strict upper contour set of his second preferred object with probability $\nicefrac {1}{3}$ whereas any other agent also gets that object with probability $\nicefrac {1}{3}$ (a probability that is not higher).

• No agent $i\in N$ would claim a higher probability for his least preferred object (because he gets an object in the strict upper contour set of his least preferred object with probability $\nicefrac {2}{3}$ whereas any other agent only gets that object with probability $\nicefrac {1}{3}$).

$\square$

Proof of Proposition 1

Let deterministic matching p be stable and note that for deterministic matchings no envy implies ex-ante stability (as also noted by Kesten and Ünver 2015). We have shown that ex-ante stability implies robust ex-post stability; that robust ex-post stability implies ex-post stability; that ex-post stability implies fractional stability; and that fractional stability implies claimwise stability. We are done if we can show that any deterministic claimwise stable matching is stable. Assume that there exists a deterministic matching q that is not stable. Then, there exist $i,j\in N$ and $o,o'\in O$ such that $q(i,o')=1$, $q(j,o)=1$, $o\succ _i o'$, and $i\succ _{o}j$. But then, $1=q(j,o)>\sum _{o'':o''\succ _i o}p(i,o'')=0$ and agent i has a claim against agent j. Afacan (2018) also proved that any deterministic claimwise stable matching is stable. $\square$

Section 2.3

For generalized random matchings the definition of Aharoni–Fleiner fractional stability (Definition 25) remains the same and its equivalence to no ex-ante envy follows as before.

The following two lemmas follow from the definition of fractional weak stability and claimwise weak stability via linear inequalities.

Lemma 5

Fractional weak stability is convex.

Lemma 6

Claimwise weak stability is convex.

Section 3

Proof of Proposition 3

We show that the generalized deterministic matching p has no envy and is non-wasteful and individually rational if and only if the associated deterministic matching $p'$ is weakly stable.

Part 1 Let p be a generalized deterministic matching and $p'$ its associated deterministic matching. We first show that p being wasteful, or individually irrational, or having envy implies that $p'$ is not weakly stable. We do so via the following table that for any of the possible violations for p lists an associated no-envy violation for $p'$. In Table 1, $(i,o_j)\in N\times O$:

Table 1 Violation for p and associated no-envy violation for $p'$. IIR individually irrational, W wasteful, E envy

Part 2 We show that $p'$ not being weakly stable implies that p is wasteful, or individually irrational, or has envy, or that the weak stability violation at $p'$ was not possible. Assume that at $p'$ some agent a envies an agent b for object c. Then, $p'(a,d)=1$, $p'(b,c)=1$, $c\succ '_{a}d$, and $a\succ '_{c}b$. Depending on the specifications of a, b, c, and d, different violations can be identified for p. The following table list all no-envy violations for $p'$ and for p associates wastefulness, individual irrationality, or envy, or explains why the no-envy violations of $p'$ cannot occur given its definition. Note that since $(a,d),(b,c)\in \{N\times O, N\times \Phi , D\times O, D\times \Phi \}$ we have in total 16 different cases to discuss (Table 2).

Table 2 No-envy violation for $p'$ and associated violation for p. IIR individually irrational, W wasteful, E envy, IP impossibility

$\square$

Proof of Proposition 4

We show that a generalized deterministic matching is weakly stable if and only if it is non-wasteful, individually rational, and has no envy.

Let p be a generalized deterministic matching that is individually rational. Assume p is weakly stable, i.e., there exists no pair $(i,o)\in N\times O$ such that $\sum _{o':o'\succsim _i o}p(i,o')=0$ and $\sum _{j:j\succsim _o i}p(j,o)=0$. Since p is deterministic, this is equivalent to there being no pair $(i,o)\in N\times O$ such that $\sum _{o':o'\succsim _i o}p(i,o')=0$ and (a) $\sum _{j\in N}p(j,o)=0$ or (b) for some agent $j\in N$, $i\succ _o j$ and $p(j,o)=1$. This in turn is equivalent to p being (a) non-wasteful and (b) having no envy. $\square$

Proof of Proposition 6

We show that the generalized random matching p has no ex-ante envy and is non-wasteful and individually rational if and only if the associated random matching $p'$ is ex-ante weakly stable.

The proof follows exactly along the lines of the proof of Proposition 3. The only difference is that in that proof no envy, non-wastefulness, individual rationality, and weak stability all are defined for probabilities 1 and 0 to receive an object and when we now consider no ex-ante envy, non-wastefulness, individual rationality, and ex-ante weak stability, these definitions pertain to any probability of receiving an object: all arguments that were using an agent receiving an object with probability 1 now apply for an agent receiving a positive probability of that object. $\square$

Proof of Proposition 7

We show that a generalized random matching is ex-ante weakly stable if and only if it has no ex-ante envy and it is non-wasteful and individually rational.

Let p be a generalized random matching that is individually rational. Assume p is ex-ante weakly stable, i.e., there exists no pair $(i,o)\in N\times O$ such that $\sum _{o':o'\succsim _i o}p(i,o')<1$ and $\sum _{j:j\succsim _o i}p(j,o)<1$. This is equivalent to there being no pair $(i,o)\in N\times O$ such that $\sum _{o':o'\succsim _i o}p(i,o')<1$ and (a) $\sum _{j\in N}p(j,o)<1$ or (b) for some agent $j\in N$, $i\succ _o j$ and $p(j,o)>0$. This in turn is equivalent to p being (a) non-wasteful and (b) having no ex-ante envy. $\square$

Proof of Lemma 1

We show that a generalized random matching is individually rational if and only if in each of its decompositions all generalized deterministic matchings are individually rational.

Part 1 Suppose that generalized random matching p is individually irrational. Then, for some $(i,o)\in N\times O$, $p(i,o)>0$ and agent i or object o considers the other unacceptable. Then, in any decomposition of p into generalized deterministic matchings, there exists a generalized deterministic matching q such that $q(i,o)=1$ and q is individually irrational.

Part 2 Suppose that at some decomposition of p there exists an individually irrational generalized deterministic matching q, i.e., for some $(i,o)\in N\times O$, $q(i,o)=1$ and agent i or object o considers the other unacceptable. Then, $p(i,o)>0$ and p is individually irrational. $\square$

Proof of Lemma 2

We show that if a generalized random matching is non-wasteful, then in each of its decompositions all generalized deterministic matchings are non-wasteful.

Suppose that at some decomposition of p there exists a wasteful generalized deterministic matching q, i.e., there exists an acceptable pair $(i,o)\in N\times O$ such that $\sum _{o':o'\succsim _i o}q(i,o')=0$ (i would like to have object o) and $\sum _{j\in N}q(j,o)=0$ (object o is not allocated). Then it follows that $\sum _{o':o'\succsim _i o}p(i,o')<1$ and $\sum _{j\in N}p(j,o)<1$. Hence, p is wasteful. $\square$

Proof of Proposition 9

We show that the generalized random matching p is ex-post weakly stable if and only if the associated random matching $p'$ is ex-post weakly stable and respects non-wastefulness.

Let p be a generalized random matching and $p'$ its associated random matching.

Part 1 Let p be an ex-post weakly stable generalized matching. Recall that the non-wastefulness of p is equivalent to $p'$ respecting non-wastefulness. Furthermore, p can be decomposed into generalized deterministic weakly stable matchings. By Proposition 5, each generalized deterministic weakly stable matching in the decomposition corresponds to an associated deterministic weakly stable matching. The induced decomposition consisting of the associated deterministic weakly stable matchings is a decomposition of the associated random matching $p'$. Hence, $p'$ is ex-post weakly stable.

Part 2 Recall that from any associated random matching $p'$ we can obtain the original generalized random matching p by taking its first n rows and its first m columns ($|N|=n$ and $|O|=m$). Let the associated random matching $p'$ of p be ex-post weakly stable and respect non-wastefulness. Then, $p'$ can be decomposed into deterministic weakly stable matchings. Note that by taking the first n rows and the first m columns of each of the deterministic weakly stable matchings in the decomposition, we can derive a decomposition of p into generalized deterministic weakly stable matchings (Proposition 5). Furthermore, since $p'$ respects non-wastefulness, p is non-wasteful. Hence, p is ex-post weakly stable. $\square$

Proof of Proposition 10

We show that the generalized random matching p is robust ex-post weakly stable if and only if the associated random matching $p'$ is robust ex-post weakly stable and respects non-wastefulness.

Let p be a generalized random matching and $p'$ its associated random matching. By Proposition 9, p is ex-post weakly stable if and only if $p'$ is ex-post weakly stable and respects non-wastefulness.

Part 1 Let p be an ex-post weakly stable generalized matching that is not robust ex-post weakly stable. Hence, p has a decomposition into generalized deterministic matchings that is not weakly stable, i.e., at least one of the generalized deterministic matchings in the decomposition is not weakly stable. By Proposition 5, each generalized deterministic matching in the decomposition corresponds to an associated deterministic matching and the weakly unstable generalized deterministic matching leads to a weakly unstable associated deterministic matching. The induced decomposition consisting of the associated deterministic matchings is a decomposition of the associated random matching $p'$. Hence, $p'$ has a decomposition into deterministic matchings that are not all weakly stable and $p'$ is not robust ex-post weakly stable.

Part 2 Recall that from any associated random matching $p'$ we can obtain the original generalized random matching p by taking its first n rows and its first m columns ($|N|=n$ and $|O|=m$). Let the associated random matching $p'$ of p respect non-wastefulness and be ex-post weakly stable but not robust ex-post weakly stable. Hence, $p'$ has a decomposition into deterministic matchings that is not weakly stable, i.e., at least one of the deterministic matchings in the decomposition is not weakly stable. Note that by taking the first n rows and the first m columns of each of the deterministic matchings in the decomposition, we can derive a decomposition of p into generalized deterministic matchings and the weakly unstable associated deterministic matching leads to a weakly unstable generalized deterministic matching (Proposition 5). The induced decomposition consisting of the generalized deterministic matchings is a decomposition of the generalized random matching p. Hence, p has a decomposition into generalized deterministic matchings that are not all weakly stable and p is not robust ex-post weakly stable. $\square$

Proof of Proposition 11

Roth et al. (1993) show that in the general model with strict preferences and priorities, any individually rational generalized random matching satisfying inequalities (23) can be decomposed into non-wasteful and stable generalized deterministic matchings. On top of that, the rural hospital theorem (Roth 1986) implies that the set of matched agents and objects is always the same in all stable generalized deterministic matchings. Now suppose that a convex combination of non-wasteful and stable generalized deterministic matchings $q^1,\ldots , q^m$ leads to a wasteful generalized random matching p. By definition of wastefulness, there is an acceptable pair $(i,o)\in N\times O$ such that $\sum _{o':o'\succsim _i o}p(i,o')<1$ (i would like to have more of o) and $\sum _{j\in N}p(j,o)<1$ (o is not fully allocated). Then, the object o that is wasted at generalized random matching p is not assigned to any agent in at least one of the generalized deterministic stable matchings $q^j$ in the convex combination. Thus, by the rural hospital theorem, o is not assigned to any agent in any stable generalized deterministic matching in $\{q^1,\ldots , q^m\}$. Since $\sum _{o':o'\succsim _i o}p(i,o')<1$, it follows that in at least one of the stable generalized deterministic matchings $q^k$, $\sum _{o':o'\succsim _i o}q^k(i,o')=0$ and $\sum _{j\in N}q^k(j,o)=0$. Hence, $q^k$ is wasteful; a contradiction. $\square$

Proof of Proposition 12

We show that the generalized random matching p is fractionally weakly stable if and only if the associated random matching $p'$ is fractionally stable and respects non-wastefulness.

Let p be a generalized random matching and $p'$ its associated random matching.

Part 1 Let p be a fractionally weakly stable generalized random matching. Thus, p is non-wasteful, individually rational, and satisfies inequalities (23). Then, $p'$ respects non-wastefulness and individual rationality. Suppose, by contradiction, that $p'$ is not fractionally weakly stable. Then, for some pair $(a,b)\in N'\times O'$,

$$\begin{aligned} \sum _{a':a'\prec '_{b} a}p'(a',b)>\sum _{b':b'\succsim '_a b;b'\ne b}p'(a,b'). \end{aligned}$$

In particular,

$$\begin{aligned} \sum _{a':a'\prec '_{b} a}p'(a',b)>0. \end{aligned}$$

Furthermore, recall that $\sum _{b':b'\succsim '_{a} b}p'(a,b')<1$ and hence,

$$\begin{aligned} \sum _{b':b'\prec '_{a} b}p'(a,b')>0. \end{aligned}$$

Case 1. Suppose that $b=o_j\in O$. Recall that

$$\begin{aligned} \succsim '_{o_j}=\succsim _{o_j}\left( \{k\in N\mathbin {:}k\succ _{o_j}\emptyset \}\right) ,\ d_j,\ \mathrm {lex}\left( D\setminus \{d_j\}\right) ,\ \succsim _{o_j}\left( \{k\in N\mathbin {:}\emptyset \succ _{o_j}k\}\right) . \end{aligned}$$

By the definition of $\succsim '$ and $p'$ and individual rationality (of p), for all $d_k\in D\setminus \{d_j\}$, $p'(d_k,o_j)=0$ (by definition of $p'$) and for all $l\in N$ such that $l\prec _{o_j}\emptyset$, $p'(l,o_j)=0$. Thus, if $a\precsim '_{o_j}d_j$, then $\sum _{a':a'\prec '_{o_j} a}p'(a',o_j)=0$; a contradiction. Hence, $a\succ '_{o_j}d_j$ and $a=i\in N$ is an acceptable agent. By a symmetric argument, starting with $a=i\in N$ and

$$\begin{aligned} \succsim '_{i}=\succsim _{i}\left( \{o\in O\mathbin {:}o\succ _{i}\emptyset \}\right) ,\ \phi _i,\ \mathrm {lex}\left( \Phi \setminus \{\phi _i\}\right) ,\ \succsim _{i}\left( \{o\in O\mathbin {:}\emptyset \succ _{i}o\}\right) , \end{aligned}$$

we obtain $b\succ '_i\phi _i$ and that $b=o_j\in O$ is an acceptable object.

Then, by the definition of $\succsim '$ and $p'$ (recall that $p'(d_j,o_j)=p(\emptyset ,o_j)$),

$$\begin{aligned} i=a\succ '_{o_j}d_j \text{ implies } \sum _{a':a'\prec '_{o_j} i}p'(a',o_j)= \sum _{k:k\prec _{o_j} i}p(k,o_j)+p(\emptyset ,o_j) \end{aligned}$$

and

$$\begin{aligned} o_j=b\succ '_i\phi _i \text{ implies } \sum _{b':b'\succsim '_{i} o_j; b'\ne o_j}p'(i,b')= \sum _{o':o'\succsim _{i} o_j; o'\ne o_j}p(i,o'). \end{aligned}$$

Hence, inequality $\sum _{a':a'\prec '_{b} a}p'(a',b)>\sum _{b':b'\succsim '_a b;b'\ne b}p'(a,b')$ for $a=i\in N$ and $b=o_j\in O$ can be rewritten as

$$\begin{aligned} \sum _{k:k\prec _{o_j} i}p(k,o_j) +p(\emptyset ,o_j) > \sum _{o':o'\succsim _{i} o_j; o'\ne o_j}p(i,o'), \end{aligned}$$

which contradicts that p was fractionally weakly stable.

Since in Case 1 we have shown that $b\in O$ implies $a\in N$ and vice versa, the only remaining case to discuss is $(a,b)\in D\times \Phi$.

Case 2. Suppose that $a=d_j\in D$ and $b=\phi _i\in \Phi$. Recall that

$$\begin{aligned} \succsim '_{d_j}= o_j,\ \mathrm {lex}\left( O\setminus \{o_j\}\right) ,\ \succsim '_{d_j}(\Phi ). \end{aligned}$$

By the definition of $\succsim '_{d_j}$ and $p'$, for all $l,m\in N$, $\phi _l\succsim '_{d_j}\phi _m$ if and only if $l\succsim _{o_j}m$ and $p'(d_j,o_j)=p(\emptyset ,o_j)$. Then, we have

$$\begin{aligned} \sum _{b':b'\succsim '_{d_j}\phi _i; b'\ne \phi _i}p'(d_j,b')= & {} \sum _{k:\phi _k\succsim '_{d_j}\phi _i; k\ne i}p'(d_j,\phi _k) +p'(d_j,o_j)\\= & {} \sum _{k:k\succsim _{o_j}i; k\ne i}p(k,o_j)+ p(\emptyset ,o_j). \end{aligned}$$

Next, recall that

$$\begin{aligned} \succsim '_{\phi _i}= i,\ \mathrm {lex}\left( N\setminus \{i\}\right) ,\ \succsim '_{\phi _i}(D). \end{aligned}$$

By the definition of $\succsim '_{\phi _i}$, for all $x,y\in O$, $d_x\succsim '_{\phi _i}d_y$ if and only if $x\succsim _{i}y$. Then, by the definition of $\succsim '_{\phi _i}$ and $p'$, we have

$$\begin{aligned} \sum _{a':a'\prec '_{\phi _i} d_j}p'(a' ,\phi _i)=\sum _{d_{l}:d_l\prec '_{\phi _i} d_j}p'(d_l,\phi _i)=\sum _{o_{l}:o_l\prec _{i} o_j}p(i,o_l). \end{aligned}$$

Hence, inequality $\sum _{a':a'\prec '_{b} a}p'(a',b)>\sum _{b':b'\succsim '_a b;b'\ne b}p'(a,b')$ for $a=d_j\in D$ and $b=\phi _i\in \Phi$ can be rewritten as

$$\begin{aligned} \sum _{o_{l}:o_l\prec _{i} o_j}p(i,o_l) > \sum _{k:k\succsim _{o_j}i; k\ne i}p(k,o_j)+ p(\emptyset ,o_j). \end{aligned}$$

This implies $\sum _{o_{l}:o_l\prec _{i} o_j}p(i,o_l)> 0$ and individual rationality implies that agent i finds object $o_j$ acceptable. Similarly it follows that $\sum _{k:k\succsim _{o_j}i; k\ne i}p(k,o_j)+ p(\emptyset ,o_j)<1$, therefore $\sum _{k:k\precsim _{o_j} i}p(k,o_j)> 0$, and by individual rationality, object $o_j$ finds agent i acceptable. Hence, $(i,o_j)\in N\times O$ is an acceptable pair. Furthermore, recall that $\sum _{b':b'\succsim '_{a} b}p'(a,b')<1$ and hence, $\sum _{o_{l}:o_l\succsim _{i} o_j}p(i,o_l) < 1$. Thus, by non-wastefulness, $p(\emptyset ,o_j)=0$. Therefore, for the acceptable pair $(i,o_j)\in N\times O$,

$$\begin{aligned} \sum _{o_{l}:o_l\prec _{i} o_j}p(i,o_l) > \sum _{k:k\succsim _{o_j}i; k\ne i}p(k,o_j) \end{aligned}$$

and therefore also

$$\begin{aligned} \sum _{o_{l}:o_l\prec _{i} o_j}p(i,o_l)+p(i,\emptyset ) > \sum _{k:k\succsim _{o_j}i; k\ne i}p(k,o_j); \end{aligned}$$

contradicting that p was fractionally weakly stable.

Part 2 Let $p'$ be a fractionally stable random matching that respects non-wastefulness. Thus, p is non-wasteful. We first show that p is individually rational. Consider an unacceptable pair $(i,o_j)\in N\times O$. Assume that object $o_j$ finds agent i unacceptable, i.e., $\emptyset \succ _{o_j} i$. Now consider the pair $(d_j,o_j)\in N'\times O'$. Fractional stability of $p'$ requires

$$\begin{aligned} \sum _{b':b'\succsim '_{d_j}o_j;b'\ne o_j}p'(d_j,b') \ge \sum _{a': a'\prec '_{o_j} d_j}p'(a',o_j). \end{aligned}$$

Since object $o_j$ is the best object for $d_j$ at $\succsim '_{d_j}$, it follows that $\sum _{b':b'\succsim '_{i}d_j;b'\ne d_j}p'(d_j,b') =0$. Hence, $\sum _{a':a'\prec '_{o_j} d_j}p'(a',o_j)=0$ and for each $a'\prec '_{o_j} d_j$, $p'(a',o_j)=0$. Next, $a'\prec '_{o_j} d_j$ if and only if $a'\in D\setminus \{d_j\}$ or [$a'\in N$ and $\emptyset \succ _{o_j} a'$]. Thus, by the definition of $p'$, for each $a'\in N$ such that $\emptyset \succ _{o_j} a'$, $p(a',o_j)=p'(a',o_j)=0$. Symmetrically, starting from agent i finding agent $o_j$ unacceptable, i.e., $\emptyset \succ _{o_j} i$, we obtain that for each $b'\in O$ such that $\emptyset \succ _i b'$, $p(i,b')=p'(i,b')=0$. Hence, the generalized random matching p is individually rational.

Next suppose, by contradiction, that p violates one of the inequalities (23). Then, for some acceptable pair $(i,o_j)\in N\times O$,

$$\begin{aligned} \sum _{k:k\prec _{o_j} i}p(k,o_j)+p(\emptyset ,o_j)>\sum _{o':o'\succsim _i o_j;o'\ne o_j}p(i,o'). \end{aligned}$$

Recall that

$$\begin{aligned} \succsim '_{o_j}=\succsim _{o_j}\left( \{k\in N\mathbin {:}k\succ _{o_j}\emptyset \}\right) ,\ d_j,\ \mathrm {lex}\left( D\setminus \{d_j\}\right) ,\ \succsim _{o_j}\left( \{k\in N\mathbin {:}\emptyset \succ _{o_j}k\}\right) \end{aligned}$$

and

$$\begin{aligned} \succsim '_{i}=\succsim _{i}\left( \{o\in O\mathbin {:}o\succ _{i}\emptyset \}\right) ,\ \phi _i,\ \mathrm {lex}\left( \Phi \setminus \{\phi _i\}\right) ,\ \succsim _{i}\left( \{o\in O\mathbin {:}\emptyset \succ _{i}o\}\right) . \end{aligned}$$

Then, by the definition of $\succsim '$ and $p'$ (recall that $p(\emptyset ,o_j)=p'(d_j,o_j)$),

$$\begin{aligned} i\succ _{o_j}\emptyset \text{ implies } \sum _{k:k\prec _{o_j} i}p(k,o_j)+p(\emptyset ,o_j)=\sum _{a':a'\prec '_{o_j} i}p'(a',o_j) \end{aligned}$$

and

$$\begin{aligned} o_j\succ _{i}\emptyset \text{ implies } \sum _{o':o'\succsim _i o_j;o'\ne o_j}p(i,o')=\sum _{b':b'\succsim '_{i} o_j; b'\ne o_j}p'(b',i). \end{aligned}$$

Hence, inequality $\sum _{k:k\prec _{o_j} i}p(k,o_j)+p(\emptyset ,o_j)>\sum _{o':o'\succsim _i o_j;o'\ne o_j}p(i,o')$ can be rewritten as

$$\begin{aligned} \sum _{a':a'\prec '_{o_j} i}p'(a',o_j) > \sum _{b':b'\succsim '_{i} o_j; b'\ne o_j}p'(b',i), \end{aligned}$$

which contradicts that $p'$ was fractionally stable. $\square$

Proof of Proposition 13

We show that the generalized random matching p is claimwise weakly stable if the associated random matching $p'$ is claimwise stable and respects non-wastefulness and individual rationality.

Let p be a generalized random matching and $p'$ its associated random matching. Let $p'$ be claimwise stable and respect non-wastefulness and individual rationality. Thus, p is non-wasteful and individual rational. Suppose, by contradiction, that p violates one of the inequalities (27). Then, for some acceptable pair $(i,o_j)\in N\times O$ and some agent $k\in N$ such that $k\prec _{o_j} i$,

$$\begin{aligned} p(k,o_j)+p(\emptyset ,o_j)>\sum _{o':o'\succsim _i o_j;o'\ne o_j}p(i,o'). \end{aligned}$$

Furthermore, $\sum _{o':o'\succsim _i o}p(i,o')<1$ and hence, by non-wastefulness, $p(\emptyset ,o_j)=0$. Recall that

$$\begin{aligned} \succsim '_{o_j}=\succsim _{o_j}\left( \{k\in N\mathbin {:}k\succ _{o_j}\emptyset \}\right) ,\ d_j,\ \mathrm {lex}\left( D\setminus \{d_j\}\right) ,\ \succsim _{o_j}\left( \{k\in N\mathbin {:}\emptyset \succ _{o_j}k\}\right) \end{aligned}$$

and

$$\begin{aligned} \succsim '_{i}=\succsim _{i}\left( \{o\in O\mathbin {:}o\succ _{i}\emptyset \}\right) ,\ \phi _i,\ \mathrm {lex}\left( \Phi \setminus \{\phi _i\}\right) ,\ \succsim _{i}\left( \{o\in O\mathbin {:}\emptyset \succ _{i}o\}\right) . \end{aligned}$$

Then, by the definition of $\succsim '$ and $p'$ (recall that $p(\emptyset ,o_j)=p'(d_j,o_j)=0$),

$$\begin{aligned} k\prec '_{o_j} i\quad \text{ and }\quad p(k,o_j)+p(\emptyset ,o_j)=p'(k,o_j) \end{aligned}$$

and

$$\begin{aligned} o_j\succ _{i}\emptyset \text{ implies } \sum _{o':o'\succsim _i o_j;o'\ne o_j}p(i,o')=\sum _{b':b'\succsim '_{i} o_j; b'\ne o_j}p'(b',i). \end{aligned}$$

Hence, inequality $p(k,o_j)+p(\emptyset ,o_j)>\sum _{o':o'\succsim _i o_j;o'\ne o_j}p(i,o')$ can be rewritten as

$$\begin{aligned} p'(k,o_j) > \sum _{b':b'\succsim '_{i} o_j; b'\ne o_j}p'(b',i), \end{aligned}$$

which contradicts that $p'$ was claimwise stable. $\square$

Example 8

(A non-wasteful, individually rational, and claimwise weakly stable generalized random matching p but ${p'}$ is not claimwise stable) We reconsider the example used in the proof for the base model that claimwise stability does not necessarily imply ex-post stability. Let $N=\{1,2,3\}$ and $O=\{x,y,z\}$. Consider the following preferences and priorities:

Let p be the uniform random matching. Thus,

Random matching p is claimwise stable, non-wasteful, and individually rational.

The associated instance is $I'=(N',O',\succsim ')$ where $N'=\{1,2,3,d_x,d_y,d_z\}$, $O'=\{x,y,z,\phi _1,\phi _2,\phi _3\}$, with preferences and priorities:

The associated random matching is

By definition, $p'$ respects non-wastefulness and individually rationality with respect to p. However, $p'$ is not claimwise stable: agent $d_x$ has a justified claim against $d_z$ for $\phi _2$ because $d_x\succ _{\phi _2} d_z$, $p'(d_z,\phi _2)=1/3$ and $\sum _{o':o'\succ _{d_x} \phi _2}p'(i,o')=0$. $\diamond$

Example 9

(An individually irrational and claimwise weakly stable associated random matching ${p'}$) We show why we had to impose that $p'$ respects individual rationality in Proposition 13.

Let $N=\{1,2\}$ and $O=\{x,y\}$. Consider the following preferences and priorities:

Let p be the uniform random matching. Thus,

Random matching p is non-wasteful, individually irrational, and satisfies inequalities (27) in the definition of claimwise weak stability.

The associated instance is $I'=(N',O',\succsim ')$ where $N'=\{1,2,d_x,d_y\}$, $O'=\{x,y,\phi _1,\phi _2\}$, with preferences and priorities:

The associated random matching is

and does not respect individual rationality. We argue that $p'$ is claimwise stable. Agent 1 gets 1 / 2 of x and thus does not have a justified claim for $\phi _1$ or $\phi _2$ against $d_x$ or $d_y$. Agent 2 gets a best possible outcome and thus has no justified claim. Agents $d_x$ or $d_y$ cannot have a justified claim against agent 1 or 2 because the latter have higher priority. Finally, agents $d_x$ and $d_y$ have no justified claim against each other. $\diamond$

Weak and strong stochastic dominance stability (Manjunath 2013) re-examined

In this section, we point out connections with weak and strong stochastic dominance (sd) stable matchings as studied by Manjunath (2013) for the base model as introduced in Sect. 2 (with an equal number of agents and objects and strict preferences/priorities). Note that our model involves ordinal preferences of agents over objects and ordinal priorities of objects over agents. These preferences/priorities can be extended to preferences/priorities over random allocations via the first order stochastic dominance relation.

Definition 26

(First order stochastic dominance) Given two random matchings p and q and an agent $i\in N$ with preference $o_1\succ _i o_2 \succ _i \ldots \succ _i o_n$ over $O=\{o_1,\ldots ,o_n\}$, we say that agent i$\varvec{ sd }$-prefers match p(i) to match q(i), denoted by $p(i) \succsim _i^{ sd } q(i)$, if and only if,

$$\begin{aligned}\begin{array}{rcl} p(i,o_1) &{} \ge &{} q(i,o_1) \\ p(i,o_1) + p(i,o_2) &{} \ge &{} q(i,o_1) + q(i,o_2) \\ p(i,o_1) + p(i,o_2) + p(i,o_3) &{} \ge &{} q(i,o_1) + q(i,o_2) + q(i,o_3)\\ &{}\vdots &{} \end{array}\end{aligned}$$

If $p(i) \succsim _i^{ sd } q(i)$ and $p(i)\ne q(i)$, then $p(i) \succ _i^{ sd } q(i)$.

Given two random matchings p and q and an object $o\in O$ with priorities $i_1\succ _o i_2 \succ _o\ldots \succ _o i_n$ over $N=\{i_1,\ldots , i_n\}$, we say that object o$\varvec{ sd }$-prioritizes match p(o) to match q(o), denoted by $p(o) \succsim _o^{ sd } q(o)$, if and only if

$$\begin{aligned}\begin{array}{rcl} p(i_1,o) &{} \ge &{} q(i_1,o) \\ p(i_1,o) + p(i_2,o) &{} \ge &{} q(i_1,o) + q(i_2,o) \\ p(i_1,o) + p(i_2,o) + p(i_3,o) &{} \ge &{} q(i_1,o) + q(i_2,o) + q(i_2,o)\\ &{}\vdots &{} \end{array} \end{aligned}$$

If $p(o) \succsim _o^{ sd } q(o)$ and $p(o)\ne q(o)$, then $p(o) \succ _o^{ sd } q(o)$.

The definitions of Manjunath’s weak and strong stochastic dominance stability are based on the following two pairwise blocking notions.

Definition 27

(Weak and strong (pairwise) sd-blocking Manjunath 2013) A random matching p is weakly sd-blocked by pair $(i,o)\in N\times O$ if there exists a corresponding deterministic matching q such that $q(i,o)=1\ne p(i,o)$ and

$$\begin{aligned} \text{ neither } p(i)\succ _i^{sd}q(i) \text{ nor } p(o)\succ _o^{sd}q(o). \end{aligned}$$

A random matching p is strongly sd-blocked by pair $(i,o)\in N\times O$ if there exists a corresponding deterministic matching $q\ne p$ such that $q(i,o)=1$ and

$$\begin{aligned} q(i)\succ _i^{sd}p(i) \text{ and } q(o)\succ _o^{sd}p(o). \end{aligned}$$

Definition 28

(Weak and strong sd-stability Manjunath 2013)

A random matching p is weakly sd-stable if there exists no pair $(i,o)\in N\times O$ that strongly sd-blocks p.

A random matching p is strongly sd-stable if there exists no pair $(i,o)\in N\times O$ that weakly sd-blocks p.

Proposition 16

A random matching is strongly sd-stable if and only if it is ex-ante stable.

Proof

Suppose random matching p has ex-ante envy. Then, there exist $i,j\in N$ and $o,o'\in O$ such that $p(i,o')>0$, $p(j,o)>0$, $o\succ _i o'$, and $i\succ _o j$. Consider a corresponding deterministic matching q such that $q(i,o)=1\ne p(i,o)$. Thus, neither $p(i)\succ _i^{sd}q(i)$ nor $p(o)\succ _o^{sd}q(o)$. Hence, p is weakly sd-blocked by pair (i, o) and not strongly sd-stable.

Suppose random matching p is not strongly sd-stable. Then, there exists a pair $(i,o)\in N\times O$ that weakly sd-blocks p, i.e., there exists a corresponding deterministic matching q such that $q(i,o)=1\ne p(i,o)$ and neither $p(i)\succ _i^{sd}q(i)$ nor $p(o)\succ _o^{sd}q(o)$. Note $\sum _{o':o'\succsim _i o}p(i,o')=1$ would imply $p(i)\succ _i^{sd}q(i)$ and $\sum _{j:j\succsim _o i}p(j,o)=1$ would imply $p(o)\succ _o^{sd}q(o)$. Thus, $\sum _{o':o'\succsim _i o}p(i,o')<1$ and $\sum _{j:j\succsim _o i}p(j,o)<1$. Then, there exist $j\in N$ and $o'\in O$ such that $p(i,o')>0$, $p(j,o)>0$, $o\succ _i o'$, and $i\succ _o j$. Hence, p is not ex-ante stable. $\square$

Proposition 17

If a random matching p is claimwise weakly stable, then it is weakly sd-stable.

Proof

Suppose random matching p is not weakly sd-stable. Then, there exists a pair $(i,o)\in N\times O$ that strongly sd-blocks p, i.e., there exists a corresponding deterministic matching $q\ne p$ such that $q(i,o)=1$ and $q(i)\succ _i^{sd}p(i)$ and $q(o)\succ _o^{sd}p(o)$. Note that $q(o)\succ _o^{sd}p(o)$ implies $\sum _{k:k\succ _o i}p(k,o)=0$ and $p(i,o)<1$. Hence, there exists an agent $j\in N$ such that $i\succ _o j$ and $p(j,o)>0$. Furthermore, $q(i)\succ _i^{sd}p(i)$ implies $\sum _{o':o'\succ _io}p(i,o')=0$. Thus, $p(j,o)>\sum _{o':o'\succ _io}p(i,o')$ and agent i has a claim against agent j and p is not claimwise stable. $\square$

Proposition 18

Weak sd-stability does not imply claimwise stability.

Proof

Let $N=\{1,2,3\}$ and $O=\{x,y,z\}$. Consider the following preferences and priorities:

Consider the random matching

First, note that agent 2 wants more of object x, $2\succ _x 3$, and $p(3,x)=\frac{1}{2}>\frac{1}{4}=\sum _{o:o\succ _2x}p(2,o)$. Hence, agent 2 has a claim against agent 3 and p is not claimwise stable.

Second, we show that random matching p is weakly sd-stable by checking that for no pair $(i,o)\in N\times O$ with corresponding deterministic matching $q\ne p$ such that $q(i,o)=1$, $q(i)\succ _i^{sd}p(i)$ and $q(o)\succ _o^{sd}p(o).$

• For an agent $i\in N$ and his most preferred object $o\in O$, $q(o)\not \succ _o^{sd}p(o)$ because all other agents have higher priority for that object.

• For an agent $i\in N$ and his second or third preferred object, $q(i)\not \succ _i^{sd}p(i)$ because agent i receives his best object with positive probability.

$\square$