What price stability? Social welfare in matching markets

Mathematical Social Sciences 67 (2014) 27–33

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What price stability? Social welfare in matching markets✩ James W. Boudreau a , Vicki Knoblauch b,∗ a

Department of Economics and Finance, University of Texas-Pan American, United States

b

Department of Economics, University of Connecticut, United States

highlights • • • •

We formalize the concept of social welfare functions for matching markets. Stability can involve a price in terms of other notions of social welfare. We show some price tags are very very likely. We also show that price tags can be substantial.

article

info

Article history: Received 22 May 2012 Received in revised form 15 May 2013 Accepted 30 September 2013 Available online 18 October 2013

abstract In two-sided matching markets, stability can be costly. We define social welfare functions for matching markets and use them to formulate a definition of the price of stability. We then show that it is common to find a price tag attached to stability, and that the price of stability can be substantial. Therefore, when choosing a matching mechanism, a social planner would be well advised to weigh the price of stability against the value of stability, which varies from market to market. © 2013 Elsevier B.V. All rights reserved.

1. Introduction We treat formally two issues that have been dealt with in an ad hoc manner in the marriage matching literature: the price of stability and welfare comparisons of matchings. In the theory and practice of marriage matching, stability has traditionally been considered a vital requirement. By definition, every unstable matching contains at least one blocking pair, a man and a woman who prefer each other to their assigned mates. The presence of such pairs can be problematic since individuals then have the incentive to leave their assigned partners to be with each other. But in some scenarios stability is not vital; for example, a strong central authority can prevent renegotiation, that is, can prevent blocking pairs from abandoning their assigned mates to form new marriages, and can thereby prevent unstable matchings from falling apart after partners are assigned. A school district, for example, with a strong central administration, could potentially forbid

✩ We are grateful to Bettina Klaus for helpful comments and suggestions. We would also like to thank seminar participants at Rice University and the UECE Lisbon Meetings 2010: Game Theory and Applications, at which a related paper was presented. ∗ Corresponding author. Tel.: +1 860 486 9076. E-mail addresses: [email protected] (J.W. Boudreau), [email protected] (V. Knoblauch).

0165-4896/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.mathsocsci.2013.09.004

schools from altering their enrollments once matchings have been made by the mechanism of choice. In fact, it is important to recognize that two-sided matching markets were originally centralized not to prevent matchings from falling apart after assignments have been made, but rather to prevent backward unraveling, that is, to prevent individuals from trying to form partnerships earlier and earlier, before adequate time is allowed to ascertain partners’ post-match potential. The requirement of stability, however, is neither necessary nor sufficient to prevent backward unraveling. Unstable matchings can survive over time and successfully limit backward unraveling (Unver, 2001, 2005), while stable mechanisms on their own do not guarantee that backward unraveling will not occur (Halaburda, 2010). If participants are made aware that they would expect to do better under a mechanism that does not guarantee stability than under any stable matching, they might be willing and able to commit as a group to not seek alternative partners before or after the unstable matching is assigned. Alternatively, even when participants are not aware that they could expect to do better under a mechanism that does not guarantee stability, they may be extremely unhappy about the results of a stable matching mechanism.1 And

1 For example, in 2002 a class-action lawsuit was brought against the National Residency Matching Program, the matching mechanism that assigns new doctors to residencies, on the grounds that the stable mechanism held down the salaries of new doctors. Bulow and Levin (2006) provide theoretical support for the

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J.W. Boudreau, V. Knoblauch / Mathematical Social Sciences 67 (2014) 27–33

even when forced or committed non-renegotiation cannot be attained, it may be so difficult for blocking pairs to find each other that match breaking is unlikely to start and once started is likely to stall. In cases such as these, where stability is not vital, it is useful to weigh the price of stability against the value of stability. Finally, even when stability is considered necessary, curiosity alone motivates an investigation of the price of stability. This paper is not the first to question the priority of stability over other welfare criteria in matching markets. Axtell and Kimbrough (2008), for example, compare the man-optimal stable matching mechanism introduced by Gale and Shapley (1962) with a decentralized algorithm that terminates in unstable matchings and show that the unstable mechanism outperforms significantly in terms of agents’ average rank of partner. Anshelevich et al. (2009, 2013) similarly point out that stability can come at a substantial cost in utilitarian terms, and investigate the potential size of such costs both theoretically and with simulation experiments for various distributions of agents’ utility functions. And in a companion paper (Boudreau and Knoblauch, 2013) we also study tradeoffs in utilitarian terms,2 also both theoretically and with simulation experiments, though in that paper we focus on how the price of stability changes with ordinal categorizations of agents’ preferences. Clearly, however, traditional utilitarianism is just one notion of welfare. This brings us to our second issue. In order to present a formal, more general definition of the price of stability, we need a general method for assigning values to matchings. Many studies assign values to matchings, but this has always been done in an ad hoc manner. We provide a formal foundation for assigning values to matchings by adapting the concept of a social welfare function for use in the marriage matching arena. As a brief and informal preview of our two key definitions, a social welfare function (SWF) assigns a non-negative real number to every ordered pair consisting of a matching and a preference profile. For a market of size n with preference profile Pn , the price of stability associated with an SWF f is the ratio of the maximum value of f over all matchings to the maximum value of f over all stable matchings. Then we will say there is a price tag attached to stability for a market of size n if that ratio is greater than one for some Pn . When defined as above, an SWF can vary not only with the outcome of some process – in our case the outcome is a matching – but also with a feature of the market, the preference profile. This makes it possible for the values we place on matchings to depend, in a variety of ways, on the levels of satisfaction of the participants. In the next section, Section 2, we define SWFs over marriage matchings, provide several families of SWFs motivated by previous literature, and argue that SWFs in general are legitimate tools for the study of marriage matching. In Section 3 we define the price of stability for marriage matching and show that for most of our examples of SWFs there is a price tag attached to stability. We show that for at least two of our examples, when markets are large, a randomly chosen preference profile will almost certainly show that the SWF in question comes with a price tag attached to stability. We also demonstrate that the price of stability for four of our examples is substantial. Section 4 concludes with a discussion of how simulation can provide information about the price of stability for many scenarios that are intractable via theory.

2. Social welfare functions The model considered here is the simple marriage matching problem first popularized by Gale and Shapley (1962). The model features two finite disjoint sets of n agents denoted M = {m1 , m2 , . . . , mn } and W = {w1 , w2 , . . . , wn }. We adopt the marriage market interpretation and refer to the two sets as men and women, but alternative interpretations categorize agents as firms and workers, or workers and machines. Each agent has a complete, strict, transitive preference ordering over the agents on the other side of the market.3 Man i’s preferences are given by a one-to-one and onto ranking function rmi : W → {1, 2, . . . , n} where wj is preferred to wk by mi if rmi (wj ) < rmi (wk ). Woman j’s preferences are similarly represented by rwj . A market’s preference profile, Pn , is then simply the collection of all agents’ preference orderings induced by their ranking functions. The outcome of a marriage matching problem is a matching of men and women given by a one-to-one and onto function µ : M → W . Let M denote the set of all matchings. A matching is said to be stable if there does not exist a blocking pair {mi , wj } such that rmi (wj ) < rmi (µ(mi )) and rwj (mi ) < rwj (µ(wj )). As proved by Gale and Shapley (1962), at least one such matching always exists. But although a matching market’s set of stable outcomes can be guaranteed non-empty, it is rarely single valued. When more than one stable matching does exist for a market, the set has a lattice structure with respect to the interests of the two sides of the market (Roth and Sotomayor, 1990, Theorem 3.8 attributed to Conway). That is, in the case of marriage matching, there will always be one stable matching that is most preferred by all men and least preferred by all women. This is known as the manoptimal matching. There will also be an analogous woman-optimal matching, with matchings in between being partially ordered by at-least-as-good-for-every-man when moving from the manoptimal toward the woman-optimal matching and vice versa. Let S denote the set of all stable matchings, and let µM and µW be the man-optimal and the woman-optimal matchings, respectively. Definition 1. A social welfare function (SWF) f assigns a positive real number to every ordered triple (n, µn , Pn ), where Pn and µn are, respectively, a preference profile and a matching for M ∪ W with |M | = |W | = n. We will write f (µn , Pn ) rather than f (n, µn , Pn ) and we will sometimes write µ rather than µn and/or P rather than Pn if n is fixed or if it is not necessary to specify n. We now introduce three examples of SWFs and nine examples of families of SWFs. The first two examples are formalizations of the priority placed on stability and Pareto efficiency which is implicit in much of the existing work on matching. Example 1. f (µ, Pn ) =

measure, referred to as a Rawlsian measure, which is discussed below.

2 1

if µ ∈ S otherwise.



2 1

if µ is Pareto efficient otherwise.

Example 2. f (µ, Pn ) =

defendants’ claim, and Crawford (2008) suggests improvements to the matching procedure to alleviate such problems. More details on the case itself are available in those papers. 2 In that paper (Boudreau and Knoblauch, 2013) we also consider another welfare



3 As in the traditional version of the model (Gale and Shapley, 1962), we assume that all agents rank the option of remaining single last, so it is omitted for brevity.

J.W. Boudreau, V. Knoblauch / Mathematical Social Sciences 67 (2014) 27–33

Example 3. The utilitarian SWF f (µ, Pn ) =

1



ra (µ(a))

.

a∈M ∪W

The third example, a utilitarian SWF, also relates to an oftenused welfare concept in the marriage matching literature. Its denominator is the sum of the ranking each agent gives her partner in a matching µ, a measure also known as the choice count of a matching (McVitie and Wilson, 1971; Gusfield and Irving, 1989). This measure is a simple measure of aggregate satisfaction, and is also related to the notion of envy described by Romero-Medina (2001) and Klaus (2009). Families of social welfare functions Our first examples of families of SWFs reflect the possibility that some priority may be given to specific individuals or groups of individuals. In particular, our notion of gender monotonicity is analogous to the prioritization of student welfare above that of schools in student assignment problems (Balinski and Sönmez, 1999; Ergin, 2002; Klaus and Klijn, 2007; Kesten, 2010). Example 4. f is a-focused for some a ∈ M ∪ W ; that is, f (µ, Pn ) > f (µ′ , Pn ) if ra (µ(a)) < ra (µ′ (a)). Example 5. f is strictly female monotonic; that is, f (µ, Pn ) > f (µ′ , Pn ) if µ female Pareto dominates µ′ , that is, if no female prefers µ′ to µ and some female prefers µ to µ′ . Example 6. f is strictly male monotonic. (Simply the analogous mirror case of the previous example.) Example 7. f is strictly monotonic; that is, f (µ, Pn ) > f (µ′ , Pn ) if µ Pareto dominates µ′ , that is, if no male or female prefers µ′ to µ and some male or female prefers µ to µ′ . Our next family of SWFs we refer to as Rawlsian SWFs, since they are based on the concept of making the worst-off individual as well off as possible, a welfare principle made famous by philosopher Rawls (1971). This maximin criterion has also been considered and applied to matching markets by Masarani and Gokturk (1989), and is defined as follows: for P fixed, µ is maximin optimal if and only if µ maximizes a Rawlsian SWF. Example 8. f is Rawlsian; that is, f (µ, Pn ) > f (µ′ , Pn ) if max{ra (µ(a)) : a ∈ M ∪ W } < max{ra (µ′ (a)) : a ∈ M ∪ W }. Our next example combines the spirit of democracy with the context of matching. Given Pn , we define a dominating set of matchings as one such that each member defeats every matching outside the set in a pairwise election. We then define the Smith set (a reference to Smith, 1973) as the smallest nonempty dominating set; that is, the smallest nonempty set of matchings such that each member defeats every matching outside the set in a head to head election where the electorate is M ∪ W . The Smith set exists for every Pn since the set of all matchings is a dominating set, the dominating sets are nested, and the number of matchings is finite.

29

Example 10. f respects stability; that is, f (µ, Pn ) > f (µ′ , Pn ) if µ is stable and µ′ is not (for example, Example 1). Finally, our last two examples allude to notions of envyfreeness4 or gender-equality addressed most explicitly by RomeroMedina (2005), but which also relate to the work of Teo and Sethuraman (1998), Sethuraman et al. (2006) Klaus and Klijn (2006, 2010), all of whom study the issue of compromise between the two sides of the market. Example 11. f respects gender balancedness; that is, f (µ, Pn ) > f (µ′ , Pn ) if

       rw (µ(w)) − rm (µ(m))  w∈W  m∈M        ′ ′ < rw (µ (w)) − rm (µ (m)) . w∈W  m∈M The concept of gender balancedness is an attempt to equalize levels of partner satisfaction across the two sides of the market. Alternatively, it could instead be the priority to maintain a balance across all individuals. Example 12. f respects balancedness across individuals; that is, f (µ, Pn ) > f (µ′ , Pn ) if





|ra (µ(a)) − rb (µ(b))|

a∈M ∪W b∈M ∪W \a

<





  ra (µ′ (a)) − rb (µ′ (b)) .

a∈M ∪W b∈M ∪W \a

With definition and examples in hand, we are ready to argue that it is appropriate, that is, reasonable and useful, to consider SWFs even though their cardinal values may, for a given matching, depend only on participants’ ordinal preferences. Here are three arguments. 1. Preference profiles are ordinal and do not express intensity of preference, but that does not prevent a central authority, social planner or mechanism designer from possessing preferences over matchings that are held with varying degrees of intensity. 2. Preference profiles do not express intensities of preference, but a participant (man or woman) probably does hold preferences over mates or even matchings with various degrees of intensity. These preferences might be expressible by an SWF which could perhaps be used by the participant to decide whether to participate in the market. For example, the preferences of participant a might be represented by an a-focused SWF, while those of a more community minded participant b might be represented by some combination of a b-focused SWF and a Rawlsian SWF. 3. SWFs are powerful tools that generate valid ordinal conclusions. For example, a study of the utilitarian SWF often leads to conclusions like ‘‘If preferences are similar to what we have seen in the past, the average man can expect a match under µ that he ranks seven places better than his match under µ′ ’’ (for example, see Boudreau and Knoblauch, 2010).

Example 9. f is Smith; that is, for every P , f is maximal on the Smith Set. Our next example is simply a generalization of the already mentioned priority of stability to a family of SWFs.

4 We acknowledge that the term ‘‘envy-free’’ has varied meanings in the game theory literature, but here refer to it within the confines of the works referenced, particularly Romero-Medina (2005).

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J.W. Boudreau, V. Knoblauch / Mathematical Social Sciences 67 (2014) 27–33

3. The price of stability A natural way to measure the price of stability is to compare the best stable matching outcome to the best matching outcome, where the best matching outcome could possibly be unstable. This approach follows that of Anshelevich et al. (2004), who first introduced the term ‘‘price of stability’’ as a measurement of inefficiency for network games,5 and is similarly used in the context of matching markets by Anshelevich et al. (2009, 2013) and Boudreau and Knoblauch (2013). Definition 2. For a matching market with preference profile Pn , the price of stability for a social welfare function is PofS (f , Pn ) =



max{f (µ, Pn ) : µ ∈ M } max{f (µ, Pn ) : µ ∈ S }

 .

Theorem 1. There is a price tag attached to stability for Examples 3– 6, 8, 11 and 12, n ≥ 3. Proof. We treat the examples one by one. Example 3. f is the utilitarian social welfare function. Define Pn by

w1 : m1 , m2 , m3 , . . . w2 : m1 , m2 , m3 , . . . w3 : m2 , m3 , m1 , . . . wk : mk , . . . for k > 3

Example 8. Suppose f is Rawlsian. For Pn , µ and µs as in the proof for Example 3, max{ra (µ(a)) : a ∈ M ∪ W } = 2 < 3 = max{ra (µs (a)) : a ∈ M ∪ W }. Therefore, f (µ, Pn ) > f (µs , Pn ) so that PofS (f , Pn ) > 1. m1 m2 m3 mk mn

: w1 , w2 , w3 , . . . , wn : w2 , w3 , w4 , . . . , wn , w1 : w3 , w4 , w5 , . . . , wn , w1 , w2 : wk , wk+1 , . . . , wn , w1 , . . . , wk−1 for 1 < k < n : wn , w1 , w2 , . . . , wn−1

w1 : m2 , m3 , m4 , . . . , mn , m1 w2 : m3 , m4 , m5 , . . . , mn , m2 , m1 w3 : m4 , m5 , m6 , . . . , mn , m2 , m3 , m1 wk : mk+1 , mk+2 , mk+3 , . . . , mn , m2 , . . . , mk , m1 for 1 < k < n wn : m2 , m3 , m4 , . . . , mn , m1 . The unique stable matching is µs = µM =µW = (mk , wk )for  all k ≤ n, which yields  w∈W rw (µs (w)) − m∈M rm (µs (m)) = n2 − 2n + 1. But, for n even, the matching µ = (mk , wk+ n −1 ) for 2

+ 1 and µ = (mk , wk− 2n −1 ) for 2n + 2 < k ≤ n yields    = n − 1. For n odd, the w∈W rw (µ(w)) − m∈M rm (µ(m)) 1 matching µ = (mk , wk+ n−1 ) for k ≤ n+ and µ = (mk , wk− n+1 ) 2 2 2    n+1   for 2 < k ≤ n yields w∈W rw (µ(w)) − m∈M rm (µ(m)) = s 0. Either way, f (µ, Pn ) > f (µ , Pn ) so that PofS (f , Pn ) > 1.

k ≤

n 2

Example 12. f respects balancedness. See previous treatment for Example 11. 

where the ellipses indicate the remaining preferences are irrelevant. Since µM is men optimal, µW is women optimal and for this Pn µM = µW , the unique stable matching is µs = (m1 , w2 ), (m2 , w3 ), (m3 , w1 ), (mk , wk ) for k > 3. Then f (µs , Pn ) = 2n1+5 . For 1 . 2n+3

Therefore PofS (f , Pn ) =

Example 4. f is a-focused. Without loss of generality, suppose f is m3 -focused. For Pn , µ, and µs as in the previous proof, rm3 (µ(m3 )) = 1 < 3 = rm3 (µs (m3 )). Therefore, f (µ, Pn ) > f (µs , Pn ) so that PofS (f , Pn ) > 1. Example 5. f is strictly female monotonic. Define Pn by m1 m2 m3 mk

Example 6. f is strictly male monotonic. Follows from the previous proof by symmetry.

 

: w2 , w1 , w3 , . . . : w2 , w3 , w1 , . . . : w3 , w2 , w1 , . . . : wk , . . . for k > 3

µ = (mk , wk ) for all k, f (µ, Pn ) = 2n+5 > 1. 2n+3

Then the unique stable matching is µs = µM = µW = (mk , wk ) for k ≥ 1. Consider µ = (m1 , w2 ), (m2 , w1 ), (mk , wk ) for k > 2. Then µ female Pareto dominates µs . Therefore f (µ, Pn ) > f (µs , Pn ) so that PofS (f , Pn ) > 1.

Example 11. f respects gender balancedness. Define Pn by

For an SWF f , we will say there is a price tag attached to stability if PofS (f , Pn ) > 1 for some Pn . Our investigation of the price of stability will proceed in three stages in which we present three types of evidence to show that it is common for stability to come with a price tag attached and that the price of stability can be substantial. First we prove that stability comes with a price tag for seven of our examples. This is an important first step, as the fact that each member of this fairly diverse group of SWFs entails a price tag when stability is required strongly suggests that many others do as well.

m1 m2 m3 mk

w1 : m2 , m1 , m3 , . . . w2 : m1 , m2 , m3 , . . . w3 : m2 , m1 , m3 , . . . wk : mk , . . . for k > 3.

: w1 , w3 , w2 , . . . : w2 , w1 , w3 , . . . : w3 , . . . : wk , . . . for k > 3

5 Anshelevich et al. define the price of stability for a network game as the ratio of the solution quality at the best Nash equilibrium relative to that of the global optimum (2004, p. 1603). Roughgarden and Tardos (2007) similarly use a quotient to measure inefficiency in the context of more general non-cooperative games.

Remark. PofS (f , Pn ) = 1 trivially for all n and Pn for f from Examples 1, 10 or 2 (this last since every stable matching is Pareto efficient). One nontrivial example for which PofS (f , Pn ) = 1 for all n and Pn is Example 9, for which the upper bound of PofS (f , Pn ) = 1 because it can be shown that the set of all stable matchings is a subset of the Smith set. Proposition 1. If an SWF f is Smith, then PofS (f , Pn ) = 1 for all n ≥ 1 and Pn . Proof. Fix Pn . Suppose µs ∈ S and µ ∈ M . Then for m ∈ M , rm (µ(m)) < rm (µs (m)) implies rµ(m) (m) > rµ(m) (µs (µ(m))) or else (m, µ(m)) would be a blocking pair for µs , contradicting its stability. It follows that if µs ∈ S then no matching defeats µs in a head to head election. Therefore, S is a subset of the Smith set. Since f is Smith, S is a subset of arg max{f (µ, Pn ) : µ ∈ M }. Therefore, PofSn(f , Pn ) = 1 for all Pn .  Concerning Example 7, the prospect of a price tag on stability holds for some but not all strictly monotonic SWFs. For instance, we have already seen that there is a price tag attached to stability for the utilitarian SWF, but this is not the case for the sum of the utilitarian SWF and the SWF of Example 1. In that case the relative importance (in cardinal terms) of stability in the combined SWF would ensure that PofS (f , Pn ) = 1 for all Pn .

J.W. Boudreau, V. Knoblauch / Mathematical Social Sciences 67 (2014) 27–33

Notice that our definition of a price tag attached to stability involves maximizing over all preference profiles. Therefore for a given SWF the price of stability might be one for all but a few preference profiles. In the second stage of our analysis we show that for two of our examples the price tag attached to stability is not due to rare and unusual preference profiles. In fact, for those two examples, for large n, the price of stability is greater than one for nearly all preference profiles, where the meaning of ‘‘nearly all’’ is made precise in the statement of Theorem 2. Furthermore, Theorem 2 also shows that for many SWFs the expected price of stability is greater than 1.

Next, each of the following inequalities holds for sufficiently large n. Q3 <

×  ≤ K

By Theorem 1, the last inequality in the statement of Theorem 2(c) holds for SWFs that are utilitarian, a-focused, Rawlsian, strictly male monotonic, strictly female monotonic, gender balanced or balanced. For the proof of Theorem 2 we will need a brief description of the men-propose McVitie and Wilson (1971) algorithm, which for any ordering mi1 , mi2 , . . . , min of men provides an nround sequence of proposals leading to µM . First mi1 proposes to a woman and they form a temporary pair. Before round k, mi1 , mi2 , . . . , mik−1 are engaged. Round k begins when mik proposes to his favorite woman. If she is engaged she either rejects mik or accepts him and rejects her current partner. The rejected man proposes to his next favorite, and so on. Round k ends when a woman receives her first proposal. At the end of round n, the matching µM has been achieved. Proof of Theorem 2(a). Without loss of generality, suppose a = m1 and rm1 (w1 ) = 1. It is sufficient to prove limn→+∞ Prob(µM (m1 ) ̸= w1 ) = 1. Given ϵ > 0 choose positive integer K > 1ϵ . Consider three procedures. Procedure 1 is the McVitie–Wilson algorithm with proposing sequence m1 , m2 , . . . , mn . Procedure 2 is exactly [(n ln n)/2] proposals long and consists of Procedure 1 truncated after [(n ln n)/2] proposals if necessary; or finished with m1 repeatedly proposing to w1 after w1 receives her K th proposal or if Procedure 1 ends in fewer than [(n ln n)/2] proposals. Procedure 3 is [(n ln n)/2] − (K − 1)n draws with replacement from an urn containing n balls. For i = 1, 2 let Qi be the probability that Procedure i ends with fewer than K proposals to w1 . Let Q3 be the probability that Procedure 3 ends with fewer than K draws of ball 1. Then Q2 < Q3 , since first of all in Procedure 2 more than [(n ln n)/2] − (K − 1)n proposals are made with w1 in play. This holds because w1 is out of play only if a man is making a proposal after proposing to w1 and he is not m1 in the repeated-proposal-to-w1 mode. There are at most K − 1 men who make such proposals and each must make fewer than n such proposals. Second, in each proposal with w1 in play, the probability that w1 is proposed to is at least 1/n.

1

n−1

n

n

[(n ln n)/2]−(K −1)n−j

[(n ln n)/2] − (K − 1)n K

 K  1

×

n

 ≤ K

n−1



[(n ln n)/2]−(K −1)n−K

n

n ln n

  K 

K

(n ln n)K ≤ K K!

Prob(max{f (µ, Pn ) : µ ∈ M } > max{f (µs , Pn ) : µs ∈ S })

E (max{f (µ, Pn ) : µ ∈ M }) > E (max{f (µs , Pn ) : µs ∈ S }).

j

 j 

(a) f is a-focused and for each n a preference profile is chosen uniform randomly, or (b) if f is utilitarian and for each n men’s preferences are identical and women’s are chosen uniform randomly, then

(c) Also, if f is a SWF and Pn is drawn randomly from a uniform distribution of preference profiles such that there exists a matching µ∗ and a preference profile Rn with non-zero probability satisfying f (µ∗ , Rn ) > max{f (µs , Rn ) : µs ∈ S }, then

 K   [(n ln n)/2] − (K − 1)n j =1

Theorem 2. If

→ 1 as n → +∞.

31

1

n−1

n

n

 K  1 n

(n ln n)/3

1

ln n

√ 4

e

1

(ln n)K √ < ϵ. 4 n

Finally, by the definitions of Procedures 1 and 2, if Procedure 1 ends with fewer than K proposals to w1 and Procedure 2 ends with at least K proposals to w1 , it must be that Procedure 1 ends after fewer than [(n ln n)/2] proposals. By a result of Pittel (1989, Theorem 2) this occurs with probability less than ϵ for sufficiently large n. In summary, given ϵ > 0, for sufficiently large n Q1 ≤ Q2 + ϵ ≤ Q3 + ϵ ≤ 2ϵ . Since Prob(µM (m1 ) ̸= w1 ) is greater than or equal to the probability that w1 receives at least 1/ϵ proposals times the probability that µM (m1 ) ̸= w1 given that w1 receives at least 1/ϵ proposals, for sufficiently large n, Prob(µM (m1 ) ̸= w1 ) ≥ (1 − 2ϵ)(1 − ϵ).  Proof of Theorem 2(b). Before beginning, we refer the reader to Pittel’s (1989) Theorem 1 for help understanding the structure of the proof of our Theorem 2(b). Since men’s preferences are identical an exchange of partners has no effect on men’s aggregate satisfaction, so we need to consider only women’s aggregate satisfaction. Without loss of generality assume rmi (wn ) = n for all i. For Pn as in the hypotheses, the probability that wn receives her kth choice under the unique stable matching µs is 1/n for k = 1, 2, . . . , n. Therefore, the probability that wn is matched with her [n/ ln n]th choice or worse (where [n/ ln n] is the integer part of n/ ln n) is 1−

[n/ ln n] − 1 n

≥1−

n/ ln n n

=1−

1 ln n

→ 1 as n → +∞.

Now suppose µs (wn ) is wn ’s [n/ ln n]th choice or worse. There are at least [n/ ln n] − 1 women matched with men wn prefers to µs (wn ). We can assume µs (wi ) = mi and rwn (mi ) = i for i = 1, 2, . . . , [n/ ln n] − 1 and µs (wn ) = mn . Then for i = 1, 2, . . . , [n/ ln n] − 1 if wn exchanges partners with wi , rwn (mn ) − rwn (mi ) ≥ [n/ ln n]− i. The trade results in an increase in aggregate satisfaction if rwi (mn ) − rwi (mi ) < [n/ ln n] − i, which will happen if rwi (mn ) < rwi (mi ) + [n/ ln n] − i. Since µs = µM under the hypotheses on preferences and since rmn (wn ) = n, mn must have been rejected by wi before proposing to wn . Therefore rwi (mi ) < rwi (mn ). The trade with wi results in an increase in aggregate satisfaction if rwi (mi ) < rwi (mn ) < rwi (mi ) + [n/ ln n] − i. The probability of this occurring is Prob(rwi (mi ) ≤ rwi (mn ) < rwi (mi )+

[n/ ln n] − i) ≥ [n/ ln nn]−(i+1) , where the inequality follows from the fact that wi ’s preferences are uniform random and the fact that if rwi (mi ) + [n/ ln n] − i > n then the probability in question is 1.

32

J.W. Boudreau, V. Knoblauch / Mathematical Social Sciences 67 (2014) 27–33

Therefore the probability that none of the trades between wn and wi increase net aggregate satisfaction for i = 1, 2, . . . , [n/ ln n] − 1 is at most [n/ ln n]−1

Πi=1

 1−

n/ ln n − (i + 1) n



 n/4 ln n n/2 ln n ≤ 1− n

where the inequality follows by considering only slightly more than quarter of the factors, each of which is bounded above  the first  n/2 ln n by 1 − n . Continuing

 1−

n/2 ln n

n/4 ln n

 =

n

 as n → +∞ since

1−

1−

1

→0

2 ln n

2 ln n

1

2 ln n n/8 ln2 n

2 ln n

→ 1/e as n → +∞.

Finally, Pr(max{f (µ, Pn ) : µ ∈ M } > max{f (µs , Pn ) : µs ∈ S })

≥ Pr(rwn (µs (wn )) ≥ [n/ ln n]) × Pr(some wn − wi trade increases f | (rwn (µs (wn ))) ≥ [n/ ln n]) → 1 as n → +∞ since we showed both factors approach 1.



Proof of Theorem 2(c). The inequality follows from the assumption f (µ∗ , Rn ) > max{f (µs , Rn ) : µs ∈ S }, the fact that for all Pn , max{f (µ, Pn ) : µ ∈ M } ≥ max{f (µs , Pn ) : µ ∈ S } and the assumption that Rn is assigned a positive probability.  In our third and final theorem, we prove that the upper bound on the price of stability can be substantial using four specific forms of SWFs. The first is the utilitarian SWF from Example 3, while the rest are specific members of the a-focused, Rawlsian, and genderbalanced families of SWFs. Theorem 3. If n ≥ 3, for the utilitarian SWF, maxPn {PofS (f , Pn )} ≥ n/3 for the a-focused SWF, f (µ, Pn ) = 1/ra (µ(a)), maxPn {PofS (f , Pn )} ≥ n for the Rawlsian SWF, f (µ, Pn ) = 1/ max{ra (µ(a)) : a ∈ M ∪ W }, maxPn {PofS (f , Pn )} ≥ n/2 for the gender-balanced SWF, 1 f (µ, Pn ) =  , maxPn {PofS (f , Pn )} r (µ(w))− r (µ(m)) +1

|

≥

n2 −2n+2 . n

w∈W w

m∈M m

Proof. Consider the preference profile Pn : m1 : w2 , w1 , . . . m2 : w2 , . . . , w3 , w1

. .

mk : wk , . . . , wk+1 , w1

. .

mn−1 : wn−1 , . . . , wn , w1 mn : wn , . . . , w 1 w1 : m1 . . . mn w2 : m1 , m2 , . . .

. . wk : mk−1 , mk , . . . . . wn : mn−1 , mn , . . .

|

where the ellipses indicate arbitrary preferences. Since µM = µW , there is a unique stable matching µs : (m1 , w2 ), (m2 , w3 ), . . . , (mn−1 , wn ), (mn , w1 ) and for the utilitarian SWF, f (µs , Pn ) = 1 1 . For µ : (m1 , w1 ), (m2 , w2 ), . . . , (mn , wn ), f (µ, Pn ) = 3n . n 2 +2 Therefore maxPn {PofS (f , Pn )} ≥

n2 +2 3n

≥ 3n . 1 For the a-focused SWF f (µ, Pn ) = r (µ( , without loss of a)) a generality we take a = mn . Then for Pn , µ and µs from the first part of this proof, f (µ, Pn ) = 1 and f (µs , Pn ) = 1n so that maxPn {PofS (f , Pn )} ≥ n. For the Rawlsian SWF f (µ, Pn ) = 1/ max{ra (µ(a)) : a ∈ M ∪ W } for Pn , µ, µs as above, f (µs , Pn ) = 1/rmn (µs (mn )) = 1n and f (µ, Pn ) = 1/rwn (µ(wn )) = 21 so that maxPn {PofS (f , Pn )} ≥ n/2.

For the gender-balanced SWF 1 f (µ, Pn ) =  | w∈W rw (µ(w))− m∈M rm (µ(m))|+1 consult the preference profile and matchings used in the proof of Theorem 1 for Example 11 for n even.  4. Concluding remarks Our first theorem demonstrates that requiring matchings to be stable can lead to potential welfare sacrifices for a wide array of SWFs, while Theorem 3 demonstrates that those sacrifices can be substantial. Theorem 2 shows that, for two of our examples of SWFs, there is a price tag attached to stability for nearly all preference profiles, and for many SWFs the expected price of stability is greater than one. In real-life markets, a social planner or mechanism designer might want a combination of Theorems 2 and 3type information; that is, she might want to know the price of stability defined as an average over a category of preference profiles rather than a maximum over all preference profiles. For example, the social planner might know the approximate level of correlation (the extent to which the preferences are similar) on each side of the market, and/or the approximate level of intercorrelation (the extent to which men prefer women who prefer them) and wish to know the expected-magnitude price of stability. In a companion study (Boudreau and Knoblauch, 2013) we use another tool to address such questions—simulation. The simulation approach involves three steps: the generation of preference profiles, the construction of matchings by the mechanisms to be tested and the evaluation of the social welfare function for each matching. The process is repeated many times to yield an approximation of the expected level of social welfare provided by each mechanism. The social planner then has all the information needed (including the price of stability) to choose a matching mechanism. The advantage of the theoretical approach used in this study is that theory is more rigorous and more transparent. On the other hand the advantage of the simulation approach is that it works for any category of preference profiles and any matching mechanism. A further consideration for social planners or market designers when considering the price of stability, in addition to the restriction of certain types of preference profiles, might be the restriction of some types of matchings. In matching markets where agents are not required to list all potential partners as acceptable, for example, stability also requires individual rationality. Further restrictions on which matchings are permissible would likely alter the existence and prevalence of welfare tradeoffs for many cases of SWFs, and we leave a more in-depth consideration that of topic for future work. References Anshelevich, E., Das, S., Naamad, Y., 2009. Anarchy, stability, and utopia: creating better matchings. In: Proc. of the 2nd International Symposium on Algorithmic Game Theory, pp. 159–170. Anshelevich, E., Das, S., Naamad, Y., 2013. Anarchy, stability, and utopia: creating better matchings. Autonomous Agents and Multi-Agent Systems 26 (1), 120–140.

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