- Email: [email protected]
A marriage matching mechanism menagerie
Accepted Manuscript A marriage matching mechanism menagerie James W. Boudreau, Vicki Knoblauch PII: DOI: Reference:
S0167-6377(16)30244-9 http://dx.doi.org/10.1016/j.orl.2016.12.001 OPERES 6175
To appear in:
Operations Research Letters
Received date: 30 May 2016 Revised date: 1 December 2016 Accepted date: 1 December 2016 Please cite this article as: J.W. Boudreau, V. Knoblauch, A marriage matching mechanism menagerie, Operations Research Letters (2016), http://dx.doi.org/10.1016/j.orl.2016.12.001 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Manuscript Click here to view linked References
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
A Marriage Matching Mechanism Menagerie James W. Boudreau∗† Dept. of Economics and Finance University of Texas Rio Grande Valley and Vicki Knoblauch‡ Dept. of Economics University of Connecticut December, 2016
Abstract For each of several measures of social welfare we present a marriage matching mechanism that produces a welfare maximizing matching, and our basic approach generalizes to many other welfare measures.
JEL Classification Codes: C78, D63. Keywords:Matching; assignment; social welfare.
Corresponding author. Email: [email protected]. ‡ Email: [email protected].
∗
†
1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
1
Introduction
Matching markets and assignment problems are popular topics with researchers involved in the field of mechanism design. In the case of the former, two distinct sets of agents, each with their own preferences over partners from the other side of the market, must be paired off; in the latter a set of agents with preferences must be matched to indivisible objects that may lack preferences or priorities of their own. Both tend to face difficulties if left unfettered due to the fact that most feature prohibitions on monetary transfers and also a distinct difficulty in terms of re-pairing over time. Along with the fact that assignments in these contexts tend to be both long-term and extremely important to those involved (classic examples include the marriage market, entry-level labor markets, public school assignment, and room or housing assignments) the process by which the allocation of prized partners or possessions is conducted can be quite controversial. It is therefore crucial to consider the social welfare of those involved when designing such procedures. In this paper we present a family of algorithmic procedures to match two sets of agents in a way that maximizes a wide variety of—or combination of—welfare criteria in polynomial time. Though we specify our results in terms of a one-to-one matching problem and in terms of several specific welfare criteria, the general approach is quite flexible and can easily be adapted for alternative criteria or for assignment problems. Loosely speaking, our approach is to first reduce the graph representing the general matching problem to a collection of subgraphs that satisfy a specified set of specified welfare criteria and then allow agents to choose their partners according to an exogenously specified ordering, resulting in a match that is Pareto-optimal among all matchings that satisfy the desired combination of welfare measures. As chronicled in the next section, our approach is motivated by the fact that different notions of social welfare are increasingly being recognized as important in the context of matching markets. Deferred acceptance algorithms based on the celebrated work of Gale and Shapley [10] have come to form the basis of most currently implemented centralized matching procedures due to their stability properties (for a history of the use of deferred acceptance algorithms see [21]). In the event that stability is not the primary goal, however, and market designers wish to maximize alternative notions of social welfare, our approach 2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
may prove quite useful.
2
Motivation and Previous Literature
It is well-known that the Gale-Shapley deferred acceptance algorithm produces a stable matching for any marriage matching market (to simplify terminology we use the marriage market interpretation to refer to one-to-one two-sided matching markets, labeling one set of agents men and the other women; many other interpretations are possible). That is, it produces a matching without blocking pairs, meaning there will never be a man-woman pair such that each prefers the other to his or her assigned mate [10]. This type of outcome has been celebrated by economists as it takes advantage of any possible mutual gains and thus results in Pareto-efficiency, but it is easy to imagine two-sided matching scenarios in which stability is unimportant, or at least not the most desired property of a matching. For example, a strong central authority or a system of binding contracts may enforce an unstable matching. Or complexity issues, incomplete information, time constraints or other barriers may make it extremely unlikely that members of a potential blocking pair will find each other. Other than stability, two popular criteria for evaluating matchings are the utilitarian and Rawlsian welfare measurements. The utilitarian measure evaluates matchings based on the unweighted sum of agents’ partner rankings in a match, while the Rawlsian measure considers only the ranking(s) of the agent(s) with the worst (meaning highest) ranked partner in a match (named for the difference principal of John Rawls, the Rawlsian criterion of social justice roughly asserts that the worst off in a society should be made as well off as possible [18]). Axtell and Kimbrough [2], for example, argue that “the nearly universal focus on stable matchings in this [matching] literature is misguided at best,” and use simulations to highlight the utilitarian losses incurred by the Gale-Shapley algorithm as compared to a distributed, decentralized matching process. Masarani and Gokturk [17], meanwhile, prove the incompatibility of stability and the Rawlsian criterion, and Brams and Kilgour [7] outline a procedure—which can be thought of as a special case of one of the algorithms we present below—to attain Rawlsian-optimal matchings. Boudreau and 3
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Knoblauch [5] demonstrate how the suboptimal performance of the Gale-Shapley algorithm, in terms of both the utilitarian and Rawlsian measurements, varies according to properties of a marriage market’s preferences, while Hafalir and Miralles [11] find mechanisms to attain utilitarian and Rawlsian optimality in special types of large assignment markets. The utilitarian measure represents aggregate welfare while the Rawlsian measure focuses on the welfare of the least advantaged. A third form of welfare criterion popular in the literature is one that focuses on the equality of outcomes, which we refer to as balancedness. Such considerations are particularly important for matching markets, since it is well known that the Gale-Shapley algorithm favors one side of the market at the expense of the other. Accordingly, much attention has been given to matchings that attempt to balance the interests of both sides of the market. For example, Klaus and Klijn [14, 15] establish the existence of “median” stable matchings in various matching formats, and Romero-Medina [19, 20] provides algorithms to attain gender-balanced stable matchings in marriage markets. Boudreau and Knoblauch [6], however, show that stability is not generally compatible with balancedness across genders or individuals. There are many other possible notions of welfare for matching markets (for example those in [6]), and the algorithms we describe below are widely applicable to any which can be described as properties of a graph. For brevity, however, we illustrate our results by focusing on the three basic measurements just described, in addition to combinations thereof. Similarly, notice that in our use of Rawlsian, utilitarian and balanced measures of social welfare, we are implicitly assuming that participants’ preferences over mates are cardinal and we are also making interpersonal comparisons of utility. For example, when using the Rawlsian measure of social welfare in a marriage matching market, a social architect who wishes to maximize the utility of the worst off participant and who does so by choosing a matching that minimizes the highest ranking number any participant attaches to his assigned mate is assuming that if k < l then the utility received by participant i upon being matched with his or her k th choice is greater than the utility received by participant j (j 6= i) upon being matched with his or her lth choice. Budish [8] emphasizes the nuances of different matching environments, and the fact
4
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
that the design of any individual market should be based on its unique goals rather than those that have been celebrated elsewhere. Our work is in the spirit of that criticism, providing mechanisms that can match agents to achieve a wide variety of goals. Moreover, the structure of our approach is similar to that of Kesten [13] and Tang and Yu [22], who also proceed by iteratively refining a matching problem to proceed toward a desired solution.
3
Preliminaries
A marriage matching (or more simply matching) for a set M = {m1 , m2 , . . . , mn } of men and a set W = {w1 , w2 , . . . , wn } of women is a 1-1, onto function µ: M ∪ W → M ∪ W such that µ(M ) = W and for all m ∈ M , µ(µ(m)) = m. Let M M denote the collection of all marriage matchings for M and W . For x ∈ M ∪ W , let x be a linear order representing x’s preferences over members of M ∪ W of the opposite gender. Then x gives rise to a ranking rx such that if yi1 x yi2 x . . . x yin , then rx (yi1 ) = 1, rx (yi2 ) = 2, . . . , rx (yin ) = n. Let be the preference profile ( m1 , m2 , . . . , mn , w1 , w2 , . . . , wn ). Let π be the collection of all preference profiles for M and W . In nearly all that follows, we begin with the assumption that M, W and are fixed. We will be dealing with certain properties of matchings, including but not limited to the following: a matching µ0 is Rawlsian if it minimizes maxx∈M ∪W rx (µ(x)), utilitarian if P it minimizes x∈M ∪W rx (µ(x)) and Pareto optimal or efficient if for no µ ∈ M M is it the case that rx (µ(x)) ≤ rx (µ(x0 )) for all x ∈ M ∪ W and that the inequality is strict for some x ∈ M ∪ W . Given ∈ π, a property P of matchings is non-null if there exists at least one matching with property P . A social welfare function (SWF) is a real-valued function f : π × M M → R. Whenever is fixed, we will write f (µ) rather than f ( , µ). It will sometimes be useful to describe a property of matchings in terms of a SWF. For example, a matching µ0 is Rawlsian if fR (µ0 ) = maxµ∈M M fR (µ) where fR (µ) = − maxx∈M ∪W rx (µ(x)). A bipartite graph G = (U, V ; E) consists of two disjoint finite sets of vertices U and 5
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
V and a collection E of edges with one endpoint in each set. The Hopcroft-Karp (HK) algorithm [12] inputs a bipartite graph, (U, V ; E) with, for our purposes, |U | = |V | and in polynomial time outputs a matching µ for U and V such that {u, µ(u)} ∈ E for all u ∈ U , or outputs “no” if no such matching exists. The Hungarian (Hun) algorithm [16] inputs a complete bipartite graph (one with an edge {u, v} for each u ∈ U, v ∈ V ) with an integer assigned to each edge and in polynomial time outputs a matching that minimizes the sum of the edge values.
4
Marriage Matching Algorithms
For a given preference profile, a property P of matchings is graph determined if there exists a bipartite graph G0 = (M, W ; E) such that for every µ ∈ M M , µ has property P if and only if µ is a subgraph of G0 . Given a preference profile, a non-null, graph-determined property P with associated graph G0 and an enumeration x1 , x2 , . . . , x2n of M ∪ W , the following algorithm uses repeated applications of the HK algorithm to produce a matching that is Pareto optimal among all matchings with property P .
4.1
Lexicographic Optimizer (LO)
Given a preference profile, a bipartite graph G0 = (M, W ; E0 ) associated with a non-null, graph-determined property P and an enumeration x1 , x2 , . . . , x2n of the men and women, we begin by initializing variables. E ← E0 \{{x1 , y}: rx1 (y) > 1}, G ← (M, W, E), µ ← ∅, c ← 1, d ← 1. Step 1. Apply HK to G. If HK finds a matching µ0 , then µ ← µ0 and go to Step 3. Else, go to Step 2. Step 2. E ← E ∪ {{xc , xi } ∈ E0 : rxc (xi ) = d + 1 and i > c}, d ← d + 1, G ← (M, W, E) and go to Step 1. 6
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Step 3. If c < 2n, then c ← c + 1, d ← 1, E ← E\{{xc , y}: rxc (y) > 1}, G ← (M, W, E) and go to Step 1. Else, Go to Step 4. Step 4. Output µ. End. Since the output µ is a subgraph of G0 , µ has property P . Since repetitions of Steps 1 and 2 first yield an optimal property P match for x1 , then for x2 with x1 ’s match fixed, then for x3 with x1 ’s match and x2 ’s match fixed, etc., the output µ is Pareto optimal among matchings with property P . Since HK is called at most twice for each edge of G0 and there are at most n2 edges, and since HK is polynomial in n, LO is polynomial in n. We begin with a simple application.
4.2
Pareto Optimal Matchings
Let P be the all-inclusive property; that is, a matching µ has property P if µ is a matching. Then P is graph determined and the unique bipartite graph G0 associated with P is the complete bipartite graph; that is, {m, w} is an edge of G0 for all m ∈ M , w ∈ W . Applying LO to G0 for a given preference profile and any enumeration of the participants produces a matching that is Pareto optimal among property P matchings and is therefore Pareto optimal (since P is all inclusive). It is well known that the Gale-Shapley (GS) men-propose algorithm produces Pareto optimal matchings. The above application of LO differs from GS in that it produces matchings that are not necessarily stable.
4.3
Rawlsian Matchings
A matching µ is Rawlsian if max {rx (µ(x)} = 0min
x∈M ∪W
max {rx (µ0 (x))}
µ ∈M M x∈M ∪W
7
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
where M M is the collection of all matchings. The quantity on the right side of the equation is called the minimax . A Rawlsian algorithm. Given a preference profile, for k with 1 ≤ k ≤ n, let Gk be the bipartite graph whose edges are {{m, w}: rm (w) ≤ k and rw (m) ≤ k}. Apply HK to G1 , G2 , . . . until HK finds a matching and therefore has found the minimax k0 . If P is the Rawlsian property, then P is graph determined and G0 = Gk0 is an associated graph. Applying LO to G0 with an arbitrary enumeration of M ∪ W produces a matching that is Pareto optimal among Rawlsian matchings and therefore Pareto optimal.
4.4
Balanced Matchings
A matching µ is balanced if max |rm (µ(m)) − rµ(m) (m)| = 0min max |rm (µ0 (m)) − rµ0 (m) (m)|.
m∈M
µ ∈M M m∈M
The quantity on the right side of the equation is called the minimax mismatch. A Balancing Algorithm. For a given preference profile and k with 0 ≤ k ≤ n − 1, let Hk be the bipartite graph whose edges are {{m, w}: |rm (w) − rw (m)| ≤ k}. Apply HK to H0 , H1 , H2 , . . . until HK finds a matching and has therefore found the minimax mismatch k0 . If P is the balanced property, then P is graph determined and G0 = Hk0 is an associated graph. Applying LO to G0 with an arbitrary enumeration of M ∪ W produces a matching that is Pareto optimal among balanced matchings.
4.5
Constrained Matchings
Definition. A collection {S(x)}x∈M ∪W of subsets of M ∪ W such that S(m) ⊆ W for all m ∈ M and S(w) ⊆ M for all w ∈ W is a constraint. Definition. A matching µ is constrained by constraint {S(x)}x∈M ∪W if µ(x) ∈ S(x) for all x ∈ M ∪ W. 8
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Definition. A matching µ has property P ({S(x)}x∈M ∪W ) if µ is constrained by {S(x)}x∈M ∪W . A constraining algorithm. Given a preference profile and constraint {S(x)}x∈M ∪W , let G0 be the bipartite graph with edges {{m, w}: m ∈ S(w) and w ∈ S(m)}. Apply LO to G0 with LO modified so that if the first application of HK fails to find a matching, then LO outputs the message that no matching has property P ({S(x)}x∈M ∪W ). Barring such a failure, LO outputs a matching that is Pareto optimal among all matchings with property P ({S(x)}x∈M ∪W ).
4.6
Utilitarian Matchings
P The utilitarian SWF, f ( , µ) = − x∈M ∪W rx (µ(x)), is maximized by the Hun algorithm acting on the bipartite graph G0 = {M, W ; {{m, w}: m ∈ M, w ∈ W }} and with edge {m, w} assigned value rm (w) + rw (m). Hun finds a matching that minimizes the sum of the edge values and therefore maximizes the utilitarian SWF. A utilitarian matching is also Pareto optimal.
4.7
Lexicographic Maximization of Ordered Pairs of Social Wel-
fare Functions For a given matching problem, we will say a matching µ lexicographically maximizes the ordered pair (f, g) of SWFs if µ0 a matching implies f (µ) > f (µ0 ) or both f (µ) = f (µ0 ) and g(µ) ≥ g(µ0 ). Here are some polynomial time algorithms that produce matchings that lexicographically maximize ordered pairs of SWFs. 4.7.1
f The Rawlsian SWF and g the Utilitarian SWF
Use HK to find k0 , the minimax as in Subsection 4.3. Use Hun to find a matching µ that maximizes the SWF
9
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
h(µ) = −
P
x∈M ∪W
j(x), where j(x) =
rx (µ(x)) if rx (µ(x)) ≤ k0 2n2 else
The matching µ is Rawlsian since any Rawlsian matching µR satisfies h(µR ) ≥ −2nk0 ≥ −2n2 while any non-Rawlsian matching µ0 satifies h(µ0 ) < −2n2 . Therefore µ maximizes g restricted to Rawlsian matchings. Therefore µ lexicographically maximizes (f, g). Similar algorithms lexicographically maximize (f, g) where g is the utilitarian SWF and f is either the balance SWF (f (µ) = − maxm∈M |rm (µ(m)) − rµ(m) (m)|) or a constraint SWF: for a given constraint {S(x): x ∈ M ∪ W }, f (µ) = 4.7.2
1 if µ(x) ∈ S(x) for all x ∈ M ∪ W 0 else
f the Utilitarian SWF and g the Rawlsian SWF
For k = n, n − 1, . . . use Hun to find a matching µk that maximizes the SWF hk (µ) = −
X
jk (x)
x∈M ∪W
rx (µ(x)) if rx (µ(x)) ≤ k 2n2 else until a k is found such that hn (µn ) > hk−1 (µk−1 ). where
jk (x) =
Then µk is utilitarian since µn is utilitarian and therefore µk maximizes (f, g). Similar algorithms maximize (f, g) if f is the utilitarian SWF and g is either the balance SWF or a constraint SWF. 4.7.3
f the Rawlsian SWF and g the Balanced SWF
For 1 ≤ k ≤ n and 0 ≤ l ≤ n − 1 let Gk,l = (M, W ; {{m, w}: rm (w) ≤ k, rw (m) ≤ k and |rm (w) − rw (m)| ≤ l}). Apply HK repeatedly to find the lexicographically minimal ordered pair (k0 , l0 ) such that there is a matching µ0 that is a subgraph of Gk0 ,l0 . Then 10
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
µ0 maximizes (f, g). Also the property “µ maximizes (f, g)” is graph determined with associated graph Gk0 ,l0 . Therefore LO can be applied to find a matching that is Pareto optimal among all matchings that maximize (f, g). Similar algorithms apply to lexicographically maximize any ordered pair formed by choosing each of f and g to be the Rawlsian or balance SWF or a constraint SWF.
5
Discussion
Our methods here are deterministic: at the end of any one of our algorithms each agent’s match is certain. Alternative approaches, however, could allow for some form of randomization in the process. If we define a probabilistic matching as a probability distribution over deterministic matchings, one direction for future research is the search for polynomialtime algorithms that produce probabilistic matchings that yield social welfare levels above those provided by the best deterministic matchings. Probabilistic assignment procedures have previously been used to bring forms of fairness to allocation problems [4] and matching problems [1, 9], for example, and a recent paper by Bogomolnaia [3] shows that the probabilistic rule developed in [4] maximizes a form of Rawlsian welfare. We have already begun to develop algorithms similar in spirit to those we have outlined here but extended to the probabilistic domain. Doing so, however, requires first extending the domain of social welfare functions from the set of all deterministic matchings to the set of all probabilistic matchings, a non-trivial step for even specific social welfare functions. We therefore leave their treatment for a separate paper.
11
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
References [1] Atila Abdulkadiroglu and Tayfun Sonmez, Random serial dictatorship and the core from random endowments house allocation problems, Econometrica 66 (1998), 689– 701. [2] Robert L. Axtell and Steven O. Kimbrough, The high cost of stability in two-sided matching: How much social welfare should be sacrificed in the pursuit of stability?, Proc. of the 2008 World Congress on Social Simulation (2008). [3] Anna Bogomolnaia, Random assignment: redefining the serial rule, Journal of Economic Theory 158 (2015), no. A, 308–318. [4] Anna Bogomolnaia and Herve Moulin, A new solution to the random assignment problem, Journal of Economic Theory 100 (2001), no. 2, 295–328. [5] James W. Boudreau and Vicki Knoblauch, Preferences and the price of stability in matching markets, Theory and Decision 74 (2013), no. 4, 565–589. [6]
, What price stability? social welfare in matching markets, Mathematical Social Sciences 67 (2014), 27–33.
[7] Steven Brams and Marc Kilgour, Two-sided matchings: an algorithm ensuring they are minimax and pareto-optimal, Working Paper MPRA Paper No. 48113, 2013. [8] Eric Budish, Matching ‘versus’ mechanism design, ACM SIGecom Exchanges 11 (2012), no. 2, 4–15. [9] David Gale, College course assignments and optimal lotteries, Working paper, University of California, Berkeley, 1987. [10] David Gale and Lloyd S. Shapley, College admissions and the stability of marriage, The American Mathematical Monthly 69 (1962), no. 1, 9–15. [11] Isa Hafalir and Antonio Miralles, Welfare-maximizing assignment of agents to hierarchical positions, Working paper, 2015. [12] John Hopcroft and Richard Karp, An n5/2 algorithm for maximum matchings in bipartite graphs, Siam Journal of Computing 2 (1973), no. 4, 225–231. 12
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
[13] Onur Kesten, School choice with consent, Quarterly Journal of Economics 125 (2010), no. 3, 1297–1348. [14] Bettina Klaus and Flip Klijn, Median stable matching for college admissions, International Journal of Game Theory 34 (2006), 1–11. [15]
, Smith and rawls share a room: stability and medians, Social Choice and Welfare 35 (2010), 647–667.
[16] H. W. Kuhn, The hungarian method for the assignment problem, Naval Research Logistics Quarterly 2 (1955), no. 7, 83–97. [17] F. Masarani and S. S. Gokturk, On the existence of fair matching algorithms, Theory and Decision 26 (1989), 305–322. [18] John Rawls, A theory of justice, Harvard University Press, Cambridge, MA, 1971. [19] Antonio Romero-Medina, Sex-equal stable matchings, Theory and Decision 50 (2001), 197–212. [20]
, Equitable selection in bilateral matching markets, Theory and Decision 58 (2005), 305–324.
[21] Alvin E. Roth, Deferred acceptance algorithms: History, theory, practice, and open questions, International Journal of Game Theory 36 (2008), 537–569. [22] Qianfeng Tang and Jingsheng Yu, A new perspective on kesten’s school choice with consent idea, Journal of Economic Theory 154 (2014), 543–561.
13
S0167-6377(16)30244-9 http://dx.doi.org/10.1016/j.orl.2016.12.001 OPERES 6175
To appear in:
Operations Research Letters
Received date: 30 May 2016 Revised date: 1 December 2016 Accepted date: 1 December 2016 Please cite this article as: J.W. Boudreau, V. Knoblauch, A marriage matching mechanism menagerie, Operations Research Letters (2016), http://dx.doi.org/10.1016/j.orl.2016.12.001 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Manuscript Click here to view linked References
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
A Marriage Matching Mechanism Menagerie James W. Boudreau∗† Dept. of Economics and Finance University of Texas Rio Grande Valley and Vicki Knoblauch‡ Dept. of Economics University of Connecticut December, 2016
Abstract For each of several measures of social welfare we present a marriage matching mechanism that produces a welfare maximizing matching, and our basic approach generalizes to many other welfare measures.
JEL Classification Codes: C78, D63. Keywords:Matching; assignment; social welfare.
Corresponding author. Email: [email protected]. ‡ Email: [email protected].
∗
†
1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
1
Introduction
Matching markets and assignment problems are popular topics with researchers involved in the field of mechanism design. In the case of the former, two distinct sets of agents, each with their own preferences over partners from the other side of the market, must be paired off; in the latter a set of agents with preferences must be matched to indivisible objects that may lack preferences or priorities of their own. Both tend to face difficulties if left unfettered due to the fact that most feature prohibitions on monetary transfers and also a distinct difficulty in terms of re-pairing over time. Along with the fact that assignments in these contexts tend to be both long-term and extremely important to those involved (classic examples include the marriage market, entry-level labor markets, public school assignment, and room or housing assignments) the process by which the allocation of prized partners or possessions is conducted can be quite controversial. It is therefore crucial to consider the social welfare of those involved when designing such procedures. In this paper we present a family of algorithmic procedures to match two sets of agents in a way that maximizes a wide variety of—or combination of—welfare criteria in polynomial time. Though we specify our results in terms of a one-to-one matching problem and in terms of several specific welfare criteria, the general approach is quite flexible and can easily be adapted for alternative criteria or for assignment problems. Loosely speaking, our approach is to first reduce the graph representing the general matching problem to a collection of subgraphs that satisfy a specified set of specified welfare criteria and then allow agents to choose their partners according to an exogenously specified ordering, resulting in a match that is Pareto-optimal among all matchings that satisfy the desired combination of welfare measures. As chronicled in the next section, our approach is motivated by the fact that different notions of social welfare are increasingly being recognized as important in the context of matching markets. Deferred acceptance algorithms based on the celebrated work of Gale and Shapley [10] have come to form the basis of most currently implemented centralized matching procedures due to their stability properties (for a history of the use of deferred acceptance algorithms see [21]). In the event that stability is not the primary goal, however, and market designers wish to maximize alternative notions of social welfare, our approach 2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
may prove quite useful.
2
Motivation and Previous Literature
It is well-known that the Gale-Shapley deferred acceptance algorithm produces a stable matching for any marriage matching market (to simplify terminology we use the marriage market interpretation to refer to one-to-one two-sided matching markets, labeling one set of agents men and the other women; many other interpretations are possible). That is, it produces a matching without blocking pairs, meaning there will never be a man-woman pair such that each prefers the other to his or her assigned mate [10]. This type of outcome has been celebrated by economists as it takes advantage of any possible mutual gains and thus results in Pareto-efficiency, but it is easy to imagine two-sided matching scenarios in which stability is unimportant, or at least not the most desired property of a matching. For example, a strong central authority or a system of binding contracts may enforce an unstable matching. Or complexity issues, incomplete information, time constraints or other barriers may make it extremely unlikely that members of a potential blocking pair will find each other. Other than stability, two popular criteria for evaluating matchings are the utilitarian and Rawlsian welfare measurements. The utilitarian measure evaluates matchings based on the unweighted sum of agents’ partner rankings in a match, while the Rawlsian measure considers only the ranking(s) of the agent(s) with the worst (meaning highest) ranked partner in a match (named for the difference principal of John Rawls, the Rawlsian criterion of social justice roughly asserts that the worst off in a society should be made as well off as possible [18]). Axtell and Kimbrough [2], for example, argue that “the nearly universal focus on stable matchings in this [matching] literature is misguided at best,” and use simulations to highlight the utilitarian losses incurred by the Gale-Shapley algorithm as compared to a distributed, decentralized matching process. Masarani and Gokturk [17], meanwhile, prove the incompatibility of stability and the Rawlsian criterion, and Brams and Kilgour [7] outline a procedure—which can be thought of as a special case of one of the algorithms we present below—to attain Rawlsian-optimal matchings. Boudreau and 3
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Knoblauch [5] demonstrate how the suboptimal performance of the Gale-Shapley algorithm, in terms of both the utilitarian and Rawlsian measurements, varies according to properties of a marriage market’s preferences, while Hafalir and Miralles [11] find mechanisms to attain utilitarian and Rawlsian optimality in special types of large assignment markets. The utilitarian measure represents aggregate welfare while the Rawlsian measure focuses on the welfare of the least advantaged. A third form of welfare criterion popular in the literature is one that focuses on the equality of outcomes, which we refer to as balancedness. Such considerations are particularly important for matching markets, since it is well known that the Gale-Shapley algorithm favors one side of the market at the expense of the other. Accordingly, much attention has been given to matchings that attempt to balance the interests of both sides of the market. For example, Klaus and Klijn [14, 15] establish the existence of “median” stable matchings in various matching formats, and Romero-Medina [19, 20] provides algorithms to attain gender-balanced stable matchings in marriage markets. Boudreau and Knoblauch [6], however, show that stability is not generally compatible with balancedness across genders or individuals. There are many other possible notions of welfare for matching markets (for example those in [6]), and the algorithms we describe below are widely applicable to any which can be described as properties of a graph. For brevity, however, we illustrate our results by focusing on the three basic measurements just described, in addition to combinations thereof. Similarly, notice that in our use of Rawlsian, utilitarian and balanced measures of social welfare, we are implicitly assuming that participants’ preferences over mates are cardinal and we are also making interpersonal comparisons of utility. For example, when using the Rawlsian measure of social welfare in a marriage matching market, a social architect who wishes to maximize the utility of the worst off participant and who does so by choosing a matching that minimizes the highest ranking number any participant attaches to his assigned mate is assuming that if k < l then the utility received by participant i upon being matched with his or her k th choice is greater than the utility received by participant j (j 6= i) upon being matched with his or her lth choice. Budish [8] emphasizes the nuances of different matching environments, and the fact
4
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
that the design of any individual market should be based on its unique goals rather than those that have been celebrated elsewhere. Our work is in the spirit of that criticism, providing mechanisms that can match agents to achieve a wide variety of goals. Moreover, the structure of our approach is similar to that of Kesten [13] and Tang and Yu [22], who also proceed by iteratively refining a matching problem to proceed toward a desired solution.
3
Preliminaries
A marriage matching (or more simply matching) for a set M = {m1 , m2 , . . . , mn } of men and a set W = {w1 , w2 , . . . , wn } of women is a 1-1, onto function µ: M ∪ W → M ∪ W such that µ(M ) = W and for all m ∈ M , µ(µ(m)) = m. Let M M denote the collection of all marriage matchings for M and W . For x ∈ M ∪ W , let x be a linear order representing x’s preferences over members of M ∪ W of the opposite gender. Then x gives rise to a ranking rx such that if yi1 x yi2 x . . . x yin , then rx (yi1 ) = 1, rx (yi2 ) = 2, . . . , rx (yin ) = n. Let be the preference profile ( m1 , m2 , . . . , mn , w1 , w2 , . . . , wn ). Let π be the collection of all preference profiles for M and W . In nearly all that follows, we begin with the assumption that M, W and are fixed. We will be dealing with certain properties of matchings, including but not limited to the following: a matching µ0 is Rawlsian if it minimizes maxx∈M ∪W rx (µ(x)), utilitarian if P it minimizes x∈M ∪W rx (µ(x)) and Pareto optimal or efficient if for no µ ∈ M M is it the case that rx (µ(x)) ≤ rx (µ(x0 )) for all x ∈ M ∪ W and that the inequality is strict for some x ∈ M ∪ W . Given ∈ π, a property P of matchings is non-null if there exists at least one matching with property P . A social welfare function (SWF) is a real-valued function f : π × M M → R. Whenever is fixed, we will write f (µ) rather than f ( , µ). It will sometimes be useful to describe a property of matchings in terms of a SWF. For example, a matching µ0 is Rawlsian if fR (µ0 ) = maxµ∈M M fR (µ) where fR (µ) = − maxx∈M ∪W rx (µ(x)). A bipartite graph G = (U, V ; E) consists of two disjoint finite sets of vertices U and 5
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
V and a collection E of edges with one endpoint in each set. The Hopcroft-Karp (HK) algorithm [12] inputs a bipartite graph, (U, V ; E) with, for our purposes, |U | = |V | and in polynomial time outputs a matching µ for U and V such that {u, µ(u)} ∈ E for all u ∈ U , or outputs “no” if no such matching exists. The Hungarian (Hun) algorithm [16] inputs a complete bipartite graph (one with an edge {u, v} for each u ∈ U, v ∈ V ) with an integer assigned to each edge and in polynomial time outputs a matching that minimizes the sum of the edge values.
4
Marriage Matching Algorithms
For a given preference profile, a property P of matchings is graph determined if there exists a bipartite graph G0 = (M, W ; E) such that for every µ ∈ M M , µ has property P if and only if µ is a subgraph of G0 . Given a preference profile, a non-null, graph-determined property P with associated graph G0 and an enumeration x1 , x2 , . . . , x2n of M ∪ W , the following algorithm uses repeated applications of the HK algorithm to produce a matching that is Pareto optimal among all matchings with property P .
4.1
Lexicographic Optimizer (LO)
Given a preference profile, a bipartite graph G0 = (M, W ; E0 ) associated with a non-null, graph-determined property P and an enumeration x1 , x2 , . . . , x2n of the men and women, we begin by initializing variables. E ← E0 \{{x1 , y}: rx1 (y) > 1}, G ← (M, W, E), µ ← ∅, c ← 1, d ← 1. Step 1. Apply HK to G. If HK finds a matching µ0 , then µ ← µ0 and go to Step 3. Else, go to Step 2. Step 2. E ← E ∪ {{xc , xi } ∈ E0 : rxc (xi ) = d + 1 and i > c}, d ← d + 1, G ← (M, W, E) and go to Step 1. 6
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Step 3. If c < 2n, then c ← c + 1, d ← 1, E ← E\{{xc , y}: rxc (y) > 1}, G ← (M, W, E) and go to Step 1. Else, Go to Step 4. Step 4. Output µ. End. Since the output µ is a subgraph of G0 , µ has property P . Since repetitions of Steps 1 and 2 first yield an optimal property P match for x1 , then for x2 with x1 ’s match fixed, then for x3 with x1 ’s match and x2 ’s match fixed, etc., the output µ is Pareto optimal among matchings with property P . Since HK is called at most twice for each edge of G0 and there are at most n2 edges, and since HK is polynomial in n, LO is polynomial in n. We begin with a simple application.
4.2
Pareto Optimal Matchings
Let P be the all-inclusive property; that is, a matching µ has property P if µ is a matching. Then P is graph determined and the unique bipartite graph G0 associated with P is the complete bipartite graph; that is, {m, w} is an edge of G0 for all m ∈ M , w ∈ W . Applying LO to G0 for a given preference profile and any enumeration of the participants produces a matching that is Pareto optimal among property P matchings and is therefore Pareto optimal (since P is all inclusive). It is well known that the Gale-Shapley (GS) men-propose algorithm produces Pareto optimal matchings. The above application of LO differs from GS in that it produces matchings that are not necessarily stable.
4.3
Rawlsian Matchings
A matching µ is Rawlsian if max {rx (µ(x)} = 0min
x∈M ∪W
max {rx (µ0 (x))}
µ ∈M M x∈M ∪W
7
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
where M M is the collection of all matchings. The quantity on the right side of the equation is called the minimax . A Rawlsian algorithm. Given a preference profile, for k with 1 ≤ k ≤ n, let Gk be the bipartite graph whose edges are {{m, w}: rm (w) ≤ k and rw (m) ≤ k}. Apply HK to G1 , G2 , . . . until HK finds a matching and therefore has found the minimax k0 . If P is the Rawlsian property, then P is graph determined and G0 = Gk0 is an associated graph. Applying LO to G0 with an arbitrary enumeration of M ∪ W produces a matching that is Pareto optimal among Rawlsian matchings and therefore Pareto optimal.
4.4
Balanced Matchings
A matching µ is balanced if max |rm (µ(m)) − rµ(m) (m)| = 0min max |rm (µ0 (m)) − rµ0 (m) (m)|.
m∈M
µ ∈M M m∈M
The quantity on the right side of the equation is called the minimax mismatch. A Balancing Algorithm. For a given preference profile and k with 0 ≤ k ≤ n − 1, let Hk be the bipartite graph whose edges are {{m, w}: |rm (w) − rw (m)| ≤ k}. Apply HK to H0 , H1 , H2 , . . . until HK finds a matching and has therefore found the minimax mismatch k0 . If P is the balanced property, then P is graph determined and G0 = Hk0 is an associated graph. Applying LO to G0 with an arbitrary enumeration of M ∪ W produces a matching that is Pareto optimal among balanced matchings.
4.5
Constrained Matchings
Definition. A collection {S(x)}x∈M ∪W of subsets of M ∪ W such that S(m) ⊆ W for all m ∈ M and S(w) ⊆ M for all w ∈ W is a constraint. Definition. A matching µ is constrained by constraint {S(x)}x∈M ∪W if µ(x) ∈ S(x) for all x ∈ M ∪ W. 8
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Definition. A matching µ has property P ({S(x)}x∈M ∪W ) if µ is constrained by {S(x)}x∈M ∪W . A constraining algorithm. Given a preference profile and constraint {S(x)}x∈M ∪W , let G0 be the bipartite graph with edges {{m, w}: m ∈ S(w) and w ∈ S(m)}. Apply LO to G0 with LO modified so that if the first application of HK fails to find a matching, then LO outputs the message that no matching has property P ({S(x)}x∈M ∪W ). Barring such a failure, LO outputs a matching that is Pareto optimal among all matchings with property P ({S(x)}x∈M ∪W ).
4.6
Utilitarian Matchings
P The utilitarian SWF, f ( , µ) = − x∈M ∪W rx (µ(x)), is maximized by the Hun algorithm acting on the bipartite graph G0 = {M, W ; {{m, w}: m ∈ M, w ∈ W }} and with edge {m, w} assigned value rm (w) + rw (m). Hun finds a matching that minimizes the sum of the edge values and therefore maximizes the utilitarian SWF. A utilitarian matching is also Pareto optimal.
4.7
Lexicographic Maximization of Ordered Pairs of Social Wel-
fare Functions For a given matching problem, we will say a matching µ lexicographically maximizes the ordered pair (f, g) of SWFs if µ0 a matching implies f (µ) > f (µ0 ) or both f (µ) = f (µ0 ) and g(µ) ≥ g(µ0 ). Here are some polynomial time algorithms that produce matchings that lexicographically maximize ordered pairs of SWFs. 4.7.1
f The Rawlsian SWF and g the Utilitarian SWF
Use HK to find k0 , the minimax as in Subsection 4.3. Use Hun to find a matching µ that maximizes the SWF
9
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
h(µ) = −
P
x∈M ∪W
j(x), where j(x) =
rx (µ(x)) if rx (µ(x)) ≤ k0 2n2 else
The matching µ is Rawlsian since any Rawlsian matching µR satisfies h(µR ) ≥ −2nk0 ≥ −2n2 while any non-Rawlsian matching µ0 satifies h(µ0 ) < −2n2 . Therefore µ maximizes g restricted to Rawlsian matchings. Therefore µ lexicographically maximizes (f, g). Similar algorithms lexicographically maximize (f, g) where g is the utilitarian SWF and f is either the balance SWF (f (µ) = − maxm∈M |rm (µ(m)) − rµ(m) (m)|) or a constraint SWF: for a given constraint {S(x): x ∈ M ∪ W }, f (µ) = 4.7.2
1 if µ(x) ∈ S(x) for all x ∈ M ∪ W 0 else
f the Utilitarian SWF and g the Rawlsian SWF
For k = n, n − 1, . . . use Hun to find a matching µk that maximizes the SWF hk (µ) = −
X
jk (x)
x∈M ∪W
rx (µ(x)) if rx (µ(x)) ≤ k 2n2 else until a k is found such that hn (µn ) > hk−1 (µk−1 ). where
jk (x) =
Then µk is utilitarian since µn is utilitarian and therefore µk maximizes (f, g). Similar algorithms maximize (f, g) if f is the utilitarian SWF and g is either the balance SWF or a constraint SWF. 4.7.3
f the Rawlsian SWF and g the Balanced SWF
For 1 ≤ k ≤ n and 0 ≤ l ≤ n − 1 let Gk,l = (M, W ; {{m, w}: rm (w) ≤ k, rw (m) ≤ k and |rm (w) − rw (m)| ≤ l}). Apply HK repeatedly to find the lexicographically minimal ordered pair (k0 , l0 ) such that there is a matching µ0 that is a subgraph of Gk0 ,l0 . Then 10
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
µ0 maximizes (f, g). Also the property “µ maximizes (f, g)” is graph determined with associated graph Gk0 ,l0 . Therefore LO can be applied to find a matching that is Pareto optimal among all matchings that maximize (f, g). Similar algorithms apply to lexicographically maximize any ordered pair formed by choosing each of f and g to be the Rawlsian or balance SWF or a constraint SWF.
5
Discussion
Our methods here are deterministic: at the end of any one of our algorithms each agent’s match is certain. Alternative approaches, however, could allow for some form of randomization in the process. If we define a probabilistic matching as a probability distribution over deterministic matchings, one direction for future research is the search for polynomialtime algorithms that produce probabilistic matchings that yield social welfare levels above those provided by the best deterministic matchings. Probabilistic assignment procedures have previously been used to bring forms of fairness to allocation problems [4] and matching problems [1, 9], for example, and a recent paper by Bogomolnaia [3] shows that the probabilistic rule developed in [4] maximizes a form of Rawlsian welfare. We have already begun to develop algorithms similar in spirit to those we have outlined here but extended to the probabilistic domain. Doing so, however, requires first extending the domain of social welfare functions from the set of all deterministic matchings to the set of all probabilistic matchings, a non-trivial step for even specific social welfare functions. We therefore leave their treatment for a separate paper.
11
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
References [1] Atila Abdulkadiroglu and Tayfun Sonmez, Random serial dictatorship and the core from random endowments house allocation problems, Econometrica 66 (1998), 689– 701. [2] Robert L. Axtell and Steven O. Kimbrough, The high cost of stability in two-sided matching: How much social welfare should be sacrificed in the pursuit of stability?, Proc. of the 2008 World Congress on Social Simulation (2008). [3] Anna Bogomolnaia, Random assignment: redefining the serial rule, Journal of Economic Theory 158 (2015), no. A, 308–318. [4] Anna Bogomolnaia and Herve Moulin, A new solution to the random assignment problem, Journal of Economic Theory 100 (2001), no. 2, 295–328. [5] James W. Boudreau and Vicki Knoblauch, Preferences and the price of stability in matching markets, Theory and Decision 74 (2013), no. 4, 565–589. [6]
, What price stability? social welfare in matching markets, Mathematical Social Sciences 67 (2014), 27–33.
[7] Steven Brams and Marc Kilgour, Two-sided matchings: an algorithm ensuring they are minimax and pareto-optimal, Working Paper MPRA Paper No. 48113, 2013. [8] Eric Budish, Matching ‘versus’ mechanism design, ACM SIGecom Exchanges 11 (2012), no. 2, 4–15. [9] David Gale, College course assignments and optimal lotteries, Working paper, University of California, Berkeley, 1987. [10] David Gale and Lloyd S. Shapley, College admissions and the stability of marriage, The American Mathematical Monthly 69 (1962), no. 1, 9–15. [11] Isa Hafalir and Antonio Miralles, Welfare-maximizing assignment of agents to hierarchical positions, Working paper, 2015. [12] John Hopcroft and Richard Karp, An n5/2 algorithm for maximum matchings in bipartite graphs, Siam Journal of Computing 2 (1973), no. 4, 225–231. 12
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
[13] Onur Kesten, School choice with consent, Quarterly Journal of Economics 125 (2010), no. 3, 1297–1348. [14] Bettina Klaus and Flip Klijn, Median stable matching for college admissions, International Journal of Game Theory 34 (2006), 1–11. [15]
, Smith and rawls share a room: stability and medians, Social Choice and Welfare 35 (2010), 647–667.
[16] H. W. Kuhn, The hungarian method for the assignment problem, Naval Research Logistics Quarterly 2 (1955), no. 7, 83–97. [17] F. Masarani and S. S. Gokturk, On the existence of fair matching algorithms, Theory and Decision 26 (1989), 305–322. [18] John Rawls, A theory of justice, Harvard University Press, Cambridge, MA, 1971. [19] Antonio Romero-Medina, Sex-equal stable matchings, Theory and Decision 50 (2001), 197–212. [20]
, Equitable selection in bilateral matching markets, Theory and Decision 58 (2005), 305–324.
[21] Alvin E. Roth, Deferred acceptance algorithms: History, theory, practice, and open questions, International Journal of Game Theory 36 (2008), 537–569. [22] Qianfeng Tang and Jingsheng Yu, A new perspective on kesten’s school choice with consent idea, Journal of Economic Theory 154 (2014), 543–561.
13