A new suggestion: Mimic the uniform distribution If you are one of the participants in a centralized matching problem, your main concern is – who will I get matched to? For the uniform distribution, let P 0 be the matrix of probabilities where the ( i , j )th entry of P 0 is the probability that m i is matched to w j when a stable matching of I is chosen uniformly at random. That is, if I has N stable matchings then N 1 � P 0 = N X µ i i =1 where µ i , i = 1 , . . . , N are the stable matchings of I . Christine T. Cheng University of Wisconsin-Milwaukee Fair Stable Matchings
According to the BvN-like Theorem for stable matchings, r � P 0 = λ i X µ i i =1 where λ i > 0, � r i =1 λ i = 1 and r ≤ n 2 . Christine T. Cheng University of Wisconsin-Milwaukee Fair Stable Matchings
According to the BvN-like Theorem for stable matchings, r � P 0 = λ i X µ i i =1 where λ i > 0, � r i =1 λ i = 1 and r ≤ n 2 . That is, we can run a lottery using r ≤ n 2 stable matchings of I and have the same expected result as the uniform distribution! So what are these stable matchings, and what probability distribution should we assign to them? Christine T. Cheng University of Wisconsin-Milwaukee Fair Stable Matchings
Teo & Sethuraman’s decomposition of P 0 Suppose SM instance I has N stable matchings. For each man m , collect his partners from the N stable matchings and arrange them from his most preferred to least preferred woman. Let p i ( m ) denote the i th woman in this sorted list. For i = 1 , . . . , N , let α i = { ( m , p i ( m )) , m ∈ M } . Christine T. Cheng University of Wisconsin-Milwaukee Fair Stable Matchings
Teo & Sethuraman’s decomposition of P 0 Suppose SM instance I has N stable matchings. For each man m , collect his partners from the N stable matchings and arrange them from his most preferred to least preferred woman. Let p i ( m ) denote the i th woman in this sorted list. For i = 1 , . . . , N , let α i = { ( m , p i ( m )) , m ∈ M } . Theorem:(T&S) For i = 1 , . . . , N , α i is a stable matching of I . Christine T. Cheng University of Wisconsin-Milwaukee Fair Stable Matchings
Example µ 1 µ 2 µ 3 µ 4 µ 5 µ 6 µ 7 µ 8 µ 9 µ 10 m 1 w 1 w 2 w 1 w 2 w 2 w 3 w 3 w 4 w 3 w 4 m 2 w 2 w 1 w 2 w 1 w 4 w 1 w 4 w 3 w 4 w 3 m 3 w 3 w 3 w 4 w 4 w 1 w 4 w 1 w 1 w 2 w 2 m 4 w 4 w 4 w 3 w 3 w 3 w 2 w 2 w 2 w 1 w 1 Christine T. Cheng University of Wisconsin-Milwaukee Fair Stable Matchings
Example µ 1 µ 2 µ 3 µ 4 µ 5 µ 6 µ 7 µ 8 µ 9 µ 10 m 1 w 1 w 2 w 1 w 2 w 2 w 3 w 3 w 4 w 3 w 4 m 2 w 2 w 1 w 2 w 1 w 4 w 1 w 4 w 3 w 4 w 3 m 3 w 3 w 3 w 4 w 4 w 1 w 4 w 1 w 1 w 2 w 2 m 4 w 4 w 4 w 3 w 3 w 3 w 2 w 2 w 2 w 1 w 1 After sorting each man’s partners, α 1 α 2 α 3 α 4 α 5 α 6 α 7 α 8 α 9 α 10 m 1 w 1 w 1 w 2 w 2 w 2 w 3 w 3 w 3 w 4 w 4 m 2 w 2 w 2 w 1 w 1 w 1 w 4 w 4 w 4 w 3 w 3 m 3 w 3 w 3 w 4 w 4 w 4 w 1 w 1 w 1 w 2 w 2 m 4 w 4 w 4 w 3 w 3 w 3 w 2 w 2 w 2 w 1 w 1 Christine T. Cheng University of Wisconsin-Milwaukee Fair Stable Matchings
Observations on the α i ’s: ◮ α 1 is the man-optimal stable matching. ◮ α N is the woman-optimal stable matching. ◮ α 1 ≥ m α 2 ≥ m α 3 . . . ≥ m α N for each man m . ◮ the α i ’s form a chain in L ( I ). ◮ hence, there are at most n 2 distinct α i ’s. WHY? Christine T. Cheng University of Wisconsin-Milwaukee Fair Stable Matchings
The decomposition: Let S = { µ : µ = α i , i ∈ { 1 , 2 , . . . , N }} . For each µ ∈ S , let π ( µ ) = |{ i : α i = µ }| / N . It’s not difficult to see that P 0 = � µ ∈ S π ( µ ) X µ . Christine T. Cheng University of Wisconsin-Milwaukee Fair Stable Matchings
Example cont’d α 1 α 2 α 3 α 4 α 5 α 6 α 7 α 8 α 9 α 10 m 1 w 1 w 1 w 2 w 2 w 2 w 3 w 3 w 3 w 4 w 4 m 2 w 2 w 2 w 1 w 1 w 1 w 4 w 4 w 4 w 3 w 3 m 3 w 3 w 3 w 4 w 4 w 4 w 1 w 1 w 1 w 2 w 2 m 4 w 4 w 4 w 3 w 3 w 3 w 2 w 2 w 2 w 1 w 1 Christine T. Cheng University of Wisconsin-Milwaukee Fair Stable Matchings
Example cont’d α 1 α 2 α 3 α 4 α 5 α 6 α 7 α 8 α 9 α 10 m 1 w 1 w 1 w 2 w 2 w 2 w 3 w 3 w 3 w 4 w 4 m 2 w 2 w 2 w 1 w 1 w 1 w 4 w 4 w 4 w 3 w 3 m 3 w 3 w 3 w 4 w 4 w 4 w 1 w 1 w 1 w 2 w 2 m 4 w 4 w 4 w 3 w 3 w 3 w 2 w 2 w 2 w 1 w 1 µ 1 = { ( m 1 , w 1 ) , ( m 2 , w 2 ) , ( m 3 , w 3 ) , ( m 4 , w 4 ) } µ 4 = { ( m 1 , w 2 ) , ( m 2 , w 1 ) , ( m 3 , w 4 ) , ( m 4 , w 3 ) } µ 7 = { ( m 1 , w 3 ) , ( m 2 , w 4 ) , ( m 3 , w 1 ) , ( m 4 , w 2 ) } µ 10 = { ( m 1 , w 4 ) , ( m 2 , w 3 ) , ( m 3 , w 2 ) , ( m 4 , w 1 ) } So P 0 = 2 10 X µ 1 + 3 10 X µ 4 + 3 10 X µ 7 + 2 10 X µ 10 . Christine T. Cheng University of Wisconsin-Milwaukee Fair Stable Matchings
U 10 U U 8 9 U 7 U U 6 5 U U 4 U 2 3 U 1 P 0 = 2 10 X µ 1 + 3 10 X µ 4 + 3 10 X µ 7 + 2 10 X µ 10 . Christine T. Cheng University of Wisconsin-Milwaukee Fair Stable Matchings
Fact No. 1: There is a Birkhoff-vonNeumann - like decomposition theorem for fractional stable matchings. Christine T. Cheng University of Wisconsin-Milwaukee Fair Stable Matchings
Fact No. 1: There is a Birkhoff-vonNeumann - like decomposition theorem for fractional stable matchings. ◮ It can be used to mimic any probability distribution on the set of stable matchings (incl. the uniform distribution) in a concise way . ◮ T & S’s decomposition of X f makes use of a set of stable matchings that form a chain. It is the only decomposition that forms a chain. Christine T. Cheng University of Wisconsin-Milwaukee Fair Stable Matchings
Geometry of Fractional Stable Matchings and its Applications by C.P. Teo and J. Sethuraman Mathematics of Operations Research, 1998 Understanding the Generalized Median Stable Matchings by C. Cheng Algorithmica, 2010 Christine T. Cheng University of Wisconsin-Milwaukee Fair Stable Matchings
The counterpart of the α i ’s Suppose SM instance I has N stable matchings. For each woman w , collect her partners from the N stable matchings and arrange them from her most preferred to least preferred man. Let p i ( w ) denote the i th man in this sorted list. For i = 1 , . . . , N , let β i = { ( p i ( w ) , w ) , w ∈ W } . Christine T. Cheng University of Wisconsin-Milwaukee Fair Stable Matchings
The counterpart of the α i ’s Suppose SM instance I has N stable matchings. For each woman w , collect her partners from the N stable matchings and arrange them from her most preferred to least preferred man. Let p i ( w ) denote the i th man in this sorted list. For i = 1 , . . . , N , let β i = { ( p i ( w ) , w ) , w ∈ W } . Theorem: (T&S) For i = 1 , . . . , N , α i = β N − i +1 . [Fleiner (2003) and Klaus and Klijn (2006) proved the existence of the α i ’s using different approaches.] Christine T. Cheng University of Wisconsin-Milwaukee Fair Stable Matchings
Example cont’d α 1 α 2 α 3 α 4 α 5 α 6 α 7 α 8 α 9 α 10 β 10 β 9 β 8 β 7 β 6 β 5 β 4 β 3 β 2 β 1 m 1 w 1 w 1 w 2 w 2 w 2 w 3 w 3 w 3 w 4 w 4 m 2 w 2 w 2 w 1 w 1 w 1 w 4 w 4 w 4 w 3 w 3 m 3 w 3 w 3 w 4 w 4 w 4 w 1 w 1 w 1 w 2 w 2 m 4 w 4 w 4 w 3 w 3 w 3 w 2 w 2 w 2 w 1 w 1 Christine T. Cheng University of Wisconsin-Milwaukee Fair Stable Matchings
Example cont’d α 1 α 2 α 3 α 4 α 5 α 6 α 7 α 8 α 9 α 10 β 10 β 9 β 8 β 7 β 6 β 5 β 4 β 3 β 2 β 1 m 1 w 1 w 1 w 2 w 2 w 2 w 3 w 3 w 3 w 4 w 4 m 2 w 2 w 2 w 1 w 1 w 1 w 4 w 4 w 4 w 3 w 3 m 3 w 3 w 3 w 4 w 4 w 4 w 1 w 1 w 1 w 2 w 2 m 4 w 4 w 4 w 3 w 3 w 3 w 2 w 2 w 2 w 1 w 1 Thus, every participant in the middle α i ’s is matched to his/her (lower or upper) median stable partner! Christine T. Cheng University of Wisconsin-Milwaukee Fair Stable Matchings
Define the median stable matching of I as - α ( N +1) / 2 when N is odd and - α N / 2 and α N / 2+1 when N is even. Christine T. Cheng University of Wisconsin-Milwaukee Fair Stable Matchings
Define the median stable matching of I as - α ( N +1) / 2 when N is odd and - α N / 2 and α N / 2+1 when N is even. Teo and Sethuraman asked the following question: Q: What is the computational complexity of finding the median stable matching of an SM instance? ◮ Using the definition will require enumerating all the stable matchings of the instance – and this can take exponential time. Christine T. Cheng University of Wisconsin-Milwaukee Fair Stable Matchings
The medians of a distributive lattice Def: Let G be a connected graph. A vertex v of G is a median of G if its total (or average) distance from all other vertices of G is the least. In the 1960’s, Barbut initiated the study of medians of distributive lattices by using the covering graphs of these lattices. He showed that they behaved “nicely.” This leads to an intriguing question: Christine T. Cheng University of Wisconsin-Milwaukee Fair Stable Matchings
The medians of a distributive lattice Def: Let G be a connected graph. A vertex v of G is a median of G if its total (or average) distance from all other vertices of G is the least. In the 1960’s, Barbut initiated the study of medians of distributive lattices by using the covering graphs of these lattices. He showed that they behaved “nicely.” This leads to an intriguing question: Q: What is the relationship between the medians of L ( I ) and the median stable matchings of I ? Christine T. Cheng University of Wisconsin-Milwaukee Fair Stable Matchings
Contributions Theorem: (Cheng, Nemoto 2000) [Characterization] For each rotation ρ in P L ( I ) , let n ρ denote the number of closed subsets that contain ρ . Then, α i corresponds to { ρ : n ρ ≥ N − i + 1 } . In particular, when N is odd, α ( N +1) / 2 corresponds to { ρ : ρ appeared in majority of the closed subsets } . Christine T. Cheng University of Wisconsin-Milwaukee Fair Stable Matchings
By relating the new characterization with the results of Barbut, we have the following: Theorem: (Cheng) [Fairness] Suppose I has N stable matchings. a. When N is odd, α ( N +1) / 2 is the unique median vertex of L ( I ). b. When N is even, a stable matching µ is a median vertex of L ( I ) if and only if α N / 2 � µ � α N / 2+1 . Christine T. Cheng University of Wisconsin-Milwaukee Fair Stable Matchings
By relating the new characterization with the results of Barbut, we have the following: Theorem: (Cheng) [Fairness] Suppose I has N stable matchings. a. When N is odd, α ( N +1) / 2 is the unique median vertex of L ( I ). b. When N is even, a stable matching µ is a median vertex of L ( I ) if and only if α N / 2 � µ � α N / 2+1 . Thus, SM instances have stable matchings that are fair “locally” and “globally”. We call this the local/global median phenomenon . Christine T. Cheng University of Wisconsin-Milwaukee Fair Stable Matchings
U 10 U U 8 9 U 7 U U 6 5 U U 4 U 2 3 U 1 For the instance I , α 5 = µ 4 , α 6 = µ 7 and every stable matching µ such that µ 4 � µ � µ 7 is a median of L ( I ). Christine T. Cheng University of Wisconsin-Milwaukee Fair Stable Matchings
Theorem: (Cheng) [Complexity] When i is O (log n ), α i can be computed efficiently. But in general, it is #P-hard. ◮ If there is an efficient algorithm for computing the median stable matching of an SM instance, then there is an efficient algorithm for counting the number of stable matchings of an SM instance. But the latter is #P-complete. Christine T. Cheng University of Wisconsin-Milwaukee Fair Stable Matchings
My approach: use poset representation of stable matchings Christine T. Cheng University of Wisconsin-Milwaukee Fair Stable Matchings
An aside: Posets and Distributive Lattices posets ⇔ distributive lattices Let P = ( P , ≤ ) be a poset. A subset P ′ is a closed subset (also down-set or order-ideal) of P if whenever y ∈ P ′ then so is x ∈ P ′ whenever x < y . Christine T. Cheng University of Wisconsin-Milwaukee Fair Stable Matchings
An aside: Posets and Distributive Lattices posets ⇔ distributive lattices Let P = ( P , ≤ ) be a poset. A subset P ′ is a closed subset (also down-set or order-ideal) of P if whenever y ∈ P ′ then so is x ∈ P ′ whenever x < y . ⇒ (Folklore) Let CS ( P ) consist of the closed subsets of P . Then ( CS ( P ) , ⊆ ) is a distributive lattice. Christine T. Cheng University of Wisconsin-Milwaukee Fair Stable Matchings
An aside: Posets and Distributive Lattices posets ⇔ distributive lattices Let P = ( P , ≤ ) be a poset. A subset P ′ is a closed subset (also down-set or order-ideal) of P if whenever y ∈ P ′ then so is x ∈ P ′ whenever x < y . ⇒ (Folklore) Let CS ( P ) consist of the closed subsets of P . Then ( CS ( P ) , ⊆ ) is a distributive lattice. ⇐ (Birkhoff) For every distributive lattice L , there is a poset P L so that ( CS ( P L ) , ⊆ ) is order isomorphic to L . Every distributive lattice L can be encoded by a poset P L . Christine T. Cheng University of Wisconsin-Milwaukee Fair Stable Matchings
U 10 U U 8 9 U 7 U U 6 5 U U 4 U 2 3 U 1 The poset P L associated with the distributive lattice L is shown on the right. Christine T. Cheng University of Wisconsin-Milwaukee Fair Stable Matchings
For stable matchings: the poset can be constructed directly from the instance. U 10 U U 8 9 U 7 U U 6 5 U U 4 U 2 3 U 1 Christine T. Cheng University of Wisconsin-Milwaukee Fair Stable Matchings
stable marriage instances ⇔ posets ⇒ (Irving et al.) Suppose I in an SM instance with n men and n women with L ( I ) as the its distributive lattice of stable matchings. Christine T. Cheng University of Wisconsin-Milwaukee Fair Stable Matchings
stable marriage instances ⇔ posets ⇒ (Irving et al.) Suppose I in an SM instance with n men and n women with L ( I ) as the its distributive lattice of stable matchings. ◮ P L ( I ) can be derived directly from the preference lists of the participants. It’s called the rotation poset of I . ◮ It has at most O ( n 2 ) elements called rotations , and it can be constructed in O ( n 2 ) time. Christine T. Cheng University of Wisconsin-Milwaukee Fair Stable Matchings
stable marriage instances ⇔ posets ⇒ (Irving et al.) Suppose I in an SM instance with n men and n women with L ( I ) as the its distributive lattice of stable matchings. ◮ P L ( I ) can be derived directly from the preference lists of the participants. It’s called the rotation poset of I . ◮ It has at most O ( n 2 ) elements called rotations , and it can be constructed in O ( n 2 ) time. ⇐ (Blair, Gusfield et al.) For every poset P , there is an SM instance I P whose rotation poset is order-isomorphic to P . Moreover, its size is O ( poly ( |P| ). Christine T. Cheng University of Wisconsin-Milwaukee Fair Stable Matchings
Q: Which closed subset of P L ( I ) corresponds to α i for i = 1 , . . . , N ? Christine T. Cheng University of Wisconsin-Milwaukee Fair Stable Matchings
Q: Which closed subset of P L ( I ) corresponds to α i for i = 1 , . . . , N ? * The new characterization answered this question. Christine T. Cheng University of Wisconsin-Milwaukee Fair Stable Matchings
Fact No. 2: Every SM instance has a stable matching that is both locally median and globally median. Unfortunately, for general instances, it is #P-hard to compute such a stable matching. Christine T. Cheng University of Wisconsin-Milwaukee Fair Stable Matchings
Fact No. 2: Every SM instance has a stable matching that is both locally median and globally median. Unfortunately, for general instances, it is #P-hard to compute such a stable matching. ◮ When the rotation poset associated with the instance is series-parallel, an interval order or 2-dimensional, computing a median stable matching can be done efficiently. ◮ The local/global median phenomenon applies to an arbitrary collection of stable matchings. Christine T. Cheng University of Wisconsin-Milwaukee Fair Stable Matchings
Two Extensions Stable Roommates Matchings, Mirror Posets, Median Graphs, and the Local/Global Median Phenomenon in Stable Matchings by C. Cheng and A. Lin SIAM Journal of Discrete Math, 2011 The center stable matchings and the centers of cover graphs of distributive lattices by C. Cheng, E. McDermid and I. Suzuki ICALP 2011 [A journal version of the paper is under submission.] Christine T. Cheng University of Wisconsin-Milwaukee Fair Stable Matchings
Extension 1. Stable Roommates T & S noted that solvable Stable Roommates (SR) instances also have median stable matchings. Are the median stable matchings also globally median? Christine T. Cheng University of Wisconsin-Milwaukee Fair Stable Matchings
Extension 1. Stable Roommates T & S noted that solvable Stable Roommates (SR) instances also have median stable matchings. Are the median stable matchings also globally median? State of knowledge at that time: SM instances ⇔ posets ⇔ distributive lattices SR instances ⇒ mirror posets ?? Christine T. Cheng University of Wisconsin-Milwaukee Fair Stable Matchings
Cheng and Lin showed ◮ Like SM instances, SR instances also had “dualities”: SR instances ⇔ mirror posets ⇔ median graphs ◮ A median stable matching of a solvable SR instance is also a median vertex of its median graph. Christine T. Cheng University of Wisconsin-Milwaukee Fair Stable Matchings
Cheng and Lin showed ◮ Like SM instances, SR instances also had “dualities”: SR instances ⇔ mirror posets ⇔ median graphs ◮ A median stable matching of a solvable SR instance is also a median vertex of its median graph. Fact No. 3a: The local/global median phenomenon extends to solvable SR instances. Christine T. Cheng University of Wisconsin-Milwaukee Fair Stable Matchings
Extension 2. Center Stable Matchings Let G be a connected graph. Def: A center of G is a node whose maximum distance from another node of G is the least. Def: Given an SM instance I , center stable matching of I is a center of the cover graph of L ( I ). Christine T. Cheng University of Wisconsin-Milwaukee Fair Stable Matchings
Extension 2. Center Stable Matchings Let G be a connected graph. Def: A center of G is a node whose maximum distance from another node of G is the least. Def: Given an SM instance I , center stable matching of I is a center of the cover graph of L ( I ). Like a median stable matching of I , a center stable matching of I is “fair” because it is a good representative of I ’s stable matchings. Q: What is the computational complexity of computing a center stable matching of L ( I )? Christine T. Cheng University of Wisconsin-Milwaukee Fair Stable Matchings
Cheng, McDermid & Suzuki showed ◮ A center stable matching of an SM instance I can be computed in polynomial time. ◮ A characterization of all the center stable matchings of I . - Some center stable matchings are the middle nodes of a longest chain of L ( I ) but the converse is not true. Christine T. Cheng University of Wisconsin-Milwaukee Fair Stable Matchings
Cheng, McDermid & Suzuki showed ◮ A center stable matching of an SM instance I can be computed in polynomial time. ◮ A characterization of all the center stable matchings of I . - Some center stable matchings are the middle nodes of a longest chain of L ( I ) but the converse is not true. Fact No. 3b: A center stable matching is another globally fair stable matching. It can be computed efficiently. Christine T. Cheng University of Wisconsin-Milwaukee Fair Stable Matchings
On the Stable Matchings that can be Reached When the Agents Go Marching in One by One by C. Cheng under submission, 2014 Christine T. Cheng University of Wisconsin-Milwaukee Fair Stable Matchings
In the Gale-Shapley Algorithm, ◮ only one group can make a proposal ◮ the output favors the proposing group A common question I get from students: Is there are algorithm where both men and women propose, and will that result in a less biased output? Christine T. Cheng University of Wisconsin-Milwaukee Fair Stable Matchings
In the Gale-Shapley Algorithm, ◮ only one group can make a proposal ◮ the output favors the proposing group A common question I get from students: Is there are algorithm where both men and women propose, and will that result in a less biased output? One possibility: Ma’s Random Order Mechanism (proposed in 1996), a sequential version of the Gale-Shapley algorithm. Christine T. Cheng University of Wisconsin-Milwaukee Fair Stable Matchings
How the Random Order Mechanism (ROM) works: ◮ Start with a random permutation of the participants π . ◮ At the beginning of each iteration i , ◮ there is a stable matching µ i − 1 for the participants in π (1 · · · i − 1). ◮ π ( i ) marches in and starts proposing to the person he or she prefers the most among those in the room. ◮ a GS-algorithm-like step ensues where the individuals on the side of µ i proposing. ◮ at the end of the iteration, there is a stable matching µ i for the i participants. Christine T. Cheng University of Wisconsin-Milwaukee Fair Stable Matchings
How the Random Order Mechanism (ROM) works: Let π = m 1 , w 2 , w 1 , m 2 , w 3 , m 3 , m 4 , w 4 . m 1 : w 1 w 2 w 3 w 4 w 1 : m 4 m 3 m 2 m 1 m 2 : w 2 w 1 w 4 w 3 w 2 : m 3 m 4 m 1 m 2 m 3 : w 3 w 4 w 1 w 2 w 3 : m 2 m 1 m 4 m 3 m 4 : w 4 w 3 w 2 w 1 w 4 : m 1 m 2 m 3 m 4 Christine T. Cheng University of Wisconsin-Milwaukee Fair Stable Matchings
How the Random Order Mechanism (ROM) works: Let π = m 1 , w 2 , w 1 , m 2 , w 3 , m 3 , m 4 , w 4 . m 1 : w 1 w 2 w 3 w 4 w 1 : m 4 m 3 m 2 m 1 m 2 : w 2 w 1 w 4 w 3 w 2 : m 3 m 4 m 1 m 2 m 3 : w 3 w 4 w 1 w 2 w 3 : m 2 m 1 m 4 m 3 m 4 : w 4 w 3 w 2 w 1 w 4 : m 1 m 2 m 3 m 4 Christine T. Cheng University of Wisconsin-Milwaukee Fair Stable Matchings
Skipping ahead... Christine T. Cheng University of Wisconsin-Milwaukee Fair Stable Matchings
How the Random Order Mechanism (ROM) works: Let π = m 1 , w 2 , w 1 , m 2 , w 3 , m 3 , m 4 , w 4 . m 1 : w 1 w 2 w 3 w 4 w 1 : m 4 m 3 m 2 m 1 m 2 : w 2 w 1 w 4 w 3 w 2 : m 3 m 4 m 1 m 2 m 3 : w 3 w 4 w 1 w 2 w 3 : m 2 m 1 m 4 m 3 m 4 : w 4 w 3 w 2 w 1 w 4 : m 1 m 2 m 3 m 4 Christine T. Cheng University of Wisconsin-Milwaukee Fair Stable Matchings
How the Random Order Mechanism (ROM) works: Let π = m 1 , w 2 , w 1 , m 2 , w 3 , m 3 , m 4 , w 4 . m 1 : w 1 w 2 w 3 w 4 w 1 : m 4 m 3 m 2 m 1 m 2 : w 2 w 1 w 4 w 3 w 2 : m 3 m 4 m 1 m 2 m 3 : w 3 w 4 w 1 w 2 w 3 : m 2 m 1 m 4 m 3 m 4 : w 4 w 3 w 2 w 1 w 4 : m 1 m 2 m 3 m 4 Christine T. Cheng University of Wisconsin-Milwaukee Fair Stable Matchings
How the Random Order Mechanism (ROM) works: Let π = m 1 , w 2 , w 1 , m 2 , w 3 , m 3 , m 4 , w 4 . m 1 : w 1 w 2 w 3 w 4 w 1 : m 4 m 3 m 2 m 1 m 2 : w 2 w 1 w 4 w 3 w 2 : m 3 m 4 m 1 m 2 m 3 : w 3 w 4 w 1 w 2 w 3 : m 2 m 1 m 4 m 3 m 4 : w 4 w 3 w 2 w 1 w 4 : m 1 m 2 m 3 m 4 Christine T. Cheng University of Wisconsin-Milwaukee Fair Stable Matchings
How the Random Order Mechanism (ROM) works: Let π = m 1 , w 2 , w 1 , m 2 , w 3 , m 3 , m 4 , w 4 . m 1 : w 1 w 2 w 3 w 4 w 1 : m 4 m 3 m 2 m 1 m 2 : w 2 w 1 w 4 w 3 w 2 : m 3 m 4 m 1 m 2 m 3 : w 3 w 4 w 1 w 2 w 3 : m 2 m 1 m 4 m 3 m 4 : w 4 w 3 w 2 w 1 w 4 : m 1 m 2 m 3 m 4 Christine T. Cheng University of Wisconsin-Milwaukee Fair Stable Matchings
Skipping ahead... Christine T. Cheng University of Wisconsin-Milwaukee Fair Stable Matchings
How the Random Order Mechanism (ROM) works: Let π = m 1 , w 2 , w 1 , m 2 , w 3 , m 3 , m 4 , w 4 . m 1 : w 1 w 2 w 3 w 4 w 1 : m 4 m 3 m 2 m 1 m 2 : w 2 w 1 w 4 w 3 w 2 : m 3 m 4 m 1 m 2 m 3 : w 3 w 4 w 1 w 2 w 3 : m 2 m 1 m 4 m 3 m 4 : w 4 w 3 w 2 w 1 w 4 : m 1 m 2 m 3 m 4 Christine T. Cheng University of Wisconsin-Milwaukee Fair Stable Matchings
How the Random Order Mechanism (ROM) works: Let π = m 1 , w 2 , w 1 , m 2 , w 3 , m 3 , m 4 , w 4 . m 1 : w 1 w 2 w 3 w 4 w 1 : m 4 m 3 m 2 m 1 m 2 : w 2 w 1 w 4 w 3 w 2 : m 3 m 4 m 1 m 2 m 3 : w 3 w 4 w 1 w 2 w 3 : m 2 m 1 m 4 m 3 m 4 : w 4 w 3 w 2 w 1 w 4 : m 1 m 2 m 3 m 4 Christine T. Cheng University of Wisconsin-Milwaukee Fair Stable Matchings
How the Random Order Mechanism (ROM) works: Let π = m 1 , w 2 , w 1 , m 2 , w 3 , m 3 , m 4 , w 4 . m 1 : w 1 w 2 w 3 w 4 w 1 : m 4 m 3 m 2 m 1 m 2 : w 2 w 1 w 4 w 3 w 2 : m 3 m 4 m 1 m 2 m 3 : w 3 w 4 w 1 w 2 w 3 : m 2 m 1 m 4 m 3 m 4 : w 4 w 3 w 2 w 1 w 4 : m 1 m 2 m 3 m 4 ROM can reach stable matchings different from the man-optimal and woman-optimal stable matchings! Christine T. Cheng University of Wisconsin-Milwaukee Fair Stable Matchings
Facts about ROM:(Ma (1996), Blum et al. (1997), Cechal´ arov´ a (2002)) ◮ ROM will output a stable matching in O ( n 3 ) time. Christine T. Cheng University of Wisconsin-Milwaukee Fair Stable Matchings
Facts about ROM:(Ma (1996), Blum et al. (1997), Cechal´ arov´ a (2002)) ◮ ROM will output a stable matching in O ( n 3 ) time. ◮ ROM can simulate the Gale-Shapley algorithm. - when π consist of all men followed by all women, ROM( π ) will output the woman-optimal SM, etc. Christine T. Cheng University of Wisconsin-Milwaukee Fair Stable Matchings
Facts about ROM:(Ma (1996), Blum et al. (1997), Cechal´ arov´ a (2002)) ◮ ROM will output a stable matching in O ( n 3 ) time. ◮ ROM can simulate the Gale-Shapley algorithm. - when π consist of all men followed by all women, ROM( π ) will output the woman-optimal SM, etc. ◮ ROM will always match the last person in π to his/her best stable partner. Consequence: If no agent in µ is matched to his/her best stable partner, ROM will never output µ . Christine T. Cheng University of Wisconsin-Milwaukee Fair Stable Matchings
Assume the permutation π , the input to ROM, was chosen uniformly at random. ◮ Klaus and Klijn argued that this is a procedurally fair mechanism for generating a stable matching. Christine T. Cheng University of Wisconsin-Milwaukee Fair Stable Matchings
Assume the permutation π , the input to ROM, was chosen uniformly at random. ◮ Klaus and Klijn argued that this is a procedurally fair mechanism for generating a stable matching. ◮ Some natural questions to ask – What is the probability distribution induced by ROM on the set of stable matchings? What is the support of this probability distribution? Christine T. Cheng University of Wisconsin-Milwaukee Fair Stable Matchings
Assume the permutation π , the input to ROM, was chosen uniformly at random. ◮ Klaus and Klijn argued that this is a procedurally fair mechanism for generating a stable matching. ◮ Some natural questions to ask – What is the probability distribution induced by ROM on the set of stable matchings? What is the support of this probability distribution? Call µ ROM-reachable if there is a permutation π of the agents so that ROM( π ) outputs µ . Which stable matchings are ROM-reachable? Christine T. Cheng University of Wisconsin-Milwaukee Fair Stable Matchings
Results: ◮ Given a stable matching µ , determining if µ is ROM-reachable is NP-complete. - the difficulty lies in the unstable partners Christine T. Cheng University of Wisconsin-Milwaukee Fair Stable Matchings
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