Matching games¶
This module implements a class for matching games (stable marriage problems) [DI1989]. At present the extended GaleShapley algorithm is implemented which can be used to obtain stable matchings.
AUTHORS:
 James Campbell and Vince Knight 062014: Original version

class
sage.game_theory.matching_game.
MatchingGame
(generator, revr=None)¶ Bases:
sage.structure.sage_object.SageObject
A matching game.
A matching game (also called a stable matching problem) models a situation in a population of \(N\) suitors and \(N\) reviewers. Suitors and reviewers rank their preferences and attempt to find a match.
Formally, a matching game of size \(N\) is defined by two disjoint sets \(S\) and \(R\) of size \(N\). Associated to each element of \(S\) and \(R\) is a preference list:
\[f : S \to R^N \text{ and } g : R \to S^N.\]Here is an example of matching game on 4 players:
\[\begin{split}S = \{J, K, L, M\}, \\ R = \{A, B, C, D\}.\end{split}\]With preference functions:
\[ \begin{align}\begin{aligned}\begin{split}f(s) = \begin{cases} (A, D, C, B) & \text{ if } s=J,\\ (A, B, C, D) & \text{ if } s=K,\\ (B, D, C, A) & \text{ if } s=L,\\ (C, A, B, D) & \text{ if } s=M,\\ \end{cases}\end{split}\\\begin{split}g(s) = \begin{cases} (L, J, K, M) & \text{ if } s=A,\\ (J, M, L, K) & \text{ if } s=B,\\ (K, M, L, J) & \text{ if } s=C,\\ (M, K, J, L) & \text{ if } s=D.\\ \end{cases}\end{split}\end{aligned}\end{align} \]INPUT:
Two potential inputs are accepted (see below to see the effect of each):
reviewer/suitors_preferences
– a dictionary containing the preferences of all players: key  each reviewer/suitors
 value  a tuple of suitors/reviewers
OR:
integer
– an integer simply representing the number of reviewers and suitors.
To implement the above game in Sage:
sage: suitr_pref = {'J': ('A', 'D', 'C', 'B'), ....: 'K': ('A', 'B', 'C', 'D'), ....: 'L': ('B', 'D', 'C', 'A'), ....: 'M': ('C', 'A', 'B', 'D')} sage: reviewr_pref = {'A': ('L', 'J', 'K', 'M'), ....: 'B': ('J', 'M', 'L', 'K'), ....: 'C': ('K', 'M', 'L', 'J'), ....: 'D': ('M', 'K', 'J', 'L')} sage: m = MatchingGame([suitr_pref, reviewr_pref]) sage: m A matching game with 4 suitors and 4 reviewers sage: m.suitors() ('J', 'K', 'L', 'M') sage: m.reviewers() ('A', 'B', 'C', 'D')
A matching \(M\) is any bijection between \(S\) and \(R\). If \(s \in S\) and \(r \in R\) are matched by \(M\) we denote:
\[M(s) = r.\]On any given matching game, one intends to find a matching that is stable. In other words, so that no one individual has an incentive to break their current match.
Formally, a stable matching is a matching that has no blocking pairs. A blocking pair is any pair \((s, r)\) such that \(M(s) \neq r\) but \(s\) prefers \(r\) to \(M(r)\) and \(r\) prefers \(s\) to \(M^{1}(r)\).
To obtain the stable matching in Sage we use the
solve
method which uses the extended GaleShapley algorithm [DI1989]:sage: m.solve() {'J': 'A', 'K': 'C', 'L': 'D', 'M': 'B'}
Matchings have a natural representations as bipartite graphs:
sage: plot(m) Graphics object consisting of 13 graphics primitives
The above plots the bipartite graph associated with the matching. This plot can be accessed directly:
sage: graph = m.bipartite_graph() sage: graph Bipartite graph on 8 vertices
It is possible to initiate a matching game without having to name each suitor and reviewer:
sage: n = 8 sage: big_game = MatchingGame(n) sage: big_game.suitors() (1, 2, 3, 4, 5, 6, 7, 8) sage: big_game.reviewers() (1, 2, 3, 4, 5, 6, 7, 8)
If we attempt to obtain the stable matching for the above game, without defining the preference function we obtain an error:
sage: big_game.solve() Traceback (most recent call last): ... ValueError: suitor preferences are not complete
To continue we have to populate the preference dictionary. Here is one example where the preferences are simply the corresponding element of the permutation group:
sage: from itertools import permutations sage: suitr_preferences = list(permutations([i1 for i in range(n)])) sage: revr_preferences = list(permutations([i+1 for i in range(n)])) sage: for player in range(n): ....: big_game.suitors()[player].pref = suitr_preferences[player] ....: big_game.reviewers()[player].pref = revr_preferences[player] sage: big_game.solve() {1: 1, 2: 8, 3: 6, 4: 7, 5: 5, 6: 4, 7: 3, 8: 2}
Note that we can also combine the two ways of creating a game. For example here is an initial matching game:
sage: suitrs = {'Romeo': ('Juliet', 'Rosaline'), ....: 'Mercutio': ('Juliet', 'Rosaline')} sage: revwrs = {'Juliet': ('Romeo', 'Mercutio'), ....: 'Rosaline': ('Mercutio', 'Romeo')} sage: g = MatchingGame(suitrs, revwrs)
Let us assume that all of a sudden a new pair of suitors and reviewers is added but their names are not known:
sage: g.add_reviewer() sage: g.add_suitor() sage: g.reviewers() (3, 'Juliet', 'Rosaline') sage: g.suitors() (3, 'Mercutio', 'Romeo')
Note that when adding a reviewer or a suitor all preferences are wiped:
sage: [s.pref for s in g.suitors()] [[], [], []] sage: [r.pref for r in g.reviewers()] [[], [], []]
If we now try to solve the game we will get an error as we have not specified the preferences which will need to be updated:
sage: g.solve() Traceback (most recent call last): ... ValueError: suitor preferences are not complete
Here we update the preferences so that the new reviewers and suitors do not affect things too much (they prefer each other and are the least preferred of the others):
sage: g.suitors()[1].pref = suitrs['Mercutio'] + (3,) sage: g.suitors()[2].pref = suitrs['Romeo'] + (3,) sage: g.suitors()[0].pref = (3, 'Juliet', 'Rosaline') sage: g.reviewers()[2].pref = revwrs['Rosaline'] + (3,) sage: g.reviewers()[1].pref = revwrs['Juliet'] + (3,) sage: g.reviewers()[0].pref = (3, 'Romeo', 'Mercutio')
Now the game can be solved:
sage: D = g.solve() sage: D['Mercutio'] 'Rosaline' sage: D['Romeo'] 'Juliet' sage: D[3] 3
Note that the above could be equivalently (and more simply) carried out by simply updated the original preference dictionaries:
sage: for key in suitrs: ....: suitrs[key] = suitrs[key] + (3,) sage: for key in revwrs: ....: revwrs[key] = revwrs[key] + (3,) sage: suitrs[3] = (3, 'Juliet', 'Rosaline') sage: revwrs[3] = (3, 'Romeo', 'Mercutio') sage: g = MatchingGame(suitrs, revwrs) sage: D = g.solve() sage: D['Mercutio'] 'Rosaline' sage: D['Romeo'] 'Juliet' sage: D[3] 3
It can be shown that the GaleShapley algorithm will return the stable matching that is optimal from the point of view of the suitors and is in fact the worst possible matching from the point of view of the reviewers. To quickly obtain the matching that is optimal for the reviewers we use the
solve
method with theinvert=True
option:sage: left_dict = {'a': ('A', 'B', 'C'), ....: 'b': ('B', 'C', 'A'), ....: 'c': ('B', 'A', 'C')} sage: right_dict = {'A': ('b', 'c', 'a'), ....: 'B': ('a', 'c', 'b'), ....: 'C': ('a', 'b', 'c')} sage: quick_game = MatchingGame([left_dict, right_dict]) sage: quick_game.solve() {'a': 'A', 'b': 'C', 'c': 'B'} sage: quick_game.solve(invert=True) {'A': 'c', 'B': 'a', 'C': 'b'}
EXAMPLES:
8 player letter game:
sage: suitr_pref = {'J': ('A', 'D', 'C', 'B'), ....: 'K': ('A', 'B', 'C', 'D'), ....: 'L': ('B', 'D', 'C', 'A'), ....: 'M': ('C', 'A', 'B', 'D')} sage: reviewr_pref = {'A': ('L', 'J', 'K', 'M'), ....: 'B': ('J', 'M', 'L', 'K'), ....: 'C': ('K', 'M', 'L', 'J'), ....: 'D': ('M', 'K', 'J', 'L')} sage: m = MatchingGame([suitr_pref, reviewr_pref]) sage: m.suitors() ('J', 'K', 'L', 'M') sage: m.reviewers() ('A', 'B', 'C', 'D')
Also works for numbers:
sage: suit = {0: (3, 4), ....: 1: (3, 4)} sage: revr = {3: (0, 1), ....: 4: (1, 0)} sage: g = MatchingGame([suit, revr])
Can create a game from an integer. This gives default set of preference functions:
sage: g = MatchingGame(3) sage: g A matching game with 3 suitors and 3 reviewers
We have an empty set of preferences for a default named set of preferences:
sage: for s in g.suitors(): ....: s, s.pref (1, []) (2, []) (3, []) sage: for r in g.reviewers(): ....: r, r.pref (1, []) (2, []) (3, [])
Before trying to solve such a game the algorithm will check if it is complete or not:
sage: g.solve() Traceback (most recent call last): ... ValueError: suitor preferences are not complete
To be able to obtain the stable matching we must input the preferences:
sage: for s in g.suitors(): ....: s.pref = (1, 2, 3) sage: for r in g.reviewers(): ....: r.pref = (1, 2, 3) sage: g.solve() {1: 1, 2: 2, 3: 3}

add_reviewer
(name=None)¶ Add a reviewer to the game.
INPUT:
name
– can be a string or number; if left blank will automatically generate an integer
EXAMPLES:
Creating a two player game:
sage: g = MatchingGame(2) sage: g.reviewers() (1, 2)
Adding a suitor without specifying a name:
sage: g.add_reviewer() sage: g.reviewers() (1, 2, 3)
Adding a suitor while specifying a name:
sage: g.add_reviewer(10) sage: g.reviewers() (1, 2, 3, 10)
Note that now our game is no longer complete:
sage: g._is_complete() Traceback (most recent call last): ... ValueError: must have the same number of reviewers as suitors
Note that an error is raised if one tries to add a reviewer with a name that already exists:
sage: g.add_reviewer(10) Traceback (most recent call last): ... ValueError: a reviewer with name "10" already exists
If we add a reviewer without passing a name then the name of the reviewer will not use one that is already chosen:
sage: suit = {0: (1, 3), ....: 1: (3, 1)} sage: revr = {1: (0, 1), ....: 3: (1, 0)} sage: g = MatchingGame([suit, revr]) sage: g.reviewers() (1, 3) sage: g.add_reviewer() sage: g.reviewers() (1, 3, 4)

add_suitor
(name=None)¶ Add a suitor to the game.
INPUT:
name
– can be a string or a number; if left blank will automatically generate an integer
EXAMPLES:
Creating a two player game:
sage: g = MatchingGame(2) sage: g.suitors() (1, 2)
Adding a suitor without specifying a name:
sage: g.add_suitor() sage: g.suitors() (1, 2, 3)
Adding a suitor while specifying a name:
sage: g.add_suitor('D') sage: g.suitors() (1, 2, 3, 'D')
Note that now our game is no longer complete:
sage: g._is_complete() Traceback (most recent call last): ... ValueError: must have the same number of reviewers as suitors
Note that an error is raised if one tries to add a suitor with a name that already exists:
sage: g.add_suitor('D') Traceback (most recent call last): ... ValueError: a suitor with name "D" already exists
If we add a suitor without passing a name then the name of the suitor will not use one that is already chosen:
sage: suit = {0: (1, 2), ....: 2: (2, 1)} sage: revr = {1: (0, 1), ....: 2: (1, 0)} sage: g = MatchingGame([suit, revr]) sage: g.suitors() (0, 2) sage: g.add_suitor() sage: g.suitors() (0, 2, 3)

bipartite_graph
()¶ Construct a
BipartiteGraph
Object of the game. This method is similar to the plot method. Note that the game must be solved for this to work.EXAMPLES:
An error is returned if the game is not solved:
sage: suit = {0: (3, 4), ....: 1: (3, 4)} sage: revr = {3: (0, 1), ....: 4: (1, 0)} sage: g = MatchingGame([suit, revr]) sage: g.bipartite_graph() Traceback (most recent call last): ... ValueError: game has not been solved yet sage: g.solve() {0: 3, 1: 4} sage: g.bipartite_graph() Bipartite graph on 4 vertices

plot
()¶ Create the plot representing the stable matching for the game. Note that the game must be solved for this to work.
EXAMPLES:
An error is returned if the game is not solved:
sage: suit = {0: (3, 4), ....: 1: (3, 4)} sage: revr = {3: (0, 1), ....: 4: (1, 0)} sage: g = MatchingGame([suit, revr]) sage: plot(g) Traceback (most recent call last): ... ValueError: game has not been solved yet sage: g.solve() {0: 3, 1: 4} sage: plot(g) Graphics object consisting of 7 graphics primitives

reviewers
()¶ Return the reviewers of
self
.EXAMPLES:
sage: g = MatchingGame(2) sage: g.reviewers() (1, 2)

solve
(invert=False)¶ Compute a stable matching for the game using the GaleShapley algorithm.
EXAMPLES:
sage: suitr_pref = {'J': ('A', 'D', 'C', 'B'), ....: 'K': ('A', 'B', 'C', 'D'), ....: 'L': ('B', 'C', 'D', 'A'), ....: 'M': ('C', 'A', 'B', 'D')} sage: reviewr_pref = {'A': ('L', 'J', 'K', 'M'), ....: 'B': ('J', 'M', 'L', 'K'), ....: 'C': ('M', 'K', 'L', 'J'), ....: 'D': ('M', 'K', 'J', 'L')} sage: m = MatchingGame([suitr_pref, reviewr_pref]) sage: m.solve() {'J': 'A', 'K': 'D', 'L': 'B', 'M': 'C'} sage: suitr_pref = {'J': ('A', 'D', 'C', 'B'), ....: 'K': ('A', 'B', 'C', 'D'), ....: 'L': ('B', 'C', 'D', 'A'), ....: 'M': ('C', 'A', 'B', 'D')} sage: reviewr_pref = {'A': ('L', 'J', 'K', 'M'), ....: 'B': ('J', 'M', 'L', 'K'), ....: 'C': ('M', 'K', 'L', 'J'), ....: 'D': ('M', 'K', 'J', 'L')} sage: m = MatchingGame([suitr_pref, reviewr_pref]) sage: m.solve(invert=True) {'A': 'L', 'B': 'J', 'C': 'M', 'D': 'K'} sage: suitr_pref = {1: (1,)} sage: reviewr_pref = {1: (1,)} sage: m = MatchingGame([suitr_pref, reviewr_pref]) sage: m.solve() {1: 1} sage: suitr_pref = {} sage: reviewr_pref = {} sage: m = MatchingGame([suitr_pref, reviewr_pref]) sage: m.solve() {}

suitors
()¶ Return the suitors of
self
.EXAMPLES:
sage: g = MatchingGame(2) sage: g.suitors() (1, 2)

class
sage.game_theory.matching_game.
Player
(name)¶ Bases:
object
A class to act as a data holder for the players used of the matching games.
These instances are used when initiating players and to keep track of whether or not partners have a preference.