# Matching games¶

This module implements a class for matching games (stable marriage problems) [DI1989]. At present the extended Gale-Shapley algorithm is implemented which can be used to obtain stable matchings.

AUTHORS:

• James Campbell and Vince Knight 06-2014: Original version

class sage.game_theory.matching_game.MatchingGame(generator, revr=None)

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 Gale-Shapley 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([-i-1 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.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 Gale-Shapley 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 the invert=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 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.reviewers()
(-1, -3, -4)


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.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 Gale-Shapley 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.