A package for solving matching games

## Project description

## A package for solving matching games.

A matching game is defined by two sets, called suitors and reviewers. Each suitor has a ranked preference list of the reviewers and vice versa. The objective of a matching game is to find a mapping between the two sets such that no pair in the mapping can do better without destablising the other matchings.

In `matching`, we divide matching games into two general problems; these are
known colloquially as stable marriage problems and hospital-resident assignment
problems, respectively.

## The stable marriage problem

Here, our sets must be of the same size and we require each suitor (and reviewer) to rank all members of the other set. In this way, we obtain a matching between our suitors and reviewers where each suitor is matched to exactly one reviewer, and so our matching is bijective.

It is known that instances of the stable marriage problem can be solved to give a unique, stable and suitor-optimal matching using an algorithm developed by David Gale and Lloyd Shapley. The algorithm is as follows:

- Assign all suitors and reviewers to be unmatched.
- Take any unmatched suitor,
**s**, and their most preferred reviewer,**r**. If all suitors are matched, end. - If
**r**is unmatched, then match**s**to**r**, and go to 1. Otherwise, go to 3. - Consider
**r**’s current match,**s’**. If**r**prefers**s**to**s’**, then unmatch**s’**from**r**and match**s**to**r**. Otherwise, leave**s**unmatched and remove**r**from the preference list of**s**. In either case, go to 1.

By unique, we mean that the result of this algorithm is invariant of which order unmatched suitors are considered. Stability is the concept that the pairings in the matching are such that nobody could be matched to someone they prefer more without their new partner then being able to be with someone better. The final matching being suitor-optimal means that every suitor has their best possible matching without making the matching unstable. A corollary of this is that, in fact, every reviewer has their worst possible matching after the algorithm terminates.

### Usage

With both forms of matching game, `matching` uses standard dictionaries to
construct and solve a given instance. In particular, for instances of the stable
marriage problem, we require a dictionary detailing the preference lists of each
suitor (and reviewer).

Consider the following stable marriage problem which is represented on a bipartite graph.

We convey the information above in the following way:

>>> suitor_prefs = {'A': ['D', 'E', 'F'], ... 'B': ['D', 'F', 'E'], ... 'C': ['F', 'D', 'E']} >>> reviewer_prefs = {'D': ['B', 'C', 'A'], ... 'E': ['A', 'C', 'B'], ... 'F': ['C', 'B', 'A']}

Then to solve this matching game, we make use of the `algorithms` submodule,
like so:

>>> from matching.algorithms import galeshapley >>> matching = galeshapley(suitor_prefs, reviewer_prefs) >>> matching {'A': 'E', 'B': 'D', 'C': 'F'}

It is easily checked - on paper or mentally - that this is the correct solution.

## The hospital-resident assignment problem

For this family of problems, we have a set of suitors (residents) and reviewers (hospitals), and ranked preferences associated with each element of these sets, as in the stable marriage problem. However, we do not require these sets to be of the same size, nor do we require any given suitor (or reviewer) to rank all elements of the other set. In addition to these sets, each reviewer has associated with it a capacity. This capacity is the maximum number of suitors that may be matched to it at any given time.

The algorithm which solves this problem is famously utilised in the USA by the National Resident Matching Program, hence the nickname. In fact, research surrounding this algorithm won Shapley, along with Alvin Roth, the Nobel Prize for Economics in 2012. In this package we refer to this algorithm as the Extended Gale-Shapley algorithm. However, it has several synonyms including: ‘The Match’, ‘the Capacitated Gale-Shapley algorithm’, ‘the Roth-Shapley algorithm’, and ‘the deferred acceptance algorithm’. This algorithm has also been used to develop donor chains for kidney transplants saving thousands of lives in the process.

The algorithm is as follows:

- Assign all suitors and reviewers to be unmatched.
- Take any unmatched suitor,
**s**, that is still up for consideration, and go to 2. If there are no such suitors, end. - If the preference list of
**s**is empty, remove them from consideration, and go to 1. Otherwise, consider their most preferred reviewer,**r**. Go to 3. - If
**s**is not ranked by**r**, remove**r**from the preference list of**s**and go to 2. Otherwise, if**r**has space, match**s**to**r**and go to 1. If not, go to 4. - Consider
**r**’s current matching, and particular their least preferable current matching,**s’**. If**r**prefers**s**to**s’**, then unmatch**s’**from**r**, match**s**to**r**, and go to 1. Otherwise, leave**s**unmatched, remove**s**from the preference list of**r**and**r**from the preference list of**s**, and go to 2.

### Usage

In a similar fashion to the stable marriage problem, we interpret hospital-resident assignment problems using dictionaries. In addition to the suitor and reviewer preference dictionaries, however, we have a capacity dictionary which takes reviewers as keys and integer capacities as values.

Consider the following example. We have five medical residents - Arthur, Sunny, Joseph, Latha and Darrius - and three hospitals, each with 2 positions available: Mercy, City and General. We display their preferences in a similar fashion to before:

In `matching` we summarise this problem in the following way:

>>> suitor_prefs = {'A': ['C'], ... 'S': ['C', 'M'], ... 'J': ['C', 'G', 'M'], ... 'L': ['M', 'C', 'G'], ... 'D': ['C', 'M', 'G']} >>> reviewer_prefs = {'M': ['D', 'J'], ... 'C': ['D', 'A', 'S', 'L', 'J'], ... 'G': ['D', 'A', 'J', 'L']} >>> capacities = {r: 2 for r in reviewer_prefs.keys()}

We then solve this problem using the `extended_galeshapley` algorithm:

>>> from matching.algorithms import extended_galeshapley >>> suitor_match, reviewer_match = extended_galeshapley(suitor_prefs, ... reviewer_prefs, ... capacities) >>> suitor_match {'A': 'C', 'S': None, 'J': 'G', 'L': 'G', 'D': 'C'}

Again, though less likely to be done in your head, you can verify that this matching is correct according to our algorithm.

## Get in contact!

I hope this package is useful, and please feel free to ping me here (or on Twitter: @daffidwilde) with any issues or recommendations. PRs always welcome!

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