A package for solving matching games.

## A package for solving matching games.

Matching games allow for the allocation of resources and partnerships in a fair way. Typically, a matching game is defined by two sets of players that each have preferences over at least some of the elements of the other set. The objective of the game is then to find a mapping between the sets of players in which everyone is happy enough with their match.

In Matching, we deal with four types of matching game:

• the stable marriage problem (SM);
• the hospital-resident assignment problem (HR);
• the student-allocation problem (SA);
• the stable roommates problem (SR).

## Installation

Matching requires Python 3.5 or above, and relies only on NumPy for general use.

The library is most easily installed using `pip`:

```\$ python -m pip install matching
```

However, if you would like to install it from source then go ahead and clone the GitHub repo:

```\$ git clone https://github.com/daffidwilde/matching.git
\$ cd matching
\$ python setup.py install
```

## Documentation

Full documentation (including tutorials and discussion material) is available here: https://matching.readthedocs.io

An academic paper on this library has been included in the Journal of Open Source Software (JOSS) and is available here: https://joss.theoj.org/papers/10.21105/joss.02169

## Playing a simple game

With all games, Matching uses a Player class to represent the members of the “applying” party, i.e. residents and students. For HR and SA, there are specific classes to represent the roles of Hospital, Project and Supervisor.

Consider the following instance of SM which is represented on a bipartite graph where the suitors and reviewers are along the left and right respectively.

We can construct these preferences using dictionaries:

```>>> suitor_preferences = {
...     "A": ["D", "E", "F"], "B": ["D", "F", "E"], "C": ["F", "D", "E"]
... }
>>> reviewer_preferences = {
...     "D": ["B", "C", "A"], "E": ["A", "C", "B"], "F": ["C", "B", "A"]
... }
```

Then to solve this matching game, we make use of the StableMarriage class, like so:

```>>> from matching.games import StableMarriage
>>> game = StableMarriage.create_from_dictionaries(
...     suitor_preferences, reviewer_preferences
... )
>>> game.solve()
{A: E, B: D, C: F}
```

### The Matching object

This matching is not a standard Python dictionary, though it does largely look and behave like one. It is in fact an instance of the SingleMatching class:

```>>> matching = game.matching
>>> type(matching)
<class 'matching.matchings.SingleMatching'>
```

This dictionary-like object is primarily useful as a teaching device that eases the process of manipulating a matching after a solution has been found.

### Player classes

Despite passing dictionaries of strings here, the matching displays instances of matching.player.Player:

```>>> matching = game.matching
>>> for suitor in matching:
...     print(type(suitor))
<class 'matching.players.player.Player'>
<class 'matching.players.player.Player'>
<class 'matching.players.player.Player'>
```

This is because create_from_dictionaries creates instances of the appropriate player classes first and passes them to the game class. Using dictionaries like this can be an efficient way of creating large games but it does require the names of the players in each party to be unique.

With all games, Matching uses a Player class to represent the members of the “applying” party, i.e. residents and students. For HR and SA, there are specific classes to represent the roles of Hospital, Project and Supervisor.

## A note on performance

One of the limitations of this library is the time complexities of the algorithm implementations. In practical terms, the running time of any of the algorithms in Matching is negligible but the theoretic complexity of each has not yet been attained. For example, an instance of HR with 400 applicants and 20 hospitals is solved in less than one tenth of a second:

```>>> from matching.games import HospitalResident
>>> import numpy as np
>>> np.random.seed(0)
>>> num_residents, num_hospitals = 400, 20
>>> resident_prefs = {
...     r: np.argsort(np.random.random(size=num_hospitals))
...     for r in range(num_residents)
... }
>>> hospital_prefs = {
...     h: np.argsort(np.random.random(size=num_residents))
...     for h in range(num_hospitals)
... }
>>> capacities = {h: num_hospitals for h in hospital_prefs}
>>> game = HospitalResident.create_from_dictionaries(
...     resident_prefs, hospital_prefs, capacities
... )
>>> _ = game.solve() # 48.6 ms ± 963 µs per loop
```

## Get in contact!

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