# Approximate Moran Process¶

Due to the high computational cost of a single Moran process, an approximate
Moran process is implemented that can make use of cached outcomes of games. The
following code snippet will generate a Moran process in which the outcomes of
the matches played by a `Random: 0.5`

are sampled from one possible
outcome against each opponent (`Defector`

and `Random: 0.5`

). First
the cache is built by passing counter objects of outcomes:

```
>>> import axelrod as axl
>>> from collections import Counter
>>> cached_outcomes = {}
>>> cached_outcomes[("Random: 0.5", "Defector")] = axl.Pdf(Counter([(1, 1)]))
>>> cached_outcomes[("Random: 0.5", "Random: 0.5")] = axl.Pdf(Counter([(3, 3)]))
>>> cached_outcomes[("Defector", "Defector")] = axl.Pdf(Counter([(1, 1)]))
```

Now let us create an Approximate Moran Process:

```
>>> players = [axl.Defector(), axl.Random(), axl.Random()]
>>> amp = axl.ApproximateMoranProcess(players, cached_outcomes, seed=5)
>>> results = amp.play()
>>> amp.population_distribution()
Counter({'Random: 0.5': 3})
```

Note that by nature of being an approximation, it’s possible that the results of an
ApproximateMoranProcess may not always match the results of a standard MoranProcess,
even for the same random seed. We see that, for this random seed, the `Random: 0.5`

won this Moran process. This is not what happens in a standard Moran process where the
`Random: 0.5`

player will not win:

```
>>> mp = axl.MoranProcess(players, seed=5)
>>> results = mp.play()
>>> mp.population_distribution()
Counter({'Defector': 3})
```