# Handle Degenerate gamesΒΆ

When dealing with degenerate games unexpected results can occur:

```>>> import nashpy as nash
>>> import numpy as np
>>> A = np.array([[0, -1, 1], [-1, 0, 1], [-1, 0, 1]])
>>> game = nash.Game(A)
```

Here is the output when using Support enumeration:

```>>> for eq in game.support_enumeration():
...     print(np.round(eq[0], 2), np.round(eq[1], 2))
[0.5 0.5 0. ] [0.5 0.5 0. ]
[0.5 0.  0.5] [0.5 0.5 0. ]
```

Here is the output when using Vertex enumeration:

```>>> for eq in game.vertex_enumeration():
...     print(np.round(eq[0], 2), np.round(eq[1], 2))
[0.5 0.  0.5] [ 0.5  0.5 -0. ]
[ 0.5  0.5 -0. ] [ 0.5  0.5 -0. ]
```

Here is the output when using the The Lemke Howson Algorithm:

```>>> for eq in game.lemke_howson_enumeration():
...     print(np.round(eq[0], 2), np.round(eq[1], 2))
[0.33... 0.33... 0.33...] [nan]
```

We see that the lemke-howson algorithm fails but also that the Support enumeration and Vertex enumeration fail to find some equilibria: there is in fact a range of strategies the row player can play against `[ 0.5 0.5 0]` that is still a best response.

The Support enumeration algorithm can be run with two optional arguments:

• `non_degenerate=True` (`False` is the default) will only consider supports of equal size. If you know your game is non degenerate this will make support enumeration make less checks.

• `tol=0` (`10 ** -16` is the default), when considering the underlying linear system `tol` is considered to be a lower bound for difference between two real numbers. Using `tol=0` ensures a strict run of the algorithm.

Here is an example:

```>>> A = np.array([[4, 9, 9], [9, 1, 6], [9, 2, 3]])
>>> B = np.array([[2, 2, 5], [7, 4, 4], [1, 6, 4]])
>>> game = nash.Game(A, B)
>>> for eq in game.support_enumeration():
...     print(np.round(eq[0], 2), np.round(eq[1], 2))
[1. 0. 0.] [0. 0. 1.]
[0. 1. 0.] [1. 0. 0.]
[0.5 0.5 0. ] [0.38 0.   0.62]
[0.2 0.5 0.3] [0.57 0.32 0.11]
>>> for eq in game.support_enumeration(non_degenerate=True):
...     print(np.round(eq[0], 2), np.round(eq[1], 2))
[1. 0. 0.] [0. 0. 1.]
[0. 1. 0.] [1. 0. 0.]
[0.2 0.5 0.3] [0.57 0.32 0.11]
>>> for eq in game.support_enumeration(non_degenerate=False, tol=0):
...     print(np.round(eq[0], 2), np.round(eq[1], 2))
[1. 0. 0.] [0. 0. 1.]
[0. 1. 0.] [1. 0. 0.]
[0.2 0.5 0.3] [0.57 0.32 0.11]
```