# Degenerate games¶

A two player game is called nondegenerate if no mixed strategy of support size $$k$$ has more than $$k$$ pure best responses.

For example, the zero sum game defined by the following matrix is degenerate:

$\begin{split}A = \begin{pmatrix} 0 & -1 & 1\\ -1 & 0 & 1\\ -1 & 1 & 0 \end{pmatrix}\end{split}$

The third column has two pure best responses.

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, 2), np.round(eq, 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, 2), np.round(eq, 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, 2), np.round(eq, 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, 2), np.round(eq, 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, 2), np.round(eq, 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, 2), np.round(eq, 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]