# Solve with Lemke Howson¶

One of the algorithms implemented in Nashpy is The Lemke Howson Algorithm. This algorithm does not return all equilibria and takes an input argument:

>>> import nashpy as nash
>>> import numpy as np
>>> A = np.array([[1, -1], [-1, 1]])
>>> matching_pennies = nash.Game(A)
>>> matching_pennies.lemke_howson(initial_dropped_label=0)
(array([ 0.5,  0.5]), array([ 0.5,  0.5]))


The initial_dropped_label is an integer between 0 and sum(A.shape) - 1. To iterate over all possible labels use the lemke_howson_enumeration which returns a generator:

>>> equilibria = matching_pennies.lemke_howson_enumeration()
>>> for eq in equilibria:
...     print(eq)
(array([ 0.5,  0.5]), array([ 0.5,  0.5]))
(array([ 0.5,  0.5]), array([ 0.5,  0.5]))
(array([ 0.5,  0.5]), array([ 0.5,  0.5]))
(array([ 0.5,  0.5]), array([ 0.5,  0.5]))


Note that this algorithm is not guaranteed to find all equilibria but is an efficient way of finding an equilibrium.