"""A class for the Lemke Howson algorithm"""
from nash.integer_pivoting import (make_tableau, non_basic_variables,
pivot_tableau)
import numpy as np
from itertools import cycle
[docs]def shift_tableau(tableau, shape):
"""
Shift a tableau to ensure labels of pairs of tableaux coincide
Parameters
----------
tableau: a numpy array
shape: a tuple
Returns
-------
tableau: a numpy array
"""
return np.append(np.roll(tableau[:,:-1], shape[0], axis=1),
np.ones((shape[0], 1)), axis=1)
[docs]def tableau_to_strategy(tableau, basic_labels, strategy_labels):
"""
Return a strategy vector from a tableau
Parameters
----------
tableau: a numpy array
basic_labels: a set
strategy_labels: a set
Returns
-------
strategy: a numpy array
"""
vertex = []
for column in strategy_labels:
if column in basic_labels:
for i, row in enumerate(tableau[:, column]):
if row != 0:
vertex.append(tableau[i, -1] / row)
else:
vertex.append(0)
strategy = np.array(vertex)
return strategy / sum(strategy)
[docs]def lemke_howson(A, B, initial_dropped_label=0):
"""
Obtain the Nash equilibria using the Lemke Howson algorithm implemented
using integer pivoting.
Algorithm implemented here is Algorithm 3.6 of [Nisan2007]_.
1. Start at the artificial equilibrium (which is fully labeled)
2. Choose an initial label to drop and move in the polytope for which
the vertex has that label to the edge
that does not share that label. (This is implemented using integer
pivoting)
3. A label will now be duplicated in the other polytope, drop it in a
similar way.
4. Repeat steps 2 and 3 until have Nash Equilibrium.
Parameters
----------
initial_dropped_label: int
Returns
-------
equilibria: A tuple.
"""
if np.min(A) <= 0:
A = A + abs(np.min(A)) + 1
if np.min(B) <= 0:
B = B + abs(np.min(B)) + 1
# build tableaux
col_tableau = make_tableau(A)
col_tableau = shift_tableau(col_tableau, A.shape)
row_tableau = make_tableau(B.transpose())
full_labels = set(range(sum(A.shape)))
if initial_dropped_label in non_basic_variables(row_tableau):
tableux = cycle((row_tableau, col_tableau))
else:
tableux = cycle((col_tableau, row_tableau))
# First pivot (to drop a label)
entering_label = pivot_tableau(next(tableux), initial_dropped_label)
while non_basic_variables(row_tableau).union(non_basic_variables(col_tableau)) != full_labels:
entering_label = pivot_tableau(next(tableux), next(iter(entering_label)))
row_strategy = tableau_to_strategy(row_tableau, non_basic_variables(col_tableau),
range(A.shape[0]))
col_strategy = tableau_to_strategy(col_tableau, non_basic_variables(row_tableau),
range(A.shape[0], sum(A.shape)))
return row_strategy, col_strategy