matrix_games_LP.py

"""
Copyright 2013 Steven Diamond

Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
You may obtain a copy of the License at

    http://www.apache.org/licenses/LICENSE-2.0

Unless required by applicable law or agreed to in writing, software
distributed under the License is distributed on an "AS IS" BASIS,
WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
See the License for the specific language governing permissions and
limitations under the License.
"""

# for decimal division
from __future__ import division

import cvxopt

from cvxpy import Maximize, Minimize, Problem, Variable

# Taken from CVX website http://cvxr.com/cvx/examples/
# Example: Section 5.2.5: Mixed strategies for matrix games (LP formulation)
# Ported from cvx matlab to cvxpy by Misrab Faizullah-Khan
# Original comments below


# Boyd & Vandenberghe, "Convex Optimization"
# Joelle Skaf - 08/24/05
#
# Player 1 wishes to choose u to minimize his expected payoff u'Pv, while
# player 2 wishes to choose v to maximize u'Pv, where P is the payoff
# matrix, u and v are the probability distributions of the choices of each
# player (i.e. u>=0, v>=0, sum(u_i)=1, sum(v_i)=1)
# LP formulation:   minimize    t
#                       s.t.    u >=0 , sum(u) = 1, P'*u <= t*1
#                   maximize    t
#                       s.t.    v >=0 , sum(v) = 1, P*v >= t*1

# Input data
n = 12
m = 12
P = cvxopt.normal(n,m)

# Variables for two players
x = Variable(n)
y = Variable(m)
t1 = Variable()
t2 = Variable()

# Note in one case we are maximizing; in the other we are minimizing
objective1 = Minimize(t1)
objective2 = Maximize(t2)

constraints1 = [ x>=0, sum(x)==1, P.T*x <= t1 ]
constraints2 = [ y>=0, sum(y)==1, P*y >= t2 ]


p1 = Problem(objective1, constraints1)
p2 = Problem(objective2, constraints2)

# Optimal strategy for Player 1
print('Computing the optimal strategy for player 1 ... ')
result1 = p1.solve()
print('Done!')

# Optimal strategy for Player 2
print('Computing the optimal strategy for player 2 ... ')
result2 = p2.solve()
print('Done!')

# Displaying results
print('------------------------------------------------------------------------')
print('The optimal strategies for players 1 and 2 are respectively: ')
print(x.value, y.value)
print('The expected payoffs for player 1 and player 2 respectively are: ')
print(result1, result2)
print('They are equal as expected!')
## ISSUE: THEY AREN'T EXACTLY EQUAL FOR SOME REASON!