from cvxopt import matrix, printing, solvers
from cones import CartesianProduct, IceCream, NonnegativeOrthant
from errors import GameUnsolvableException
-from matrices import append_col, append_row, identity, inner_product
+from matrices import append_col, append_row, identity, inner_product, norm
import options
printing.options['dformat'] = options.FLOAT_FORMAT
Tests for the SymmetricLinearGame and Solution classes.
"""
+ def random_square_matrix(self, dims):
+ """
+ Generate a random square (``dims``-by-``dims``) matrix,
+ represented as a list of rows.
+ """
+ return [[uniform(-10, 10) for i in range(dims)] for j in range(dims)]
+
+
def random_orthant_params(self):
"""
Generate the ``L``, ``K``, ``e1``, and ``e2`` parameters for a
"""
ambient_dim = randint(1, 10)
K = NonnegativeOrthant(ambient_dim)
- e1 = [uniform(0.1, 10) for idx in range(K.dimension())]
- e2 = [uniform(0.1, 10) for idx in range(K.dimension())]
- L = [[uniform(-10, 10) for i in range(K.dimension())]
- for j in range(K.dimension())]
+ e1 = [uniform(0.5, 10) for idx in range(K.dimension())]
+ e2 = [uniform(0.5, 10) for idx in range(K.dimension())]
+ L = self.random_square_matrix(K.dimension())
return (L, K, e1, e2)
fudge_factor = 1.0 / (2.0*sqrt(K.dimension() - 1.0))
e1 += [fudge_factor*uniform(0, 1) for idx in range(K.dimension() - 1)]
e2 += [fudge_factor*uniform(0, 1) for idx in range(K.dimension() - 1)]
- L = [[uniform(-10, 10) for i in range(K.dimension())]
- for j in range(K.dimension())]
+ L = self.random_square_matrix(K.dimension())
return (L, K, e1, e2)
self.assertTrue(abs(first - second) < options.ABS_TOL)
+ def assert_norm_within_tol(self, first, second):
+ """
+ Test that ``first`` and ``second`` vectors are equal in the
+ sense that the norm of their difference is within our default
+ tolerance.
+ """
+ self.assert_within_tol(norm(first - second), 0)
+
+
def assert_solution_exists(self, L, K, e1, e2):
"""
Given the parameters needed to construct a SymmetricLinearGame,
"""
G = SymmetricLinearGame(L, K, e1, e2)
soln = G.solution()
+
+ # The matrix() constructor assumes that ``L`` is a list of
+ # columns, so we transpose it to agree with what
+ # SymmetricLinearGame() thinks.
L_matrix = matrix(L).trans()
expected = inner_product(L_matrix*soln.player1_optimal(),
soln.player2_optimal())
self.assert_within_tol(soln.game_value(), expected)
+
def test_solution_exists_nonnegative_orthant(self):
"""
Every linear game has a solution, so we should be able to solve
(L, K, e1, e2) = self.random_orthant_params()
self.assert_solution_exists(L, K, e1, e2)
+
def test_solution_exists_ice_cream(self):
"""
Like :meth:`test_solution_exists_nonnegative_orthant`, except
our cone.
"""
(L, K, e1, e2) = self.random_orthant_params()
- L = matrix(L) # So that we can scale it by alpha below.
+ # Make ``L`` a matrix so that we can scale it by alpha. Its
+ # random, so who cares if it gets transposed.
+ L = matrix(L)
G1 = SymmetricLinearGame(L, K, e1, e2)
value1 = G1.solution().game_value()
- alpha = uniform(0.1, 10)
+ alpha = uniform(0.1, 10)
G2 = SymmetricLinearGame(alpha*L, K, e1, e2)
value2 = G2.solution().game_value()
self.assert_within_tol(alpha*value1, value2)
except over the ice cream cone.
"""
(L, K, e1, e2) = self.random_icecream_params()
- L = matrix(L) # So that we can scale it by alpha below.
-
+ # Make ``L`` a matrix so that we can scale it by alpha. Its
+ # random, so who cares if it gets transposed.
+ L = matrix(L)
G1 = SymmetricLinearGame(L, K, e1, e2)
value1 = G1.solution().game_value()
- alpha = uniform(0.1, 10)
+ alpha = uniform(0.1, 10)
G2 = SymmetricLinearGame(alpha*L, K, e1, e2)
value2 = G2.solution().game_value()
self.assert_within_tol(alpha*value1, value2)
+
+ def assert_translation_works(self, L, K, e1, e2):
+ """
+ Check that translating ``L`` by alpha*(e1*e2.trans()) increases
+ the value of the associated game by alpha.
+ """
+ e1 = matrix(e1, (K.dimension(), 1))
+ e2 = matrix(e2, (K.dimension(), 1))
+ G = SymmetricLinearGame(L, K, e1, e2)
+ G_soln = G.solution()
+ value_G = G_soln.game_value()
+ x_bar = G_soln.player1_optimal()
+ y_bar = G_soln.player2_optimal()
+
+ alpha = uniform(-10, 10)
+ # Make ``L`` a CVXOPT matrix so that we can do math with
+ # it. Note that this gives us the "correct" representation of
+ # ``L`` (in agreement with what G has), but COLUMN indexed.
+ L = matrix(L).trans()
+ E = e1*e2.trans()
+ # Likewise, this is the "correct" representation of ``M``, but
+ # COLUMN indexed...
+ M = L + alpha*E
+
+ # so we have to transpose it when we feed it to the constructor.
+ H = SymmetricLinearGame(M.trans(), K, e1, e2)
+ value_H = H.solution().game_value()
+
+ # Make sure the same optimal pair works.
+ H_payoff = inner_product(M*x_bar, y_bar)
+
+ self.assert_within_tol(value_G + alpha, value_H)
+ self.assert_within_tol(value_H, H_payoff)
+
+
+ def test_translation_orthant(self):
+ """
+ Test that translation works over the nonnegative orthant.
+ """
+ (L, K, e1, e2) = self.random_orthant_params()
+ self.assert_translation_works(L, K, e1, e2)
+
+
+ def test_translation_icecream(self):
+ """
+ The same as :meth:`test_translation_orthant`, except over the
+ ice cream cone.
+ """
+ (L, K, e1, e2) = self.random_icecream_params()
+ self.assert_translation_works(L, K, e1, e2)
+
+
+ def assert_opposite_game_works(self, L, K, e1, e2):
+ e1 = matrix(e1, (K.dimension(), 1))
+ e2 = matrix(e2, (K.dimension(), 1))
+ G = SymmetricLinearGame(L, K, e1, e2)
+
+ # Make ``L`` a CVXOPT matrix so that we can do math with
+ # it. Note that this gives us the "correct" representation of
+ # ``L`` (in agreement with what G has), but COLUMN indexed.
+ L = matrix(L).trans()
+
+ # Likewise, this is the "correct" representation of ``M``, but
+ # COLUMN indexed...
+ M = -L.trans()
+
+ # so we have to transpose it when we feed it to the constructor.
+ H = SymmetricLinearGame(M.trans(), K, e2, e1)
+
+ G_soln = G.solution()
+ x_bar = G_soln.player1_optimal()
+ y_bar = G_soln.player2_optimal()
+ H_soln = H.solution()
+
+ # Make sure the switched optimal pair works.
+ H_payoff = inner_product(M*y_bar, x_bar)
+
+ self.assert_within_tol(-G_soln.game_value(), H_soln.game_value())
+ self.assert_within_tol(H_soln.game_value(), H_payoff)
+
+
+ def test_opposite_game_orthant(self):
+ """
+ Check the value of the "opposite" game that gives rise to a
+ value that is the negation of the original game. Comes from
+ some corollary.
+ """
+ (L, K, e1, e2) = self.random_orthant_params()
+ self.assert_opposite_game_works(L, K, e1, e2)
+
+
+ def test_opposite_game_icecream(self):
+ """
+ Like :meth:`test_opposite_game_orthant`, except over the
+ ice-cream cone.
+ """
+ (L, K, e1, e2) = self.random_icecream_params()
+ self.assert_opposite_game_works(L, K, e1, e2)
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