"""
Unit tests for the :class:`SymmetricLinearGame` class.
"""
from unittest import TestCase
from dunshire.games import SymmetricLinearGame
from dunshire.matrices import eigenvalues_re, inner_product, norm
from dunshire import options
from .randomgen import (random_icecream_game, random_ll_icecream_game,
random_ll_orthant_game, random_nn_scaling,
random_orthant_game, random_positive_orthant_game,
random_translation)
# Tell pylint to shut up about the large number of methods.
[docs]
class SymmetricLinearGameTest(TestCase): # pylint: disable=R0904
"""
Tests for the SymmetricLinearGame and Solution classes.
"""
[docs]
def assert_within_tol(self, first, second, modifier=1):
"""
Test that ``first`` and ``second`` are equal within a multiple of
our default tolerances.
Parameters
----------
first : float
The first number to compare.
second : float
The second number to compare.
modifier : float
A scaling factor (default: 1) applied to the default
tolerance for this comparison. If you have a poorly-
conditioned matrix, for example, you may want to set this
greater than one.
"""
self.assertTrue(abs(first - second) < options.ABS_TOL*modifier)
[docs]
def test_solutions_dont_change_orthant(self):
"""
If we solve the same game twice over the nonnegative orthant,
then we should get the same solution both times. The solution to
a game is not unique, but the process we use is (as far as we
know) deterministic.
"""
G = random_orthant_game()
self.assert_solutions_dont_change(G)
[docs]
def test_solutions_dont_change_icecream(self):
"""
If we solve the same game twice over the ice-cream cone, then we
should get the same solution both times. The solution to a game
is not unique, but the process we use is (as far as we know)
deterministic.
"""
G = random_icecream_game()
self.assert_solutions_dont_change(G)
[docs]
def assert_solutions_dont_change(self, G):
"""
Solve ``G`` twice and check that the solutions agree.
"""
soln1 = G.solution()
soln2 = G.solution()
p1_diff = norm(soln1.player1_optimal() - soln2.player1_optimal())
p2_diff = norm(soln1.player2_optimal() - soln2.player2_optimal())
gv_diff = abs(soln1.game_value() - soln2.game_value())
p1_close = p1_diff < options.ABS_TOL
p2_close = p2_diff < options.ABS_TOL
gv_close = gv_diff < options.ABS_TOL
self.assertTrue(p1_close and p2_close and gv_close)
[docs]
def assert_player1_start_valid(self, G):
"""
Ensure that player one's starting point satisfies both the
equality and cone inequality in the CVXOPT primal problem.
"""
x = G.player1_start()['x']
s = G.player1_start()['s']
s1 = s[0:G.dimension()]
s2 = s[G.dimension():]
self.assert_within_tol(norm(G.A()*x - G.b()), 0)
self.assertTrue((s1, s2) in G.C())
[docs]
def test_player1_start_valid_orthant(self):
"""
Ensure that player one's starting point is feasible over the
nonnegative orthant.
"""
G = random_orthant_game()
self.assert_player1_start_valid(G)
[docs]
def test_player1_start_valid_icecream(self):
"""
Ensure that player one's starting point is feasible over the
ice-cream cone.
"""
G = random_icecream_game()
self.assert_player1_start_valid(G)
[docs]
def assert_player2_start_valid(self, G):
"""
Check that player two's starting point satisfies both the
cone inequality in the CVXOPT dual problem.
"""
z = G.player2_start()['z']
z1 = z[0:G.dimension()]
z2 = z[G.dimension():]
self.assertTrue((z1, z2) in G.C())
[docs]
def test_player2_start_valid_orthant(self):
"""
Ensure that player two's starting point is feasible over the
nonnegative orthant.
"""
G = random_orthant_game()
self.assert_player2_start_valid(G)
[docs]
def test_player2_start_valid_icecream(self):
"""
Ensure that player two's starting point is feasible over the
ice-cream cone.
"""
G = random_icecream_game()
self.assert_player2_start_valid(G)
[docs]
def test_condition_lower_bound(self):
"""
Ensure that the condition number of a game is greater than or
equal to one.
It should be safe to compare these floats directly: we compute
the condition number as the ratio of one nonnegative real number
to a smaller nonnegative real number.
"""
G = random_orthant_game()
self.assertTrue(G.condition() >= 1.0)
G = random_icecream_game()
self.assertTrue(G.condition() >= 1.0)
[docs]
def assert_scaling_works(self, G):
"""
Test that scaling ``L`` by a nonnegative number scales the value
of the game by the same number.
"""
(alpha, H) = random_nn_scaling(G)
soln1 = G.solution()
soln2 = H.solution()
value1 = soln1.game_value()
value2 = soln2.game_value()
modifier1 = G.tolerance_scale(soln1)
modifier2 = H.tolerance_scale(soln2)
modifier = max(modifier1, modifier2)
self.assert_within_tol(alpha*value1, value2, modifier)
[docs]
def test_scaling_orthant(self):
"""
Test that scaling ``L`` by a nonnegative number scales the value
of the game by the same number over the nonnegative orthant.
"""
G = random_orthant_game()
self.assert_scaling_works(G)
[docs]
def test_scaling_icecream(self):
"""
The same test as :meth:`test_nonnegative_scaling_orthant`,
except over the ice cream cone.
"""
G = random_icecream_game()
self.assert_scaling_works(G)
[docs]
def assert_translation_works(self, G):
"""
Check that translating ``L`` by alpha*(e1*e2.trans()) increases
the value of the associated game by alpha.
"""
# We need to use ``L`` later, so make sure we transpose it
# before passing it in as a column-indexed matrix.
soln1 = G.solution()
value1 = soln1.game_value()
x_bar = soln1.player1_optimal()
y_bar = soln1.player2_optimal()
# This is the "correct" representation of ``M``, but COLUMN
# indexed...
(alpha, H) = random_translation(G)
value2 = H.solution().game_value()
modifier = G.tolerance_scale(soln1)
self.assert_within_tol(value1 + alpha, value2, modifier)
# Make sure the same optimal pair works.
self.assert_within_tol(value2, H.payoff(x_bar, y_bar), modifier)
[docs]
def test_translation_orthant(self):
"""
Test that translation works over the nonnegative orthant.
"""
G = random_orthant_game()
self.assert_translation_works(G)
[docs]
def test_translation_icecream(self):
"""
The same as :meth:`test_translation_orthant`, except over the
ice cream cone.
"""
G = random_icecream_game()
self.assert_translation_works(G)
[docs]
def assert_opposite_game_works(self, G):
"""
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.
"""
# Since L is a CVXOPT matrix, it will be transposed automatically.
# Note: the condition number of ``H`` should be comparable to ``G``.
H = SymmetricLinearGame(-G.L(), G.K(), G.e2(), G.e1())
soln1 = G.solution()
x_bar = soln1.player1_optimal()
y_bar = soln1.player2_optimal()
soln2 = H.solution()
modifier = G.tolerance_scale(soln1)
self.assert_within_tol(-soln1.game_value(),
soln2.game_value(),
modifier)
# Make sure the switched optimal pair works. Since x_bar and
# y_bar come from G, we use the same modifier.
self.assert_within_tol(soln2.game_value(),
H.payoff(y_bar, x_bar),
modifier)
[docs]
def test_opposite_game_orthant(self):
"""
Test the value of the "opposite" game over the nonnegative
orthant.
"""
G = random_orthant_game()
self.assert_opposite_game_works(G)
[docs]
def test_opposite_game_icecream(self):
"""
Like :meth:`test_opposite_game_orthant`, except over the
ice-cream cone.
"""
G = random_icecream_game()
self.assert_opposite_game_works(G)
[docs]
def assert_orthogonality(self, G):
"""
Two orthogonality relations hold at an optimal solution, and we
check them here.
"""
soln = G.solution()
x_bar = soln.player1_optimal()
y_bar = soln.player2_optimal()
value = soln.game_value()
ip1 = inner_product(y_bar, G.L()*x_bar - value*G.e1())
ip2 = inner_product(value*G.e2() - G.L().trans()*y_bar, x_bar)
modifier = G.tolerance_scale(soln)
self.assert_within_tol(ip1, 0, modifier)
self.assert_within_tol(ip2, 0, modifier)
[docs]
def test_orthogonality_orthant(self):
"""
Check the orthgonality relationships that hold for a solution
over the nonnegative orthant.
"""
G = random_orthant_game()
self.assert_orthogonality(G)
[docs]
def test_orthogonality_icecream(self):
"""
Check the orthgonality relationships that hold for a solution
over the ice-cream cone.
"""
G = random_icecream_game()
self.assert_orthogonality(G)
[docs]
def test_positive_operator_value(self):
"""
Test that a positive operator on the nonnegative orthant gives
rise to a a game with a nonnegative value.
This test theoretically applies to the ice-cream cone as well,
but we don't know how to make positive operators on that cone.
"""
G = random_positive_orthant_game()
self.assertTrue(G.solution().game_value() >= -options.ABS_TOL)
[docs]
def assert_lyapunov_works(self, G):
"""
Check that Lyapunov games act the way we expect.
"""
soln = G.solution()
# We only check for positive/negative stability if the game
# value is not basically zero. If the value is that close to
# zero, we just won't check any assertions.
#
# See :meth:`assert_within_tol` for an explanation of the
# fudge factors.
eigs = eigenvalues_re(G.L())
if soln.game_value() > options.ABS_TOL:
# L should be positive stable
positive_stable = all([eig > -options.ABS_TOL for eig in eigs])
self.assertTrue(positive_stable)
elif soln.game_value() < -options.ABS_TOL:
# L should be negative stable
negative_stable = all([eig < options.ABS_TOL for eig in eigs])
self.assertTrue(negative_stable)
dualsoln = G.dual().solution()
mod = G.tolerance_scale(soln)
self.assert_within_tol(dualsoln.game_value(), soln.game_value(), mod)
[docs]
def test_lyapunov_orthant(self):
"""
Test that a Lyapunov game on the nonnegative orthant works.
"""
G = random_ll_orthant_game()
self.assert_lyapunov_works(G)
[docs]
def test_lyapunov_icecream(self):
"""
Test that a Lyapunov game on the ice-cream cone works.
"""
G = random_ll_icecream_game()
self.assert_lyapunov_works(G)