test/symmetric_linear_game_test.py

   1 """
   2 Unit tests for the :class:`SymmetricLinearGame` class.
   3 """
   4
   5 from unittest import TestCase
   6
   7 from dunshire.games import SymmetricLinearGame
   8 from dunshire.matrices import eigenvalues_re, inner_product, norm
   9 from dunshire import options
  10 from .randomgen import (random_icecream_game, random_ll_icecream_game,
  11                         random_ll_orthant_game, random_nn_scaling,
  12                         random_orthant_game, random_positive_orthant_game,
  13                         random_translation)
  14
  15
  16 # Tell pylint to shut up about the large number of methods.
  17 class SymmetricLinearGameTest(TestCase): # pylint: disable=R0904
  18     """
  19     Tests for the SymmetricLinearGame and Solution classes.
  20     """
  21     def assert_within_tol(self, first, second, modifier=1):
  22         """
  23         Test that ``first`` and ``second`` are equal within a multiple of
  24         our default tolerances.
  25
  26         Parameters
  27         ----------
  28
  29         first : float
  30             The first number to compare.
  31
  32         second : float
  33             The second number to compare.
  34
  35         modifier : float
  36             A scaling factor (default: 1) applied to the default
  37             tolerance for this comparison. If you have a poorly-
  38             conditioned matrix, for example, you may want to set this
  39             greater than one.
  40
  41         """
  42         self.assertTrue(abs(first - second) < options.ABS_TOL*modifier)
  43
  44
  45     def test_solutions_dont_change_orthant(self):
  46         G = random_orthant_game()
  47         self.assert_solutions_dont_change(G)
  48
  49     def test_solutions_dont_change_icecream(self):
  50         G = random_icecream_game()
  51         self.assert_solutions_dont_change(G)
  52
  53     def assert_solutions_dont_change(self, G):
  54         """
  55         If we solve the same problem twice, we should get
  56         the same answer both times.
  57         """
  58         soln1 = G.solution()
  59         soln2 = G.solution()
  60         p1_diff = norm(soln1.player1_optimal() - soln2.player1_optimal())
  61         p2_diff = norm(soln1.player2_optimal() - soln2.player2_optimal())
  62         gv_diff = abs(soln1.game_value() - soln2.game_value())
  63
  64         p1_close = p1_diff < options.ABS_TOL
  65         p2_close = p2_diff < options.ABS_TOL
  66         gv_close = gv_diff < options.ABS_TOL
  67
  68         self.assertTrue(p1_close and p2_close and gv_close)
  69
  70
  71     def assert_player1_start_valid(self, G):
  72         x = G.player1_start()['x']
  73         s = G.player1_start()['s']
  74         s1 = s[0:G.dimension()]
  75         s2 = s[G.dimension():]
  76         self.assert_within_tol(norm(G.A()*x - G.b()), 0)
  77         self.assertTrue((s1,s2) in G.C())
  78
  79
  80     def test_player1_start_valid_orthant(self):
  81         """
  82         Ensure that player one's starting point is in the orthant.
  83         """
  84         G = random_orthant_game()
  85         self.assert_player1_start_valid(G)
  86
  87
  88     def test_player1_start_valid_icecream(self):
  89         """
  90         Ensure that player one's starting point is in the ice-cream cone.
  91         """
  92         G = random_icecream_game()
  93         self.assert_player1_start_valid(G)
  94
  95
  96     def test_condition_lower_bound(self):
  97         """
  98         Ensure that the condition number of a game is greater than or
  99         equal to one.
 100
 101         It should be safe to compare these floats directly: we compute
 102         the condition number as the ratio of one nonnegative real number
 103         to a smaller nonnegative real number.
 104         """
 105         G = random_orthant_game()
 106         self.assertTrue(G.condition() >= 1.0)
 107         G = random_icecream_game()
 108         self.assertTrue(G.condition() >= 1.0)
 109
 110
 111     def assert_scaling_works(self, G):
 112         """
 113         Test that scaling ``L`` by a nonnegative number scales the value
 114         of the game by the same number.
 115         """
 116         (alpha, H) = random_nn_scaling(G)
 117         value1 = G.solution().game_value()
 118         value2 = H.solution().game_value()
 119         modifier = 4*max(abs(alpha), 1)
 120         self.assert_within_tol(alpha*value1, value2, modifier)
 121
 122
 123     def test_scaling_orthant(self):
 124         """
 125         Test that scaling ``L`` by a nonnegative number scales the value
 126         of the game by the same number over the nonnegative orthant.
 127         """
 128         G = random_orthant_game()
 129         self.assert_scaling_works(G)
 130
 131
 132     def test_scaling_icecream(self):
 133         """
 134         The same test as :meth:`test_nonnegative_scaling_orthant`,
 135         except over the ice cream cone.
 136         """
 137         G = random_icecream_game()
 138         self.assert_scaling_works(G)
 139
 140
 141     def assert_translation_works(self, G):
 142         """
 143         Check that translating ``L`` by alpha*(e1*e2.trans()) increases
 144         the value of the associated game by alpha.
 145         """
 146         # We need to use ``L`` later, so make sure we transpose it
 147         # before passing it in as a column-indexed matrix.
 148         soln1 = G.solution()
 149         value1 = soln1.game_value()
 150         x_bar = soln1.player1_optimal()
 151         y_bar = soln1.player2_optimal()
 152
 153         # This is the "correct" representation of ``M``, but COLUMN
 154         # indexed...
 155         (alpha, H) = random_translation(G)
 156         value2 = H.solution().game_value()
 157
 158         modifier = 4*max(abs(alpha), 1)
 159         self.assert_within_tol(value1 + alpha, value2, modifier)
 160
 161         # Make sure the same optimal pair works.
 162         self.assert_within_tol(value2, H.payoff(x_bar, y_bar), modifier)
 163
 164
 165     def test_translation_orthant(self):
 166         """
 167         Test that translation works over the nonnegative orthant.
 168         """
 169         G = random_orthant_game()
 170         self.assert_translation_works(G)
 171
 172
 173     def test_translation_icecream(self):
 174         """
 175         The same as :meth:`test_translation_orthant`, except over the
 176         ice cream cone.
 177         """
 178         G = random_icecream_game()
 179         self.assert_translation_works(G)
 180
 181
 182     def assert_opposite_game_works(self, G):
 183         """
 184         Check the value of the "opposite" game that gives rise to a
 185         value that is the negation of the original game. Comes from
 186         some corollary.
 187         """
 188         # This is the "correct" representation of ``M``, but
 189         # COLUMN indexed...
 190         M = -G.L().trans()
 191
 192         # so we have to transpose it when we feed it to the constructor.
 193         # Note: the condition number of ``H`` should be comparable to ``G``.
 194         H = SymmetricLinearGame(M.trans(), G.K(), G.e2(), G.e1())
 195
 196         soln1 = G.solution()
 197         x_bar = soln1.player1_optimal()
 198         y_bar = soln1.player2_optimal()
 199         soln2 = H.solution()
 200
 201         # The modifier of 4 is because each could be off by 2*ABS_TOL,
 202         # which is how far apart the primal/dual objectives have been
 203         # observed being.
 204         self.assert_within_tol(-soln1.game_value(), soln2.game_value(), 4)
 205
 206         # Make sure the switched optimal pair works. Since x_bar and
 207         # y_bar come from G, we use the same modifier.
 208         self.assert_within_tol(soln2.game_value(), H.payoff(y_bar, x_bar), 4)
 209
 210
 211
 212     def test_opposite_game_orthant(self):
 213         """
 214         Test the value of the "opposite" game over the nonnegative
 215         orthant.
 216         """
 217         G = random_orthant_game()
 218         self.assert_opposite_game_works(G)
 219
 220
 221     def test_opposite_game_icecream(self):
 222         """
 223         Like :meth:`test_opposite_game_orthant`, except over the
 224         ice-cream cone.
 225         """
 226         G = random_icecream_game()
 227         self.assert_opposite_game_works(G)
 228
 229
 230     def assert_orthogonality(self, G):
 231         """
 232         Two orthogonality relations hold at an optimal solution, and we
 233         check them here.
 234         """
 235         soln = G.solution()
 236         x_bar = soln.player1_optimal()
 237         y_bar = soln.player2_optimal()
 238         value = soln.game_value()
 239
 240         ip1 = inner_product(y_bar, G.L()*x_bar - value*G.e1())
 241         ip2 = inner_product(value*G.e2() - G.L().trans()*y_bar, x_bar)
 242
 243         # Huh.. well, y_bar and x_bar can each be epsilon away, but
 244         # x_bar is scaled by L, so that's (norm(L) + 1), and then
 245         # value could be off by epsilon, so that's another norm(e1) or
 246         # norm(e2). On the other hand, this test seems to pass most of
 247         # the time even with a modifier of one. How about.. four?
 248         self.assert_within_tol(ip1, 0, 4)
 249         self.assert_within_tol(ip2, 0, 4)
 250
 251
 252     def test_orthogonality_orthant(self):
 253         """
 254         Check the orthgonality relationships that hold for a solution
 255         over the nonnegative orthant.
 256         """
 257         G = random_orthant_game()
 258         self.assert_orthogonality(G)
 259
 260
 261     def test_orthogonality_icecream(self):
 262         """
 263         Check the orthgonality relationships that hold for a solution
 264         over the ice-cream cone.
 265         """
 266         G = random_icecream_game()
 267         self.assert_orthogonality(G)
 268
 269
 270     def test_positive_operator_value(self):
 271         """
 272         Test that a positive operator on the nonnegative orthant gives
 273         rise to a a game with a nonnegative value.
 274
 275         This test theoretically applies to the ice-cream cone as well,
 276         but we don't know how to make positive operators on that cone.
 277         """
 278         G = random_positive_orthant_game()
 279         self.assertTrue(G.solution().game_value() >= -options.ABS_TOL)
 280
 281
 282     def assert_lyapunov_works(self, G):
 283         """
 284         Check that Lyapunov games act the way we expect.
 285         """
 286         soln = G.solution()
 287
 288         # We only check for positive/negative stability if the game
 289         # value is not basically zero. If the value is that close to
 290         # zero, we just won't check any assertions.
 291         #
 292         # See :meth:`assert_within_tol` for an explanation of the
 293         # fudge factors.
 294         eigs = eigenvalues_re(G.L())
 295
 296         if soln.game_value() > options.ABS_TOL:
 297             # L should be positive stable
 298             positive_stable = all([eig > -options.ABS_TOL for eig in eigs])
 299             self.assertTrue(positive_stable)
 300         elif soln.game_value() < -options.ABS_TOL:
 301             # L should be negative stable
 302             negative_stable = all([eig < options.ABS_TOL for eig in eigs])
 303             self.assertTrue(negative_stable)
 304
 305         # The dual game's value should always equal the primal's.
 306         # The modifier of 4 is because even though the games are dual,
 307         # CVXOPT doesn't know that, and each could be off by 2*ABS_TOL.
 308         dualsoln = G.dual().solution()
 309         self.assert_within_tol(dualsoln.game_value(), soln.game_value(), 4)
 310
 311
 312     def test_lyapunov_orthant(self):
 313         """
 314         Test that a Lyapunov game on the nonnegative orthant works.
 315         """
 316         G = random_ll_orthant_game()
 317         self.assert_lyapunov_works(G)
 318
 319
 320     def test_lyapunov_icecream(self):
 321         """
 322         Test that a Lyapunov game on the ice-cream cone works.
 323         """
 324         G = random_ll_icecream_game()
 325         self.assert_lyapunov_works(G)