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 assert_player2_start_valid(self, G):
  97         z = G.player2_start()['z']
  98         z1 = z[0:G.dimension()]
  99         z2 = z[G.dimension():]
 100         self.assertTrue((z1,z2) in G.C())
 101
 102
 103     def test_player2_start_valid_orthant(self):
 104         """
 105         Ensure that player two's starting point is in the orthant.
 106         """
 107         G = random_orthant_game()
 108         self.assert_player2_start_valid(G)
 109
 110
 111     def test_player2_start_valid_icecream(self):
 112         """
 113         Ensure that player two's starting point is in the ice-cream cone.
 114         """
 115         G = random_icecream_game()
 116         self.assert_player2_start_valid(G)
 117
 118
 119     def test_condition_lower_bound(self):
 120         """
 121         Ensure that the condition number of a game is greater than or
 122         equal to one.
 123
 124         It should be safe to compare these floats directly: we compute
 125         the condition number as the ratio of one nonnegative real number
 126         to a smaller nonnegative real number.
 127         """
 128         G = random_orthant_game()
 129         self.assertTrue(G.condition() >= 1.0)
 130         G = random_icecream_game()
 131         self.assertTrue(G.condition() >= 1.0)
 132
 133
 134     def assert_scaling_works(self, G):
 135         """
 136         Test that scaling ``L`` by a nonnegative number scales the value
 137         of the game by the same number.
 138         """
 139         (alpha, H) = random_nn_scaling(G)
 140         value1 = G.solution().game_value()
 141         value2 = H.solution().game_value()
 142         modifier = 4*max(abs(alpha), 1)
 143         self.assert_within_tol(alpha*value1, value2, modifier)
 144
 145
 146     def test_scaling_orthant(self):
 147         """
 148         Test that scaling ``L`` by a nonnegative number scales the value
 149         of the game by the same number over the nonnegative orthant.
 150         """
 151         G = random_orthant_game()
 152         self.assert_scaling_works(G)
 153
 154
 155     def test_scaling_icecream(self):
 156         """
 157         The same test as :meth:`test_nonnegative_scaling_orthant`,
 158         except over the ice cream cone.
 159         """
 160         G = random_icecream_game()
 161         self.assert_scaling_works(G)
 162
 163
 164     def assert_translation_works(self, G):
 165         """
 166         Check that translating ``L`` by alpha*(e1*e2.trans()) increases
 167         the value of the associated game by alpha.
 168         """
 169         # We need to use ``L`` later, so make sure we transpose it
 170         # before passing it in as a column-indexed matrix.
 171         soln1 = G.solution()
 172         value1 = soln1.game_value()
 173         x_bar = soln1.player1_optimal()
 174         y_bar = soln1.player2_optimal()
 175
 176         # This is the "correct" representation of ``M``, but COLUMN
 177         # indexed...
 178         (alpha, H) = random_translation(G)
 179         value2 = H.solution().game_value()
 180
 181         modifier = 4*max(abs(alpha), 1)
 182         self.assert_within_tol(value1 + alpha, value2, modifier)
 183
 184         # Make sure the same optimal pair works.
 185         self.assert_within_tol(value2, H.payoff(x_bar, y_bar), modifier)
 186
 187
 188     def test_translation_orthant(self):
 189         """
 190         Test that translation works over the nonnegative orthant.
 191         """
 192         G = random_orthant_game()
 193         self.assert_translation_works(G)
 194
 195
 196     def test_translation_icecream(self):
 197         """
 198         The same as :meth:`test_translation_orthant`, except over the
 199         ice cream cone.
 200         """
 201         G = random_icecream_game()
 202         self.assert_translation_works(G)
 203
 204
 205     def assert_opposite_game_works(self, G):
 206         """
 207         Check the value of the "opposite" game that gives rise to a
 208         value that is the negation of the original game. Comes from
 209         some corollary.
 210         """
 211         # This is the "correct" representation of ``M``, but
 212         # COLUMN indexed...
 213         M = -G.L().trans()
 214
 215         # so we have to transpose it when we feed it to the constructor.
 216         # Note: the condition number of ``H`` should be comparable to ``G``.
 217         H = SymmetricLinearGame(M.trans(), G.K(), G.e2(), G.e1())
 218
 219         soln1 = G.solution()
 220         x_bar = soln1.player1_optimal()
 221         y_bar = soln1.player2_optimal()
 222         soln2 = H.solution()
 223
 224         # The modifier of 4 is because each could be off by 2*ABS_TOL,
 225         # which is how far apart the primal/dual objectives have been
 226         # observed being.
 227         self.assert_within_tol(-soln1.game_value(), soln2.game_value(), 4)
 228
 229         # Make sure the switched optimal pair works. Since x_bar and
 230         # y_bar come from G, we use the same modifier.
 231         self.assert_within_tol(soln2.game_value(), H.payoff(y_bar, x_bar), 4)
 232
 233
 234
 235     def test_opposite_game_orthant(self):
 236         """
 237         Test the value of the "opposite" game over the nonnegative
 238         orthant.
 239         """
 240         G = random_orthant_game()
 241         self.assert_opposite_game_works(G)
 242
 243
 244     def test_opposite_game_icecream(self):
 245         """
 246         Like :meth:`test_opposite_game_orthant`, except over the
 247         ice-cream cone.
 248         """
 249         G = random_icecream_game()
 250         self.assert_opposite_game_works(G)
 251
 252
 253     def assert_orthogonality(self, G):
 254         """
 255         Two orthogonality relations hold at an optimal solution, and we
 256         check them here.
 257         """
 258         soln = G.solution()
 259         x_bar = soln.player1_optimal()
 260         y_bar = soln.player2_optimal()
 261         value = soln.game_value()
 262
 263         ip1 = inner_product(y_bar, G.L()*x_bar - value*G.e1())
 264         ip2 = inner_product(value*G.e2() - G.L().trans()*y_bar, x_bar)
 265
 266         # Huh.. well, y_bar and x_bar can each be epsilon away, but
 267         # x_bar is scaled by L, so that's (norm(L) + 1), and then
 268         # value could be off by epsilon, so that's another norm(e1) or
 269         # norm(e2). On the other hand, this test seems to pass most of
 270         # the time even with a modifier of one. How about.. four?
 271         self.assert_within_tol(ip1, 0, 4)
 272         self.assert_within_tol(ip2, 0, 4)
 273
 274
 275     def test_orthogonality_orthant(self):
 276         """
 277         Check the orthgonality relationships that hold for a solution
 278         over the nonnegative orthant.
 279         """
 280         G = random_orthant_game()
 281         self.assert_orthogonality(G)
 282
 283
 284     def test_orthogonality_icecream(self):
 285         """
 286         Check the orthgonality relationships that hold for a solution
 287         over the ice-cream cone.
 288         """
 289         G = random_icecream_game()
 290         self.assert_orthogonality(G)
 291
 292
 293     def test_positive_operator_value(self):
 294         """
 295         Test that a positive operator on the nonnegative orthant gives
 296         rise to a a game with a nonnegative value.
 297
 298         This test theoretically applies to the ice-cream cone as well,
 299         but we don't know how to make positive operators on that cone.
 300         """
 301         G = random_positive_orthant_game()
 302         self.assertTrue(G.solution().game_value() >= -options.ABS_TOL)
 303
 304
 305     def assert_lyapunov_works(self, G):
 306         """
 307         Check that Lyapunov games act the way we expect.
 308         """
 309         soln = G.solution()
 310
 311         # We only check for positive/negative stability if the game
 312         # value is not basically zero. If the value is that close to
 313         # zero, we just won't check any assertions.
 314         #
 315         # See :meth:`assert_within_tol` for an explanation of the
 316         # fudge factors.
 317         eigs = eigenvalues_re(G.L())
 318
 319         if soln.game_value() > options.ABS_TOL:
 320             # L should be positive stable
 321             positive_stable = all([eig > -options.ABS_TOL for eig in eigs])
 322             self.assertTrue(positive_stable)
 323         elif soln.game_value() < -options.ABS_TOL:
 324             # L should be negative stable
 325             negative_stable = all([eig < options.ABS_TOL for eig in eigs])
 326             self.assertTrue(negative_stable)
 327
 328         # The dual game's value should always equal the primal's.
 329         # The modifier of 4 is because even though the games are dual,
 330         # CVXOPT doesn't know that, and each could be off by 2*ABS_TOL.
 331         dualsoln = G.dual().solution()
 332         self.assert_within_tol(dualsoln.game_value(), soln.game_value(), 4)
 333
 334
 335     def test_lyapunov_orthant(self):
 336         """
 337         Test that a Lyapunov game on the nonnegative orthant works.
 338         """
 339         G = random_ll_orthant_game()
 340         self.assert_lyapunov_works(G)
 341
 342
 343     def test_lyapunov_icecream(self):
 344         """
 345         Test that a Lyapunov game on the ice-cream cone works.
 346         """
 347         G = random_ll_icecream_game()
 348         self.assert_lyapunov_works(G)