Fix a pylint warning in the matrices module.

[dunshire.git] / test / symmetric_linear_game_test.py

diff --git a/test/symmetric_linear_game_test.py b/test/symmetric_linear_game_test.py
index 5adbb2ddd11942faf3c31354513b62ab6877e853..12783f300078939c14db7f4b26db196c24166fe0 100644 (file)
--- a/test/symmetric_linear_game_test.py
+++ b/test/symmetric_linear_game_test.py
@@ -1,4 +1,19 @@
-# These few are used only for tests.
+"""
+Unit tests for the :class:`SymmetricLinearGame` class.
+"""
+
+MAX_COND = 250
+"""
+The maximum condition number of a randomly-generated game.
+"""
+
+RANDOM_MAX = 10
+"""
+When generating uniform random real numbers, this will be used as the
+largest allowed magnitude. It keeps our condition numbers down and other
+properties within reason.
+"""
+

 from math import sqrt
 from random import randint, uniform
 from unittest import TestCase
 from math import sqrt
 from random import randint, uniform
 from unittest import TestCase


@@ -10,42 +25,130 @@ from dunshire.matrices import (append_col, append_row, eigenvalues_re,
                                identity, inner_product)
 from dunshire import options
 
                                identity, inner_product)
 from dunshire import options
 

-def _random_matrix(dims):
+
+def random_matrix(dims):

     """
     """

-    Generate a random square (``dims``-by-``dims``) matrix. This is used
-    only by the :class:`SymmetricLinearGameTest` class.
+    Generate a random square matrix.
+
+    Parameters
+    ----------
+
+    dims : int
+        The number of rows/columns you want in the returned matrix.
+
+    Returns
+    -------
+
+    matrix
+        A new matrix whose entries are random floats chosen uniformly from
+        the interval [-RANDOM_MAX, RANDOM_MAX].
+
+    Examples
+    --------
+
+        >>> A = random_matrix(3)
+        >>> A.size
+        (3, 3)
+

     """
     """

-    return matrix([[uniform(-10, 10) for i in range(dims)]
-                   for j in range(dims)])
+    return matrix([[uniform(-RANDOM_MAX, RANDOM_MAX) for _ in range(dims)]
+                   for _ in range(dims)])
+

 
 

-def _random_nonnegative_matrix(dims):
+def random_nonnegative_matrix(dims):

     """
     """

-    Generate a random square (``dims``-by-``dims``) matrix with
-    nonnegative entries. This is used only by the
-    :class:`SymmetricLinearGameTest` class.
+    Generate a random square matrix with nonnegative entries.
+
+    Parameters
+    ----------
+
+    dims : int
+        The number of rows/columns you want in the returned matrix.
+
+    Returns
+    -------
+
+    matrix
+        A new matrix whose entries are random floats chosen uniformly from
+        the interval [0, RANDOM_MAX].
+
+    Examples
+    --------
+
+        >>> A = random_nonnegative_matrix(3)
+        >>> A.size
+        (3, 3)
+        >>> all([entry >= 0 for entry in A])
+        True
+

     """
     """

-    L = _random_matrix(dims)
+    L = random_matrix(dims)

     return matrix([abs(entry) for entry in L], (dims, dims))
 
     return matrix([abs(entry) for entry in L], (dims, dims))
 

-def _random_diagonal_matrix(dims):
+
+def random_diagonal_matrix(dims):

     """
     """

-    Generate a random square (``dims``-by-``dims``) matrix with nonzero
-    entries only on the diagonal. This is used only by the
-    :class:`SymmetricLinearGameTest` class.
+    Generate a random square matrix with zero off-diagonal entries.
+
+    These matrices are Lyapunov-like on the nonnegative orthant, as is
+    fairly easy to see.
+
+    Parameters
+    ----------
+
+    dims : int
+        The number of rows/columns you want in the returned matrix.
+
+    Returns
+    -------
+
+    matrix
+        A new matrix whose diagonal entries are random floats chosen
+        uniformly from the interval [-RANDOM_MAX, RANDOM_MAX] and whose
+        off-diagonal entries are zero.
+
+    Examples
+    --------
+
+        >>> A = random_diagonal_matrix(3)
+        >>> A.size
+        (3, 3)
+        >>> A[0,1] == A[0,2] == A[1,0] == A[2,0] == A[1,2] == A[2,1] == 0
+        True
+

     """
     """

-    return matrix([[uniform(-10, 10)*int(i == j) for i in range(dims)]
+    return matrix([[uniform(-RANDOM_MAX, RANDOM_MAX)*int(i == j)
+                   for i in range(dims)]

                    for j in range(dims)])
 
 
                    for j in range(dims)])
 
 

-def _random_skew_symmetric_matrix(dims):
+def random_skew_symmetric_matrix(dims):

     """
     """

-    Generate a random skew-symmetrix (``dims``-by-``dims``) matrix.
+    Generate a random skew-symmetrix matrix.
+
+    Parameters
+    ----------
+
+    dims : int
+        The number of rows/columns you want in the returned matrix.
+
+    Returns
+    -------
+
+    matrix
+        A new skew-matrix whose strictly above-diagonal entries are
+        random floats chosen uniformly from the interval
+        [-RANDOM_MAX, RANDOM_MAX].

 
     Examples
     --------
 
 
     Examples
     --------
 

+        >>> A = random_skew_symmetric_matrix(3)
+        >>> A.size
+        (3, 3)
+

         >>> from dunshire.matrices import norm
         >>> from dunshire.matrices import norm

-        >>> A = _random_skew_symmetric_matrix(randint(1, 10))
+        >>> A = random_skew_symmetric_matrix(randint(1, 10))

         >>> norm(A + A.trans()) < options.ABS_TOL
         True
 
         >>> norm(A + A.trans()) < options.ABS_TOL
         True
 


@@ -57,38 +160,88 @@ def _random_skew_symmetric_matrix(dims):
     return strict_ut - strict_ut.trans()
 
 
     return strict_ut - strict_ut.trans()
 
 

-def _random_lyapunov_like_icecream(dims):
-    """
-    Generate a random Lyapunov-like matrix over the ice-cream cone in
-    ``dims`` dimensions.
+def random_lyapunov_like_icecream(dims):
+    r"""
+    Generate a random matrix Lyapunov-like on the ice-cream cone.
+
+    The form of these matrices is cited in Gowda and Tao
+    [GowdaTao]_. The scalar ``a`` and the vector ``b`` (using their
+    notation) are easy to generate. The submatrix ``D`` is a little
+    trickier, but it can be found noticing that :math:`C + C^{T} = 0`
+    for a skew-symmetric matrix :math:`C` implying that :math:`C + C^{T}
+    + \left(2a\right)I = \left(2a\right)I`. Thus we can stick an
+    :math:`aI` with each of :math:`C,C^{T}` and let those be our
+    :math:`D,D^{T}`.
+
+    Parameters
+    ----------
+
+    dims : int
+        The dimension of the ice-cream cone (not of the matrix you want!)
+        on which the returned matrix should be Lyapunov-like.
+
+    Returns
+    -------
+
+    matrix
+        A new matrix, Lyapunov-like on the ice-cream cone in ``dims``
+        dimensions, whose free entries are random floats chosen uniformly
+        from the interval [-10, 10].
+
+    References
+    ----------
+
+    .. [GowdaTao] M. S. Gowda and J. Tao. On the bilinearity rank of a
+       proper cone and Lyapunov-like transformations. Mathematical
+       Programming, 147:155-170, 2014.
+
+    Examples
+    --------
+
+        >>> L = random_lyapunov_like_icecream(3)
+        >>> L.size
+        (3, 3)
+        >>> x = matrix([1,1,0])
+        >>> s = matrix([1,-1,0])
+        >>> abs(inner_product(L*x, s)) < options.ABS_TOL
+        True
+

     """
     a = matrix([uniform(-10, 10)], (1, 1))
     """
     a = matrix([uniform(-10, 10)], (1, 1))

-    b = matrix([uniform(-10, 10) for idx in range(dims-1)], (dims-1, 1))
-    D = _random_skew_symmetric_matrix(dims-1) + a*identity(dims-1)
+    b = matrix([uniform(-10, 10) for _ in range(dims-1)], (dims-1, 1))
+    D = random_skew_symmetric_matrix(dims-1) + a*identity(dims-1)

     row1 = append_col(a, b.trans())
     row2 = append_col(b, D)
     return append_row(row1, row2)
 
 
     row1 = append_col(a, b.trans())
     row2 = append_col(b, D)
     return append_row(row1, row2)
 
 

-def _random_orthant_params():
+def random_orthant_game():

     """
     Generate the ``L``, ``K``, ``e1``, and ``e2`` parameters for a
     """
     Generate the ``L``, ``K``, ``e1``, and ``e2`` parameters for a

-    random game over the nonnegative orthant. This is only used by
-    the :class:`SymmetricLinearGameTest` class.
+    random game over the nonnegative orthant, and return the
+    corresponding :class:`SymmetricLinearGame`.
+
+    We keep going until we generate a game with a condition number under
+    5000.

     """
     ambient_dim = randint(1, 10)
     K = NonnegativeOrthant(ambient_dim)
     """
     ambient_dim = randint(1, 10)
     K = NonnegativeOrthant(ambient_dim)

-    e1 = [uniform(0.5, 10) for idx in range(K.dimension())]
-    e2 = [uniform(0.5, 10) for idx in range(K.dimension())]
-    L = _random_matrix(K.dimension())
-    return (L, K, matrix(e1), matrix(e2))
+    e1 = [uniform(0.5, 10) for _ in range(K.dimension())]
+    e2 = [uniform(0.5, 10) for _ in range(K.dimension())]
+    L = random_matrix(K.dimension())
+    G = SymmetricLinearGame(L, K, e1, e2)
+
+    if G._condition() <= MAX_COND:
+        return G
+    else:
+        return random_orthant_game()

 
 
 
 

-def _random_icecream_params():
+def random_icecream_game():

     """
     Generate the ``L``, ``K``, ``e1``, and ``e2`` parameters for a
     """
     Generate the ``L``, ``K``, ``e1``, and ``e2`` parameters for a

-    random game over the ice cream cone. This is only used by
-    the :class:`SymmetricLinearGameTest` class.
+    random game over the ice-cream cone, and return the corresponding
+    :class:`SymmetricLinearGame`.

     """
     # Use a minimum dimension of two to avoid divide-by-zero in
     # the fudge factor we make up later.
     """
     # Use a minimum dimension of two to avoid divide-by-zero in
     # the fudge factor we make up later.


@@ -105,11 +258,15 @@ def _random_icecream_params():
     # non-height part is sqrt(dim(K) - 1), and we can divide by
     # twice that.
     fudge_factor = 1.0 / (2.0*sqrt(K.dimension() - 1.0))
     # non-height part is sqrt(dim(K) - 1), and we can divide by
     # twice that.
     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 = _random_matrix(K.dimension())
+    e1 += [fudge_factor*uniform(0, 1) for _ in range(K.dimension() - 1)]
+    e2 += [fudge_factor*uniform(0, 1) for _ in range(K.dimension() - 1)]
+    L = random_matrix(K.dimension())
+    G = SymmetricLinearGame(L, K, e1, e2)

 
 

-    return (L, K, matrix(e1), matrix(e2))
+    if G._condition() <= MAX_COND:
+        return G
+    else:
+        return random_icecream_game()

 
 
 # Tell pylint to shut up about the large number of methods.
 
 
 # Tell pylint to shut up about the large number of methods.


@@ -119,37 +276,47 @@ class SymmetricLinearGameTest(TestCase): # pylint: disable=R0904
     """
     def assert_within_tol(self, first, second):
         """
     """
     def assert_within_tol(self, first, second):
         """

-        Test that ``first`` and ``second`` are equal within our default
-        tolerance.
-        """
-        self.assertTrue(abs(first - second) < options.ABS_TOL)
-
+        Test that ``first`` and ``second`` are equal within a multiple of
+        our default tolerances.

 
 

-    def assert_norm_within_tol(self, first, second):
+        The factor of two is because if we compare two solutions, both
+        of which may be off by ``ABS_TOL``, then the result could be off
+        by ``2*ABS_TOL``. The factor of ``RANDOM_MAX`` allows for
+        scaling a result (by ``RANDOM_MAX``) that may be off by
+        ``ABS_TOL``. The final factor of two is to allow for the edge
+        cases where we get an "unknown" result and need to lower the
+        CVXOPT tolerance by a factor of two.

         """
         """

-        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)
+        self.assertTrue(abs(first - second) < 2*2*RANDOM_MAX*options.ABS_TOL)

 
 
 
 

-    def assert_solution_exists(self, L, K, e1, e2):
+    def assert_solution_exists(self, G):

         """
         """

-        Given the parameters needed to construct a SymmetricLinearGame,
-        ensure that that game has a solution.
+        Given  a SymmetricLinearGame, ensure that it has a solution.

         """
         """

-        # The matrix() constructor assumes that ``L`` is a list of
-        # columns, so we transpose it to agree with what
-        # SymmetricLinearGame() thinks.
-        G = SymmetricLinearGame(L.trans(), K, e1, e2)

         soln = G.solution()
 
         soln = G.solution()
 

-        expected = inner_product(L*soln.player1_optimal(),
+        expected = inner_product(G._L*soln.player1_optimal(),

                                  soln.player2_optimal())
         self.assert_within_tol(soln.game_value(), expected)
 
 
                                  soln.player2_optimal())
         self.assert_within_tol(soln.game_value(), expected)
 
 

+
+    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)
+
+

     def test_solution_exists_orthant(self):
         """
         Every linear game has a solution, so we should be able to solve
     def test_solution_exists_orthant(self):
         """
         Every linear game has a solution, so we should be able to solve


@@ -158,8 +325,8 @@ class SymmetricLinearGameTest(TestCase): # pylint: disable=R0904
         optimal solutions should give us the optimal game value when we
         apply the payoff operator to them.
         """
         optimal solutions should give us the optimal game value when we
         apply the payoff operator to them.
         """

-        (L, K, e1, e2) = _random_orthant_params()
-        self.assert_solution_exists(L, K, e1, e2)
+        G = random_orthant_game()
+        self.assert_solution_exists(G)

 
 
     def test_solution_exists_icecream(self):
 
 
     def test_solution_exists_icecream(self):


@@ -167,8 +334,8 @@ class SymmetricLinearGameTest(TestCase): # pylint: disable=R0904
         Like :meth:`test_solution_exists_nonnegative_orthant`, except
         over the ice cream cone.
         """
         Like :meth:`test_solution_exists_nonnegative_orthant`, except
         over the ice cream cone.
         """

-        (L, K, e1, e2) = _random_icecream_params()
-        self.assert_solution_exists(L, K, e1, e2)
+        G = random_icecream_game()
+        self.assert_solution_exists(G)

 
 
     def test_negative_value_z_operator(self):
 
 
     def test_negative_value_z_operator(self):


@@ -184,16 +351,27 @@ class SymmetricLinearGameTest(TestCase): # pylint: disable=R0904
         self.assertTrue(G.solution().game_value() < -options.ABS_TOL)
 
 
         self.assertTrue(G.solution().game_value() < -options.ABS_TOL)
 
 

-    def assert_scaling_works(self, L, K, e1, e2):
+    def assert_scaling_works(self, game1):

         """
         Test that scaling ``L`` by a nonnegative number scales the value
         of the game by the same number.
         """
         """
         Test that scaling ``L`` by a nonnegative number scales the value
         of the game by the same number.
         """

-        game1 = SymmetricLinearGame(L, K, e1, e2)

         value1 = game1.solution().game_value()
 
         alpha = uniform(0.1, 10)
         value1 = game1.solution().game_value()
 
         alpha = uniform(0.1, 10)

-        game2 = SymmetricLinearGame(alpha*L, K, e1, e2)
+        game2 = SymmetricLinearGame(alpha*game1._L.trans(),
+                                    game1._K,
+                                    game1._e1,
+                                    game1._e2)
+
+        while game2._condition() > MAX_COND:
+            # Loop until the condition number of game2 doesn't suck.
+            alpha = uniform(0.1, 10)
+            game2 = SymmetricLinearGame(alpha*game1._L.trans(),
+                                        game1._K,
+                                        game1._e1,
+                                        game1._e2)
+

         value2 = game2.solution().game_value()
         self.assert_within_tol(alpha*value1, value2)
 
         value2 = game2.solution().game_value()
         self.assert_within_tol(alpha*value1, value2)
 


@@ -203,8 +381,8 @@ class SymmetricLinearGameTest(TestCase): # pylint: disable=R0904
         Test that scaling ``L`` by a nonnegative number scales the value
         of the game by the same number over the nonnegative orthant.
         """
         Test that scaling ``L`` by a nonnegative number scales the value
         of the game by the same number over the nonnegative orthant.
         """

-        (L, K, e1, e2) = _random_orthant_params()
-        self.assert_scaling_works(L, K, e1, e2)
+        G = random_orthant_game()
+        self.assert_scaling_works(G)

 
 
     def test_scaling_icecream(self):
 
 
     def test_scaling_icecream(self):


@@ -212,32 +390,39 @@ class SymmetricLinearGameTest(TestCase): # pylint: disable=R0904
         The same test as :meth:`test_nonnegative_scaling_orthant`,
         except over the ice cream cone.
         """
         The same test as :meth:`test_nonnegative_scaling_orthant`,
         except over the ice cream cone.
         """

-        (L, K, e1, e2) = _random_icecream_params()
-        self.assert_scaling_works(L, K, e1, e2)
+        G = random_icecream_game()
+        self.assert_scaling_works(G)

 
 
 
 

-    def assert_translation_works(self, L, K, e1, e2):
+    def assert_translation_works(self, game1):

         """
         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.
         """
         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.

-        game1 = SymmetricLinearGame(L.trans(), K, e1, e2)

         soln1 = game1.solution()
         value1 = soln1.game_value()
         x_bar = soln1.player1_optimal()
         y_bar = soln1.player2_optimal()
         soln1 = game1.solution()
         value1 = soln1.game_value()
         x_bar = soln1.player1_optimal()
         y_bar = soln1.player2_optimal()

-
-        alpha = uniform(-10, 10)
-        tensor_prod = e1*e2.trans()
+        tensor_prod = game1._e1*game1._e2.trans()

 
         # This is the "correct" representation of ``M``, but COLUMN
         # indexed...
 
         # This is the "correct" representation of ``M``, but COLUMN
         # indexed...

-        M = L + alpha*tensor_prod
+        alpha = uniform(-10, 10)
+        M = game1._L + alpha*tensor_prod

 
         # so we have to transpose it when we feed it to the constructor.
 
         # so we have to transpose it when we feed it to the constructor.

-        game2 = SymmetricLinearGame(M.trans(), K, e1, e2)
+        game2 = SymmetricLinearGame(M.trans(), game1._K, game1._e1, game1._e2)
+        while game2._condition() > MAX_COND:
+            # Loop until the condition number of game2 doesn't suck.
+            alpha = uniform(-10, 10)
+            M = game1._L + alpha*tensor_prod
+            game2 = SymmetricLinearGame(M.trans(),
+                                        game1._K,
+                                        game1._e1,
+                                        game1._e2)
+

         value2 = game2.solution().game_value()
 
         self.assert_within_tol(value1 + alpha, value2)
         value2 = game2.solution().game_value()
 
         self.assert_within_tol(value1 + alpha, value2)


@@ -250,8 +435,8 @@ class SymmetricLinearGameTest(TestCase): # pylint: disable=R0904
         """
         Test that translation works over the nonnegative orthant.
         """
         """
         Test that translation works over the nonnegative orthant.
         """

-        (L, K, e1, e2) = _random_orthant_params()
-        self.assert_translation_works(L, K, e1, e2)
+        G = random_orthant_game()
+        self.assert_translation_works(G)

 
 
     def test_translation_icecream(self):
 
 
     def test_translation_icecream(self):


@@ -259,26 +444,23 @@ class SymmetricLinearGameTest(TestCase): # pylint: disable=R0904
         The same as :meth:`test_translation_orthant`, except over the
         ice cream cone.
         """
         The same as :meth:`test_translation_orthant`, except over the
         ice cream cone.
         """

-        (L, K, e1, e2) = _random_icecream_params()
-        self.assert_translation_works(L, K, e1, e2)
+        G = random_icecream_game()
+        self.assert_translation_works(G)

 
 
 
 

-    def assert_opposite_game_works(self, L, K, e1, e2):
+    def assert_opposite_game_works(self, game1):

         """
         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.
         """
         """
         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.
         """

-        # We need to use ``L`` later, so make sure we transpose it
-        # before passing it in as a column-indexed matrix.
-        game1 = SymmetricLinearGame(L.trans(), K, e1, e2)
-

         # This is the "correct" representation of ``M``, but
         # COLUMN indexed...
         # This is the "correct" representation of ``M``, but
         # COLUMN indexed...

-        M = -L.trans()
+        M = -game1._L.trans()

 
         # so we have to transpose it when we feed it to the constructor.
 
         # so we have to transpose it when we feed it to the constructor.

-        game2 = SymmetricLinearGame(M.trans(), K, e2, e1)
+        # Note: the condition number of game2 should be comparable to game1.
+        game2 = SymmetricLinearGame(M.trans(), game1._K, game1._e2, game1._e1)

 
         soln1 = game1.solution()
         x_bar = soln1.player1_optimal()
 
         soln1 = game1.solution()
         x_bar = soln1.player1_optimal()


@@ -297,8 +479,8 @@ class SymmetricLinearGameTest(TestCase): # pylint: disable=R0904
         Test the value of the "opposite" game over the nonnegative
         orthant.
         """
         Test the value of the "opposite" game over the nonnegative
         orthant.
         """

-        (L, K, e1, e2) = _random_orthant_params()
-        self.assert_opposite_game_works(L, K, e1, e2)
+        G = random_orthant_game()
+        self.assert_opposite_game_works(G)

 
 
     def test_opposite_game_icecream(self):
 
 
     def test_opposite_game_icecream(self):


@@ -306,27 +488,24 @@ class SymmetricLinearGameTest(TestCase): # pylint: disable=R0904
         Like :meth:`test_opposite_game_orthant`, except over the
         ice-cream cone.
         """
         Like :meth:`test_opposite_game_orthant`, except over the
         ice-cream cone.
         """

-        (L, K, e1, e2) = _random_icecream_params()
-        self.assert_opposite_game_works(L, K, e1, e2)
+        G = random_icecream_game()
+        self.assert_opposite_game_works(G)

 
 
 
 

-    def assert_orthogonality(self, L, K, e1, e2):
+    def assert_orthogonality(self, G):

         """
         Two orthogonality relations hold at an optimal solution, and we
         check them here.
         """
         """
         Two orthogonality relations hold at an optimal solution, and we
         check them here.
         """

-        # We need to use ``L`` later, so make sure we transpose it
-        # before passing it in as a column-indexed matrix.
-        game = SymmetricLinearGame(L.trans(), K, e1, e2)
-        soln = game.solution()
+        soln = G.solution()

         x_bar = soln.player1_optimal()
         y_bar = soln.player2_optimal()
         value = soln.game_value()
 
         x_bar = soln.player1_optimal()
         y_bar = soln.player2_optimal()
         value = soln.game_value()
 

-        ip1 = inner_product(y_bar, L*x_bar - value*e1)
+        ip1 = inner_product(y_bar, G._L*x_bar - value*G._e1)

         self.assert_within_tol(ip1, 0)
 
         self.assert_within_tol(ip1, 0)
 

-        ip2 = inner_product(value*e2 - L.trans()*y_bar, x_bar)
+        ip2 = inner_product(value*G._e2 - G._L.trans()*y_bar, x_bar)

         self.assert_within_tol(ip2, 0)
 
 
         self.assert_within_tol(ip2, 0)
 
 


@@ -335,8 +514,8 @@ class SymmetricLinearGameTest(TestCase): # pylint: disable=R0904
         Check the orthgonality relationships that hold for a solution
         over the nonnegative orthant.
         """
         Check the orthgonality relationships that hold for a solution
         over the nonnegative orthant.
         """

-        (L, K, e1, e2) = _random_orthant_params()
-        self.assert_orthogonality(L, K, e1, e2)
+        G = random_orthant_game()
+        self.assert_orthogonality(G)

 
 
     def test_orthogonality_icecream(self):
 
 
     def test_orthogonality_icecream(self):


@@ -344,8 +523,8 @@ class SymmetricLinearGameTest(TestCase): # pylint: disable=R0904
         Check the orthgonality relationships that hold for a solution
         over the ice-cream cone.
         """
         Check the orthgonality relationships that hold for a solution
         over the ice-cream cone.
         """

-        (L, K, e1, e2) = _random_icecream_params()
-        self.assert_orthogonality(L, K, e1, e2)
+        G = random_icecream_game()
+        self.assert_orthogonality(G)

 
 
     def test_positive_operator_value(self):
 
 
     def test_positive_operator_value(self):


@@ -356,35 +535,46 @@ class SymmetricLinearGameTest(TestCase): # pylint: disable=R0904
         This test theoretically applies to the ice-cream cone as well,
         but we don't know how to make positive operators on that cone.
         """
         This test theoretically applies to the ice-cream cone as well,
         but we don't know how to make positive operators on that cone.
         """

-        (K, e1, e2) = _random_orthant_params()[1:]
-        L = _random_nonnegative_matrix(K.dimension())
+        G = random_orthant_game()
+        L = random_nonnegative_matrix(G._K.dimension())
+
+        # Replace the totally-random ``L`` with the random nonnegative one.
+        G = SymmetricLinearGame(L, G._K, G._e1, G._e2)
+
+        while G._condition() > MAX_COND:
+            # Try again until the condition number is satisfactory.
+            G = random_orthant_game()
+            L = random_nonnegative_matrix(G._K.dimension())
+            G = SymmetricLinearGame(L, G._K, G._e1, G._e2)

 
 

-        game = SymmetricLinearGame(L, K, e1, e2)
-        self.assertTrue(game.solution().game_value() >= -options.ABS_TOL)
+        self.assertTrue(G.solution().game_value() >= -options.ABS_TOL)

 
 
 
 

-    def assert_lyapunov_works(self, L, K, e1, e2):
+    def assert_lyapunov_works(self, G):

         """
         Check that Lyapunov games act the way we expect.
         """
         """
         Check that Lyapunov games act the way we expect.
         """

-        game = SymmetricLinearGame(L, K, e1, e2)
-        soln = game.solution()
+        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.
 
         # 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.

-        eigs = eigenvalues_re(L)
-        if soln.game_value() > options.ABS_TOL:
+        #
+        # See :meth:`assert_within_tol` for an explanation of the
+        # fudge factors.
+        eigs = eigenvalues_re(G._L)
+        epsilon = 2*2*RANDOM_MAX*options.ABS_TOL
+        if soln.game_value() > epsilon:

             # L should be positive stable
             positive_stable = all([eig > -options.ABS_TOL for eig in eigs])
             self.assertTrue(positive_stable)
             # 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:
+        elif soln.game_value() < -epsilon:

             # L should be negative stable
             negative_stable = all([eig < options.ABS_TOL for eig in eigs])
             self.assertTrue(negative_stable)
 
         # The dual game's value should always equal the primal's.
             # L should be negative stable
             negative_stable = all([eig < options.ABS_TOL for eig in eigs])
             self.assertTrue(negative_stable)
 
         # The dual game's value should always equal the primal's.

-        dualsoln = game.dual().solution()
+        dualsoln = G.dual().solution()

         self.assert_within_tol(dualsoln.game_value(), soln.game_value())
 
 
         self.assert_within_tol(dualsoln.game_value(), soln.game_value())
 
 


@@ -392,17 +582,35 @@ class SymmetricLinearGameTest(TestCase): # pylint: disable=R0904
         """
         Test that a Lyapunov game on the nonnegative orthant works.
         """
         """
         Test that a Lyapunov game on the nonnegative orthant works.
         """

-        (K, e1, e2) = _random_orthant_params()[1:]
-        L = _random_diagonal_matrix(K.dimension())
+        G = random_orthant_game()
+        L = random_diagonal_matrix(G._K.dimension())
+
+        # Replace the totally-random ``L`` with random Lyapunov-like one.
+        G = SymmetricLinearGame(L, G._K, G._e1, G._e2)
+
+        while G._condition() > MAX_COND:
+            # Try again until the condition number is satisfactory.
+            G = random_orthant_game()
+            L = random_diagonal_matrix(G._K.dimension())
+            G = SymmetricLinearGame(L, G._K, G._e1, G._e2)

 
 

-        self.assert_lyapunov_works(L, K, e1, e2)
+        self.assert_lyapunov_works(G)

 
 
     def test_lyapunov_icecream(self):
         """
         Test that a Lyapunov game on the ice-cream cone works.
         """
 
 
     def test_lyapunov_icecream(self):
         """
         Test that a Lyapunov game on the ice-cream cone works.
         """

-        (K, e1, e2) = _random_icecream_params()[1:]
-        L = _random_lyapunov_like_icecream(K.dimension())
+        G = random_icecream_game()
+        L = random_lyapunov_like_icecream(G._K.dimension())
+
+        # Replace the totally-random ``L`` with random Lyapunov-like one.
+        G = SymmetricLinearGame(L, G._K, G._e1, G._e2)
+
+        while G._condition() > MAX_COND:
+            # Try again until the condition number is satisfactory.
+            G = random_icecream_game()
+            L = random_lyapunov_like_icecream(G._K.dimension())
+            G = SymmetricLinearGame(L, G._K, G._e1, G._e2)

 
 

-        self.assert_lyapunov_works(L, K, e1, e2)
+        self.assert_lyapunov_works(G)