to solve a linear game.
"""
-from cvxopt import matrix, printing, solvers
+# These few are used only for tests.
+from math import sqrt
+from random import randint, uniform
+from unittest import TestCase
-from cones import CartesianProduct
+# These are mostly actually needed.
+from cvxopt import matrix, printing, solvers
+from cones import CartesianProduct, IceCream, NonnegativeOrthant
from errors import GameUnsolvableException
-from matrices import append_col, append_row, identity
+from matrices import append_col, append_row, identity, inner_product
+import options
-printing.options['dformat'] = '%.7f'
-solvers.options['show_progress'] = False
+printing.options['dformat'] = options.FLOAT_FORMAT
+solvers.options['show_progress'] = options.VERBOSE
class Solution:
"""
INPUT:
- - ``L`` -- an n-by-b matrix represented as a list of lists
- of real numbers.
+ - ``L`` -- an square matrix represented as a list of lists
+ of real numbers. ``L`` itself is interpreted as a list of
+ ROWS, which agrees with (for example) SageMath and NumPy,
+ but not with CVXOPT (whose matrix constructor accepts a
+ list of columns).
- ``K`` -- a SymmetricCone instance.
- - ``e1`` -- the interior point of ``K`` belonging to player one,
- as a column vector.
+ - ``e1`` -- the interior point of ``K`` belonging to player one;
+ it can be of any enumerable type having the correct length.
+
+ - ``e2`` -- the interior point of ``K`` belonging to player two;
+ it can be of any enumerable type having the correct length.
+
+ EXAMPLES:
+
+ Lists can (and probably should) be used for every argument:
+
+ >>> from cones import NonnegativeOrthant
+ >>> K = NonnegativeOrthant(2)
+ >>> L = [[1,0],[0,1]]
+ >>> e1 = [1,1]
+ >>> e2 = [1,1]
+ >>> G = SymmetricLinearGame(L, K, e1, e2)
+ >>> print(G)
+ The linear game (L, K, e1, e2) where
+ L = [ 1 0]
+ [ 0 1],
+ K = Nonnegative orthant in the real 2-space,
+ e1 = [ 1]
+ [ 1],
+ e2 = [ 1]
+ [ 1].
- - ``e2`` -- the interior point of ``K`` belonging to player two,
- as a column vector.
+ The points ``e1`` and ``e2`` can also be passed as some other
+ enumerable type (of the correct length) without much harm, since
+ there is no row/column ambiguity:
+ >>> import cvxopt
+ >>> import numpy
+ >>> from cones import NonnegativeOrthant
+ >>> K = NonnegativeOrthant(2)
+ >>> L = [[1,0],[0,1]]
+ >>> e1 = cvxopt.matrix([1,1])
+ >>> e2 = numpy.matrix([1,1])
+ >>> G = SymmetricLinearGame(L, K, e1, e2)
+ >>> print(G)
+ The linear game (L, K, e1, e2) where
+ L = [ 1 0]
+ [ 0 1],
+ K = Nonnegative orthant in the real 2-space,
+ e1 = [ 1]
+ [ 1],
+ e2 = [ 1]
+ [ 1].
+
+ However, ``L`` will always be intepreted as a list of rows, even
+ if it is passed as a ``cvxopt.base.matrix`` which is otherwise
+ indexed by columns:
+
+ >>> import cvxopt
+ >>> from cones import NonnegativeOrthant
+ >>> K = NonnegativeOrthant(2)
+ >>> L = [[1,2],[3,4]]
+ >>> e1 = [1,1]
+ >>> e2 = e1
+ >>> G = SymmetricLinearGame(L, K, e1, e2)
+ >>> print(G)
+ The linear game (L, K, e1, e2) where
+ L = [ 1 2]
+ [ 3 4],
+ K = Nonnegative orthant in the real 2-space,
+ e1 = [ 1]
+ [ 1],
+ e2 = [ 1]
+ [ 1].
+ >>> L = cvxopt.matrix(L)
+ >>> print(L)
+ [ 1 3]
+ [ 2 4]
+ <BLANKLINE>
+ >>> G = SymmetricLinearGame(L, K, e1, e2)
+ >>> print(G)
+ The linear game (L, K, e1, e2) where
+ L = [ 1 2]
+ [ 3 4],
+ K = Nonnegative orthant in the real 2-space,
+ e1 = [ 1]
+ [ 1],
+ e2 = [ 1]
+ [ 1].
"""
self._K = K
self._e1 = matrix(e1, (K.dimension(), 1))
self._e2 = matrix(e2, (K.dimension(), 1))
- self._L = matrix(L, (K.dimension(), K.dimension()))
+
+ # Our input ``L`` is indexed by rows but CVXOPT matrices are
+ # indexed by columns, so we need to transpose the input before
+ # feeding it to CVXOPT.
+ self._L = matrix(L, (K.dimension(), K.dimension())).trans()
if not K.contains_strict(self._e1):
raise ValueError('the point e1 must lie in the interior of K')
>>> from cones import NonnegativeOrthant
>>> K = NonnegativeOrthant(3)
- >>> L = [[1,-1,-12],[-5,2,-15],[-15,-3,1]]
+ >>> L = [[1,-5,-15],[-1,2,-3],[-12,-15,1]]
>>> e1 = [1,1,1]
>>> e2 = [1,2,3]
>>> SLG = SymmetricLinearGame(L, K, e1, e2)
' K = {!s},\n' \
' e1 = {:s},\n' \
' e2 = {:s}.'
- L_str = '\n '.join(str(self._L).splitlines())
- e1_str = '\n '.join(str(self._e1).splitlines())
- e2_str = '\n '.join(str(self._e2).splitlines())
- return tpl.format(L_str, str(self._K), e1_str, e2_str)
+ indented_L = '\n '.join(str(self._L).splitlines())
+ indented_e1 = '\n '.join(str(self._e1).splitlines())
+ indented_e2 = '\n '.join(str(self._e2).splitlines())
+ return tpl.format(indented_L, str(self._K), indented_e1, indented_e2)
def solution(self):
>>> from cones import NonnegativeOrthant
>>> K = NonnegativeOrthant(3)
- >>> L = [[1,-1,-12],[-5,2,-15],[-15,-3,1]]
+ >>> L = [[1,-5,-15],[-1,2,-3],[-12,-15,1]]
>>> e1 = [1,1,1]
>>> e2 = [1,1,1]
>>> SLG = SymmetricLinearGame(L, K, e1, e2)
>>> from cones import NonnegativeOrthant
>>> K = NonnegativeOrthant(3)
- >>> L = [[1,-1,-12],[-5,2,-15],[-15,-3,1]]
+ >>> L = [[1,-5,-15],[-1,2,-3],[-12,-15,1]]
>>> e1 = [1,1,1]
>>> e2 = [1,2,3]
>>> SLG = SymmetricLinearGame(L, K, e1, e2)
[ 1].
"""
- return SymmetricLinearGame(self._L.trans(),
+ return SymmetricLinearGame(self._L, # It will be transposed in __init__().
self._K, # Since "K" is symmetric.
self._e2,
self._e1)
+
+
+class SymmetricLinearGameTest(TestCase):
+ """
+ Tests for the SymmetricLinearGame and Solution classes.
+ """
+
+ 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)
+
+
+ def assert_solution_exists(self, L, K, e1, e2):
+ """
+ Given the parameters needed to construct a SymmetricLinearGame,
+ ensure that that game has a solution.
+ """
+ G = SymmetricLinearGame(L, K, e1, e2)
+ soln = G.solution()
+ 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
+ every symmetric linear game over the NonnegativeOrthant. Pick
+ some parameters randomly and give it a shot. The resulting
+ optimal solutions should give us the optimal game value when we
+ apply the payoff operator to them.
+ """
+ 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())]
+ self.assert_solution_exists(L, K, e1, e2)
+
+ def test_solution_exists_ice_cream(self):
+ """
+ Like :meth:`test_solution_exists_nonnegative_orthant`, except
+ over the ice cream cone.
+ """
+ # Use a minimum dimension of two to avoid divide-by-zero in
+ # the fudge factor we make up later.
+ ambient_dim = randint(2, 10)
+ K = IceCream(ambient_dim)
+ e1 = [1]
+ e2 = [1]
+ # If we choose the rest of the components of e1,e2 randomly
+ # between 0 and 1, then the largest the squared norm of the
+ # non-height part of e1,e2 could be is the 1*(dim(K) - 1). We
+ # need to make it less than one (the height of the cone) so
+ # that the whole thing is in the cone. The norm of the
+ # 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 = [[uniform(-10, 10) for i in range(K.dimension())]
+ for j in range(K.dimension())]
+ self.assert_solution_exists(L, K, e1, e2)
+
projects / dunshire.git / blobdiff