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:
the value of the game, and both players' strategies.
"""
def __init__(self, game_value, p1_optimal, p2_optimal):
+ """
+ Create a new Solution object from a game value and two optimal
+ strategies for the players.
+ """
self._game_value = game_value
self._player1_optimal = p1_optimal
self._player2_optimal = p2_optimal
* The optimal strategy of player one.
* The optimal strategy of player two.
- """
+ EXAMPLES:
+
+ >>> print(Solution(10, matrix([1,2]), matrix([3,4])))
+ Game value: 10.0000000
+ Player 1 optimal:
+ [ 1]
+ [ 2]
+ Player 2 optimal:
+ [ 3]
+ [ 4]
+ """
tpl = 'Game value: {:.7f}\n' \
'Player 1 optimal:{:s}\n' \
- 'Player 2 optimal:{:s}\n'
+ 'Player 2 optimal:{:s}'
p1_str = '\n{!s}'.format(self.player1_optimal())
p1_str = '\n '.join(p1_str.splitlines())
def game_value(self):
+ """
+ Return the game value for this solution.
+ """
return self._game_value
def player1_optimal(self):
+ """
+ Return player one's optimal strategy in this solution.
+ """
return self._player1_optimal
def player2_optimal(self):
+ """
+ Return player two's optimal strategy in this solution.
+ """
return self._player2_optimal
The ambient space is assumed to be the span of ``K``.
"""
-
def __init__(self, L, K, e1, e2):
"""
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.
-
- - ``e2`` -- the interior point of ``K`` belonging to player two,
- 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].
+
+ 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')
if not K.contains_strict(self._e2):
raise ValueError('the point e2 must lie in the interior of K')
+ def __str__(self):
+ """
+ Return a string representatoin of this game.
+
+ EXAMPLES:
+
+ >>> from cones import NonnegativeOrthant
+ >>> K = NonnegativeOrthant(3)
+ >>> L = [[1,-5,-15],[-1,2,-3],[-12,-15,1]]
+ >>> e1 = [1,1,1]
+ >>> e2 = [1,2,3]
+ >>> SLG = SymmetricLinearGame(L, K, e1, e2)
+ >>> print(SLG)
+ The linear game (L, K, e1, e2) where
+ L = [ 1 -5 -15]
+ [ -1 2 -3]
+ [-12 -15 1],
+ K = Nonnegative orthant in the real 3-space,
+ e1 = [ 1]
+ [ 1]
+ [ 1],
+ e2 = [ 1]
+ [ 2]
+ [ 3].
+
+ """
+ tpl = 'The linear game (L, K, e1, e2) where\n' \
+ ' L = {:s},\n' \
+ ' K = {!s},\n' \
+ ' e1 = {:s},\n' \
+ ' e2 = {:s}.'
+ 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):
+ """
+ Solve this linear game and return a Solution object.
+
+ OUTPUT:
+
+ If the cone program associated with this game could be
+ successfully solved, then a Solution object containing the
+ game's value and optimal strategies is returned. If the game
+ could *not* be solved -- which should never happen -- then a
+ GameUnsolvableException is raised. It can be printed to get the
+ raw output from CVXOPT.
+
+ EXAMPLES:
+
+ This example is computed in Gowda and Ravindran in the section
+ "The value of a Z-transformation":
+
+ >>> from cones import NonnegativeOrthant
+ >>> K = NonnegativeOrthant(3)
+ >>> L = [[1,-5,-15],[-1,2,-3],[-12,-15,1]]
+ >>> e1 = [1,1,1]
+ >>> e2 = [1,1,1]
+ >>> SLG = SymmetricLinearGame(L, K, e1, e2)
+ >>> print(SLG.solution())
+ Game value: -6.1724138
+ Player 1 optimal:
+ [ 0.5517241]
+ [-0.0000000]
+ [ 0.4482759]
+ Player 2 optimal:
+ [0.4482759]
+ [0.0000000]
+ [0.5517241]
+
+ The value of the following game can be computed using the fact
+ that the identity is invertible:
+
+ >>> from cones import NonnegativeOrthant
+ >>> K = NonnegativeOrthant(3)
+ >>> L = [[1,0,0],[0,1,0],[0,0,1]]
+ >>> e1 = [1,2,3]
+ >>> e2 = [4,5,6]
+ >>> SLG = SymmetricLinearGame(L, K, e1, e2)
+ >>> print(SLG.solution())
+ Game value: 0.0312500
+ Player 1 optimal:
+ [0.0312500]
+ [0.0625000]
+ [0.0937500]
+ Player 2 optimal:
+ [0.1250000]
+ [0.1562500]
+ [0.1875000]
+
+ """
+ # The cone "C" that appears in the statement of the CVXOPT
+ # conelp program.
C = CartesianProduct(self._K, self._K)
+
+ # The column vector "b" that appears on the right-hand side of
+ # Ax = b in the statement of the CVXOPT conelp program.
b = matrix([1], tc='d')
+
# A column of zeros that fits K.
zero = matrix(0, (self._K.dimension(), 1), tc='d')
+
+ # The column vector "h" that appears on the right-hand side of
+ # Gx + s = h in the statement of the CVXOPT conelp program.
h = matrix([zero, zero])
+
+ # The column vector "c" that appears in the objective function
+ # value <c,x> in the statement of the CVXOPT conelp program.
c = matrix([-1, zero])
+
+ # The matrix "G" that appears on the left-hand side of Gx + s = h
+ # in the statement of the CVXOPT conelp program.
G = append_row(append_col(zero, -identity(self._K.dimension())),
append_col(self._e1, -self._L))
- A = matrix([0, self._e1], (1, self._K.dimension() + 1), 'd')
+ # The matrix "A" that appears on the right-hand side of Ax = b
+ # in the statement of the CVXOPT conelp program.
+ A = matrix([0, self._e2], (1, self._K.dimension() + 1), 'd')
+
+ # Actually solve the thing and obtain a dictionary describing
+ # what happened.
soln_dict = solvers.conelp(c, G, h, C.cvxopt_dims(), A, b)
+ # The "status" field contains "optimal" if everything went
+ # according to plan. Other possible values are "primal
+ # infeasible", "dual infeasible", "unknown", all of which
+ # mean we didn't get a solution. That should never happen,
+ # because by construction our game has a solution, and thus
+ # the cone program should too.
if soln_dict['status'] != 'optimal':
raise GameUnsolvableException(soln_dict)
p2_optimal = soln_dict['z'][self._K.dimension():]
return Solution(p1_value, p1_optimal, p2_optimal)
+
+ def dual(self):
+ """
+ Return the dual game to this game.
+
+ EXAMPLES:
+
+ >>> from cones import NonnegativeOrthant
+ >>> K = NonnegativeOrthant(3)
+ >>> L = [[1,-5,-15],[-1,2,-3],[-12,-15,1]]
+ >>> e1 = [1,1,1]
+ >>> e2 = [1,2,3]
+ >>> SLG = SymmetricLinearGame(L, K, e1, e2)
+ >>> print(SLG.dual())
+ The linear game (L, K, e1, e2) where
+ L = [ 1 -1 -12]
+ [ -5 2 -15]
+ [-15 -3 1],
+ K = Nonnegative orthant in the real 3-space,
+ e1 = [ 1]
+ [ 2]
+ [ 3],
+ e2 = [ 1]
+ [ 1]
+ [ 1].
+
+ """
+ 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