2 Symmetric linear games and their solutions.
4 This module contains the main :class:`SymmetricLinearGame` class that
5 knows how to solve a linear game.
8 from cvxopt
import matrix
, printing
, solvers
9 from .cones
import CartesianProduct
10 from .errors
import GameUnsolvableException
, PoorScalingException
11 from .matrices
import (append_col
, append_row
, condition_number
, identity
,
15 printing
.options
['dformat'] = options
.FLOAT_FORMAT
19 A representation of the solution of a linear game. It should contain
20 the value of the game, and both players' strategies.
25 >>> print(Solution(10, matrix([1,2]), matrix([3,4])))
26 Game value: 10.0000000
35 def __init__(self
, game_value
, p1_optimal
, p2_optimal
):
37 Create a new Solution object from a game value and two optimal
38 strategies for the players.
40 self
._game
_value
= game_value
41 self
._player
1_optimal
= p1_optimal
42 self
._player
2_optimal
= p2_optimal
46 Return a string describing the solution of a linear game.
48 The three data that are described are,
50 * The value of the game.
51 * The optimal strategy of player one.
52 * The optimal strategy of player two.
54 The two optimal strategy vectors are indented by two spaces.
56 tpl
= 'Game value: {:.7f}\n' \
57 'Player 1 optimal:{:s}\n' \
58 'Player 2 optimal:{:s}'
60 p1_str
= '\n{!s}'.format(self
.player1_optimal())
61 p1_str
= '\n '.join(p1_str
.splitlines())
62 p2_str
= '\n{!s}'.format(self
.player2_optimal())
63 p2_str
= '\n '.join(p2_str
.splitlines())
65 return tpl
.format(self
.game_value(), p1_str
, p2_str
)
70 Return the game value for this solution.
75 >>> s = Solution(10, matrix([1,2]), matrix([3,4]))
80 return self
._game
_value
83 def player1_optimal(self
):
85 Return player one's optimal strategy in this solution.
90 >>> s = Solution(10, matrix([1,2]), matrix([3,4]))
91 >>> print(s.player1_optimal())
97 return self
._player
1_optimal
100 def player2_optimal(self
):
102 Return player two's optimal strategy in this solution.
107 >>> s = Solution(10, matrix([1,2]), matrix([3,4]))
108 >>> print(s.player2_optimal())
114 return self
._player
2_optimal
117 class SymmetricLinearGame
:
119 A representation of a symmetric linear game.
121 The data for a symmetric linear game are,
123 * A "payoff" operator ``L``.
124 * A symmetric cone ``K``.
125 * Two points ``e1`` and ``e2`` in the interior of ``K``.
127 The ambient space is assumed to be the span of ``K``.
129 With those data understood, the game is played as follows. Players
130 one and two choose points :math:`x` and :math:`y` respectively, from
131 their respective strategy sets,
138 x \in K \ \middle|\ \left\langle x, e_{2} \right\rangle = 1
143 y \in K \ \middle|\ \left\langle y, e_{1} \right\rangle = 1
147 Afterwards, a "payout" is computed as :math:`\left\langle
148 L\left(x\right), y \right\rangle` and is paid to player one out of
149 player two's pocket. The game is therefore zero sum, and we suppose
150 that player one would like to guarantee himself the largest minimum
151 payout possible. That is, player one wishes to,
156 &\underset{y \in \Delta_{2}}{\min}\left(
157 \left\langle L\left(x\right), y \right\rangle
159 \text{subject to } & x \in \Delta_{1}.
162 Player two has the simultaneous goal to,
167 &\underset{x \in \Delta_{1}}{\max}\left(
168 \left\langle L\left(x\right), y \right\rangle
170 \text{subject to } & y \in \Delta_{2}.
173 These goals obviously conflict (the game is zero sum), but an
174 existence theorem guarantees at least one optimal min-max solution
175 from which neither player would like to deviate. This class is
176 able to find such a solution.
181 L : list of list of float
182 A matrix represented as a list of ROWS. This representation
183 agrees with (for example) SageMath and NumPy, but not with CVXOPT
184 (whose matrix constructor accepts a list of columns).
186 K : :class:`SymmetricCone`
187 The symmetric cone instance over which the game is played.
190 The interior point of ``K`` belonging to player one; it
191 can be of any iterable type having the correct length.
194 The interior point of ``K`` belonging to player two; it
195 can be of any enumerable type having the correct length.
201 If either ``e1`` or ``e2`` lie outside of the cone ``K``.
206 >>> from dunshire import *
207 >>> K = NonnegativeOrthant(3)
208 >>> L = [[1,-5,-15],[-1,2,-3],[-12,-15,1]]
211 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
213 The linear game (L, K, e1, e2) where
217 K = Nonnegative orthant in the real 3-space,
224 Condition((L, K, e1, e2)) = 31.834...
226 Lists can (and probably should) be used for every argument::
228 >>> from dunshire import *
229 >>> K = NonnegativeOrthant(2)
230 >>> L = [[1,0],[0,1]]
233 >>> G = SymmetricLinearGame(L, K, e1, e2)
235 The linear game (L, K, e1, e2) where
238 K = Nonnegative orthant in the real 2-space,
243 Condition((L, K, e1, e2)) = 1.707...
245 The points ``e1`` and ``e2`` can also be passed as some other
246 enumerable type (of the correct length) without much harm, since
247 there is no row/column ambiguity::
251 >>> from dunshire import *
252 >>> K = NonnegativeOrthant(2)
253 >>> L = [[1,0],[0,1]]
254 >>> e1 = cvxopt.matrix([1,1])
255 >>> e2 = numpy.matrix([1,1])
256 >>> G = SymmetricLinearGame(L, K, e1, e2)
258 The linear game (L, K, e1, e2) where
261 K = Nonnegative orthant in the real 2-space,
266 Condition((L, K, e1, e2)) = 1.707...
268 However, ``L`` will always be intepreted as a list of rows, even
269 if it is passed as a :class:`cvxopt.base.matrix` which is
270 otherwise indexed by columns::
273 >>> from dunshire import *
274 >>> K = NonnegativeOrthant(2)
275 >>> L = [[1,2],[3,4]]
278 >>> G = SymmetricLinearGame(L, K, e1, e2)
280 The linear game (L, K, e1, e2) where
283 K = Nonnegative orthant in the real 2-space,
288 Condition((L, K, e1, e2)) = 6.073...
289 >>> L = cvxopt.matrix(L)
294 >>> G = SymmetricLinearGame(L, K, e1, e2)
296 The linear game (L, K, e1, e2) where
299 K = Nonnegative orthant in the real 2-space,
304 Condition((L, K, e1, e2)) = 6.073...
307 def __init__(self
, L
, K
, e1
, e2
):
309 Create a new SymmetricLinearGame object.
312 self
._e
1 = matrix(e1
, (K
.dimension(), 1))
313 self
._e
2 = matrix(e2
, (K
.dimension(), 1))
315 # Our input ``L`` is indexed by rows but CVXOPT matrices are
316 # indexed by columns, so we need to transpose the input before
317 # feeding it to CVXOPT.
318 self
._L = matrix(L
, (K
.dimension(), K
.dimension())).trans()
320 if not self
._e
1 in K
:
321 raise ValueError('the point e1 must lie in the interior of K')
323 if not self
._e
2 in K
:
324 raise ValueError('the point e2 must lie in the interior of K')
330 Return a string representation of this game.
332 tpl
= 'The linear game (L, K, e1, e2) where\n' \
337 ' Condition((L, K, e1, e2)) = {:f}.'
338 indented_L
= '\n '.join(str(self
._L).splitlines())
339 indented_e1
= '\n '.join(str(self
._e
1).splitlines())
340 indented_e2
= '\n '.join(str(self
._e
2).splitlines())
342 return tpl
.format(indented_L
,
351 Return the matrix ``L`` passed to the constructor.
357 The matrix that defines this game's :meth:`payoff` operator.
362 >>> from dunshire import *
363 >>> K = NonnegativeOrthant(3)
364 >>> L = [[1,-5,-15],[-1,2,-3],[-12,-15,1]]
367 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
380 Return the cone over which this game is played.
386 The :class:`SymmetricCone` over which this game is played.
391 >>> from dunshire import *
392 >>> K = NonnegativeOrthant(3)
393 >>> L = [[1,-5,-15],[-1,2,-3],[-12,-15,1]]
396 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
398 Nonnegative orthant in the real 3-space
406 Return player one's interior point.
412 The point interior to :meth:`K` affiliated with player one.
417 >>> from dunshire import *
418 >>> K = NonnegativeOrthant(3)
419 >>> L = [[1,-5,-15],[-1,2,-3],[-12,-15,1]]
422 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
435 Return player two's interior point.
441 The point interior to :meth:`K` affiliated with player one.
446 >>> from dunshire import *
447 >>> K = NonnegativeOrthant(3)
448 >>> L = [[1,-5,-15],[-1,2,-3],[-12,-15,1]]
451 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
462 def payoff(self
, strategy1
, strategy2
):
464 Return the payoff associated with ``strategy1`` and ``strategy2``.
466 The payoff operator takes pairs of strategies to a real
467 number. For example, if player one's strategy is :math:`x` and
468 player two's strategy is :math:`y`, then the associated payoff
469 is :math:`\left\langle L\left(x\right),y \right\rangle` \in
470 \mathbb{R}. Here, :math:`L` denotes the same linear operator as
471 :meth:`L`. This method computes the payoff given the two
478 Player one's strategy.
481 Player two's strategy.
487 The payoff for the game when player one plays ``strategy1``
488 and player two plays ``strategy2``.
493 The value of the game should be the payoff at the optimal
496 >>> from dunshire import *
497 >>> from dunshire.options import ABS_TOL
498 >>> K = NonnegativeOrthant(3)
499 >>> L = [[1,-5,-15],[-1,2,-3],[-12,-15,1]]
502 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
503 >>> soln = SLG.solution()
504 >>> x_bar = soln.player1_optimal()
505 >>> y_bar = soln.player2_optimal()
506 >>> abs(SLG.payoff(x_bar, y_bar) - soln.game_value()) < ABS_TOL
510 return inner_product(self
.L()*strategy1
, strategy2
)
515 Return the dimension of this game.
517 The dimension of a game is not needed for the theory, but it is
518 useful for the implementation. We define the dimension of a game
519 to be the dimension of its underlying cone. Or what is the same,
520 the dimension of the space from which the strategies are chosen.
526 The dimension of the cone :meth:`K`, or of the space where
532 The dimension of a game over the nonnegative quadrant in the
533 plane should be two (the dimension of the plane)::
535 >>> from dunshire import *
536 >>> K = NonnegativeOrthant(2)
537 >>> L = [[1,-5],[-1,2]]
540 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
545 return self
.K().dimension()
550 Return a column of zeros that fits ``K``.
552 This is used in our CVXOPT construction.
556 It is not safe to cache any of the matrices passed to
557 CVXOPT, because it can clobber them.
563 A ``self.dimension()``-by-``1`` column vector of zeros.
568 >>> from dunshire import *
569 >>> K = NonnegativeOrthant(3)
573 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
574 >>> print(SLG._zero())
581 return matrix(0, (self
.dimension(), 1), tc
='d')
586 Return the matrix ``A`` used in our CVXOPT construction.
588 This matrix ``A`` appears on the right-hand side of ``Ax = b``
589 in the statement of the CVXOPT conelp program.
593 It is not safe to cache any of the matrices passed to
594 CVXOPT, because it can clobber them.
600 A ``1``-by-``(1 + self.dimension())`` row vector. Its first
601 entry is zero, and the rest are the entries of ``e2``.
606 >>> from dunshire import *
607 >>> K = NonnegativeOrthant(3)
608 >>> L = [[1,1,1],[1,1,1],[1,1,1]]
611 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
613 [0.0000000 1.0000000 2.0000000 3.0000000]
617 return matrix([0, self
._e
2], (1, self
.dimension() + 1), 'd')
623 Return the matrix ``G`` used in our CVXOPT construction.
625 Thus matrix ``G`` appears on the left-hand side of ``Gx + s = h``
626 in the statement of the CVXOPT conelp program.
630 It is not safe to cache any of the matrices passed to
631 CVXOPT, because it can clobber them.
637 A ``2*self.dimension()``-by-``(1 + self.dimension())`` matrix.
642 >>> from dunshire import *
643 >>> K = NonnegativeOrthant(3)
644 >>> L = [[4,5,6],[7,8,9],[10,11,12]]
647 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
649 [ 0.0000000 -1.0000000 0.0000000 0.0000000]
650 [ 0.0000000 0.0000000 -1.0000000 0.0000000]
651 [ 0.0000000 0.0000000 0.0000000 -1.0000000]
652 [ 1.0000000 -4.0000000 -5.0000000 -6.0000000]
653 [ 2.0000000 -7.0000000 -8.0000000 -9.0000000]
654 [ 3.0000000 -10.0000000 -11.0000000 -12.0000000]
658 identity_matrix
= identity(self
.dimension())
659 return append_row(append_col(self
._zero
(), -identity_matrix
),
660 append_col(self
._e
1, -self
._L))
665 Return the vector ``c`` used in our CVXOPT construction.
667 The column vector ``c`` appears in the objective function
668 value ``<c,x>`` in the statement of the CVXOPT conelp program.
672 It is not safe to cache any of the matrices passed to
673 CVXOPT, because it can clobber them.
679 A ``self.dimension()``-by-``1`` column vector.
684 >>> from dunshire import *
685 >>> K = NonnegativeOrthant(3)
686 >>> L = [[4,5,6],[7,8,9],[10,11,12]]
689 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
698 return matrix([-1, self
._zero
()])
703 Return the cone ``C`` used in our CVXOPT construction.
705 The cone ``C`` is the cone over which the conelp program takes
712 The cartesian product of ``K`` with itself.
717 >>> from dunshire import *
718 >>> K = NonnegativeOrthant(3)
719 >>> L = [[4,5,6],[7,8,9],[10,11,12]]
722 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
724 Cartesian product of dimension 6 with 2 factors:
725 * Nonnegative orthant in the real 3-space
726 * Nonnegative orthant in the real 3-space
729 return CartesianProduct(self
._K
, self
._K
)
733 Return the ``h`` vector used in our CVXOPT construction.
735 The ``h`` vector appears on the right-hand side of :math:`Gx + s
736 = h` in the statement of the CVXOPT conelp program.
740 It is not safe to cache any of the matrices passed to
741 CVXOPT, because it can clobber them.
747 A ``2*self.dimension()``-by-``1`` column vector of zeros.
752 >>> from dunshire import *
753 >>> K = NonnegativeOrthant(3)
754 >>> L = [[4,5,6],[7,8,9],[10,11,12]]
757 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
769 return matrix([self
._zero
(), self
._zero
()])
775 Return the ``b`` vector used in our CVXOPT construction.
777 The vector ``b`` appears on the right-hand side of :math:`Ax =
778 b` in the statement of the CVXOPT conelp program.
780 This method is static because the dimensions and entries of
781 ``b`` are known beforehand, and don't depend on any other
782 properties of the game.
786 It is not safe to cache any of the matrices passed to
787 CVXOPT, because it can clobber them.
793 A ``1``-by-``1`` matrix containing a single entry ``1``.
798 >>> from dunshire import *
799 >>> K = NonnegativeOrthant(3)
800 >>> L = [[4,5,6],[7,8,9],[10,11,12]]
803 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
809 return matrix([1], tc
='d')
815 Solve this linear game and return a :class:`Solution`.
821 A :class:`Solution` object describing the game's value and
822 the optimal strategies of both players.
826 GameUnsolvableException
827 If the game could not be solved (if an optimal solution to its
828 associated cone program was not found).
831 If the game could not be solved because CVXOPT crashed while
832 trying to take the square root of a negative number.
837 This example is computed in Gowda and Ravindran in the section
838 "The value of a Z-transformation"::
840 >>> from dunshire import *
841 >>> K = NonnegativeOrthant(3)
842 >>> L = [[1,-5,-15],[-1,2,-3],[-12,-15,1]]
845 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
846 >>> print(SLG.solution())
847 Game value: -6.1724138
857 The value of the following game can be computed using the fact
858 that the identity is invertible::
860 >>> from dunshire import *
861 >>> K = NonnegativeOrthant(3)
862 >>> L = [[1,0,0],[0,1,0],[0,0,1]]
865 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
866 >>> print(SLG.solution())
867 Game value: 0.0312500
877 This is another Gowda/Ravindran example that is supposed to have
878 a negative game value::
880 >>> from dunshire import *
881 >>> from dunshire.options import ABS_TOL
882 >>> L = [[1, -2], [-2, 1]]
883 >>> K = NonnegativeOrthant(2)
886 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
887 >>> SLG.solution().game_value() < -ABS_TOL
893 The following two games are problematic numerically, but we
894 should be able to solve them::
896 >>> from dunshire import *
897 >>> L = [[-0.95237953890954685221, 1.83474556206462535712],
898 ... [ 1.30481749924621448500, 1.65278664543326403447]]
899 >>> K = NonnegativeOrthant(2)
900 >>> e1 = [0.95477167524644313001, 0.63270781756540095397]
901 >>> e2 = [0.39633793037154141370, 0.10239281495640320530]
902 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
903 >>> print(SLG.solution())
904 Game value: 18.767...
914 >>> from dunshire import *
915 >>> L = [[1.54159395026049472754, 2.21344728574316684799],
916 ... [1.33147433507846657541, 1.17913616272988108769]]
917 >>> K = NonnegativeOrthant(2)
918 >>> e1 = [0.39903040089404784307, 0.12377403622479113410]
919 >>> e2 = [0.15695181142215544612, 0.85527381344651265405]
920 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
921 >>> print(SLG.solution())
922 Game value: 24.614...
932 opts
= {'show_progress': options.VERBOSE}
933 soln_dict
= solvers
.conelp(self
._c
(),
936 self
._C
().cvxopt_dims(),
940 except ValueError as error
:
941 if str(error
) == 'math domain error':
942 # Oops, CVXOPT tried to take the square root of a
943 # negative number. Report some details about the game
944 # rather than just the underlying CVXOPT crash.
945 raise PoorScalingException(self
)
949 # The optimal strategies are named ``p`` and ``q`` in the
950 # background documentation, and we need to extract them from
951 # the CVXOPT ``x`` and ``z`` variables. The objective values
952 # :math:`nu` and :math:`omega` can also be found in the CVXOPT
953 # ``x`` and ``y`` variables; however, they're stored
954 # conveniently as separate entries in the solution dictionary.
955 p1_value
= -soln_dict
['primal objective']
956 p2_value
= -soln_dict
['dual objective']
957 p1_optimal
= soln_dict
['x'][1:]
958 p2_optimal
= soln_dict
['z'][self
.dimension():]
960 # The "status" field contains "optimal" if everything went
961 # according to plan. Other possible values are "primal
962 # infeasible", "dual infeasible", "unknown", all of which mean
963 # we didn't get a solution.
965 # The "infeasible" ones are the worst, since they indicate
966 # that CVXOPT is convinced the problem is infeasible (and that
968 if soln_dict
['status'] in ['primal infeasible', 'dual infeasible']:
969 raise GameUnsolvableException(self
, soln_dict
)
971 # The "optimal" and "unknown" results, we actually treat the
972 # same. Even if CVXOPT bails out due to numerical difficulty,
973 # it will have some candidate points in mind. If those
974 # candidates are good enough, we take them. We do the same
975 # check (perhaps pointlessly so) for "optimal" results.
977 # First we check that the primal/dual objective values are
978 # close enough (one could be low by ABS_TOL, the other high by
979 # it) because otherwise CVXOPT might return "unknown" and give
980 # us two points in the cone that are nowhere near optimal.
981 if abs(p1_value
- p2_value
) > 2*options
.ABS_TOL
:
982 raise GameUnsolvableException(self
, soln_dict
)
984 # And we also check that the points it gave us belong to the
985 # cone, just in case...
986 if (p1_optimal
not in self
._K
) or (p2_optimal
not in self
._K
):
987 raise GameUnsolvableException(self
, soln_dict
)
989 # For the game value, we could use any of:
993 # * (p1_value + p2_value)/2
996 # We want the game value to be the payoff, however, so it
997 # makes the most sense to just use that, even if it means we
998 # can't test the fact that p1_value/p2_value are close to the
1000 payoff
= self
.payoff(p1_optimal
,p2_optimal
)
1001 return Solution(payoff
, p1_optimal
, p2_optimal
)
1004 def condition(self
):
1006 Return the condition number of this game.
1008 In the CVXOPT construction of this game, two matrices ``G`` and
1009 ``A`` appear. When those matrices are nasty, numerical problems
1010 can show up. We define the condition number of this game to be
1011 the average of the condition numbers of ``G`` and ``A`` in the
1012 CVXOPT construction. If the condition number of this game is
1013 high, then you can expect numerical difficulty (such as
1014 :class:`PoorScalingException`).
1020 A real number greater than or equal to one that measures how
1021 bad this game is numerically.
1026 >>> from dunshire import *
1027 >>> K = NonnegativeOrthant(1)
1031 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
1032 >>> actual = SLG.condition()
1033 >>> expected = 1.8090169943749477
1034 >>> abs(actual - expected) < options.ABS_TOL
1038 return (condition_number(self
._G
()) + condition_number(self
._A
()))/2
1043 Return the dual game to this game.
1045 If :math:`G = \left(L,K,e_{1},e_{2}\right)` is a linear game,
1046 then its dual is :math:`G^{*} =
1047 \left(L^{*},K^{*},e_{2},e_{1}\right)`. However, since this cone
1048 is symmetric, :math:`K^{*} = K`.
1053 >>> from dunshire import *
1054 >>> K = NonnegativeOrthant(3)
1055 >>> L = [[1,-5,-15],[-1,2,-3],[-12,-15,1]]
1058 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
1059 >>> print(SLG.dual())
1060 The linear game (L, K, e1, e2) where
1064 K = Nonnegative orthant in the real 3-space,
1071 Condition((L, K, e1, e2)) = 44.476...
1074 # We pass ``self._L`` right back into the constructor, because
1075 # it will be transposed there. And keep in mind that ``self._K``
1077 return SymmetricLinearGame(self
._L,