]>
gitweb.michael.orlitzky.com - dunshire.git/blob - dunshire/games.py
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.
7 from cvxopt
import matrix
, printing
, solvers
8 from .cones
import CartesianProduct
9 from .errors
import GameUnsolvableException
, PoorScalingException
10 from .matrices
import (append_col
, append_row
, condition_number
, identity
,
11 inner_product
, norm
, specnorm
)
14 printing
.options
['dformat'] = options
.FLOAT_FORMAT
18 A representation of the solution of a linear game. It should contain
19 the value of the game, and both players' strategies.
24 >>> print(Solution(10, matrix([1,2]), matrix([3,4])))
34 def __init__(self
, game_value
, p1_optimal
, p2_optimal
):
36 Create a new Solution object from a game value and two optimal
37 strategies for the players.
39 self
._game
_value
= game_value
40 self
._player
1_optimal
= p1_optimal
41 self
._player
2_optimal
= p2_optimal
45 Return a string describing the solution of a linear game.
47 The three data that are described are,
49 * The value of the game.
50 * The optimal strategy of player one.
51 * The optimal strategy of player two.
53 The two optimal strategy vectors are indented by two spaces.
55 tpl
= 'Game value: {:.7f}\n' \
56 'Player 1 optimal:{:s}\n' \
57 'Player 2 optimal:{:s}'
59 p1_str
= '\n{!s}'.format(self
.player1_optimal())
60 p1_str
= '\n '.join(p1_str
.splitlines())
61 p2_str
= '\n{!s}'.format(self
.player2_optimal())
62 p2_str
= '\n '.join(p2_str
.splitlines())
64 return tpl
.format(self
.game_value(), p1_str
, p2_str
)
69 Return the game value for this solution.
74 >>> s = Solution(10, matrix([1,2]), matrix([3,4]))
79 return self
._game
_value
82 def player1_optimal(self
):
84 Return player one's optimal strategy in this solution.
89 >>> s = Solution(10, matrix([1,2]), matrix([3,4]))
90 >>> print(s.player1_optimal())
96 return self
._player
1_optimal
99 def player2_optimal(self
):
101 Return player two's optimal strategy in this solution.
106 >>> s = Solution(10, matrix([1,2]), matrix([3,4]))
107 >>> print(s.player2_optimal())
113 return self
._player
2_optimal
116 class SymmetricLinearGame
:
118 A representation of a symmetric linear game.
120 The data for a symmetric linear game are,
122 * A "payoff" operator ``L``.
123 * A symmetric cone ``K``.
124 * Two points ``e1`` and ``e2`` in the interior of ``K``.
126 The ambient space is assumed to be the span of ``K``.
128 With those data understood, the game is played as follows. Players
129 one and two choose points :math:`x` and :math:`y` respectively, from
130 their respective strategy sets,
137 x \in K \ \middle|\ \left\langle x, e_{2} \right\rangle = 1
142 y \in K \ \middle|\ \left\langle y, e_{1} \right\rangle = 1
146 Afterwards, a "payout" is computed as :math:`\left\langle
147 L\left(x\right), y \right\rangle` and is paid to player one out of
148 player two's pocket. The game is therefore zero sum, and we suppose
149 that player one would like to guarantee himself the largest minimum
150 payout possible. That is, player one wishes to,
155 &\underset{y \in \Delta_{2}}{\min}\left(
156 \left\langle L\left(x\right), y \right\rangle
158 \text{subject to } & x \in \Delta_{1}.
161 Player two has the simultaneous goal to,
166 &\underset{x \in \Delta_{1}}{\max}\left(
167 \left\langle L\left(x\right), y \right\rangle
169 \text{subject to } & y \in \Delta_{2}.
172 These goals obviously conflict (the game is zero sum), but an
173 existence theorem guarantees at least one optimal min-max solution
174 from which neither player would like to deviate. This class is
175 able to find such a solution.
180 L : list of list of float
181 A matrix represented as a list of ROWS. This representation
182 agrees with (for example) SageMath and NumPy, but not with CVXOPT
183 (whose matrix constructor accepts a list of columns).
185 K : :class:`SymmetricCone`
186 The symmetric cone instance over which the game is played.
189 The interior point of ``K`` belonging to player one; it
190 can be of any iterable type having the correct length.
193 The interior point of ``K`` belonging to player two; it
194 can be of any enumerable type having the correct length.
200 If either ``e1`` or ``e2`` lie outside of the cone ``K``.
205 >>> from dunshire import *
206 >>> K = NonnegativeOrthant(3)
207 >>> L = [[1,-5,-15],[-1,2,-3],[-12,-15,1]]
210 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
212 The linear game (L, K, e1, e2) where
216 K = Nonnegative orthant in the real 3-space,
223 Condition((L, K, e1, e2)) = 31.834...
225 Lists can (and probably should) be used for every argument::
227 >>> from dunshire import *
228 >>> K = NonnegativeOrthant(2)
229 >>> L = [[1,0],[0,1]]
232 >>> G = SymmetricLinearGame(L, K, e1, e2)
234 The linear game (L, K, e1, e2) where
237 K = Nonnegative orthant in the real 2-space,
242 Condition((L, K, e1, e2)) = 1.707...
244 The points ``e1`` and ``e2`` can also be passed as some other
245 enumerable type (of the correct length) without much harm, since
246 there is no row/column ambiguity::
250 >>> from dunshire import *
251 >>> K = NonnegativeOrthant(2)
252 >>> L = [[1,0],[0,1]]
253 >>> e1 = cvxopt.matrix([1,1])
254 >>> e2 = numpy.matrix([1,1])
255 >>> G = SymmetricLinearGame(L, K, e1, e2)
257 The linear game (L, K, e1, e2) where
260 K = Nonnegative orthant in the real 2-space,
265 Condition((L, K, e1, e2)) = 1.707...
267 However, ``L`` will always be intepreted as a list of rows, even
268 if it is passed as a :class:`cvxopt.base.matrix` which is
269 otherwise indexed by columns::
272 >>> from dunshire import *
273 >>> K = NonnegativeOrthant(2)
274 >>> L = [[1,2],[3,4]]
277 >>> G = SymmetricLinearGame(L, K, e1, e2)
279 The linear game (L, K, e1, e2) where
282 K = Nonnegative orthant in the real 2-space,
287 Condition((L, K, e1, e2)) = 6.073...
288 >>> L = cvxopt.matrix(L)
293 >>> G = SymmetricLinearGame(L, K, e1, e2)
295 The linear game (L, K, e1, e2) where
298 K = Nonnegative orthant in the real 2-space,
303 Condition((L, K, e1, e2)) = 6.073...
306 def __init__(self
, L
, K
, e1
, e2
):
308 Create a new SymmetricLinearGame object.
311 self
._e
1 = matrix(e1
, (K
.dimension(), 1))
312 self
._e
2 = matrix(e2
, (K
.dimension(), 1))
314 # Our input ``L`` is indexed by rows but CVXOPT matrices are
315 # indexed by columns, so we need to transpose the input before
316 # feeding it to CVXOPT.
317 self
._L = matrix(L
, (K
.dimension(), K
.dimension())).trans()
319 if not self
._e
1 in K
:
320 raise ValueError('the point e1 must lie in the interior of K')
322 if not self
._e
2 in K
:
323 raise ValueError('the point e2 must lie in the interior of K')
329 Return a string representation of this game.
331 tpl
= 'The linear game (L, K, e1, e2) where\n' \
336 ' Condition((L, K, e1, e2)) = {:f}.'
337 indented_L
= '\n '.join(str(self
.L()).splitlines())
338 indented_e1
= '\n '.join(str(self
.e1()).splitlines())
339 indented_e2
= '\n '.join(str(self
.e2()).splitlines())
341 return tpl
.format(indented_L
,
350 Return the matrix ``L`` passed to the constructor.
356 The matrix that defines this game's :meth:`payoff` operator.
361 >>> from dunshire import *
362 >>> K = NonnegativeOrthant(3)
363 >>> L = [[1,-5,-15],[-1,2,-3],[-12,-15,1]]
366 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
379 Return the cone over which this game is played.
385 The :class:`SymmetricCone` over which this game is played.
390 >>> from dunshire import *
391 >>> K = NonnegativeOrthant(3)
392 >>> L = [[1,-5,-15],[-1,2,-3],[-12,-15,1]]
395 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
397 Nonnegative orthant in the real 3-space
405 Return player one's interior point.
411 The point interior to :meth:`K` affiliated with player one.
416 >>> from dunshire import *
417 >>> K = NonnegativeOrthant(3)
418 >>> L = [[1,-5,-15],[-1,2,-3],[-12,-15,1]]
421 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
434 Return player two's interior point.
440 The point interior to :meth:`K` affiliated with player one.
445 >>> from dunshire import *
446 >>> K = NonnegativeOrthant(3)
447 >>> L = [[1,-5,-15],[-1,2,-3],[-12,-15,1]]
450 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
461 def payoff(self
, strategy1
, strategy2
):
463 Return the payoff associated with ``strategy1`` and ``strategy2``.
465 The payoff operator takes pairs of strategies to a real
466 number. For example, if player one's strategy is :math:`x` and
467 player two's strategy is :math:`y`, then the associated payoff
468 is :math:`\left\langle L\left(x\right),y \right\rangle` \in
469 \mathbb{R}. Here, :math:`L` denotes the same linear operator as
470 :meth:`L`. This method computes the payoff given the two
477 Player one's strategy.
480 Player two's strategy.
486 The payoff for the game when player one plays ``strategy1``
487 and player two plays ``strategy2``.
492 The value of the game should be the payoff at the optimal
495 >>> from dunshire import *
496 >>> from dunshire.options import ABS_TOL
497 >>> K = NonnegativeOrthant(3)
498 >>> L = [[1,-5,-15],[-1,2,-3],[-12,-15,1]]
501 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
502 >>> soln = SLG.solution()
503 >>> x_bar = soln.player1_optimal()
504 >>> y_bar = soln.player2_optimal()
505 >>> abs(SLG.payoff(x_bar, y_bar) - soln.game_value()) < ABS_TOL
509 return inner_product(self
.L()*strategy1
, strategy2
)
514 Return the dimension of this game.
516 The dimension of a game is not needed for the theory, but it is
517 useful for the implementation. We define the dimension of a game
518 to be the dimension of its underlying cone. Or what is the same,
519 the dimension of the space from which the strategies are chosen.
525 The dimension of the cone :meth:`K`, or of the space where
531 The dimension of a game over the nonnegative quadrant in the
532 plane should be two (the dimension of the plane)::
534 >>> from dunshire import *
535 >>> K = NonnegativeOrthant(2)
536 >>> L = [[1,-5],[-1,2]]
539 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
544 return self
.K().dimension()
549 Return a column of zeros that fits ``K``.
551 This is used in our CVXOPT construction.
555 It is not safe to cache any of the matrices passed to
556 CVXOPT, because it can clobber them.
562 A ``self.dimension()``-by-``1`` column vector of zeros.
567 >>> from dunshire import *
568 >>> K = NonnegativeOrthant(3)
572 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
573 >>> print(SLG._zero())
580 return matrix(0, (self
.dimension(), 1), tc
='d')
585 Return the matrix ``A`` used in our CVXOPT construction.
587 This matrix ``A`` appears on the right-hand side of ``Ax = b``
588 in the statement of the CVXOPT conelp program.
592 It is not safe to cache any of the matrices passed to
593 CVXOPT, because it can clobber them.
599 A ``1``-by-``(1 + self.dimension())`` row vector. Its first
600 entry is zero, and the rest are the entries of ``e2``.
605 >>> from dunshire import *
606 >>> K = NonnegativeOrthant(3)
607 >>> L = [[1,1,1],[1,1,1],[1,1,1]]
610 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
612 [0.0000000 1.0000000 2.0000000 3.0000000]
616 return matrix([0, self
.e2()], (1, self
.dimension() + 1), 'd')
622 Return the matrix ``G`` used in our CVXOPT construction.
624 Thus matrix ``G`` appears on the left-hand side of ``Gx + s = h``
625 in the statement of the CVXOPT conelp program.
629 It is not safe to cache any of the matrices passed to
630 CVXOPT, because it can clobber them.
636 A ``2*self.dimension()``-by-``(1 + self.dimension())`` matrix.
641 >>> from dunshire import *
642 >>> K = NonnegativeOrthant(3)
643 >>> L = [[4,5,6],[7,8,9],[10,11,12]]
646 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
648 [ 0.0000000 -1.0000000 0.0000000 0.0000000]
649 [ 0.0000000 0.0000000 -1.0000000 0.0000000]
650 [ 0.0000000 0.0000000 0.0000000 -1.0000000]
651 [ 1.0000000 -4.0000000 -5.0000000 -6.0000000]
652 [ 2.0000000 -7.0000000 -8.0000000 -9.0000000]
653 [ 3.0000000 -10.0000000 -11.0000000 -12.0000000]
657 identity_matrix
= identity(self
.dimension())
658 return append_row(append_col(self
._zero
(), -identity_matrix
),
659 append_col(self
.e1(), -self
.L()))
664 Return the vector ``c`` used in our CVXOPT construction.
666 The column vector ``c`` appears in the objective function
667 value ``<c,x>`` in the statement of the CVXOPT conelp program.
671 It is not safe to cache any of the matrices passed to
672 CVXOPT, because it can clobber them.
678 A ``self.dimension()``-by-``1`` column vector.
683 >>> from dunshire import *
684 >>> K = NonnegativeOrthant(3)
685 >>> L = [[4,5,6],[7,8,9],[10,11,12]]
688 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
697 return matrix([-1, self
._zero
()])
702 Return the cone ``C`` used in our CVXOPT construction.
704 The cone ``C`` is the cone over which the conelp program takes
711 The cartesian product of ``K`` with itself.
716 >>> from dunshire import *
717 >>> K = NonnegativeOrthant(3)
718 >>> L = [[4,5,6],[7,8,9],[10,11,12]]
721 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
723 Cartesian product of dimension 6 with 2 factors:
724 * Nonnegative orthant in the real 3-space
725 * Nonnegative orthant in the real 3-space
728 return CartesianProduct(self
._K
, self
._K
)
732 Return the ``h`` vector used in our CVXOPT construction.
734 The ``h`` vector appears on the right-hand side of :math:`Gx + s
735 = h` in the statement of the CVXOPT conelp program.
739 It is not safe to cache any of the matrices passed to
740 CVXOPT, because it can clobber them.
746 A ``2*self.dimension()``-by-``1`` column vector of zeros.
751 >>> from dunshire import *
752 >>> K = NonnegativeOrthant(3)
753 >>> L = [[4,5,6],[7,8,9],[10,11,12]]
756 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
768 return matrix([self
._zero
(), self
._zero
()])
774 Return the ``b`` vector used in our CVXOPT construction.
776 The vector ``b`` appears on the right-hand side of :math:`Ax =
777 b` in the statement of the CVXOPT conelp program.
779 This method is static because the dimensions and entries of
780 ``b`` are known beforehand, and don't depend on any other
781 properties of the game.
785 It is not safe to cache any of the matrices passed to
786 CVXOPT, because it can clobber them.
792 A ``1``-by-``1`` matrix containing a single entry ``1``.
797 >>> from dunshire import *
798 >>> K = NonnegativeOrthant(3)
799 >>> L = [[4,5,6],[7,8,9],[10,11,12]]
802 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
808 return matrix([1], tc
='d')
811 def player1_start(self
):
813 Return a feasible starting point for player one.
815 This starting point is for the CVXOPT formulation and not for
816 the original game. The basic premise is that if you normalize
817 :meth:`e2`, then you get a point in :meth:`K` that makes a unit
818 inner product with :meth:`e2`. We then get to choose the primal
819 objective function value such that the constraint involving
820 :meth:`L` is satisfied.
822 p
= self
.e2() / (norm(self
.e2()) ** 2)
823 dist
= self
.K().ball_radius(self
.e1())
824 nu
= - specnorm(self
.L())/(dist
*norm(self
.e2()))
825 x
= matrix([nu
, p
], (self
.dimension() + 1, 1))
828 return {'x': x, 's': s}
831 def player2_start(self
):
833 Return a feasible starting point for player two.
835 q
= self
.e1() / (norm(self
.e1()) ** 2)
836 dist
= self
.K().ball_radius(self
.e2())
837 omega
= specnorm(self
.L())/(dist
*norm(self
.e1()))
840 z1
= y
*self
.e2() - self
.L().trans()*z2
841 z
= matrix([z1
, z2
], (self
.dimension()*2, 1))
843 return {'y': y, 'z': z}
848 Solve this linear game and return a :class:`Solution`.
854 A :class:`Solution` object describing the game's value and
855 the optimal strategies of both players.
859 GameUnsolvableException
860 If the game could not be solved (if an optimal solution to its
861 associated cone program was not found).
864 If the game could not be solved because CVXOPT crashed while
865 trying to take the square root of a negative number.
870 This example is computed in Gowda and Ravindran in the section
871 "The value of a Z-transformation"::
873 >>> from dunshire import *
874 >>> K = NonnegativeOrthant(3)
875 >>> L = [[1,-5,-15],[-1,2,-3],[-12,-15,1]]
878 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
879 >>> print(SLG.solution())
880 Game value: -6.172...
890 The value of the following game can be computed using the fact
891 that the identity is invertible::
893 >>> from dunshire import *
894 >>> K = NonnegativeOrthant(3)
895 >>> L = [[1,0,0],[0,1,0],[0,0,1]]
898 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
899 >>> print(SLG.solution())
910 This is another Gowda/Ravindran example that is supposed to have
911 a negative game value::
913 >>> from dunshire import *
914 >>> from dunshire.options import ABS_TOL
915 >>> L = [[1, -2], [-2, 1]]
916 >>> K = NonnegativeOrthant(2)
919 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
920 >>> SLG.solution().game_value() < -ABS_TOL
923 The following two games are problematic numerically, but we
924 should be able to solve them::
926 >>> from dunshire import *
927 >>> L = [[-0.95237953890954685221, 1.83474556206462535712],
928 ... [ 1.30481749924621448500, 1.65278664543326403447]]
929 >>> K = NonnegativeOrthant(2)
930 >>> e1 = [0.95477167524644313001, 0.63270781756540095397]
931 >>> e2 = [0.39633793037154141370, 0.10239281495640320530]
932 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
933 >>> print(SLG.solution())
934 Game value: 18.767...
944 >>> from dunshire import *
945 >>> L = [[1.54159395026049472754, 2.21344728574316684799],
946 ... [1.33147433507846657541, 1.17913616272988108769]]
947 >>> K = NonnegativeOrthant(2)
948 >>> e1 = [0.39903040089404784307, 0.12377403622479113410]
949 >>> e2 = [0.15695181142215544612, 0.85527381344651265405]
950 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
951 >>> print(SLG.solution())
952 Game value: 24.614...
962 opts
= {'show_progress': False}
963 soln_dict
= solvers
.conelp(self
._c
(),
966 self
.C().cvxopt_dims(),
969 primalstart
=self
.player1_start(),
971 except ValueError as error
:
972 if str(error
) == 'math domain error':
973 # Oops, CVXOPT tried to take the square root of a
974 # negative number. Report some details about the game
975 # rather than just the underlying CVXOPT crash.
976 printing
.options
['dformat'] = options
.DEBUG_FLOAT_FORMAT
977 raise PoorScalingException(self
)
981 # The optimal strategies are named ``p`` and ``q`` in the
982 # background documentation, and we need to extract them from
983 # the CVXOPT ``x`` and ``z`` variables. The objective values
984 # :math:`nu` and :math:`omega` can also be found in the CVXOPT
985 # ``x`` and ``y`` variables; however, they're stored
986 # conveniently as separate entries in the solution dictionary.
987 p1_value
= -soln_dict
['primal objective']
988 p2_value
= -soln_dict
['dual objective']
989 p1_optimal
= soln_dict
['x'][1:]
990 p2_optimal
= soln_dict
['z'][self
.dimension():]
992 # The "status" field contains "optimal" if everything went
993 # according to plan. Other possible values are "primal
994 # infeasible", "dual infeasible", "unknown", all of which mean
995 # we didn't get a solution.
997 # The "infeasible" ones are the worst, since they indicate
998 # that CVXOPT is convinced the problem is infeasible (and that
1000 if soln_dict
['status'] in ['primal infeasible', 'dual infeasible']:
1001 printing
.options
['dformat'] = options
.DEBUG_FLOAT_FORMAT
1002 raise GameUnsolvableException(self
, soln_dict
)
1004 # The "optimal" and "unknown" results, we actually treat the
1005 # same. Even if CVXOPT bails out due to numerical difficulty,
1006 # it will have some candidate points in mind. If those
1007 # candidates are good enough, we take them. We do the same
1008 # check (perhaps pointlessly so) for "optimal" results.
1010 # First we check that the primal/dual objective values are
1011 # close enough (one could be low by ABS_TOL, the other high by
1012 # it) because otherwise CVXOPT might return "unknown" and give
1013 # us two points in the cone that are nowhere near optimal.
1014 if abs(p1_value
- p2_value
) > 2*options
.ABS_TOL
:
1015 printing
.options
['dformat'] = options
.DEBUG_FLOAT_FORMAT
1016 raise GameUnsolvableException(self
, soln_dict
)
1018 # And we also check that the points it gave us belong to the
1019 # cone, just in case...
1020 if (p1_optimal
not in self
._K
) or (p2_optimal
not in self
._K
):
1021 printing
.options
['dformat'] = options
.DEBUG_FLOAT_FORMAT
1022 raise GameUnsolvableException(self
, soln_dict
)
1024 # For the game value, we could use any of:
1028 # * (p1_value + p2_value)/2
1031 # We want the game value to be the payoff, however, so it
1032 # makes the most sense to just use that, even if it means we
1033 # can't test the fact that p1_value/p2_value are close to the
1035 payoff
= self
.payoff(p1_optimal
, p2_optimal
)
1036 return Solution(payoff
, p1_optimal
, p2_optimal
)
1039 def condition(self
):
1041 Return the condition number of this game.
1043 In the CVXOPT construction of this game, two matrices ``G`` and
1044 ``A`` appear. When those matrices are nasty, numerical problems
1045 can show up. We define the condition number of this game to be
1046 the average of the condition numbers of ``G`` and ``A`` in the
1047 CVXOPT construction. If the condition number of this game is
1048 high, then you can expect numerical difficulty (such as
1049 :class:`PoorScalingException`).
1055 A real number greater than or equal to one that measures how
1056 bad this game is numerically.
1061 >>> from dunshire import *
1062 >>> K = NonnegativeOrthant(1)
1066 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
1067 >>> actual = SLG.condition()
1068 >>> expected = 1.8090169943749477
1069 >>> abs(actual - expected) < options.ABS_TOL
1073 return (condition_number(self
._G
()) + condition_number(self
.A()))/2
1078 Return the dual game to this game.
1080 If :math:`G = \left(L,K,e_{1},e_{2}\right)` is a linear game,
1081 then its dual is :math:`G^{*} =
1082 \left(L^{*},K^{*},e_{2},e_{1}\right)`. However, since this cone
1083 is symmetric, :math:`K^{*} = K`.
1088 >>> from dunshire import *
1089 >>> K = NonnegativeOrthant(3)
1090 >>> L = [[1,-5,-15],[-1,2,-3],[-12,-15,1]]
1093 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
1094 >>> print(SLG.dual())
1095 The linear game (L, K, e1, e2) where
1099 K = Nonnegative orthant in the real 3-space,
1106 Condition((L, K, e1, e2)) = 44.476...
1109 # We pass ``self.L()`` right back into the constructor, because
1110 # it will be transposed there. And keep in mind that ``self._K``
1112 return SymmetricLinearGame(self
.L(),