]>
gitweb.michael.orlitzky.com - dunshire.git/blob - games.py
75f5329bb973e05d9070609fcb77e5eeba80fdee
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
)
12 from .options
import ABS_TOL
, FLOAT_FORMAT
, DEBUG_FLOAT_FORMAT
14 printing
.options
['dformat'] = 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])))
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,
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,
243 The points ``e1`` and ``e2`` can also be passed as some other
244 enumerable type (of the correct length) without much harm, since
245 there is no row/column ambiguity::
249 >>> from dunshire import *
250 >>> K = NonnegativeOrthant(2)
251 >>> L = [[1,0],[0,1]]
252 >>> e1 = cvxopt.matrix([1,1])
253 >>> e2 = numpy.matrix([1,1])
254 >>> G = SymmetricLinearGame(L, K, e1, e2)
256 The linear game (L, K, e1, e2) where
259 K = Nonnegative orthant in the real 2-space,
265 However, ``L`` will always be intepreted as a list of rows, even
266 if it is passed as a :class:`cvxopt.base.matrix` which is
267 otherwise indexed by columns::
270 >>> from dunshire import *
271 >>> K = NonnegativeOrthant(2)
272 >>> L = [[1,2],[3,4]]
275 >>> G = SymmetricLinearGame(L, K, e1, e2)
277 The linear game (L, K, e1, e2) where
280 K = Nonnegative orthant in the real 2-space,
285 >>> L = cvxopt.matrix(L)
290 >>> G = SymmetricLinearGame(L, K, e1, e2)
292 The linear game (L, K, e1, e2) where
295 K = Nonnegative orthant in the real 2-space,
302 def __init__(self
, L
, K
, e1
, e2
):
304 Create a new SymmetricLinearGame object.
307 self
._e
1 = matrix(e1
, (K
.dimension(), 1))
308 self
._e
2 = matrix(e2
, (K
.dimension(), 1))
310 # Our input ``L`` is indexed by rows but CVXOPT matrices are
311 # indexed by columns, so we need to transpose the input before
312 # feeding it to CVXOPT.
313 self
._L = matrix(L
, (K
.dimension(), K
.dimension())).trans()
315 if not self
._e
1 in K
:
316 raise ValueError('the point e1 must lie in the interior of K')
318 if not self
._e
2 in K
:
319 raise ValueError('the point e2 must lie in the interior of K')
321 # Initial value of cached method.
322 self
._L_specnorm
_value
= None
327 Return a string representation of this game.
329 tpl
= 'The linear game (L, K, e1, e2) where\n' \
334 indented_L
= '\n '.join(str(self
.L()).splitlines())
335 indented_e1
= '\n '.join(str(self
.e1()).splitlines())
336 indented_e2
= '\n '.join(str(self
.e2()).splitlines())
338 return tpl
.format(indented_L
,
346 Return the matrix ``L`` passed to the constructor.
352 The matrix that defines this game's :meth:`payoff` operator.
357 >>> from dunshire import *
358 >>> K = NonnegativeOrthant(3)
359 >>> L = [[1,-5,-15],[-1,2,-3],[-12,-15,1]]
362 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
375 Return the cone over which this game is played.
381 The :class:`SymmetricCone` over which this game is played.
386 >>> from dunshire import *
387 >>> K = NonnegativeOrthant(3)
388 >>> L = [[1,-5,-15],[-1,2,-3],[-12,-15,1]]
391 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
393 Nonnegative orthant in the real 3-space
401 Return player one's interior point.
407 The point interior to :meth:`K` affiliated with player one.
412 >>> from dunshire import *
413 >>> K = NonnegativeOrthant(3)
414 >>> L = [[1,-5,-15],[-1,2,-3],[-12,-15,1]]
417 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
430 Return player two's interior point.
436 The point interior to :meth:`K` affiliated with player one.
441 >>> from dunshire import *
442 >>> K = NonnegativeOrthant(3)
443 >>> L = [[1,-5,-15],[-1,2,-3],[-12,-15,1]]
446 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
457 def payoff(self
, strategy1
, strategy2
):
459 Return the payoff associated with ``strategy1`` and ``strategy2``.
461 The payoff operator takes pairs of strategies to a real
462 number. For example, if player one's strategy is :math:`x` and
463 player two's strategy is :math:`y`, then the associated payoff
464 is :math:`\left\langle L\left(x\right),y \right\rangle` \in
465 \mathbb{R}. Here, :math:`L` denotes the same linear operator as
466 :meth:`L`. This method computes the payoff given the two
473 Player one's strategy.
476 Player two's strategy.
482 The payoff for the game when player one plays ``strategy1``
483 and player two plays ``strategy2``.
488 The value of the game should be the payoff at the optimal
491 >>> from dunshire import *
492 >>> from dunshire.options import ABS_TOL
493 >>> K = NonnegativeOrthant(3)
494 >>> L = [[1,-5,-15],[-1,2,-3],[-12,-15,1]]
497 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
498 >>> soln = SLG.solution()
499 >>> x_bar = soln.player1_optimal()
500 >>> y_bar = soln.player2_optimal()
501 >>> abs(SLG.payoff(x_bar, y_bar) - soln.game_value()) < ABS_TOL
505 return inner_product(self
.L()*strategy1
, strategy2
)
510 Return the dimension of this game.
512 The dimension of a game is not needed for the theory, but it is
513 useful for the implementation. We define the dimension of a game
514 to be the dimension of its underlying cone. Or what is the same,
515 the dimension of the space from which the strategies are chosen.
521 The dimension of the cone :meth:`K`, or of the space where
527 The dimension of a game over the nonnegative quadrant in the
528 plane should be two (the dimension of the plane)::
530 >>> from dunshire import *
531 >>> K = NonnegativeOrthant(2)
532 >>> L = [[1,-5],[-1,2]]
535 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
540 return self
.K().dimension()
545 Return a column of zeros that fits ``K``.
547 This is used in our CVXOPT construction.
551 It is not safe to cache any of the matrices passed to
552 CVXOPT, because it can clobber them.
558 A ``self.dimension()``-by-``1`` column vector of zeros.
563 >>> from dunshire import *
564 >>> K = NonnegativeOrthant(3)
568 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
569 >>> print(SLG._zero())
576 return matrix(0, (self
.dimension(), 1), tc
='d')
581 Return the matrix ``A`` used in our CVXOPT construction.
583 This matrix ``A`` appears on the right-hand side of ``Ax = b``
584 in the statement of the CVXOPT conelp program.
588 It is not safe to cache any of the matrices passed to
589 CVXOPT, because it can clobber them.
595 A ``1``-by-``(1 + self.dimension())`` row vector. Its first
596 entry is zero, and the rest are the entries of ``e2``.
601 >>> from dunshire import *
602 >>> K = NonnegativeOrthant(3)
603 >>> L = [[1,1,1],[1,1,1],[1,1,1]]
606 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
608 [0.0000000 1.0000000 2.0000000 3.0000000]
612 return matrix([0, self
.e2()], (1, self
.dimension() + 1), 'd')
618 Return the matrix ``G`` used in our CVXOPT construction.
620 Thus matrix ``G`` appears on the left-hand side of ``Gx + s = h``
621 in the statement of the CVXOPT conelp program.
625 It is not safe to cache any of the matrices passed to
626 CVXOPT, because it can clobber them.
632 A ``2*self.dimension()``-by-``(1 + self.dimension())`` matrix.
637 >>> from dunshire import *
638 >>> K = NonnegativeOrthant(3)
639 >>> L = [[4,5,6],[7,8,9],[10,11,12]]
642 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
644 [ 0.0000000 -1.0000000 0.0000000 0.0000000]
645 [ 0.0000000 0.0000000 -1.0000000 0.0000000]
646 [ 0.0000000 0.0000000 0.0000000 -1.0000000]
647 [ 1.0000000 -4.0000000 -5.0000000 -6.0000000]
648 [ 2.0000000 -7.0000000 -8.0000000 -9.0000000]
649 [ 3.0000000 -10.0000000 -11.0000000 -12.0000000]
653 identity_matrix
= identity(self
.dimension())
654 return append_row(append_col(self
._zero
(), -identity_matrix
),
655 append_col(self
.e1(), -self
.L()))
660 Return the vector ``c`` used in our CVXOPT construction.
662 The column vector ``c`` appears in the objective function
663 value ``<c,x>`` in the statement of the CVXOPT conelp program.
667 It is not safe to cache any of the matrices passed to
668 CVXOPT, because it can clobber them.
674 A ``self.dimension()``-by-``1`` column vector.
679 >>> from dunshire import *
680 >>> K = NonnegativeOrthant(3)
681 >>> L = [[4,5,6],[7,8,9],[10,11,12]]
684 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
693 return matrix([-1, self
._zero
()])
698 Return the cone ``C`` used in our CVXOPT construction.
700 The cone ``C`` is the cone over which the conelp program takes
707 The cartesian product of ``K`` with itself.
712 >>> from dunshire import *
713 >>> K = NonnegativeOrthant(3)
714 >>> L = [[4,5,6],[7,8,9],[10,11,12]]
717 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
719 Cartesian product of dimension 6 with 2 factors:
720 * Nonnegative orthant in the real 3-space
721 * Nonnegative orthant in the real 3-space
724 return CartesianProduct(self
._K
, self
._K
)
728 Return the ``h`` vector used in our CVXOPT construction.
730 The ``h`` vector appears on the right-hand side of :math:`Gx + s
731 = h` in the statement of the CVXOPT conelp program.
735 It is not safe to cache any of the matrices passed to
736 CVXOPT, because it can clobber them.
742 A ``2*self.dimension()``-by-``1`` column vector of zeros.
747 >>> from dunshire import *
748 >>> K = NonnegativeOrthant(3)
749 >>> L = [[4,5,6],[7,8,9],[10,11,12]]
752 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
764 return matrix([self
._zero
(), self
._zero
()])
770 Return the ``b`` vector used in our CVXOPT construction.
772 The vector ``b`` appears on the right-hand side of :math:`Ax =
773 b` in the statement of the CVXOPT conelp program.
775 This method is static because the dimensions and entries of
776 ``b`` are known beforehand, and don't depend on any other
777 properties of the game.
781 It is not safe to cache any of the matrices passed to
782 CVXOPT, because it can clobber them.
788 A ``1``-by-``1`` matrix containing a single entry ``1``.
793 >>> from dunshire import *
794 >>> K = NonnegativeOrthant(3)
795 >>> L = [[4,5,6],[7,8,9],[10,11,12]]
798 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
804 return matrix([1], tc
='d')
807 def player1_start(self
):
809 Return a feasible starting point for player one.
811 This starting point is for the CVXOPT formulation and not for
812 the original game. The basic premise is that if you normalize
813 :meth:`e2`, then you get a point in :meth:`K` that makes a unit
814 inner product with :meth:`e2`. We then get to choose the primal
815 objective function value such that the constraint involving
816 :meth:`L` is satisfied.
818 p
= self
.e2() / (norm(self
.e2()) ** 2)
819 dist
= self
.K().ball_radius(self
.e1())
820 nu
= - self
._L_specnorm
()/(dist
*norm(self
.e2()))
821 x
= matrix([nu
, p
], (self
.dimension() + 1, 1))
824 return {'x': x, 's': s}
827 def player2_start(self
):
829 Return a feasible starting point for player two.
831 q
= self
.e1() / (norm(self
.e1()) ** 2)
832 dist
= self
.K().ball_radius(self
.e2())
833 omega
= self
._L_specnorm
()/(dist
*norm(self
.e1()))
836 z1
= y
*self
.e2() - self
.L().trans()*z2
837 z
= matrix([z1
, z2
], (self
.dimension()*2, 1))
839 return {'y': y, 'z': z}
842 def _L_specnorm(self
):
844 Compute the spectral norm of :meth:`L` and cache it.
846 The spectral norm of the matrix :meth:`L` is used in a few
847 places. Since it can be expensive to compute, we want to cache
848 its value. That is not possible in :func:`specnorm`, which lies
849 outside of a class, so this is the place to do it.
855 A nonnegative real number; the largest singular value of
856 the matrix :meth:`L`.
859 if self
._L_specnorm
_value
is None:
860 self
._L_specnorm
_value
= specnorm(self
.L())
861 return self
._L_specnorm
_value
864 def tolerance_scale(self
, solution
):
866 Return a scaling factor that should be applied to ``ABS_TOL``
869 When performing certain comparisons, the default tolernace
870 ``ABS_TOL`` may not be appropriate. For example, if we expect
871 ``x`` and ``y`` to be within ``ABS_TOL`` of each other, than the
872 inner product of ``L*x`` and ``y`` can be as far apart as the
873 spectral norm of ``L`` times the sum of the norms of ``x`` and
874 ``y``. Such a comparison is made in :meth:`solution`, and in
875 many of our unit tests.
877 The returned scaling factor found from the inner product mentioned
882 \left\lVert L \right\rVert_{2}
883 \left( \left\lVert \bar{x} \right\rVert
884 + \left\lVert \bar{y} \right\rVert
887 where :math:`\bar{x}` and :math:`\bar{y}` are optimal solutions
888 for players one and two respectively. This scaling factor is not
889 formally justified, but attempting anything smaller leads to
894 Optimal solutions are not unique, so the scaling factor
895 obtained from ``solution`` may not work when comparing other
902 A solution of this game, used to obtain the norms of the
909 A scaling factor to be multiplied by ``ABS_TOL`` when
910 making comparisons involving solutions of this game.
913 norm_p1_opt
= norm(solution
.player1_optimal())
914 norm_p2_opt
= norm(solution
.player2_optimal())
915 scale
= self
._L_specnorm
()*(norm_p1_opt
+ norm_p2_opt
)
917 # Don't return anything smaller than 1... we can't go below
918 # out "minimum tolerance."
924 Solve this linear game and return a :class:`Solution`.
930 A :class:`Solution` object describing the game's value and
931 the optimal strategies of both players.
935 GameUnsolvableException
936 If the game could not be solved (if an optimal solution to its
937 associated cone program was not found).
940 If the game could not be solved because CVXOPT crashed while
941 trying to take the square root of a negative number.
946 This example is computed in Gowda and Ravindran in the section
947 "The value of a Z-transformation"::
949 >>> from dunshire import *
950 >>> K = NonnegativeOrthant(3)
951 >>> L = [[1,-5,-15],[-1,2,-3],[-12,-15,1]]
954 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
955 >>> print(SLG.solution())
956 Game value: -6.172...
966 The value of the following game can be computed using the fact
967 that the identity is invertible::
969 >>> from dunshire import *
970 >>> K = NonnegativeOrthant(3)
971 >>> L = [[1,0,0],[0,1,0],[0,0,1]]
974 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
975 >>> print(SLG.solution())
986 This is another Gowda/Ravindran example that is supposed to have
987 a negative game value::
989 >>> from dunshire import *
990 >>> from dunshire.options import ABS_TOL
991 >>> L = [[1, -2], [-2, 1]]
992 >>> K = NonnegativeOrthant(2)
995 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
996 >>> SLG.solution().game_value() < -ABS_TOL
999 The following two games are problematic numerically, but we
1000 should be able to solve them::
1002 >>> from dunshire import *
1003 >>> L = [[-0.95237953890954685221, 1.83474556206462535712],
1004 ... [ 1.30481749924621448500, 1.65278664543326403447]]
1005 >>> K = NonnegativeOrthant(2)
1006 >>> e1 = [0.95477167524644313001, 0.63270781756540095397]
1007 >>> e2 = [0.39633793037154141370, 0.10239281495640320530]
1008 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
1009 >>> print(SLG.solution())
1010 Game value: 18.767...
1020 >>> from dunshire import *
1021 >>> L = [[1.54159395026049472754, 2.21344728574316684799],
1022 ... [1.33147433507846657541, 1.17913616272988108769]]
1023 >>> K = NonnegativeOrthant(2)
1024 >>> e1 = [0.39903040089404784307, 0.12377403622479113410]
1025 >>> e2 = [0.15695181142215544612, 0.85527381344651265405]
1026 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
1027 >>> print(SLG.solution())
1028 Game value: 24.614...
1036 This is another one that was difficult numerically, and caused
1037 trouble even after we fixed the first two::
1039 >>> from dunshire import *
1040 >>> L = [[57.22233908627052301199, 41.70631373437460354126],
1041 ... [83.04512571985074487202, 57.82581810406928468637]]
1042 >>> K = NonnegativeOrthant(2)
1043 >>> e1 = [7.31887017043399268346, 0.89744171905822367474]
1044 >>> e2 = [0.11099824781179848388, 6.12564670639315345113]
1045 >>> SLG = SymmetricLinearGame(L,K,e1,e2)
1046 >>> print(SLG.solution())
1047 Game value: 70.437...
1055 And finally, here's one that returns an "optimal" solution, but
1056 whose primal/dual objective function values are far apart::
1058 >>> from dunshire import *
1059 >>> L = [[ 6.49260076597376212248, -0.60528030227678542019],
1060 ... [ 2.59896077096751731972, -0.97685530240286766457]]
1062 >>> e1 = [1, 0.43749513972645248661]
1063 >>> e2 = [1, 0.46008379832200291260]
1064 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
1065 >>> print(SLG.solution())
1066 Game value: 11.596...
1076 opts
= {'show_progress': False}
1077 soln_dict
= solvers
.conelp(self
.c(),
1080 self
.C().cvxopt_dims(),
1083 primalstart
=self
.player1_start(),
1084 dualstart
=self
.player2_start(),
1086 except ValueError as error
:
1087 if str(error
) == 'math domain error':
1088 # Oops, CVXOPT tried to take the square root of a
1089 # negative number. Report some details about the game
1090 # rather than just the underlying CVXOPT crash.
1091 printing
.options
['dformat'] = DEBUG_FLOAT_FORMAT
1092 raise PoorScalingException(self
)
1096 # The optimal strategies are named ``p`` and ``q`` in the
1097 # background documentation, and we need to extract them from
1098 # the CVXOPT ``x`` and ``z`` variables. The objective values
1099 # :math:`nu` and :math:`omega` can also be found in the CVXOPT
1100 # ``x`` and ``y`` variables; however, they're stored
1101 # conveniently as separate entries in the solution dictionary.
1102 p1_value
= -soln_dict
['primal objective']
1103 p2_value
= -soln_dict
['dual objective']
1104 p1_optimal
= soln_dict
['x'][1:]
1105 p2_optimal
= soln_dict
['z'][self
.dimension():]
1107 # The "status" field contains "optimal" if everything went
1108 # according to plan. Other possible values are "primal
1109 # infeasible", "dual infeasible", "unknown", all of which mean
1110 # we didn't get a solution.
1112 # The "infeasible" ones are the worst, since they indicate
1113 # that CVXOPT is convinced the problem is infeasible (and that
1115 if soln_dict
['status'] in ['primal infeasible', 'dual infeasible']:
1116 printing
.options
['dformat'] = DEBUG_FLOAT_FORMAT
1117 raise GameUnsolvableException(self
, soln_dict
)
1119 # For the game value, we could use any of:
1123 # * (p1_value + p2_value)/2
1126 # We want the game value to be the payoff, however, so it
1127 # makes the most sense to just use that, even if it means we
1128 # can't test the fact that p1_value/p2_value are close to the
1130 payoff
= self
.payoff(p1_optimal
, p2_optimal
)
1131 soln
= Solution(payoff
, p1_optimal
, p2_optimal
)
1133 # The "optimal" and "unknown" results, we actually treat the
1134 # same. Even if CVXOPT bails out due to numerical difficulty,
1135 # it will have some candidate points in mind. If those
1136 # candidates are good enough, we take them. We do the same
1137 # check (perhaps pointlessly so) for "optimal" results.
1139 # First we check that the primal/dual objective values are
1140 # close enough (one could be low by ABS_TOL, the other high by
1141 # it) because otherwise CVXOPT might return "unknown" and give
1142 # us two points in the cone that are nowhere near optimal.
1144 if abs(p1_value
- p2_value
) > self
.tolerance_scale(soln
)*ABS_TOL
:
1145 printing
.options
['dformat'] = DEBUG_FLOAT_FORMAT
1146 raise GameUnsolvableException(self
, soln_dict
)
1148 # And we also check that the points it gave us belong to the
1149 # cone, just in case...
1150 if (p1_optimal
not in self
._K
) or (p2_optimal
not in self
._K
):
1151 printing
.options
['dformat'] = DEBUG_FLOAT_FORMAT
1152 raise GameUnsolvableException(self
, soln_dict
)
1157 def condition(self
):
1159 Return the condition number of this game.
1161 In the CVXOPT construction of this game, two matrices ``G`` and
1162 ``A`` appear. When those matrices are nasty, numerical problems
1163 can show up. We define the condition number of this game to be
1164 the average of the condition numbers of ``G`` and ``A`` in the
1165 CVXOPT construction. If the condition number of this game is
1166 high, then you can expect numerical difficulty (such as
1167 :class:`PoorScalingException`).
1173 A real number greater than or equal to one that measures how
1174 bad this game is numerically.
1179 >>> from dunshire import *
1180 >>> K = NonnegativeOrthant(1)
1184 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
1189 return (condition_number(self
.G()) + condition_number(self
.A()))/2
1194 Return the dual game to this game.
1196 If :math:`G = \left(L,K,e_{1},e_{2}\right)` is a linear game,
1197 then its dual is :math:`G^{*} =
1198 \left(L^{*},K^{*},e_{2},e_{1}\right)`. However, since this cone
1199 is symmetric, :math:`K^{*} = K`.
1204 >>> from dunshire import *
1205 >>> K = NonnegativeOrthant(3)
1206 >>> L = [[1,-5,-15],[-1,2,-3],[-12,-15,1]]
1209 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
1210 >>> print(SLG.dual())
1211 The linear game (L, K, e1, e2) where
1215 K = Nonnegative orthant in the real 3-space,
1224 # We pass ``self.L()`` right back into the constructor, because
1225 # it will be transposed there. And keep in mind that ``self._K``
1227 return SymmetricLinearGame(self
.L(),