]>
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
)
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 <http://www.sagemath.org/>`_
184 and `NumPy <http://www.numpy.org/>`_, but not with CVXOPT (whose
185 matrix constructor accepts a list of columns). In reality, ``L``
186 can be any iterable type of the correct length; however, you
187 should be extremely wary of the way we interpret anything other
190 K : dunshire.cones.SymmetricCone
191 The symmetric cone instance over which the game is played.
194 The interior point of ``K`` belonging to player one; it
195 can be of any iterable type having the correct length.
198 The interior point of ``K`` belonging to player two; it
199 can be of any enumerable type having the correct length.
205 If either ``e1`` or ``e2`` lie outside of the cone ``K``.
210 >>> from dunshire import *
211 >>> K = NonnegativeOrthant(3)
212 >>> L = [[1,-5,-15],[-1,2,-3],[-12,-15,1]]
215 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
217 The linear game (L, K, e1, e2) where
221 K = Nonnegative orthant in the real 3-space,
229 Lists can (and probably should) be used for every argument::
231 >>> from dunshire import *
232 >>> K = NonnegativeOrthant(2)
233 >>> L = [[1,0],[0,1]]
236 >>> G = SymmetricLinearGame(L, K, e1, e2)
238 The linear game (L, K, e1, e2) where
241 K = Nonnegative orthant in the real 2-space,
247 The points ``e1`` and ``e2`` can also be passed as some other
248 enumerable type (of the correct length) without much harm, since
249 there is no row/column ambiguity::
253 >>> from dunshire import *
254 >>> K = NonnegativeOrthant(2)
255 >>> L = [[1,0],[0,1]]
256 >>> e1 = cvxopt.matrix([1,1])
257 >>> e2 = numpy.matrix([1,1])
258 >>> G = SymmetricLinearGame(L, K, e1, e2)
260 The linear game (L, K, e1, e2) where
263 K = Nonnegative orthant in the real 2-space,
269 However, ``L`` will always be intepreted as a list of rows, even
270 if it is passed as a :class:`cvxopt.base.matrix` which is
271 otherwise indexed by columns::
274 >>> from dunshire import *
275 >>> K = NonnegativeOrthant(2)
276 >>> L = [[1,2],[3,4]]
279 >>> G = SymmetricLinearGame(L, K, e1, e2)
281 The linear game (L, K, e1, e2) where
284 K = Nonnegative orthant in the real 2-space,
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,
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')
325 # Initial value of cached method.
326 self
._L_specnorm
_value
= None
331 Return a string representation of this game.
333 tpl
= 'The linear game (L, K, e1, e2) where\n' \
338 indented_L
= '\n '.join(str(self
.L()).splitlines())
339 indented_e1
= '\n '.join(str(self
.e1()).splitlines())
340 indented_e2
= '\n '.join(str(self
.e2()).splitlines())
342 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 :math:`A` appears on the right-hand side of :math:`Ax
588 = b` 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 :meth:`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 :math:`G` appears on the left-hand side of :math:`Gx
625 + s = h` 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 :math:`c` appears in the objective function
667 value :math:`\left\langle c,x \right\rangle` in the statement of
668 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 :meth:`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 This is the cone over which the conelp program takes place.
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 :math:`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 scale
817 :meth:`e2` by the reciprocal of its squared norm, then you get a
818 point in :meth:`K` that makes a unit inner product with
819 :meth:`e2`. We then get to choose the primal objective function
820 value such that the constraint involving :meth:`L` is satisfied.
826 A dictionary with two keys, ``'x'`` and ``'s'``, which
827 contain the vectors of the same name in the CVXOPT primal
830 The vector ``x`` consists of the primal objective function
831 value concatenated with the strategy (for player one) that
832 achieves it. The vector ``s`` is essentially a dummy
833 variable, and is computed from the equality constraing in
834 the CVXOPT primal problem.
837 p
= self
.e2() / (norm(self
.e2()) ** 2)
838 dist
= self
.K().ball_radius(self
.e1())
839 nu
= - self
._L_specnorm
()/(dist
*norm(self
.e2()))
840 x
= matrix([nu
, p
], (self
.dimension() + 1, 1))
843 return {'x': x, 's': s}
846 def player2_start(self
):
848 Return a feasible starting point for player two.
850 This starting point is for the CVXOPT formulation and not for
851 the original game. The basic premise is that if you scale
852 :meth:`e1` by the reciprocal of its squared norm, then you get a
853 point in :meth:`K` that makes a unit inner product with
854 :meth:`e1`. We then get to choose the dual objective function
855 value such that the constraint involving :meth:`L` is satisfied.
861 A dictionary with two keys, ``'y'`` and ``'z'``, which
862 contain the vectors of the same name in the CVXOPT dual
865 The ``1``-by-``1`` vector ``y`` consists of the dual
866 objective function value. The last :meth:`dimension` entries
867 of the vector ``z`` contain the strategy (for player two)
868 that achieves it. The remaining entries of ``z`` are
869 essentially dummy variables, computed from the equality
870 constraint in the CVXOPT dual problem.
873 q
= self
.e1() / (norm(self
.e1()) ** 2)
874 dist
= self
.K().ball_radius(self
.e2())
875 omega
= self
._L_specnorm
()/(dist
*norm(self
.e1()))
878 z1
= y
*self
.e2() - self
.L().trans()*z2
879 z
= matrix([z1
, z2
], (self
.dimension()*2, 1))
881 return {'y': y, 'z': z}
884 def _L_specnorm(self
):
886 Compute the spectral norm of :meth:`L` and cache it.
888 The spectral norm of the matrix :meth:`L` is used in a few
889 places. Since it can be expensive to compute, we want to cache
890 its value. That is not possible in :func:`specnorm`, which lies
891 outside of a class, so this is the place to do it.
897 A nonnegative real number; the largest singular value of
898 the matrix :meth:`L`.
903 >>> from dunshire import *
904 >>> from dunshire.matrices import specnorm
905 >>> L = [[1,2],[3,4]]
906 >>> K = NonnegativeOrthant(2)
909 >>> SLG = SymmetricLinearGame(L,K,e1,e2)
910 >>> specnorm(SLG.L()) == SLG._L_specnorm()
914 if self
._L_specnorm
_value
is None:
915 self
._L_specnorm
_value
= specnorm(self
.L())
916 return self
._L_specnorm
_value
919 def tolerance_scale(self
, solution
):
921 Return a scaling factor that should be applied to :const:`ABS_TOL`
924 When performing certain comparisons, the default tolerance
925 :const:`ABS_TOL` may not be appropriate. For example, if we expect
926 ``x`` and ``y`` to be within :const:`ABS_TOL` of each other,
927 than the inner product of ``L*x`` and ``y`` can be as far apart
928 as the spectral norm of ``L`` times the sum of the norms of
929 ``x`` and ``y``. Such a comparison is made in :meth:`solution`,
930 and in many of our unit tests.
932 The returned scaling factor found from the inner product
937 \left\lVert L \right\rVert_{2}
938 \left( \left\lVert \bar{x} \right\rVert
939 + \left\lVert \bar{y} \right\rVert
942 where :math:`\bar{x}` and :math:`\bar{y}` are optimal solutions
943 for players one and two respectively. This scaling factor is not
944 formally justified, but attempting anything smaller leads to
949 Optimal solutions are not unique, so the scaling factor
950 obtained from ``solution`` may not work when comparing other
957 A solution of this game, used to obtain the norms of the
964 A scaling factor to be multiplied by :const:`ABS_TOL` when
965 making comparisons involving solutions of this game.
970 The spectral norm of ``L`` in this case is around ``5.464``, and
971 the optimal strategies both have norm one, so we expect the
972 tolerance scale to be somewhere around ``2 * 5.464``, or
975 >>> from dunshire import *
976 >>> L = [[1,2],[3,4]]
977 >>> K = NonnegativeOrthant(2)
980 >>> SLG = SymmetricLinearGame(L,K,e1,e2)
981 >>> SLG.tolerance_scale(SLG.solution())
985 norm_p1_opt
= norm(solution
.player1_optimal())
986 norm_p2_opt
= norm(solution
.player2_optimal())
987 scale
= self
._L_specnorm
()*(norm_p1_opt
+ norm_p2_opt
)
989 # Don't return anything smaller than 1... we can't go below
990 # out "minimum tolerance."
996 Solve this linear game and return a :class:`Solution`.
1002 A :class:`Solution` object describing the game's value and
1003 the optimal strategies of both players.
1007 GameUnsolvableException
1008 If the game could not be solved (if an optimal solution to its
1009 associated cone program was not found).
1011 PoorScalingException
1012 If the game could not be solved because CVXOPT crashed while
1013 trying to take the square root of a negative number.
1018 This example is computed in Gowda and Ravindran in the section
1019 "The value of a Z-transformation"::
1021 >>> from dunshire import *
1022 >>> K = NonnegativeOrthant(3)
1023 >>> L = [[1,-5,-15],[-1,2,-3],[-12,-15,1]]
1026 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
1027 >>> print(SLG.solution())
1028 Game value: -6.172...
1038 The value of the following game can be computed using the fact
1039 that the identity is invertible::
1041 >>> from dunshire import *
1042 >>> K = NonnegativeOrthant(3)
1043 >>> L = [[1,0,0],[0,1,0],[0,0,1]]
1046 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
1047 >>> print(SLG.solution())
1048 Game value: 0.031...
1058 This is another Gowda/Ravindran example that is supposed to have
1059 a negative game value::
1061 >>> from dunshire import *
1062 >>> from dunshire.options import ABS_TOL
1063 >>> L = [[1, -2], [-2, 1]]
1064 >>> K = NonnegativeOrthant(2)
1067 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
1068 >>> SLG.solution().game_value() < -ABS_TOL
1071 The following two games are problematic numerically, but we
1072 should be able to solve them::
1074 >>> from dunshire import *
1075 >>> L = [[-0.95237953890954685221, 1.83474556206462535712],
1076 ... [ 1.30481749924621448500, 1.65278664543326403447]]
1077 >>> K = NonnegativeOrthant(2)
1078 >>> e1 = [0.95477167524644313001, 0.63270781756540095397]
1079 >>> e2 = [0.39633793037154141370, 0.10239281495640320530]
1080 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
1081 >>> print(SLG.solution())
1082 Game value: 18.767...
1092 >>> from dunshire import *
1093 >>> L = [[1.54159395026049472754, 2.21344728574316684799],
1094 ... [1.33147433507846657541, 1.17913616272988108769]]
1095 >>> K = NonnegativeOrthant(2)
1096 >>> e1 = [0.39903040089404784307, 0.12377403622479113410]
1097 >>> e2 = [0.15695181142215544612, 0.85527381344651265405]
1098 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
1099 >>> print(SLG.solution())
1100 Game value: 24.614...
1108 This is another one that was difficult numerically, and caused
1109 trouble even after we fixed the first two::
1111 >>> from dunshire import *
1112 >>> L = [[57.22233908627052301199, 41.70631373437460354126],
1113 ... [83.04512571985074487202, 57.82581810406928468637]]
1114 >>> K = NonnegativeOrthant(2)
1115 >>> e1 = [7.31887017043399268346, 0.89744171905822367474]
1116 >>> e2 = [0.11099824781179848388, 6.12564670639315345113]
1117 >>> SLG = SymmetricLinearGame(L,K,e1,e2)
1118 >>> print(SLG.solution())
1119 Game value: 70.437...
1127 And finally, here's one that returns an "optimal" solution, but
1128 whose primal/dual objective function values are far apart::
1130 >>> from dunshire import *
1131 >>> L = [[ 6.49260076597376212248, -0.60528030227678542019],
1132 ... [ 2.59896077096751731972, -0.97685530240286766457]]
1134 >>> e1 = [1, 0.43749513972645248661]
1135 >>> e2 = [1, 0.46008379832200291260]
1136 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
1137 >>> print(SLG.solution())
1138 Game value: 11.596...
1148 opts
= {'show_progress': False}
1149 soln_dict
= solvers
.conelp(self
.c(),
1152 self
.C().cvxopt_dims(),
1155 primalstart
=self
.player1_start(),
1156 dualstart
=self
.player2_start(),
1158 except ValueError as error
:
1159 if str(error
) == 'math domain error':
1160 # Oops, CVXOPT tried to take the square root of a
1161 # negative number. Report some details about the game
1162 # rather than just the underlying CVXOPT crash.
1163 printing
.options
['dformat'] = DEBUG_FLOAT_FORMAT
1164 raise PoorScalingException(self
)
1168 # The optimal strategies are named ``p`` and ``q`` in the
1169 # background documentation, and we need to extract them from
1170 # the CVXOPT ``x`` and ``z`` variables. The objective values
1171 # :math:`nu` and :math:`omega` can also be found in the CVXOPT
1172 # ``x`` and ``y`` variables; however, they're stored
1173 # conveniently as separate entries in the solution dictionary.
1174 p1_value
= -soln_dict
['primal objective']
1175 p2_value
= -soln_dict
['dual objective']
1176 p1_optimal
= soln_dict
['x'][1:]
1177 p2_optimal
= soln_dict
['z'][self
.dimension():]
1179 # The "status" field contains "optimal" if everything went
1180 # according to plan. Other possible values are "primal
1181 # infeasible", "dual infeasible", "unknown", all of which mean
1182 # we didn't get a solution.
1184 # The "infeasible" ones are the worst, since they indicate
1185 # that CVXOPT is convinced the problem is infeasible (and that
1187 if soln_dict
['status'] in ['primal infeasible', 'dual infeasible']:
1188 printing
.options
['dformat'] = DEBUG_FLOAT_FORMAT
1189 raise GameUnsolvableException(self
, soln_dict
)
1191 # For the game value, we could use any of:
1195 # * (p1_value + p2_value)/2
1198 # We want the game value to be the payoff, however, so it
1199 # makes the most sense to just use that, even if it means we
1200 # can't test the fact that p1_value/p2_value are close to the
1202 payoff
= self
.payoff(p1_optimal
, p2_optimal
)
1203 soln
= Solution(payoff
, p1_optimal
, p2_optimal
)
1205 # The "optimal" and "unknown" results, we actually treat the
1206 # same. Even if CVXOPT bails out due to numerical difficulty,
1207 # it will have some candidate points in mind. If those
1208 # candidates are good enough, we take them. We do the same
1209 # check for "optimal" results.
1211 # First we check that the primal/dual objective values are
1212 # close enough because otherwise CVXOPT might return "unknown"
1213 # and give us two points in the cone that are nowhere near
1214 # optimal. And in fact, we need to ensure that they're close
1215 # for "optimal" results, too, because we need to know how
1216 # lenient to be in our testing.
1218 if abs(p1_value
- p2_value
) > self
.tolerance_scale(soln
)*ABS_TOL
:
1219 printing
.options
['dformat'] = DEBUG_FLOAT_FORMAT
1220 raise GameUnsolvableException(self
, soln_dict
)
1222 # And we also check that the points it gave us belong to the
1223 # cone, just in case...
1224 if (p1_optimal
not in self
._K
) or (p2_optimal
not in self
._K
):
1225 printing
.options
['dformat'] = DEBUG_FLOAT_FORMAT
1226 raise GameUnsolvableException(self
, soln_dict
)
1231 def condition(self
):
1233 Return the condition number of this game.
1235 In the CVXOPT construction of this game, two matrices ``G`` and
1236 ``A`` appear. When those matrices are nasty, numerical problems
1237 can show up. We define the condition number of this game to be
1238 the average of the condition numbers of ``G`` and ``A`` in the
1239 CVXOPT construction. If the condition number of this game is
1240 high, then you can expect numerical difficulty (such as
1241 :class:`PoorScalingException`).
1247 A real number greater than or equal to one that measures how
1248 bad this game is numerically.
1253 >>> from dunshire import *
1254 >>> K = NonnegativeOrthant(1)
1258 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
1263 return (condition_number(self
.G()) + condition_number(self
.A()))/2
1268 Return the dual game to this game.
1270 If :math:`G = \left(L,K,e_{1},e_{2}\right)` is a linear game,
1271 then its dual is :math:`G^{*} =
1272 \left(L^{*},K^{*},e_{2},e_{1}\right)`. However, since this cone
1273 is symmetric, :math:`K^{*} = K`.
1278 >>> from dunshire import *
1279 >>> K = NonnegativeOrthant(3)
1280 >>> L = [[1,-5,-15],[-1,2,-3],[-12,-15,1]]
1283 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
1284 >>> print(SLG.dual())
1285 The linear game (L, K, e1, e2) where
1289 K = Nonnegative orthant in the real 3-space,
1298 # We pass ``self.L()`` right back into the constructor, because
1299 # it will be transposed there. And keep in mind that ``self._K``
1301 return SymmetricLinearGame(self
.L(),