]>
gitweb.michael.orlitzky.com - dunshire.git/blob - src/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.
8 # These few are used only for tests.
10 from random
import randint
, uniform
11 from unittest
import TestCase
13 # These are mostly actually needed.
14 from cvxopt
import matrix
, printing
, solvers
15 from cones
import CartesianProduct
, IceCream
, NonnegativeOrthant
16 from errors
import GameUnsolvableException
17 from matrices
import append_col
, append_row
, identity
, inner_product
20 printing
.options
['dformat'] = options
.FLOAT_FORMAT
21 solvers
.options
['show_progress'] = options
.VERBOSE
26 A representation of the solution of a linear game. It should contain
27 the value of the game, and both players' strategies.
32 >>> print(Solution(10, matrix([1,2]), matrix([3,4])))
33 Game value: 10.0000000
42 def __init__(self
, game_value
, p1_optimal
, p2_optimal
):
44 Create a new Solution object from a game value and two optimal
45 strategies for the players.
47 self
._game
_value
= game_value
48 self
._player
1_optimal
= p1_optimal
49 self
._player
2_optimal
= p2_optimal
53 Return a string describing the solution of a linear game.
55 The three data that are described are,
57 * The value of the game.
58 * The optimal strategy of player one.
59 * The optimal strategy of player two.
61 The two optimal strategy vectors are indented by two spaces.
63 tpl
= 'Game value: {:.7f}\n' \
64 'Player 1 optimal:{:s}\n' \
65 'Player 2 optimal:{:s}'
67 p1_str
= '\n{!s}'.format(self
.player1_optimal())
68 p1_str
= '\n '.join(p1_str
.splitlines())
69 p2_str
= '\n{!s}'.format(self
.player2_optimal())
70 p2_str
= '\n '.join(p2_str
.splitlines())
72 return tpl
.format(self
.game_value(), p1_str
, p2_str
)
77 Return the game value for this solution.
82 >>> s = Solution(10, matrix([1,2]), matrix([3,4]))
87 return self
._game
_value
90 def player1_optimal(self
):
92 Return player one's optimal strategy in this solution.
97 >>> s = Solution(10, matrix([1,2]), matrix([3,4]))
98 >>> print(s.player1_optimal())
104 return self
._player
1_optimal
107 def player2_optimal(self
):
109 Return player two's optimal strategy in this solution.
114 >>> s = Solution(10, matrix([1,2]), matrix([3,4]))
115 >>> print(s.player2_optimal())
121 return self
._player
2_optimal
124 class SymmetricLinearGame
:
126 A representation of a symmetric linear game.
128 The data for a symmetric linear game are,
130 * A "payoff" operator ``L``.
131 * A symmetric cone ``K``.
132 * Two points ``e1`` and ``e2`` in the interior of ``K``.
134 The ambient space is assumed to be the span of ``K``.
136 With those data understood, the game is played as follows. Players
137 one and two choose points :math:`x` and :math:`y` respectively, from
138 their respective strategy sets,
145 x \in K \ \middle|\ \left\langle x, e_{2} \right\rangle = 1
150 y \in K \ \middle|\ \left\langle y, e_{1} \right\rangle = 1
154 Afterwards, a "payout" is computed as :math:`\left\langle
155 L\left(x\right), y \right\rangle` and is paid to player one out of
156 player two's pocket. The game is therefore zero sum, and we suppose
157 that player one would like to guarantee himself the largest minimum
158 payout possible. That is, player one wishes to,
163 &\underset{y \in \Delta_{2}}{\min}\left(
164 \left\langle L\left(x\right), y \right\rangle
166 \text{subject to } & x \in \Delta_{1}.
169 Player two has the simultaneous goal to,
174 &\underset{x \in \Delta_{1}}{\max}\left(
175 \left\langle L\left(x\right), y \right\rangle
177 \text{subject to } & y \in \Delta_{2}.
180 These goals obviously conflict (the game is zero sum), but an
181 existence theorem guarantees at least one optimal min-max solution
182 from which neither player would like to deviate. This class is
183 able to find such a solution.
188 L : list of list of float
189 A matrix represented as a list of ROWS. This representation
190 agrees with (for example) SageMath and NumPy, but not with CVXOPT
191 (whose matrix constructor accepts a list of columns).
193 K : :class:`SymmetricCone`
194 The symmetric cone instance over which the game is played.
197 The interior point of ``K`` belonging to player one; it
198 can be of any iterable type having the correct length.
201 The interior point of ``K`` belonging to player two; it
202 can be of any enumerable type having the correct length.
208 If either ``e1`` or ``e2`` lie outside of the cone ``K``.
213 >>> from cones import NonnegativeOrthant
214 >>> K = NonnegativeOrthant(3)
215 >>> L = [[1,-5,-15],[-1,2,-3],[-12,-15,1]]
218 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
220 The linear game (L, K, e1, e2) where
224 K = Nonnegative orthant in the real 3-space,
232 Lists can (and probably should) be used for every argument::
234 >>> from cones import NonnegativeOrthant
235 >>> K = NonnegativeOrthant(2)
236 >>> L = [[1,0],[0,1]]
239 >>> G = SymmetricLinearGame(L, K, e1, e2)
241 The linear game (L, K, e1, e2) where
244 K = Nonnegative orthant in the real 2-space,
250 The points ``e1`` and ``e2`` can also be passed as some other
251 enumerable type (of the correct length) without much harm, since
252 there is no row/column ambiguity::
256 >>> from cones import NonnegativeOrthant
257 >>> K = NonnegativeOrthant(2)
258 >>> L = [[1,0],[0,1]]
259 >>> e1 = cvxopt.matrix([1,1])
260 >>> e2 = numpy.matrix([1,1])
261 >>> G = SymmetricLinearGame(L, K, e1, e2)
263 The linear game (L, K, e1, e2) where
266 K = Nonnegative orthant in the real 2-space,
272 However, ``L`` will always be intepreted as a list of rows, even
273 if it is passed as a :class:`cvxopt.base.matrix` which is
274 otherwise indexed by columns::
277 >>> from cones import NonnegativeOrthant
278 >>> K = NonnegativeOrthant(2)
279 >>> L = [[1,2],[3,4]]
282 >>> G = SymmetricLinearGame(L, K, e1, e2)
284 The linear game (L, K, e1, e2) where
287 K = Nonnegative orthant in the real 2-space,
292 >>> L = cvxopt.matrix(L)
297 >>> G = SymmetricLinearGame(L, K, e1, e2)
299 The linear game (L, K, e1, e2) where
302 K = Nonnegative orthant in the real 2-space,
309 def __init__(self
, L
, K
, e1
, e2
):
311 Create a new SymmetricLinearGame object.
314 self
._e
1 = matrix(e1
, (K
.dimension(), 1))
315 self
._e
2 = matrix(e2
, (K
.dimension(), 1))
317 # Our input ``L`` is indexed by rows but CVXOPT matrices are
318 # indexed by columns, so we need to transpose the input before
319 # feeding it to CVXOPT.
320 self
._L = matrix(L
, (K
.dimension(), K
.dimension())).trans()
322 if not K
.contains_strict(self
._e
1):
323 raise ValueError('the point e1 must lie in the interior of K')
325 if not K
.contains_strict(self
._e
2):
326 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 indented_L
= '\n '.join(str(self
._L).splitlines())
338 indented_e1
= '\n '.join(str(self
._e
1).splitlines())
339 indented_e2
= '\n '.join(str(self
._e
2).splitlines())
340 return tpl
.format(indented_L
, str(self
._K
), indented_e1
, indented_e2
)
345 Solve this linear game and return a :class:`Solution`.
351 A :class:`Solution` object describing the game's value and
352 the optimal strategies of both players.
356 GameUnsolvableException
357 If the game could not be solved (if an optimal solution to its
358 associated cone program was not found).
363 This example is computed in Gowda and Ravindran in the section
364 "The value of a Z-transformation"::
366 >>> from cones import NonnegativeOrthant
367 >>> K = NonnegativeOrthant(3)
368 >>> L = [[1,-5,-15],[-1,2,-3],[-12,-15,1]]
371 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
372 >>> print(SLG.solution())
373 Game value: -6.1724138
383 The value of the following game can be computed using the fact
384 that the identity is invertible::
386 >>> from cones import NonnegativeOrthant
387 >>> K = NonnegativeOrthant(3)
388 >>> L = [[1,0,0],[0,1,0],[0,0,1]]
391 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
392 >>> print(SLG.solution())
393 Game value: 0.0312500
404 # The cone "C" that appears in the statement of the CVXOPT
406 C
= CartesianProduct(self
._K
, self
._K
)
408 # The column vector "b" that appears on the right-hand side of
409 # Ax = b in the statement of the CVXOPT conelp program.
410 b
= matrix([1], tc
='d')
412 # A column of zeros that fits K.
413 zero
= matrix(0, (self
._K
.dimension(), 1), tc
='d')
415 # The column vector "h" that appears on the right-hand side of
416 # Gx + s = h in the statement of the CVXOPT conelp program.
417 h
= matrix([zero
, zero
])
419 # The column vector "c" that appears in the objective function
420 # value <c,x> in the statement of the CVXOPT conelp program.
421 c
= matrix([-1, zero
])
423 # The matrix "G" that appears on the left-hand side of Gx + s = h
424 # in the statement of the CVXOPT conelp program.
425 G
= append_row(append_col(zero
, -identity(self
._K
.dimension())),
426 append_col(self
._e
1, -self
._L))
428 # The matrix "A" that appears on the right-hand side of Ax = b
429 # in the statement of the CVXOPT conelp program.
430 A
= matrix([0, self
._e
2], (1, self
._K
.dimension() + 1), 'd')
432 # Actually solve the thing and obtain a dictionary describing
434 soln_dict
= solvers
.conelp(c
, G
, h
, C
.cvxopt_dims(), A
, b
)
436 # The "status" field contains "optimal" if everything went
437 # according to plan. Other possible values are "primal
438 # infeasible", "dual infeasible", "unknown", all of which
439 # mean we didn't get a solution. That should never happen,
440 # because by construction our game has a solution, and thus
441 # the cone program should too.
442 if soln_dict
['status'] != 'optimal':
443 raise GameUnsolvableException(soln_dict
)
445 p1_value
= soln_dict
['x'][0]
446 p1_optimal
= soln_dict
['x'][1:]
447 p2_optimal
= soln_dict
['z'][self
._K
.dimension():]
449 return Solution(p1_value
, p1_optimal
, p2_optimal
)
453 Return the dual game to this game.
455 If :math:`G = \left(L,K,e_{1},e_{2}\right)` is a linear game,
456 then its dual is :math:`G^{*} =
457 \left(L^{*},K^{*},e_{2},e_{1}\right)`. However, since this cone
458 is symmetric, :math:`K^{*} = K`.
463 >>> from cones import NonnegativeOrthant
464 >>> K = NonnegativeOrthant(3)
465 >>> L = [[1,-5,-15],[-1,2,-3],[-12,-15,1]]
468 >>> SLG = SymmetricLinearGame(L, K, e1, e2)
469 >>> print(SLG.dual())
470 The linear game (L, K, e1, e2) where
474 K = Nonnegative orthant in the real 3-space,
483 # We pass ``self._L`` right back into the constructor, because
484 # it will be transposed there. And keep in mind that ``self._K``
486 return SymmetricLinearGame(self
._L,
492 class SymmetricLinearGameTest(TestCase
):
494 Tests for the SymmetricLinearGame and Solution classes.
497 def assert_within_tol(self
, first
, second
):
499 Test that ``first`` and ``second`` are equal within our default
502 self
.assertTrue(abs(first
- second
) < options
.ABS_TOL
)
505 def assert_solution_exists(self
, L
, K
, e1
, e2
):
507 Given the parameters needed to construct a SymmetricLinearGame,
508 ensure that that game has a solution.
510 G
= SymmetricLinearGame(L
, K
, e1
, e2
)
512 L_matrix
= matrix(L
).trans()
513 expected
= inner_product(L_matrix
*soln
.player1_optimal(),
514 soln
.player2_optimal())
515 self
.assert_within_tol(soln
.game_value(), expected
)
517 def test_solution_exists_nonnegative_orthant(self
):
519 Every linear game has a solution, so we should be able to solve
520 every symmetric linear game over the NonnegativeOrthant. Pick
521 some parameters randomly and give it a shot. The resulting
522 optimal solutions should give us the optimal game value when we
523 apply the payoff operator to them.
525 ambient_dim
= randint(1, 10)
526 K
= NonnegativeOrthant(ambient_dim
)
527 e1
= [uniform(0.1, 10) for idx
in range(K
.dimension())]
528 e2
= [uniform(0.1, 10) for idx
in range(K
.dimension())]
529 L
= [[uniform(-10, 10) for i
in range(K
.dimension())]
530 for j
in range(K
.dimension())]
531 self
.assert_solution_exists(L
, K
, e1
, e2
)
533 def test_solution_exists_ice_cream(self
):
535 Like :meth:`test_solution_exists_nonnegative_orthant`, except
536 over the ice cream cone.
538 # Use a minimum dimension of two to avoid divide-by-zero in
539 # the fudge factor we make up later.
540 ambient_dim
= randint(2, 10)
541 K
= IceCream(ambient_dim
)
542 e1
= [1] # Set the "height" of e1 to one
543 e2
= [1] # And the same for e2
545 # If we choose the rest of the components of e1,e2 randomly
546 # between 0 and 1, then the largest the squared norm of the
547 # non-height part of e1,e2 could be is the 1*(dim(K) - 1). We
548 # need to make it less than one (the height of the cone) so
549 # that the whole thing is in the cone. The norm of the
550 # non-height part is sqrt(dim(K) - 1), and we can divide by
552 fudge_factor
= 1.0 / (2.0*sqrt(K
.dimension() - 1.0))
553 e1
+= [fudge_factor
*uniform(0, 1) for idx
in range(K
.dimension() - 1)]
554 e2
+= [fudge_factor
*uniform(0, 1) for idx
in range(K
.dimension() - 1)]
555 L
= [[uniform(-10, 10) for i
in range(K
.dimension())]
556 for j
in range(K
.dimension())]
557 self
.assert_solution_exists(L
, K
, e1
, e2
)