from cvxopt import matrix, printing, solvers
from .cones import CartesianProduct
from .errors import GameUnsolvableException, PoorScalingException
-from .matrices import append_col, append_row, condition_number, identity
+from .matrices import (append_col, append_row, condition_number, identity,
+ inner_product)
from . import options
printing.options['dformat'] = options.FLOAT_FORMAT
' e1 = {:s},\n' \
' e2 = {:s},\n' \
' Condition((L, K, e1, e2)) = {:f}.'
- indented_L = '\n '.join(str(self._L).splitlines())
- indented_e1 = '\n '.join(str(self._e1).splitlines())
- indented_e2 = '\n '.join(str(self._e2).splitlines())
+ indented_L = '\n '.join(str(self.L()).splitlines())
+ indented_e1 = '\n '.join(str(self.e1()).splitlines())
+ indented_e2 = '\n '.join(str(self.e2()).splitlines())
return tpl.format(indented_L,
- str(self._K),
+ str(self.K()),
indented_e1,
indented_e2,
self.condition())
+ def L(self):
+ """
+ Return the matrix ``L`` passed to the constructor.
+
+ Returns
+ -------
+
+ matrix
+ The matrix that defines this game's :meth:`payoff` operator.
+
+ Examples
+ --------
+
+ >>> from dunshire import *
+ >>> K = NonnegativeOrthant(3)
+ >>> L = [[1,-5,-15],[-1,2,-3],[-12,-15,1]]
+ >>> e1 = [1,1,1]
+ >>> e2 = [1,2,3]
+ >>> SLG = SymmetricLinearGame(L, K, e1, e2)
+ >>> print(SLG.L())
+ [ 1 -5 -15]
+ [ -1 2 -3]
+ [-12 -15 1]
+ <BLANKLINE>
+
+ """
+ return self._L
+
+
+ def K(self):
+ """
+ Return the cone over which this game is played.
+
+ Returns
+ -------
+
+ SymmetricCone
+ The :class:`SymmetricCone` over which this game is played.
+
+ Examples
+ --------
+
+ >>> from dunshire import *
+ >>> K = NonnegativeOrthant(3)
+ >>> L = [[1,-5,-15],[-1,2,-3],[-12,-15,1]]
+ >>> e1 = [1,1,1]
+ >>> e2 = [1,2,3]
+ >>> SLG = SymmetricLinearGame(L, K, e1, e2)
+ >>> print(SLG.K())
+ Nonnegative orthant in the real 3-space
+
+ """
+ return self._K
+
+
+ def e1(self):
+ """
+ Return player one's interior point.
+
+ Returns
+ -------
+
+ matrix
+ The point interior to :meth:`K` affiliated with player one.
+
+ Examples
+ --------
+
+ >>> from dunshire import *
+ >>> K = NonnegativeOrthant(3)
+ >>> L = [[1,-5,-15],[-1,2,-3],[-12,-15,1]]
+ >>> e1 = [1,1,1]
+ >>> e2 = [1,2,3]
+ >>> SLG = SymmetricLinearGame(L, K, e1, e2)
+ >>> print(SLG.e1())
+ [ 1]
+ [ 1]
+ [ 1]
+ <BLANKLINE>
+
+ """
+ return self._e1
+
+
+ def e2(self):
+ """
+ Return player two's interior point.
+
+ Returns
+ -------
+
+ matrix
+ The point interior to :meth:`K` affiliated with player one.
+
+ Examples
+ --------
+
+ >>> from dunshire import *
+ >>> K = NonnegativeOrthant(3)
+ >>> L = [[1,-5,-15],[-1,2,-3],[-12,-15,1]]
+ >>> e1 = [1,1,1]
+ >>> e2 = [1,2,3]
+ >>> SLG = SymmetricLinearGame(L, K, e1, e2)
+ >>> print(SLG.e2())
+ [ 1]
+ [ 2]
+ [ 3]
+ <BLANKLINE>
+
+ """
+ return self._e2
+
+
+ def payoff(self, strategy1, strategy2):
+ r"""
+ Return the payoff associated with ``strategy1`` and ``strategy2``.
+
+ The payoff operator takes pairs of strategies to a real
+ number. For example, if player one's strategy is :math:`x` and
+ player two's strategy is :math:`y`, then the associated payoff
+ is :math:`\left\langle L\left(x\right),y \right\rangle` \in
+ \mathbb{R}. Here, :math:`L` denotes the same linear operator as
+ :meth:`L`. This method computes the payoff given the two
+ players' strategies.
+
+ Parameters
+ ----------
+
+ strategy1 : matrix
+ Player one's strategy.
+
+ strategy2 : matrix
+ Player two's strategy.
+
+ Returns
+ -------
+
+ float
+ The payoff for the game when player one plays ``strategy1``
+ and player two plays ``strategy2``.
+
+ Examples
+ --------
+
+ The value of the game should be the payoff at the optimal
+ strategies::
+
+ >>> from dunshire import *
+ >>> from dunshire.options import ABS_TOL
+ >>> K = NonnegativeOrthant(3)
+ >>> L = [[1,-5,-15],[-1,2,-3],[-12,-15,1]]
+ >>> e1 = [1,1,1]
+ >>> e2 = [1,1,1]
+ >>> SLG = SymmetricLinearGame(L, K, e1, e2)
+ >>> soln = SLG.solution()
+ >>> x_bar = soln.player1_optimal()
+ >>> y_bar = soln.player2_optimal()
+ >>> abs(SLG.payoff(x_bar, y_bar) - soln.game_value()) < ABS_TOL
+ True
+
+ """
+ return inner_product(self.L()*strategy1, strategy2)
+
+
+ def dimension(self):
+ """
+ Return the dimension of this game.
+
+ The dimension of a game is not needed for the theory, but it is
+ useful for the implementation. We define the dimension of a game
+ to be the dimension of its underlying cone. Or what is the same,
+ the dimension of the space from which the strategies are chosen.
+
+ Returns
+ -------
+
+ int
+ The dimension of the cone :meth:`K`, or of the space where
+ this game is played.
+
+ Examples
+ --------
+
+ The dimension of a game over the nonnegative quadrant in the
+ plane should be two (the dimension of the plane)::
+
+ >>> from dunshire import *
+ >>> K = NonnegativeOrthant(2)
+ >>> L = [[1,-5],[-1,2]]
+ >>> e1 = [1,1]
+ >>> e2 = [1,4]
+ >>> SLG = SymmetricLinearGame(L, K, e1, e2)
+ >>> SLG.dimension()
+ 2
+
+ """
+ return self.K().dimension()
+
+
def _zero(self):
"""
Return a column of zeros that fits ``K``.
-------
matrix
- A ``K.dimension()``-by-``1`` column vector of zeros.
+ A ``self.dimension()``-by-``1`` column vector of zeros.
Examples
--------
<BLANKLINE>
"""
- return matrix(0, (self._K.dimension(), 1), tc='d')
+ return matrix(0, (self.dimension(), 1), tc='d')
- def _A(self):
+ def A(self):
"""
Return the matrix ``A`` used in our CVXOPT construction.
-------
matrix
- A ``1``-by-``(1 + K.dimension())`` row vector. Its first
+ A ``1``-by-``(1 + self.dimension())`` row vector. Its first
entry is zero, and the rest are the entries of ``e2``.
Examples
>>> e1 = [1,1,1]
>>> e2 = [1,2,3]
>>> SLG = SymmetricLinearGame(L, K, e1, e2)
- >>> print(SLG._A())
+ >>> print(SLG.A())
[0.0000000 1.0000000 2.0000000 3.0000000]
<BLANKLINE>
"""
- return matrix([0, self._e2], (1, self._K.dimension() + 1), 'd')
+ return matrix([0, self.e2()], (1, self.dimension() + 1), 'd')
-------
matrix
- A ``2*K.dimension()``-by-``1 + K.dimension()`` matrix.
+ A ``2*self.dimension()``-by-``(1 + self.dimension())`` matrix.
Examples
--------
<BLANKLINE>
"""
- I = identity(self._K.dimension())
- return append_row(append_col(self._zero(), -I),
- append_col(self._e1, -self._L))
+ identity_matrix = identity(self.dimension())
+ return append_row(append_col(self._zero(), -identity_matrix),
+ append_col(self.e1(), -self.L()))
def _c(self):
-------
matrix
- A ``K.dimension()``-by-``1`` column vector.
+ A ``self.dimension()``-by-``1`` column vector.
Examples
--------
return matrix([-1, self._zero()])
- def _C(self):
+ def C(self):
"""
Return the cone ``C`` used in our CVXOPT construction.
>>> e1 = [1,2,3]
>>> e2 = [1,1,1]
>>> SLG = SymmetricLinearGame(L, K, e1, e2)
- >>> print(SLG._C())
+ >>> print(SLG.C())
Cartesian product of dimension 6 with 2 factors:
* Nonnegative orthant in the real 3-space
* Nonnegative orthant in the real 3-space
-------
matrix
- A ``2*K.dimension()``-by-``1`` column vector of zeros.
+ A ``2*self.dimension()``-by-``1`` column vector of zeros.
Examples
--------
return matrix([self._zero(), self._zero()])
- def _b(self):
+
+ @staticmethod
+ def b():
"""
Return the ``b`` vector used in our CVXOPT construction.
The vector ``b`` appears on the right-hand side of :math:`Ax =
b` in the statement of the CVXOPT conelp program.
+ This method is static because the dimensions and entries of
+ ``b`` are known beforehand, and don't depend on any other
+ properties of the game.
+
.. warning::
It is not safe to cache any of the matrices passed to
>>> e1 = [1,2,3]
>>> e2 = [1,1,1]
>>> SLG = SymmetricLinearGame(L, K, e1, e2)
- >>> print(SLG._b())
+ >>> print(SLG.b())
[1.0000000]
<BLANKLINE>
return matrix([1], tc='d')
- def _try_solution(self, tolerance):
- """
- Solve this linear game within ``tolerance``, if possible.
-
- This private function is the one that does all of the actual
- work for :meth:`solution`. This method accepts a ``tolerance``,
- and what :meth:`solution` does is call this method twice with
- two different tolerances. First it tries a strict tolerance, and
- then it tries a looser one.
-
- .. warning::
-
- If you try to be smart and precompute the matrices used by
- this function (the ones passed to ``conelp``), then you're
- going to shoot yourself in the foot. CVXOPT can and will
- clobber some (but not all) of its input matrices. This isn't
- performance sensitive, so play it safe.
-
- Parameters
- ----------
-
- tolerance : float
- The absolute tolerance to pass to the CVXOPT solver.
-
- Returns
- -------
-
- :class:`Solution`
- A :class:`Solution` object describing the game's value and
- the optimal strategies of both players.
-
- Raises
- ------
- GameUnsolvableException
- If the game could not be solved (if an optimal solution to its
- associated cone program was not found).
-
- PoorScalingException
- If the game could not be solved because CVXOPT crashed while
- trying to take the square root of a negative number.
-
- Examples
- --------
-
- This game can be solved easily, so the first attempt in
- :meth:`solution` should succeed::
-
- >>> from dunshire import *
- >>> from dunshire.matrices import norm
- >>> from dunshire.options import ABS_TOL
- >>> K = NonnegativeOrthant(3)
- >>> L = [[1,-5,-15],[-1,2,-3],[-12,-15,1]]
- >>> e1 = [1,1,1]
- >>> e2 = [1,1,1]
- >>> SLG = SymmetricLinearGame(L, K, e1, e2)
- >>> s1 = SLG.solution()
- >>> s2 = SLG._try_solution(options.ABS_TOL)
- >>> abs(s1.game_value() - s2.game_value()) < ABS_TOL
- True
- >>> norm(s1.player1_optimal() - s2.player1_optimal()) < ABS_TOL
- True
- >>> norm(s1.player2_optimal() - s2.player2_optimal()) < ABS_TOL
- True
-
- """
- try:
- solvers.options['show_progress'] = options.VERBOSE
- solvers.options['abs_tol'] = tolerance
- soln_dict = solvers.conelp(self._c(),
- self._G(),
- self._h(),
- self._C().cvxopt_dims(),
- self._A(),
- self._b())
- except ValueError as e:
- if str(e) == 'math domain error':
- # Oops, CVXOPT tried to take the square root of a
- # negative number. Report some details about the game
- # rather than just the underlying CVXOPT crash.
- raise PoorScalingException(self)
- else:
- raise e
-
- # The optimal strategies are named ``p`` and ``q`` in the
- # background documentation, and we need to extract them from
- # the CVXOPT ``x`` and ``z`` variables. The objective values
- # :math:`nu` and :math:`omega` can also be found in the CVXOPT
- # ``x`` and ``y`` variables; however, they're stored
- # conveniently as separate entries in the solution dictionary.
- p1_value = -soln_dict['primal objective']
- p2_value = -soln_dict['dual objective']
- p1_optimal = soln_dict['x'][1:]
- p2_optimal = soln_dict['z'][self._K.dimension():]
-
- # The "status" field contains "optimal" if everything went
- # according to plan. Other possible values are "primal
- # infeasible", "dual infeasible", "unknown", all of which mean
- # we didn't get a solution. The "infeasible" ones are the
- # worst, since they indicate that CVXOPT is convinced the
- # problem is infeasible (and that cannot happen).
- if soln_dict['status'] in ['primal infeasible', 'dual infeasible']:
- raise GameUnsolvableException(self, soln_dict)
- elif soln_dict['status'] == 'unknown':
- # When we get a status of "unknown", we may still be able
- # to salvage a solution out of the returned
- # dictionary. Often this is the result of numerical
- # difficulty and we can simply check that the primal/dual
- # objectives match (within a tolerance) and that the
- # primal/dual optimal solutions are within the cone (to a
- # tolerance as well).
- #
- # The fudge factor of two is basically unjustified, but
- # makes intuitive sense when you imagine that the primal
- # value could be under the true optimal by ``ABS_TOL``
- # and the dual value could be over by the same amount.
- #
- if abs(p1_value - p2_value) > tolerance:
- raise GameUnsolvableException(self, soln_dict)
- if (p1_optimal not in self._K) or (p2_optimal not in self._K):
- raise GameUnsolvableException(self, soln_dict)
-
- return Solution(p1_value, p1_optimal, p2_optimal)
-
def solution(self):
"""
[0.156...]
[0.187...]
+ This is another Gowda/Ravindran example that is supposed to have
+ a negative game value::
+
+ >>> from dunshire import *
+ >>> from dunshire.options import ABS_TOL
+ >>> L = [[1, -2], [-2, 1]]
+ >>> K = NonnegativeOrthant(2)
+ >>> e1 = [1, 1]
+ >>> e2 = e1
+ >>> SLG = SymmetricLinearGame(L, K, e1, e2)
+ >>> SLG.solution().game_value() < -ABS_TOL
+ True
+
+ The following two games are problematic numerically, but we
+ should be able to solve them::
+
+ >>> from dunshire import *
+ >>> L = [[-0.95237953890954685221, 1.83474556206462535712],
+ ... [ 1.30481749924621448500, 1.65278664543326403447]]
+ >>> K = NonnegativeOrthant(2)
+ >>> e1 = [0.95477167524644313001, 0.63270781756540095397]
+ >>> e2 = [0.39633793037154141370, 0.10239281495640320530]
+ >>> SLG = SymmetricLinearGame(L, K, e1, e2)
+ >>> print(SLG.solution())
+ Game value: 18.767...
+ Player 1 optimal:
+ [-0.000...]
+ [ 9.766...]
+ Player 2 optimal:
+ [1.047...]
+ [0.000...]
+
+ ::
+
+ >>> from dunshire import *
+ >>> L = [[1.54159395026049472754, 2.21344728574316684799],
+ ... [1.33147433507846657541, 1.17913616272988108769]]
+ >>> K = NonnegativeOrthant(2)
+ >>> e1 = [0.39903040089404784307, 0.12377403622479113410]
+ >>> e2 = [0.15695181142215544612, 0.85527381344651265405]
+ >>> SLG = SymmetricLinearGame(L, K, e1, e2)
+ >>> print(SLG.solution())
+ Game value: 24.614...
+ Player 1 optimal:
+ [ 6.371...]
+ [-0.000...]
+ Player 2 optimal:
+ [2.506...]
+ [0.000...]
+
"""
try:
- # First try with a stricter tolerance. Who knows, it might
- # work. If it does, we prefer that solution.
- return self._try_solution(options.ABS_TOL / 10)
+ opts = {'show_progress': False}
+ soln_dict = solvers.conelp(self._c(),
+ self._G(),
+ self._h(),
+ self.C().cvxopt_dims(),
+ self.A(),
+ self.b(),
+ options=opts)
+ except ValueError as error:
+ if str(error) == 'math domain error':
+ # Oops, CVXOPT tried to take the square root of a
+ # negative number. Report some details about the game
+ # rather than just the underlying CVXOPT crash.
+ printing.options['dformat'] = options.DEBUG_FLOAT_FORMAT
+ raise PoorScalingException(self)
+ else:
+ raise error
+
+ # The optimal strategies are named ``p`` and ``q`` in the
+ # background documentation, and we need to extract them from
+ # the CVXOPT ``x`` and ``z`` variables. The objective values
+ # :math:`nu` and :math:`omega` can also be found in the CVXOPT
+ # ``x`` and ``y`` variables; however, they're stored
+ # conveniently as separate entries in the solution dictionary.
+ p1_value = -soln_dict['primal objective']
+ p2_value = -soln_dict['dual objective']
+ p1_optimal = soln_dict['x'][1:]
+ p2_optimal = soln_dict['z'][self.dimension():]
+
+ # The "status" field contains "optimal" if everything went
+ # according to plan. Other possible values are "primal
+ # infeasible", "dual infeasible", "unknown", all of which mean
+ # we didn't get a solution.
+ #
+ # The "infeasible" ones are the worst, since they indicate
+ # that CVXOPT is convinced the problem is infeasible (and that
+ # cannot happen).
+ if soln_dict['status'] in ['primal infeasible', 'dual infeasible']:
+ printing.options['dformat'] = options.DEBUG_FLOAT_FORMAT
+ raise GameUnsolvableException(self, soln_dict)
+
+ # The "optimal" and "unknown" results, we actually treat the
+ # same. Even if CVXOPT bails out due to numerical difficulty,
+ # it will have some candidate points in mind. If those
+ # candidates are good enough, we take them. We do the same
+ # check (perhaps pointlessly so) for "optimal" results.
+ #
+ # First we check that the primal/dual objective values are
+ # close enough (one could be low by ABS_TOL, the other high by
+ # it) because otherwise CVXOPT might return "unknown" and give
+ # us two points in the cone that are nowhere near optimal.
+ if abs(p1_value - p2_value) > 2*options.ABS_TOL:
+ printing.options['dformat'] = options.DEBUG_FLOAT_FORMAT
+ raise GameUnsolvableException(self, soln_dict)
+
+ # And we also check that the points it gave us belong to the
+ # cone, just in case...
+ if (p1_optimal not in self._K) or (p2_optimal not in self._K):
+ printing.options['dformat'] = options.DEBUG_FLOAT_FORMAT
+ raise GameUnsolvableException(self, soln_dict)
- except (PoorScalingException, GameUnsolvableException):
- # Ok, that didn't work. Let's try it with the default
- # tolerance, and whatever happens, happens.
- return self._try_solution(options.ABS_TOL)
+ # For the game value, we could use any of:
+ #
+ # * p1_value
+ # * p2_value
+ # * (p1_value + p2_value)/2
+ # * the game payoff
+ #
+ # We want the game value to be the payoff, however, so it
+ # makes the most sense to just use that, even if it means we
+ # can't test the fact that p1_value/p2_value are close to the
+ # payoff.
+ payoff = self.payoff(p1_optimal, p2_optimal)
+ return Solution(payoff, p1_optimal, p2_optimal)
def condition(self):
True
"""
- return (condition_number(self._G()) + condition_number(self._A()))/2
+ return (condition_number(self._G()) + condition_number(self.A()))/2
def dual(self):
Condition((L, K, e1, e2)) = 44.476...
"""
- # We pass ``self._L`` right back into the constructor, because
+ # We pass ``self.L()`` right back into the constructor, because
# it will be transposed there. And keep in mind that ``self._K``
# is its own dual.
- return SymmetricLinearGame(self._L,
- self._K,
- self._e2,
- self._e1)
+ return SymmetricLinearGame(self.L(),
+ self.K(),
+ self.e2(),
+ self.e1())
projects / dunshire.git / blobdiff