X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=dunshire%2Fgames.py;h=e9ae21dd9fca7803ce0f67d5ba60f3d3555419c7;hb=7d7775ac5300884e4e87fad322daa98469e3531c;hp=75f5329bb973e05d9070609fcb77e5eeba80fdee;hpb=a35db50836050e28ee4e06a12caeaa30ebbb4b11;p=dunshire.git
diff --git a/dunshire/games.py b/dunshire/games.py
index 75f5329..e9ae21d 100644
--- a/dunshire/games.py
+++ b/dunshire/games.py
@@ -179,11 +179,15 @@ class SymmetricLinearGame:
----------
L : list of list of float
- A matrix represented as a list of ROWS. This representation
- agrees with (for example) SageMath and NumPy, but not with CVXOPT
- (whose matrix constructor accepts a list of columns).
-
- K : :class:`SymmetricCone`
+ A matrix represented as a list of **rows**. This representation
+ agrees with (for example) `SageMath `_
+ and `NumPy `_, but not with CVXOPT (whose
+ matrix constructor accepts a list of columns). In reality, ``L``
+ can be any iterable type of the correct length; however, you
+ should be extremely wary of the way we interpret anything other
+ than a list of rows.
+
+ K : dunshire.cones.SymmetricCone
The symmetric cone instance over which the game is played.
e1 : iterable float
@@ -461,8 +465,8 @@ class SymmetricLinearGame:
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
+ 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.
@@ -577,11 +581,11 @@ class SymmetricLinearGame:
def A(self):
- """
+ r"""
Return the matrix ``A`` used in our CVXOPT construction.
- This matrix ``A`` appears on the right-hand side of ``Ax = b``
- in the statement of the CVXOPT conelp program.
+ This matrix :math:`A` appears on the right-hand side of :math:`Ax
+ = b` in the statement of the CVXOPT conelp program.
.. warning::
@@ -593,7 +597,7 @@ class SymmetricLinearGame:
matrix
A ``1``-by-``(1 + self.dimension())`` row vector. Its first
- entry is zero, and the rest are the entries of ``e2``.
+ entry is zero, and the rest are the entries of :meth:`e2`.
Examples
--------
@@ -617,8 +621,8 @@ class SymmetricLinearGame:
r"""
Return the matrix ``G`` used in our CVXOPT construction.
- Thus matrix ``G`` appears on the left-hand side of ``Gx + s = h``
- in the statement of the CVXOPT conelp program.
+ Thus matrix :math:`G` appears on the left-hand side of :math:`Gx
+ + s = h` in the statement of the CVXOPT conelp program.
.. warning::
@@ -656,11 +660,12 @@ class SymmetricLinearGame:
def c(self):
- """
+ r"""
Return the vector ``c`` used in our CVXOPT construction.
- The column vector ``c`` appears in the objective function
- value ```` in the statement of the CVXOPT conelp program.
+ The column vector :math:`c` appears in the objective function
+ value :math:`\left\langle c,x \right\rangle` in the statement of
+ the CVXOPT conelp program.
.. warning::
@@ -671,7 +676,7 @@ class SymmetricLinearGame:
-------
matrix
- A ``self.dimension()``-by-``1`` column vector.
+ A :meth:`dimension`-by-``1`` column vector.
Examples
--------
@@ -697,8 +702,7 @@ class SymmetricLinearGame:
"""
Return the cone ``C`` used in our CVXOPT construction.
- The cone ``C`` is the cone over which the conelp program takes
- place.
+ This is the cone over which the conelp program takes place.
Returns
-------
@@ -727,7 +731,7 @@ class SymmetricLinearGame:
r"""
Return the ``h`` vector used in our CVXOPT construction.
- The ``h`` vector appears on the right-hand side of :math:`Gx + s
+ The :math:`h` vector appears on the right-hand side of :math:`Gx + s
= h` in the statement of the CVXOPT conelp program.
.. warning::
@@ -809,11 +813,26 @@ class SymmetricLinearGame:
Return a feasible starting point for player one.
This starting point is for the CVXOPT formulation and not for
- the original game. The basic premise is that if you normalize
- :meth:`e2`, then you get a point in :meth:`K` that makes a unit
- inner product with :meth:`e2`. We then get to choose the primal
- objective function value such that the constraint involving
- :meth:`L` is satisfied.
+ the original game. The basic premise is that if you scale
+ :meth:`e2` by the reciprocal of its squared norm, then you get a
+ point in :meth:`K` that makes a unit inner product with
+ :meth:`e2`. We then get to choose the primal objective function
+ value such that the constraint involving :meth:`L` is satisfied.
+
+ Returns
+ -------
+
+ dict
+ A dictionary with two keys, ``'x'`` and ``'s'``, which
+ contain the vectors of the same name in the CVXOPT primal
+ problem formulation.
+
+ The vector ``x`` consists of the primal objective function
+ value concatenated with the strategy (for player one) that
+ achieves it. The vector ``s`` is essentially a dummy
+ variable, and is computed from the equality constraing in
+ the CVXOPT primal problem.
+
"""
p = self.e2() / (norm(self.e2()) ** 2)
dist = self.K().ball_radius(self.e1())
@@ -827,6 +846,29 @@ class SymmetricLinearGame:
def player2_start(self):
"""
Return a feasible starting point for player two.
+
+ This starting point is for the CVXOPT formulation and not for
+ the original game. The basic premise is that if you scale
+ :meth:`e1` by the reciprocal of its squared norm, then you get a
+ point in :meth:`K` that makes a unit inner product with
+ :meth:`e1`. We then get to choose the dual objective function
+ value such that the constraint involving :meth:`L` is satisfied.
+
+ Returns
+ -------
+
+ dict
+ A dictionary with two keys, ``'y'`` and ``'z'``, which
+ contain the vectors of the same name in the CVXOPT dual
+ problem formulation.
+
+ The ``1``-by-``1`` vector ``y`` consists of the dual
+ objective function value. The last :meth:`dimension` entries
+ of the vector ``z`` contain the strategy (for player two)
+ that achieves it. The remaining entries of ``z`` are
+ essentially dummy variables, computed from the equality
+ constraint in the CVXOPT dual problem.
+
"""
q = self.e1() / (norm(self.e1()) ** 2)
dist = self.K().ball_radius(self.e2())
@@ -855,6 +897,19 @@ class SymmetricLinearGame:
A nonnegative real number; the largest singular value of
the matrix :meth:`L`.
+ Examples
+ --------
+
+ >>> from dunshire import *
+ >>> from dunshire.matrices import specnorm
+ >>> L = [[1,2],[3,4]]
+ >>> K = NonnegativeOrthant(2)
+ >>> e1 = [1,1]
+ >>> e2 = e1
+ >>> SLG = SymmetricLinearGame(L,K,e1,e2)
+ >>> specnorm(SLG.L()) == SLG._L_specnorm()
+ True
+
"""
if self._L_specnorm_value is None:
self._L_specnorm_value = specnorm(self.L())
@@ -863,19 +918,19 @@ class SymmetricLinearGame:
def tolerance_scale(self, solution):
r"""
- Return a scaling factor that should be applied to ``ABS_TOL``
+ Return a scaling factor that should be applied to :const:`ABS_TOL`
for this game.
- When performing certain comparisons, the default tolernace
- ``ABS_TOL`` may not be appropriate. For example, if we expect
- ``x`` and ``y`` to be within ``ABS_TOL`` of each other, than the
- inner product of ``L*x`` and ``y`` can be as far apart as the
- spectral norm of ``L`` times the sum of the norms of ``x`` and
- ``y``. Such a comparison is made in :meth:`solution`, and in
- many of our unit tests.
+ When performing certain comparisons, the default tolerance
+ :const:`ABS_TOL` may not be appropriate. For example, if we expect
+ ``x`` and ``y`` to be within :const:`ABS_TOL` of each other,
+ than the inner product of ``L*x`` and ``y`` can be as far apart
+ as the spectral norm of ``L`` times the sum of the norms of
+ ``x`` and ``y``. Such a comparison is made in :meth:`solution`,
+ and in many of our unit tests.
- The returned scaling factor found from the inner product mentioned
- above is
+ The returned scaling factor found from the inner product
+ mentioned above is
.. math::
@@ -906,9 +961,26 @@ class SymmetricLinearGame:
-------
float
- A scaling factor to be multiplied by ``ABS_TOL`` when
+ A scaling factor to be multiplied by :const:`ABS_TOL` when
making comparisons involving solutions of this game.
+ Examples
+ --------
+
+ The spectral norm of ``L`` in this case is around ``5.464``, and
+ the optimal strategies both have norm one, so we expect the
+ tolerance scale to be somewhere around ``2 * 5.464``, or
+ ``10.929``::
+
+ >>> from dunshire import *
+ >>> L = [[1,2],[3,4]]
+ >>> K = NonnegativeOrthant(2)
+ >>> e1 = [1,1]
+ >>> e2 = e1
+ >>> SLG = SymmetricLinearGame(L,K,e1,e2)
+ >>> SLG.tolerance_scale(SLG.solution())
+ 10.929...
+
"""
norm_p1_opt = norm(solution.player1_optimal())
norm_p2_opt = norm(solution.player2_optimal())
@@ -1134,12 +1206,14 @@ class SymmetricLinearGame:
# 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.
+ # check 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.
+ # close enough because otherwise CVXOPT might return "unknown"
+ # and give us two points in the cone that are nowhere near
+ # optimal. And in fact, we need to ensure that they're close
+ # for "optimal" results, too, because we need to know how
+ # lenient to be in our testing.
#
if abs(p1_value - p2_value) > self.tolerance_scale(soln)*ABS_TOL:
printing.options['dformat'] = DEBUG_FLOAT_FORMAT