X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=dunshire%2Fgames.py;h=0a473915716de7f4be4ce7d99cbc64d87c960795;hb=8371d92c42c7faded26d8ef327129ad6d8c72d73;hp=130176b63bf9a276541609ad70b25f2c0b7a7d79;hpb=13b585b28aaacb1d603c3ae41614bacf613afa14;p=dunshire.git diff --git a/dunshire/games.py b/dunshire/games.py index 130176b..0a47391 100644 --- a/dunshire/games.py +++ b/dunshire/games.py @@ -4,12 +4,13 @@ Symmetric linear games and their solutions. This module contains the main :class:`SymmetricLinearGame` class that knows how to solve a linear game. """ +from math import sqrt from cvxopt import matrix, printing, solvers -from .cones import CartesianProduct +from .cones import CartesianProduct, IceCream, NonnegativeOrthant from .errors import GameUnsolvableException, PoorScalingException from .matrices import (append_col, append_row, condition_number, identity, - inner_product) + inner_product, norm, specnorm) from . import options printing.options['dformat'] = options.FLOAT_FORMAT @@ -23,7 +24,7 @@ class Solution: -------- >>> print(Solution(10, matrix([1,2]), matrix([3,4]))) - Game value: 10.0000000 + Game value: 10.000... Player 1 optimal: [ 1] [ 2] @@ -335,12 +336,12 @@ class SymmetricLinearGame: ' 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()) @@ -581,7 +582,7 @@ class SymmetricLinearGame: return matrix(0, (self.dimension(), 1), tc='d') - def _A(self): + def A(self): """ Return the matrix ``A`` used in our CVXOPT construction. @@ -609,12 +610,12 @@ class SymmetricLinearGame: >>> 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] """ - return matrix([0, self._e2], (1, self.dimension() + 1), 'd') + return matrix([0, self.e2()], (1, self.dimension() + 1), 'd') @@ -657,7 +658,7 @@ class SymmetricLinearGame: """ identity_matrix = identity(self.dimension()) return append_row(append_col(self._zero(), -identity_matrix), - append_col(self._e1, -self._L)) + append_col(self.e1(), -self.L())) def _c(self): @@ -698,7 +699,7 @@ class SymmetricLinearGame: return matrix([-1, self._zero()]) - def _C(self): + def C(self): """ Return the cone ``C`` used in our CVXOPT construction. @@ -720,7 +721,7 @@ class SymmetricLinearGame: >>> 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 @@ -770,7 +771,7 @@ class SymmetricLinearGame: @staticmethod - def _b(): + def b(): """ Return the ``b`` vector used in our CVXOPT construction. @@ -801,7 +802,7 @@ class SymmetricLinearGame: >>> e1 = [1,2,3] >>> e2 = [1,1,1] >>> SLG = SymmetricLinearGame(L, K, e1, e2) - >>> print(SLG._b()) + >>> print(SLG.b()) [1.0000000] @@ -809,6 +810,72 @@ class SymmetricLinearGame: return matrix([1], tc='d') + def player1_start(self): + """ + 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. + """ + p = self.e2() / (norm(self.e2()) ** 2) + + # Compute the distance from p to the outside of K. + if isinstance(self.K(), NonnegativeOrthant): + # How far is it to a wall? + dist = min(list(self.e1())) + elif isinstance(self.K(), IceCream): + # How far is it to the boundary of the ball that defines + # the ice-cream cone at a given height? Now draw a + # 45-45-90 triangle and the shortest distance to the + # outside of the cone should be 1/sqrt(2) of that. + # It works in R^2, so it works everywhere, right? + height = self.e1()[0] + radius = norm(self.e1()[1:]) + dist = (height - radius) / sqrt(2) + else: + raise NotImplementedError + + nu = - specnorm(self.L())/(dist*norm(self.e2())) + x = matrix([nu,p], (self.dimension() + 1, 1)) + s = - self._G()*x + + return {'x': x, 's': s} + + + def player2_start(self): + """ + Return a feasible starting point for player two. + """ + q = self.e1() / (norm(self.e1()) ** 2) + + # Compute the distance from p to the outside of K. + if isinstance(self.K(), NonnegativeOrthant): + # How far is it to a wall? + dist = min(list(self.e2())) + elif isinstance(self.K(), IceCream): + # How far is it to the boundary of the ball that defines + # the ice-cream cone at a given height? Now draw a + # 45-45-90 triangle and the shortest distance to the + # outside of the cone should be 1/sqrt(2) of that. + # It works in R^2, so it works everywhere, right? + height = self.e2()[0] + radius = norm(self.e2()[1:]) + dist = (height - radius) / sqrt(2) + else: + raise NotImplementedError + + omega = specnorm(self.L())/(dist*norm(self.e1())) + y = matrix([omega]) + z2 = q + z1 = y*self.e2() - self.L().trans()*z2 + z = matrix([z1,z2], (self.dimension()*2, 1)) + + return {'y': y, 'z': z} + def solution(self): """ @@ -844,11 +911,11 @@ class SymmetricLinearGame: >>> e2 = [1,1,1] >>> SLG = SymmetricLinearGame(L, K, e1, e2) >>> print(SLG.solution()) - Game value: -6.1724138 + Game value: -6.172... Player 1 optimal: - [ 0.551...] - [-0.000...] - [ 0.448...] + [0.551...] + [0.000...] + [0.448...] Player 2 optimal: [0.448...] [0.000...] @@ -864,7 +931,7 @@ class SymmetricLinearGame: >>> e2 = [4,5,6] >>> SLG = SymmetricLinearGame(L, K, e1, e2) >>> print(SLG.solution()) - Game value: 0.0312500 + Game value: 0.031... Player 1 optimal: [0.031...] [0.062...] @@ -887,61 +954,60 @@ class SymmetricLinearGame: >>> SLG.solution().game_value() < -ABS_TOL True - Tests - ----- - 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 = [[-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...] + >>> 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: - opts = {'show_progress': options.VERBOSE} + opts = {'show_progress': False} soln_dict = solvers.conelp(self._c(), self._G(), self._h(), - self._C().cvxopt_dims(), - self._A(), - self._b(), + self.C().cvxopt_dims(), + self.A(), + self.b(), + primalstart=self.player1_start(), 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 @@ -966,6 +1032,7 @@ class SymmetricLinearGame: # 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 @@ -979,11 +1046,13 @@ class SymmetricLinearGame: # 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) # For the game value, we could use any of: @@ -997,7 +1066,7 @@ class SymmetricLinearGame: # 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) + payoff = self.payoff(p1_optimal, p2_optimal) return Solution(payoff, p1_optimal, p2_optimal) @@ -1035,7 +1104,7 @@ class SymmetricLinearGame: True """ - return (condition_number(self._G()) + condition_number(self._A()))/2 + return (condition_number(self._G()) + condition_number(self.A()))/2 def dual(self): @@ -1071,10 +1140,10 @@ class SymmetricLinearGame: 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())