X-Git-Url: https://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=src%2Fdunshire%2Fgames.py;h=43fa007c61077c90ef627e9738afbaa09dcde9c1;hb=fa8fa4d690c5f30f7d5fee1818a9b4c15f52c5ff;hp=e2d29788f2b2612fe0d23e12be07010902c93451;hpb=ff576f2eddae5554e4888d99b092d73e19619a91;p=dunshire.git diff --git a/src/dunshire/games.py b/src/dunshire/games.py index e2d2978..43fa007 100644 --- a/src/dunshire/games.py +++ b/src/dunshire/games.py @@ -5,17 +5,11 @@ This module contains the main :class:`SymmetricLinearGame` class that knows how to solve a linear game. """ -# These few are used only for tests. -from math import sqrt -from random import randint, uniform -from unittest import TestCase - -# These are mostly actually needed. from cvxopt import matrix, printing, solvers -from cones import CartesianProduct, IceCream, NonnegativeOrthant -from errors import GameUnsolvableException -from matrices import append_col, append_row, identity, inner_product -import options +from .cones import CartesianProduct +from .errors import GameUnsolvableException +from .matrices import append_col, append_row, identity +from . import options printing.options['dformat'] = options.FLOAT_FORMAT solvers.options['show_progress'] = options.VERBOSE @@ -122,27 +116,95 @@ class Solution: class SymmetricLinearGame: - """ + r""" A representation of a symmetric linear game. - The data for a linear game are, + The data for a symmetric linear game are, * A "payoff" operator ``L``. - * A cone ``K``. - * A point ``e`` in the interior of ``K``. - * A point ``f`` in the interior of the dual of ``K``. - - In a symmetric game, the cone ``K`` is be self-dual. We therefore - name the two interior points ``e1`` and ``e2`` to indicate that - they come from the same cone but are "chosen" by players one and - two respectively. + * A symmetric cone ``K``. + * Two points ``e1`` and ``e2`` in the interior of ``K``. The ambient space is assumed to be the span of ``K``. + With those data understood, the game is played as follows. Players + one and two choose points :math:`x` and :math:`y` respectively, from + their respective strategy sets, + + .. math:: + \begin{aligned} + \Delta_{1} + &= + \left\{ + x \in K \ \middle|\ \left\langle x, e_{2} \right\rangle = 1 + \right\}\\ + \Delta_{2} + &= + \left\{ + y \in K \ \middle|\ \left\langle y, e_{1} \right\rangle = 1 + \right\}. + \end{aligned} + + Afterwards, a "payout" is computed as :math:`\left\langle + L\left(x\right), y \right\rangle` and is paid to player one out of + player two's pocket. The game is therefore zero sum, and we suppose + that player one would like to guarantee himself the largest minimum + payout possible. That is, player one wishes to, + + .. math:: + \begin{aligned} + \text{maximize } + &\underset{y \in \Delta_{2}}{\min}\left( + \left\langle L\left(x\right), y \right\rangle + \right)\\ + \text{subject to } & x \in \Delta_{1}. + \end{aligned} + + Player two has the simultaneous goal to, + + .. math:: + \begin{aligned} + \text{minimize } + &\underset{x \in \Delta_{1}}{\max}\left( + \left\langle L\left(x\right), y \right\rangle + \right)\\ + \text{subject to } & y \in \Delta_{2}. + \end{aligned} + + These goals obviously conflict (the game is zero sum), but an + existence theorem guarantees at least one optimal min-max solution + from which neither player would like to deviate. This class is + able to find such a solution. + + Parameters + ---------- + + 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` + The symmetric cone instance over which the game is played. + + e1 : iterable float + The interior point of ``K`` belonging to player one; it + can be of any iterable type having the correct length. + + e2 : iterable float + The interior point of ``K`` belonging to player two; it + can be of any enumerable type having the correct length. + + Raises + ------ + + ValueError + If either ``e1`` or ``e2`` lie outside of the cone ``K``. + Examples -------- - >>> from cones import NonnegativeOrthant + >>> from dunshire import * >>> K = NonnegativeOrthant(3) >>> L = [[1,-5,-15],[-1,2,-3],[-12,-15,1]] >>> e1 = [1,1,1] @@ -161,10 +223,9 @@ class SymmetricLinearGame: [ 2] [ 3]. - Lists can (and probably should) be used for every argument:: - >>> from cones import NonnegativeOrthant + >>> from dunshire import * >>> K = NonnegativeOrthant(2) >>> L = [[1,0],[0,1]] >>> e1 = [1,1] @@ -186,7 +247,7 @@ class SymmetricLinearGame: >>> import cvxopt >>> import numpy - >>> from cones import NonnegativeOrthant + >>> from dunshire import * >>> K = NonnegativeOrthant(2) >>> L = [[1,0],[0,1]] >>> e1 = cvxopt.matrix([1,1]) @@ -207,7 +268,7 @@ class SymmetricLinearGame: otherwise indexed by columns:: >>> import cvxopt - >>> from cones import NonnegativeOrthant + >>> from dunshire import * >>> K = NonnegativeOrthant(2) >>> L = [[1,2],[3,4]] >>> e1 = [1,1] @@ -242,23 +303,6 @@ class SymmetricLinearGame: def __init__(self, L, K, e1, e2): """ Create a new SymmetricLinearGame object. - - INPUT: - - - ``L`` -- an square matrix represented as a list of lists - of real numbers. ``L`` itself is interpreted as a list of - ROWS, which agrees with (for example) SageMath and NumPy, - but not with CVXOPT (whose matrix constructor accepts a - list of columns). - - - ``K`` -- a SymmetricCone instance. - - - ``e1`` -- the interior point of ``K`` belonging to player one; - it can be of any enumerable type having the correct length. - - - ``e2`` -- the interior point of ``K`` belonging to player two; - it can be of any enumerable type having the correct length. - """ self._K = K self._e1 = matrix(e1, (K.dimension(), 1)) @@ -269,10 +313,10 @@ class SymmetricLinearGame: # feeding it to CVXOPT. self._L = matrix(L, (K.dimension(), K.dimension())).trans() - if not K.contains_strict(self._e1): + if not self._e1 in K: raise ValueError('the point e1 must lie in the interior of K') - if not K.contains_strict(self._e2): + if not self._e2 in K: raise ValueError('the point e2 must lie in the interior of K') def __str__(self): @@ -292,16 +336,20 @@ class SymmetricLinearGame: def solution(self): """ - Solve this linear game and return a Solution object. + Solve this linear game and return a :class:`Solution`. + + Returns + ------- - OUTPUT: + :class:`Solution` + A :class:`Solution` object describing the game's value and + the optimal strategies of both players. - If the cone program associated with this game could be - successfully solved, then a Solution object containing the - game's value and optimal strategies is returned. If the game - could *not* be solved -- which should never happen -- then a - GameUnsolvableException is raised. It can be printed to get the - raw output from CVXOPT. + Raises + ------ + GameUnsolvableException + If the game could not be solved (if an optimal solution to its + associated cone program was not found). Examples -------- @@ -309,7 +357,7 @@ class SymmetricLinearGame: This example is computed in Gowda and Ravindran in the section "The value of a Z-transformation":: - >>> from cones import NonnegativeOrthant + >>> from dunshire import * >>> K = NonnegativeOrthant(3) >>> L = [[1,-5,-15],[-1,2,-3],[-12,-15,1]] >>> e1 = [1,1,1] @@ -329,7 +377,7 @@ class SymmetricLinearGame: The value of the following game can be computed using the fact that the identity is invertible:: - >>> from cones import NonnegativeOrthant + >>> from dunshire import * >>> K = NonnegativeOrthant(3) >>> L = [[1,0,0],[0,1,0],[0,0,1]] >>> e1 = [1,2,3] @@ -379,29 +427,54 @@ class SymmetricLinearGame: # what happened. soln_dict = solvers.conelp(c, G, h, C.cvxopt_dims(), A, b) + # 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. That should never happen, - # because by construction our game has a solution, and thus - # the cone program should too. - if soln_dict['status'] != 'optimal': + # 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(soln_dict) - - p1_value = soln_dict['x'][0] - p1_optimal = soln_dict['x'][1:] - p2_optimal = soln_dict['z'][self._K.dimension():] + 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). + if abs(p1_value - p2_value) > options.ABS_TOL: + raise GameUnsolvableException(soln_dict) + if (p1_optimal not in self._K) or (p2_optimal not in self._K): + raise GameUnsolvableException(soln_dict) return Solution(p1_value, p1_optimal, p2_optimal) + def dual(self): - """ + r""" Return the dual game to this game. + If :math:`G = \left(L,K,e_{1},e_{2}\right)` is a linear game, + then its dual is :math:`G^{*} = + \left(L^{*},K^{*},e_{2},e_{1}\right)`. However, since this cone + is symmetric, :math:`K^{*} = K`. + Examples -------- - >>> from cones import NonnegativeOrthant + >>> from dunshire import * >>> K = NonnegativeOrthant(3) >>> L = [[1,-5,-15],[-1,2,-3],[-12,-15,1]] >>> e1 = [1,1,1] @@ -428,72 +501,3 @@ class SymmetricLinearGame: self._K, self._e2, self._e1) - - -class SymmetricLinearGameTest(TestCase): - """ - Tests for the SymmetricLinearGame and Solution classes. - """ - - def assert_within_tol(self, first, second): - """ - Test that ``first`` and ``second`` are equal within our default - tolerance. - """ - self.assertTrue(abs(first - second) < options.ABS_TOL) - - - def assert_solution_exists(self, L, K, e1, e2): - """ - Given the parameters needed to construct a SymmetricLinearGame, - ensure that that game has a solution. - """ - G = SymmetricLinearGame(L, K, e1, e2) - soln = G.solution() - L_matrix = matrix(L).trans() - expected = inner_product(L_matrix*soln.player1_optimal(), - soln.player2_optimal()) - self.assert_within_tol(soln.game_value(), expected) - - def test_solution_exists_nonnegative_orthant(self): - """ - Every linear game has a solution, so we should be able to solve - every symmetric linear game over the NonnegativeOrthant. Pick - some parameters randomly and give it a shot. The resulting - optimal solutions should give us the optimal game value when we - apply the payoff operator to them. - """ - ambient_dim = randint(1, 10) - K = NonnegativeOrthant(ambient_dim) - e1 = [uniform(0.1, 10) for idx in range(K.dimension())] - e2 = [uniform(0.1, 10) for idx in range(K.dimension())] - L = [[uniform(-10, 10) for i in range(K.dimension())] - for j in range(K.dimension())] - self.assert_solution_exists(L, K, e1, e2) - - def test_solution_exists_ice_cream(self): - """ - Like :meth:`test_solution_exists_nonnegative_orthant`, except - over the ice cream cone. - """ - # Use a minimum dimension of two to avoid divide-by-zero in - # the fudge factor we make up later. - ambient_dim = randint(2, 10) - K = IceCream(ambient_dim) - e1 = [1] # Set the "height" of e1 to one - e2 = [1] # And the same for e2 - - # If we choose the rest of the components of e1,e2 randomly - # between 0 and 1, then the largest the squared norm of the - # non-height part of e1,e2 could be is the 1*(dim(K) - 1). We - # need to make it less than one (the height of the cone) so - # that the whole thing is in the cone. The norm of the - # non-height part is sqrt(dim(K) - 1), and we can divide by - # twice that. - fudge_factor = 1.0 / (2.0*sqrt(K.dimension() - 1.0)) - e1 += [fudge_factor*uniform(0, 1) for idx in range(K.dimension() - 1)] - e2 += [fudge_factor*uniform(0, 1) for idx in range(K.dimension() - 1)] - L = [[uniform(-10, 10) for i in range(K.dimension())] - for j in range(K.dimension())] - self.assert_solution_exists(L, K, e1, e2) -