diff --git a/source/rst/robustness.rst b/source/rst/robustness.rst index 3f28d2d..b816079 100644 --- a/source/rst/robustness.rst +++ b/source/rst/robustness.rst @@ -100,9 +100,9 @@ Here * *Entropy* is a non-negative number that measures the size of a set of models surrounding the decision-maker's approximating model. - * Entropy is zero when the set includes only the approximating model, indicating that the decision-maker completely trusts the approximating model. + * Entropy is zero when the set includes only the approximating model, indicating that the decision-maker completely trusts the approximating model. - * Entropy is bigger, and the set of surrounding models is bigger, the less the decision-maker trusts the approximating model. + * Entropy is bigger, and the set of surrounding models is bigger, the less the decision-maker trusts the approximating model. The shaded region indicates that for **all** models having entropy less than or equal to the number on the horizontal axis, the value obtained will be somewhere within the indicated set of values. @@ -132,9 +132,9 @@ Below we'll explain in detail how to construct these sets of values for a given Here is a hint about the *secret weapons* we'll use to construct these sets - * We'll use some min problems to construct the lower bounds +* We'll use some min problems to construct the lower bounds - * We'll use some max problems to construct the upper bounds +* We'll use some max problems to construct the upper bounds We will also describe how to choose :math:`F` to shape the sets of values. @@ -540,15 +540,16 @@ In particular, the lower bound on the left side of :eq:`rob_bound` is attained w To construct the *lower bound* on the set of values associated with all perturbations :math:`{\mathbf w}` satisfying the entropy constraint :eq:`rb_dec` at a given entropy level, we proceed as follows: - * For a given :math:`\theta`, solve the minimization problem :eq:`rb_a2o`. +* For a given :math:`\theta`, solve the minimization problem :eq:`rb_a2o`. - * Compute the minimizer :math:`R_\theta(x_0, F)` and the associated entropy using :eq:`rb_pdt22`. +* Compute the minimizer :math:`R_\theta(x_0, F)` and the associated entropy using :eq:`rb_pdt22`. - * Compute the lower bound on the value function :math:`R_\theta(x_0, F) - \theta \ {\rm ent}` and plot it against :math:`{\rm ent}`. +* Compute the lower bound on the value function :math:`R_\theta(x_0, F) - \theta \ {\rm ent}` and plot it against :math:`{\rm ent}`. - * Repeat the preceding three steps for a range of values of :math:`\theta` to trace out the lower bound. +* Repeat the preceding three steps for a range of values of :math:`\theta` to trace out the lower bound. .. note:: + This procedure sweeps out a set of separating hyperplanes indexed by different values for the Lagrange multiplier :math:`\theta`. The Upper Bound @@ -608,13 +609,13 @@ The upper bound on the left side of :eq:`robboundmax` is attained when To construct the *upper bound* on the set of values associated all perturbations :math:`{\mathbf w}` with a given entropy we proceed much as we did for the lower bound - * For a given :math:`\tilde \theta`, solve the maximization problem :eq:`rba2omax`. +* For a given :math:`\tilde \theta`, solve the maximization problem :eq:`rba2omax`. - * Compute the maximizer :math:`V_{\tilde \theta}(x_0, F)` and the associated entropy using :eq:`rbpdt223`. +* Compute the maximizer :math:`V_{\tilde \theta}(x_0, F)` and the associated entropy using :eq:`rbpdt223`. - * Compute the upper bound on the value function :math:`V_{\tilde \theta}(x_0, F) + \tilde \theta \ {\rm ent}` and plot it against :math:`{\rm ent}`. +* Compute the upper bound on the value function :math:`V_{\tilde \theta}(x_0, F) + \tilde \theta \ {\rm ent}` and plot it against :math:`{\rm ent}`. - * Repeat the preceding three steps for a range of values of :math:`\tilde \theta` to trace out the upper bound. +* Repeat the preceding three steps for a range of values of :math:`\tilde \theta` to trace out the upper bound. Reshaping the Set of Values @@ -922,9 +923,9 @@ Here is a brief description of the methods of the class * ``robust_rule()`` and ``robust_rule_simple()`` both solve for the triple :math:`\hat F, \hat K, \hat P`, as described in equations :eq:`rb_oc_ih` -- :eq:`rb_kd` and the surrounding discussion - * ``robust_rule()`` is more efficient + * ``robust_rule()`` is more efficient - * ``robust_rule_simple()`` is more transparent and easier to follow + * ``robust_rule_simple()`` is more transparent and easier to follow * ``K_to_F()`` and ``F_to_K()`` solve the decision problems of :ref:`agent 1 ` and :ref:`agent 2 ` respectively