Add some more docs for the player[12]_start() methods.

diff --git a/dunshire/games.py b/dunshire/games.py
index 75f5329bb973e05d9070609fcb77e5eeba80fdee..6ab420d1b0c26cc6de561caf85629bae28f0462a 100644 (file)
--- a/dunshire/games.py
+++ b/dunshire/games.py
@@ -809,11 +809,26 @@ 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 +842,29 @@ 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())