Add exercises to docs and clean up pdf output.

.gitignore

@@ -11,3 +11,10 @@ build/
 env/
 .tox/
 *.DS_Store
+*.aux
+*.log
+*.fdb_latexmk
+*.fls
+*.xdv
+env*
+

docs/_static/discussion/extensive-form-game-example-with-perfect-information_one_step_backwards_induction/README.md

@@ -0,0 +1,8 @@
+Source files for an image used in the description of extensive form games.
+
+Compilation instructions:
+
+```
+$ latexmk --xelatex main.tex
+$ convert -density 300 main.pdf -quality 90 main.png
+```

docs/_static/discussion/extensive-form-game-example-with-perfect-information_one_step_backwards_induction/main.tex

@@ -0,0 +1,29 @@
+\documentclass{standalone}
+
+\usepackage{tikz}
+\usetikzlibrary{calc}
+
+\begin{document}
+
+\begin{tikzpicture}
+    \node [draw, color=red] (A1) at (0, 0) {\(A_1\)};
+        \node [draw, color=blue] (B1) at ($(A1) + (-2.5, -2)$) {\(B_1\)};
+            \node [draw] (O1) at ($(B1) + (-1, -2)$) {\tiny{\(O_1\)}};
+            \node (SS) at ($(O1) + (0, -.5)$) {\((4, 1)\)};
+            \node [draw] (O2) at ($(B1) + (1, -2)$) {\tiny{\(O_2\)}};
+            \node (SC) at ($(O2) + (0, -.5)$) {\((3, 5)\)};
+        \node [draw, color=blue] (B2) at ($(A1) + (2.5, -2)$) {\(B_2\)};
+                \node [draw] (O3) at ($(B2) + (-1, -2)$) {\tiny{\(O_3\)}};
+                \node (CS) at ($(O3) + (0, -.5)$) {\((4, 2)\)};
+            \node [draw] (O5) at ($(B2) + (1, -2)$) {\tiny{\(O_5\)}};
+            \node (CC) at ($(O5) + (0, -.5)$) {\((2, 0)\)};
+
+    \draw (A1) -- node[left=3mm] {X} (B1);
+    \draw (A1) -- node[right=3mm] {W} (B2);
+        \draw (B1)  -- node[left=3mm] {a} (O1);
+        \draw (B1) -- node[right=3mm] {b} (O2);
+        \draw (B2) -- node[left=3mm] {c} (O3);
+        \draw (B2) -- node[right=3mm] {d} (O5);
+\end{tikzpicture}
+
+\end{document}

docs/_static/discussion/extensive-form-game-example-with-perfect-information_three_steps_backwards_induction/README.md

@@ -0,0 +1,8 @@
+Source files for an image used in the description of extensive form games.
+
+Compilation instructions:
+
+```
+$ latexmk --xelatex main.tex
+$ convert -density 300 main.pdf -quality 90 main.png
+```

docs/_static/discussion/extensive-form-game-example-with-perfect-information_three_steps_backwards_induction/main.tex

@@ -0,0 +1,19 @@
+\documentclass{standalone}
+
+\usepackage{tikz}
+\usetikzlibrary{calc}
+
+\begin{document}
+
+\begin{tikzpicture}
+    \node [draw, color=red] (A1) at (0, 0) {\(A_1\)};
+            \node [draw] (O2) at ($(A1) + (-2.5, -2)$) {\tiny{\(O_2\)}};
+            \node (SC) at ($(O2) + (0, -.5)$) {\((3, 5)\)};
+                \node [draw] (O3) at ($(A1) + (2.5, -2)$) {\tiny{\(O_3\)}};
+                \node (CS) at ($(O3) + (0, -.5)$) {\((4, 2)\)};
+
+    \draw (A1) -- node[left=3mm] {X} (O2);
+    \draw (A1) -- node[right=3mm] {W} (O3);
+\end{tikzpicture}
+
+\end{document}

docs/_static/discussion/extensive-form-game-example-with-perfect-information_two_steps_backwards_induction/README.md

@@ -0,0 +1,8 @@
+Source files for an image used in the description of extensive form games.
+
+Compilation instructions:
+
+```
+$ latexmk --xelatex main.tex
+$ convert -density 300 main.pdf -quality 90 main.png
+```

docs/_static/discussion/extensive-form-game-example-with-perfect-information_two_steps_backwards_induction/main.tex

@@ -0,0 +1,24 @@
+\documentclass{standalone}
+
+\usepackage{tikz}
+\usetikzlibrary{calc}
+
+\begin{document}
+
+\begin{tikzpicture}
+    \node [draw, color=red] (A1) at (0, 0) {\(A_1\)};
+        \node [draw, color=blue] (B1) at ($(A1) + (-2.5, -2)$) {\(B_1\)};
+            \node [draw] (O1) at ($(B1) + (-1, -2)$) {\tiny{\(O_1\)}};
+            \node (SS) at ($(O1) + (0, -.5)$) {\((4, 1)\)};
+            \node [draw] (O2) at ($(B1) + (1, -2)$) {\tiny{\(O_2\)}};
+            \node (SC) at ($(O2) + (0, -.5)$) {\((3, 5)\)};
+                \node [draw] (O3) at ($(A1) + (2.5, -2)$) {\tiny{\(O_3\)}};
+                \node (CS) at ($(O3) + (0, -.5)$) {\((4, 2)\)};
+
+    \draw (A1) -- node[left=3mm] {X} (B1);
+    \draw (A1) -- node[right=3mm] {W} (O3);
+        \draw (B1)  -- node[left=3mm] {a} (O1);
+        \draw (B1) -- node[right=3mm] {b} (O2);
+\end{tikzpicture}
+
+\end{document}

docs/_static/discussion/extensive-form-game-exercise-four/main.tex

@@ -0,0 +1,19 @@
+\documentclass[tikz]{standalone}
+\usepackage{tikz,amsmath}
+\begin{document}
+\tikzstyle{level 1}=[level distance=3.5cm, sibling distance=3.5cm]
+\tikzstyle{level 2}=[level distance=3.5cm, sibling distance=2cm]
+\tikzstyle{player} = [text width=5em, draw, text centered, rectangle, fill=blue!20, inner sep=1pt]
+\tikzstyle{nature} = [minimum width=3pt,circle,  draw, fill=red!20, inner sep=1pt]
+\tikzstyle{end} = [circle, minimum width=3pt, fill, inner sep=0pt, right]
+\begin{tikzpicture}[grow=right, sloped]
+    \node[player]{a: 1}
+        child  {node (C2)  [player] {b: 1}
+                child {node[end] {} node[right] {$(3,2)$} edge from parent node[below] {A}}
+                child {node[end] {} node[right] {$(1,1)$} edge from parent node[below] {B}} edge from parent node[below] {A}}
+        child {node (1stC) (C1) [player] {c: 2}
+                child {node[end] {} node[right] {$(0,0)$} edge from parent node[below] {A}}
+                child {node[end] {} node[right] {$(4,0)$} edge from parent node[below] {B}}
+                child {node[end] {} node[right] {$(2,3)$} edge from parent node[below] {C}} edge from parent node[below] {B}};
+\end{tikzpicture}
+\end{document}

docs/_static/discussion/extensive-form-game-exercise-one/main.tex

@@ -0,0 +1,29 @@
+\documentclass[tikz]{standalone}
+\usepackage{tikz,amsmath}
+\begin{document}
+\tikzstyle{level 1}=[level distance=3.5cm, sibling distance=3.5cm]
+\tikzstyle{level 2}=[level distance=3.5cm, sibling distance=2cm]
+\tikzstyle{player} = [text width=5em, draw, text centered, rectangle, fill=blue!20, inner sep=1pt]
+\tikzstyle{nature} = [minimum width=3pt,circle,  draw, fill=red!20, inner sep=1pt]
+\tikzstyle{end} = [circle, minimum width=3pt, fill, inner sep=0pt, right]
+\begin{tikzpicture}[grow=right, sloped]
+    \node[player]{a: 1}
+        child {node[player] {b: 2}
+            child {node[player] {c: 1}
+                child {node[player] {d: 2}
+                    child {node[player] {e: 1}
+                        child {node[end] {} node [right] {$(2,17)$} edge from parent node[below] {W}}
+                        child {node[end] {} node [right] {$(6,4)$} edge from parent node[below] {Y}}
+                    edge from parent node[below] {J}
+                    }
+                    child {node[end] {} node [right] {$(7,30)$} edge from parent node[below] {X}}
+                    edge from parent node[below] {F}}
+                child {node[end] {} node [right] {$(70,3)$} edge from parent node[below] {Z}}
+                edge from parent node[below] {D}
+            edge from parent node[below] {D}}
+            child {node[end] {} node [right] {$(1,6)$}
+            edge from parent node[below] {C}}
+        edge from parent node[below] {B}}
+        child {node[end] {} node [right] {$(3,2)$} edge from parent node[below] {A}};
+\end{tikzpicture}
+\end{document}

docs/_static/discussion/extensive-form-game-exercise-three/main.tex

@@ -0,0 +1,34 @@
+\documentclass[tikz]{standalone}
+\usepackage{tikz,amsmath}
+\begin{document}
+\tikzstyle{level 1}=[level distance=3.5cm, sibling distance=3.5cm]
+\tikzstyle{level 2}=[level distance=3.5cm, sibling distance=1cm]
+\tikzstyle{player} = [text width=5em, draw, text centered, rectangle, fill=blue!20, inner sep=1pt]
+\tikzstyle{nature} = [minimum width=3pt,circle,  draw, fill=red!20, inner sep=1pt]
+\tikzstyle{end} = [circle, minimum width=3pt, fill, inner sep=0pt, right]
+\begin{tikzpicture}[grow=right, sloped]
+    \node[player]{a: 1}
+        child  {node (C2)  [player] {b: 2}
+                child {node[end] {} node[right] {$(3,2,1)$} edge from parent node[below] {B}}
+                child {node (C3)[player] {e: 3}
+                    child {node [end] {} node[right] {$(60,1,53)$}
+                edge from parent node[below] {B}}
+                    child {node [end] {} node[right] {$(6,0,2)$}
+                edge from parent node[below] {A}}
+                edge from parent node[below] {A}
+        }
+        edge from parent node[below] {B}}
+        child {node (1stC) (C1) [player] {c: 2}
+                child {node (C4) [player] {d: 3}
+                    child {node [end] {} node[right] {$(20,50,80)$}
+                edge from parent node[below] {B}}
+                    child {node [end] {} node[right] {$(30,40,50)$}
+                edge from parent node[below] {A}}
+            edge from parent node[below] {B}
+                }
+                child {node[end] {} node[right] {$(4,0,3)$} edge from parent node[below] {A}}
+                 edge from parent node[below] {A}};
+        \draw [dashed] (C2) -- (C1);
+        \draw [dashed] (C3) -- (C4);
+\end{tikzpicture}
+\end{document}

docs/_static/discussion/extensive-form-game-exercise-two/main.tex

@@ -0,0 +1,31 @@
+\documentclass[tikz]{standalone}
+\usepackage{tikz,amsmath}
+\begin{document}
+\tikzstyle{level 1}=[level distance=3.5cm, sibling distance=3.5cm]
+\tikzstyle{level 2}=[level distance=3.5cm, sibling distance=2cm]
+\tikzstyle{level 3}=[level distance=3.5cm, sibling distance=1cm]
+\tikzstyle{player} = [text width=5em, draw, text centered, rectangle, fill=blue!20, inner sep=1pt]
+\tikzstyle{nature} = [minimum width=3pt,circle,  draw, fill=red!20, inner sep=1pt]
+\tikzstyle{end} = [circle, minimum width=3pt, fill, inner sep=0pt, right]
+\begin{tikzpicture}[grow=right, sloped]
+    \node[player]{a: 1}
+        child {node[player] {c: 2}
+                child {node[player] {g: 1}
+                    child{node[end] {} node[right]{$(3,2)$} edge from parent node[below]{K}}
+                    child{node [end] {} node[right]{$(6,7)$} edge from parent node[below]{L}}edge from parent node[below] {B}}
+                child {node[player] {f: 1}
+                    child{node[end] {} node[right]{$(1,3)$} edge from parent node[below]{K}}
+                    child{node [end] {} node[right]{$(5,10)$} edge from parent node[below]{L}}edge from parent node[below] {A}}
+                edge from parent node[below] {W}}
+        child {node[player] {b: 2}
+                child {node[player] {e: 1}
+                    child {node[end] {} node [right] {$(2,3)$} edge from parent node[below] {Y}}
+                    child {node[end] {} node [right] {$(1,2)$} edge from parent node[below] {Z}}
+                        edge from parent node[below] {C}}
+                child {node [player] {d: 1}
+                    child{node[end] {} node [right] {$(10,1)$} edge from parent node[below] {M}}
+                    child{node[end] {} node [right] {$(5,2)$} edge from parent node[below] {N}}
+                edge from parent node[below] {D}}
+edge from parent node[below] {Y}};
+\end{tikzpicture}
+\end{document}

docs/text-book/degenerate-games.rst → docs/how-to/handle-degenerate-games.rst

@@ -1,23 +1,7 @@
 .. _degenerate-games-discussion:
 
-Degenerate games
-================
-
-A two player game is called nondegenerate if no mixed strategy of support size
-:math:`k` has more than :math:`k` pure best responses.
-
-For example, the zero sum game defined by the following matrix is degenerate:
-
-.. math::
-
-   A =
-   \begin{pmatrix}
-        0 & -1 &  1\\
-       -1 &  0 &  1\\
-       -1 &  1 &  0
-   \end{pmatrix}
-
-The third column has two pure best responses.
+Handle Degenerate games
+=======================
 
 When dealing with *degenerate* games unexpected results can occur::
 

docs/how-to/index.rst

@@ -10,6 +10,7 @@ How to:
    create-a-game.rst
    calculate-utilities.rst
    check-best-responses.rst
+   handle-degenerate-games.rst
    use-minimax.rst
    solve-with-support-enumeration.rst
    solve-with-vertex-enumeration.rst

docs/text-book/best-responses.rst

@@ -100,6 +100,33 @@ response to a row players' strategy :math:`\sigma_r` if and only if:
 
       \text{argmax}_{i\in\mathcal{A}_1}A_{ij}=2\text{ for all }j\in\mathcal{A}_2
 
+.. _definition_of_a_degenerate_game:
+
+Definition of a degenerate game
+-------------------------------
+
+A two player game is called non degenerate if no mixed strategy of support size
+:math:`k` has more than :math:`k` pure best responses.
+
+.. admonition:: Question
+   :class: note
+
+   Show that the following game is degenerate:
+
+   .. math::
+
+      A =
+      \begin{pmatrix}
+           0 & -1 &  1\\
+          -1 &  0 &  1\\
+          -1 &  1 &  0
+      \end{pmatrix}
+
+.. admonition:: Answer
+   :class: caution, dropdown
+
+   The third column has two pure best responses.
+
 .. _best_responses_in_2_by_2_games:
 
 Generic best responses in 2 by 2 games
@@ -325,6 +352,94 @@ In a two player game :math:`(A, B)\in {\mathbb{R}^{m \times n}} ^ 2`,
 response to :math:`\sigma_c` and :math:`\sigma_c` is a best response to
 :math:`\sigma_r`.
 
+Exercises
+---------
+
+
+
+1. For the following games identify the best responses:
+
+   1. 
+
+   .. math:: 
+
+      A = 
+      \begin{pmatrix}
+      2 & 1\\
+      1 & 1\end{pmatrix}
+      \qquad
+      B =
+      \begin{pmatrix}
+         1 & 1\\
+         1 & 3\end{pmatrix}
+
+   2. 
+
+   .. math::
+
+      A = 
+      \begin{pmatrix}
+      2  & 1 & 3 & 17\\
+      27 & 3 & 1 & 1\\
+      4  & 6 & 7 & 18
+      \end{pmatrix}
+      \qquad
+      B =
+      \begin{pmatrix}
+      11 & 9  & 10 & 22\\
+      0  & 1  & 1  & 0\\
+      2  & 10 & 12 & 0
+      \end{pmatrix}
+
+   3. 
+
+   .. math:: 
+
+      A =
+      \begin{pmatrix}
+      3  & 3 & 2 \\
+      2  & 1 & 3 
+      \end{pmatrix}
+      \qquad
+      B =
+      \begin{pmatrix}
+      2  & 1 & 3 \\
+      2  & 3 & 2 
+      \end{pmatrix}
+
+   4. 
+
+   .. math::
+
+      A =
+      \begin{pmatrix}
+      3 & -1\\
+      2 & 7\end{pmatrix}
+      \qquad
+      B =
+      \begin{pmatrix}
+      -3 & 1\\
+      1 & -6\end{pmatrix}
+
+2. Represent the following game in normal form:
+
+      Assume two neighbouring countries have at their disposal very
+      destructive armies. If both countries attack each other the
+      countries’ civilian population will suffer 10 thousand casualties.
+      If one country attacks whilst the other remains peaceful, the
+      peaceful country will lose 15 thousand casualties but would also
+      retaliate causing the offensive country 13 thousand casualties. If
+      both countries remain peaceful then there are no casualties.
+
+   1. Clearly state the players and strategy sets.
+   2. Plot the utilities to both countries assuming that they play a
+      mixed strategy while the other country remains peaceful.
+   3. Plot the utilities to both countries assuming that they play a
+      mixed strategy while the other country attacks.
+   4. Obtain the best responses of each player.
+
+3. Construct a degenerate :math:`3\times 3` game.
+
 Using Nashpy
 ------------