diff --git a/doc/conf.py b/doc/conf.py
index 07a6759..4e0c586 100644
--- a/doc/conf.py
+++ b/doc/conf.py
@@ -5,6 +5,8 @@
 extensions = [
     "sphinx.ext.autodoc",
     "sphinx.ext.autosummary",
+    "sphinx.ext.doctest",
+    "sphinx.ext.mathjax",
     "sphinx.ext.napoleon",
     "sphinx.ext.viewcode",
 ]
@@ -30,4 +32,5 @@
             "type": "fontawesome",
         }
     ],
+    "shorten_urls": False,
 }
diff --git a/doc/index.rst b/doc/index.rst
index 5720933..597118f 100644
--- a/doc/index.rst
+++ b/doc/index.rst
@@ -3,15 +3,139 @@ lemke
 
 **lemke** is a pure Python implementation of Lemke's algorithm for linear
 complementarity problems (LCPs), including its application to computing a
-Nash equilibrium of a bimatrix game.
+Nash equilibrium of a bimatrix game. It uses exact rational arithmetic
+and is built for teaching and research, where you want to see each pivot.
+It can be used as both a callable library and a CLI tool.
+
+lemke is part of the `Gambit project <https://www.gambit-project.org>`_ for
+computation in game theory, and it builds on the work of
+`Bernhard von Stengel <http://www.maths.lse.ac.uk/personal/stengel/>`_.
+
+
+Features
+--------
+- Lemke's algorithm for linear complementarity problems,
+  with a covering vector :math:`d`;
+- Lexicographic minimum ratio test to handle degenerate problems;
+- Arbitrary precision with integer pivoting;
+- Lemke–Howson and the tracing procedure for bimatrix games;
+- Verbose tableau output that shows every pivot;
+- Command-line tools: ``lemke``, ``bimatrix`` and ``randomstart``.
+
+
+Usage examples
+--------------
+
+Here are two quick examples of using lemke.
+You can view more detailed explanations in the :doc:`tutorials/index`.
+
+1) Solve a 2×2 LCP in Python
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+Consider LCP given in ``examples/lcp_2x2``: find :math:`z \ge 0`
+with :math:`w = q + Mz \ge 0` and :math:`z^\top w = 0`, where
+
+.. math::
+
+   M = \begin{pmatrix} \frac{1}{2} & 1 \\ -1 & \frac{5}{2} \end{pmatrix},
+   \qquad
+   q = \begin{pmatrix} -1 \\ -1 \end{pmatrix}.
+
+.. doctest::
+
+   >>> from fractions import Fraction as F
+   >>> from lemke.lemke import lcp, runlemke
+   >>> problem = lcp(
+   ...    M=[
+   ...       [F(1, 2), F(1)],
+   ...       [F(-1), F(5, 2)],
+   ...    ],
+   ...    q=[F(-1), F(-1)],
+   ...    d=[F(1), F(1)],
+   ... )
+   >>> result = runlemke(lcp=problem)
+   >>> result.success
+   True
+   >>> result.z
+   (Fraction(2, 3), Fraction(2, 3))
+
+The solution is exact: :math:`z = (\frac{2}{3}, \frac{2}{3})`, with :math:`w = 0`.
+
+
+2) Find a Nash equilibrium from the CLI
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+Find Nash equilibria with the Lemke–Howson method for the bimatrix game
+in ``examples/game``, where the payoff matrices are
+
+.. math::
+
+   A = \begin{pmatrix} 3 & 0 \\ 0 & 2 \end{pmatrix},
+   \qquad
+   B = \begin{pmatrix} 2 & 0 \\ 0 & 7 \end{pmatrix}.
+
+Save the following as ``game_file``::
+
+   # m,n=
+   2 2
+
+   # A=
+   3 0
+   0 2
+
+   # B=
+   2 0
+   0 7
+
+Then run:
+
+.. code-block:: console
+
+   $ bimatrix lh game_file
+
+The algorithm runs for all labels by default and finds 2 equilibria:
+
+.. code-block:: text
+
+    label 1 found eq (1,0),(1,0)
+        supports: [0][0]
+    label 2 found eq (0,1),(0,1)
+        supports: [1][1]
+    -------- equilibria found: --------
+    (1,0),(1,0)
+        supports: [0][0] found by labels [1, 3]
+    (0,1),(0,1)
+        supports: [1][1] found by labels [2, 4]
+
 
 Installation
 ------------
 
+lemke needs Python 3.10 or later.
+
+It is **not** yet available on PyPI, but can be installed directly from GitHub:
+
 .. code-block:: console
 
    pip install git+https://github.com/gambitproject/lemke.git
 
+
+Citing lemke
+------------
+
+lemke is Free/Open Source software, released under the terms of
+the GNU General Public Licence, version 3 (GPL-3.0).
+
+If you use lemke in a scientific publication, you can use the following citation:
+
+   Von Stengel, B. (2026). *lemke: Lemke's algorithm for linear complementarity
+   problems and bimatrix games*.
+   https://github.com/gambitproject/lemke.
+
+
+Useful links
+------------
+
 .. toctree::
    :maxdepth: 2
 
diff --git a/examples/lcp_2x2 b/examples/lcp_2x2
new file mode 100644
index 0000000..a7c76a5
--- /dev/null
+++ b/examples/lcp_2x2
@@ -0,0 +1,5 @@
+n= 2
+M= 1/2 1
+  -1 2.5
+q= -1 -1
+d= 1 1