.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/probability/classical/plot_02_st_petersburg_paradox.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_probability_classical_plot_02_st_petersburg_paradox.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_probability_classical_plot_02_st_petersburg_paradox.py:


The St. Petersburg paradox
========================================================

The game pays :math:`2^k` if the first head comes on toss :math:`k`. Its
expected payoff is infinite, so the running average of simulated
payoffs never settles; it keeps jumping upward whenever a rare long run
of tails occurs. Daniel Bernoulli's logarithmic utility values the game
at a small, finite amount.

.. GENERATED FROM PYTHON SOURCE LINES 13-18

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.probability import st_petersburg_certainty_equivalent, st_petersburg_payoffs








.. GENERATED FROM PYTHON SOURCE LINES 19-21

Running average payoff
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 21-33

.. code-block:: Python


    n = np.arange(1, 1000001)
    fig, ax = plt.subplots()
    for seed in range(4):
        payoffs = st_petersburg_payoffs(n.size, seed=seed)
        ax.semilogx(n, np.cumsum(payoffs) / n, lw=1)
    ax.semilogx(n, np.log2(n), "k--", label=r"$\log_2 n$ growth")
    ax.set_xlabel("games played")
    ax.set_ylabel("average payoff")
    ax.legend()
    ax.set_title("St. Petersburg game: the average never converges")




.. image-sg:: /api/gallery/probability/classical/images/sphx_glr_plot_02_st_petersburg_paradox_001.png
   :alt: St. Petersburg game: the average never converges
   :srcset: /api/gallery/probability/classical/images/sphx_glr_plot_02_st_petersburg_paradox_001.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    Text(0.5, 1.0, 'St. Petersburg game: the average never converges')



.. GENERATED FROM PYTHON SOURCE LINES 34-36

Bernoulli's certainty equivalent under log utility
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 36-39

.. code-block:: Python


    for wealth in (0.0, 10.0, 100.0, 1000.0, 1e6):
        print(f"wealth {wealth:>9.0f}: worth {st_petersburg_certainty_equivalent(wealth):.3f} ducats")




.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    wealth         0: worth 4.000 ducats
    wealth        10: worth 5.466 ducats
    wealth       100: worth 7.892 ducats
    wealth      1000: worth 10.969 ducats
    wealth   1000000: worth 20.874 ducats





.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.540 seconds)


.. _sphx_glr_download_api_gallery_probability_classical_plot_02_st_petersburg_paradox.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_02_st_petersburg_paradox.ipynb <plot_02_st_petersburg_paradox.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_02_st_petersburg_paradox.py <plot_02_st_petersburg_paradox.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_02_st_petersburg_paradox.zip <plot_02_st_petersburg_paradox.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_