.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/economics/auctions/plot_03_hanson_lmsr.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_economics_auctions_plot_03_hanson_lmsr.py>`
        to download the full example code or to run this example in your browser via JupyterLite.

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

.. _sphx_glr_api_gallery_economics_auctions_plot_03_hanson_lmsr.py:


Hanson's logarithmic market scoring rule (2003)
===============================================

A prediction market pays 1 for each share of the outcome that happens, so
a share's price reads as a probability. With few traders, an order book is
empty and nobody can trade. Hanson's automated market maker always quotes a
price: it keeps the shares :math:`q_i` sold of each outcome and charges each
trade the change in

.. math::

   C(q) = b \ln \sum_i e^{q_i / b}, \qquad p_i = \frac{e^{q_i/b}}{\sum_j e^{q_j/b}}.

The prices always sum to 1, and the market maker can lose at most
:math:`b \ln n`, the price it pays for the information traders bring. The
liquidity :math:`b` trades depth against that subsidy.

.. GENERATED FROM PYTHON SOURCE LINES 21-25

.. code-block:: Python

    import matplotlib.pyplot as plt

    import blockchainkit as bk








.. GENERATED FROM PYTHON SOURCE LINES 26-31

A trader who knows better
-------------------------

The market opens at even odds on two outcomes. A trader who believes
outcome 0 has a 90% chance buys until the price reaches 0.9.

.. GENERATED FROM PYTHON SOURCE LINES 31-52

.. code-block:: Python


    fig, ax = plt.subplots(figsize=(7, 4.5))
    for b, color in ((50, "#dc2626"), (200, "#2563eb")):
        quantities, shares, prices = [0.0, 0.0], [], []
        paid = 0.0
        while bk.economics.lmsr_prices(quantities, b)[0] < 0.9:
            paid += bk.economics.lmsr_trade(quantities, 0, 5, b)
            quantities[0] += 5
            shares.append(quantities[0])
            prices.append(bk.economics.lmsr_prices(quantities, b)[0])
        print(f"b = {b}: bought {quantities[0]:.0f} shares for {paid:.1f}")
        ax.plot(shares, prices, color=color, label=f"b = {b}")
        if b == 200:
            assert quantities[0] > 4 * 100  # Deeper market, more shares to move the price.

    ax.axhline(0.9, color="black", linestyle=":")
    ax.set(xlabel="shares of outcome 0 bought", ylabel="price of outcome 0")
    ax.set_title("The price follows the purchases")
    ax.legend()
    fig.tight_layout()




.. image-sg:: /api/gallery/economics/auctions/images/sphx_glr_plot_03_hanson_lmsr_001.png
   :alt: The price follows the purchases
   :srcset: /api/gallery/economics/auctions/images/sphx_glr_plot_03_hanson_lmsr_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    b = 50: bought 110 shares for 80.6
    b = 200: bought 440 shares for 322.4




.. GENERATED FROM PYTHON SOURCE LINES 53-55

The market maker's worst case
-----------------------------

.. GENERATED FROM PYTHON SOURCE LINES 55-74

.. code-block:: Python


    fig, ax = plt.subplots(figsize=(7, 4.5))
    for n in (2, 4, 8):
        losses = []
        for bought in range(0, 2_001, 100):
            quantities = [0.0] * n
            paid = bk.economics.lmsr_trade(quantities, 0, bought, 100)
            losses.append(bought - paid)  # If outcome 0 happens, the maker pays 1 per share.
        bound = bk.economics.lmsr_max_loss(n, 100)
        assert max(losses) <= bound
        ax.plot(range(0, 2_001, 100), losses, "o-", label=f"{n} outcomes")
        ax.axhline(bound, linestyle=":", color=ax.lines[-1].get_color())
    ax.set(xlabel="shares of the winning outcome sold", ylabel="market maker's loss (b = 100)")
    ax.set_title("The loss approaches b ln n and never exceeds it")
    ax.legend()
    fig.tight_layout()

    plt.show()




.. image-sg:: /api/gallery/economics/auctions/images/sphx_glr_plot_03_hanson_lmsr_002.png
   :alt: The loss approaches b ln n and never exceeds it
   :srcset: /api/gallery/economics/auctions/images/sphx_glr_plot_03_hanson_lmsr_002.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 75-81

Exercise
--------
Two traders disagree: one thinks outcome 0 has a 70% chance, the other
40%. Each trades until the price matches its belief. In which order do
they trade to the final price, and what does the market maker pay if
outcome 0 happens?


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

   **Total running time of the script:** (0 minutes 0.085 seconds)


.. _sphx_glr_download_api_gallery_economics_auctions_plot_03_hanson_lmsr.py:

.. only:: html

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

    .. container:: lite-badge

      .. image:: images/jupyterlite_badge_logo.svg
        :target: ../../../../lite/lab/index.html?path=api/gallery/economics/auctions/plot_03_hanson_lmsr.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_03_hanson_lmsr.ipynb <plot_03_hanson_lmsr.ipynb>`

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

      :download:`Download Python source code: plot_03_hanson_lmsr.py <plot_03_hanson_lmsr.py>`

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

      :download:`Download zipped: plot_03_hanson_lmsr.zip <plot_03_hanson_lmsr.zip>`


.. only:: html

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

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