.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/economics/auctions/plot_01_vickrey_second_price.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_01_vickrey_second_price.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_01_vickrey_second_price.py:


Vickrey's second-price auction: truthful bidding (1961)
=======================================================

In a sealed-bid *first-price* auction, the winner pays its own bid, so
every bidder shades its bid below its value, by an amount that depends on
guesses about the others. Vickrey proposed charging the winner the
*second-highest* bid instead. A bid then decides only whether one wins,
never what one pays, and bidding one's true value is a dominant strategy:

.. math::

   u_i(b_i) = \begin{cases} v_i - \max_{j \ne i} b_j & b_i > \max_{j \ne i} b_j,\\
   0 & \text{otherwise}, \end{cases}

is maximized by :math:`b_i = v_i` whatever the others bid. Blockchains use
the idea where honest bidding matters: fee mechanisms, block-builder
auctions, and on-chain auctions of names and collateral.

.. GENERATED FROM PYTHON SOURCE LINES 22-28

.. code-block:: Python

    from random import Random

    import matplotlib.pyplot as plt

    import blockchainkit as bk








.. GENERATED FROM PYTHON SOURCE LINES 29-31

One bidder's utility against its own bid
----------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 31-41

.. code-block:: Python


    VALUE = 60
    others = {"rival0": 35, "rival1": 48, "rival2": 22}
    bids = range(0, 101)
    second = [bk.economics.second_price_auction(others | {"me": b}).utility("me", VALUE) for b in bids]
    first = [bk.economics.first_price_auction(others | {"me": b}).utility("me", VALUE) for b in bids]
    assert max(second) == second[VALUE]  # The true value is a best bid.
    assert max(first) > first[VALUE] == 0  # Bidding the value earns nothing in a first-price auction.
    print("best first-price bid:", first.index(max(first)), "earning", max(first))





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

 .. code-block:: none

    best first-price bid: 48 earning 12.0




.. GENERATED FROM PYTHON SOURCE LINES 42-44

Truthfulness against every rival profile
----------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 44-67

.. code-block:: Python


    rng = Random(11)
    worst_gain = 0
    for _ in range(300):
        rivals = {f"r{i}": rng.randint(0, 100) for i in range(rng.randint(1, 5))}
        value = rng.randint(0, 100)
        truthful = bk.economics.second_price_auction(rivals | {"me": value}).utility("me", value)
        for deviation in range(0, 101, 5):
            result = bk.economics.second_price_auction(rivals | {"me": deviation})
            worst_gain = max(worst_gain, result.utility("me", value) - truthful)
    print("largest gain from lying over 300 random auctions:", worst_gain)
    assert worst_gain == 0

    fig, ax = plt.subplots(figsize=(7, 4.5))
    ax.plot(bids, second, color="#2563eb", label="second-price")
    ax.plot(bids, first, color="#dc2626", label="first-price")
    ax.axvline(VALUE, color="black", linestyle=":", label=f"true value {VALUE}")
    ax.set(xlabel="my bid", ylabel="my utility", title="Only the second-price auction rewards honesty")
    ax.legend()
    fig.tight_layout()

    plt.show()




.. image-sg:: /api/gallery/economics/auctions/images/sphx_glr_plot_01_vickrey_second_price_001.png
   :alt: Only the second-price auction rewards honesty
   :srcset: /api/gallery/economics/auctions/images/sphx_glr_plot_01_vickrey_second_price_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    largest gain from lying over 300 random auctions: 0




.. GENERATED FROM PYTHON SOURCE LINES 68-73

Exercise
--------
Two bidders could collude in a second-price auction: the lower one stays
out. How much does that save the winner, and who pays for it? Repeat with
three bidders where the two highest collude.


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

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


.. _sphx_glr_download_api_gallery_economics_auctions_plot_01_vickrey_second_price.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_01_vickrey_second_price.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_vickrey_second_price.ipynb <plot_01_vickrey_second_price.ipynb>`

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

      :download:`Download Python source code: plot_01_vickrey_second_price.py <plot_01_vickrey_second_price.py>`

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

      :download:`Download zipped: plot_01_vickrey_second_price.zip <plot_01_vickrey_second_price.zip>`


.. only:: html

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

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