.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/economics/rewards/plot_04_kiayias_mining_games.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_rewards_plot_04_kiayias_mining_games.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_rewards_plot_04_kiayias_mining_games.py:


Mining games: when honest mining is an equilibrium (Kiayias et al. 2016)
========================================================================

Kiayias, Koutsoupias, Kyropoulou and Tselekounis modeled Bitcoin mining as
a stochastic game in which each miner chooses which block to extend and
when to publish. Honest mining, extending the longest chain and publishing
at once, is the designer's intended behavior; they showed it is a best
response to the others' honesty when every miner is small, but not for a
large one, and that other equilibria then arise.

The experiment computes that best response. Against honest miners, a miner
of share :math:`\alpha` may withhold blocks, publish them to override or tie
the public chain, or give up. As a Markov decision process over the lengths
of the private and public branches (Sapirshtein, Sompolinsky and Zohar),
its best long-run share of the chain is

.. math::

   \rho^*(\alpha, \gamma) = \max_\pi \frac{\text{its blocks}}{\text{all blocks}} \ge \alpha,

and honest mining is an equilibrium exactly when :math:`\rho^* = \alpha`.
Here :math:`\gamma` is the fraction of the others who build on its block
during a tie.

.. GENERATED FROM PYTHON SOURCE LINES 28-33

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    import blockchainkit as bk








.. GENERATED FROM PYTHON SOURCE LINES 34-36

The best response against honest miners
---------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 36-50

.. code-block:: Python


    alphas = np.round(np.arange(0.05, 0.46, 0.05), 2)
    fig, ax = plt.subplots(figsize=(7.5, 4.5))
    ax.plot(alphas, alphas, "--", color="black", label="honest: share = alpha")
    for gamma, color in ((0.0, "#2563eb"), (0.5, "#16a34a")):
        best = [bk.economics.optimal_mining_revenue(a, gamma, max_lead=8) for a in alphas]
        selfish = [bk.consensus.selfish_mining_revenue(a, gamma) for a in alphas]
        ax.plot(alphas, best, "o-", color=color, label=f"best response, gamma = {gamma}")
        ax.plot(alphas, selfish, ":", color=color, label=f"selfish mining, gamma = {gamma}")
    ax.set(xlabel="hashrate alpha", ylabel="share of the chain's blocks")
    ax.set_title("Small miners mine honestly; large ones do better by deviating")
    ax.legend(fontsize=8)
    fig.tight_layout()




.. image-sg:: /api/gallery/economics/rewards/images/sphx_glr_plot_04_kiayias_mining_games_001.png
   :alt: Small miners mine honestly; large ones do better by deviating
   :srcset: /api/gallery/economics/rewards/images/sphx_glr_plot_04_kiayias_mining_games_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 51-53

Where honesty stops being a best response
-----------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 53-79

.. code-block:: Python



    def threshold(gamma, low=0.05, high=0.45):
        while high - low > 0.005:
            middle = (low + high) / 2
            if bk.economics.honest_mining_is_equilibrium(middle, gamma, max_lead=8):
                low = middle
            else:
                high = middle
        return low


    for gamma in (0.0, 0.5):
        found = threshold(gamma)
        eyal_sirer = bk.consensus.selfish_mining_threshold(gamma)
        print(
            f"gamma = {gamma}: honest up to alpha ~ {found:.3f} "
            f"(selfish mining pays above {eyal_sirer:.3f})"
        )
        assert found <= eyal_sirer + 0.005

    assert bk.economics.honest_mining_is_equilibrium(0.2, 0.0, max_lead=8)
    assert not bk.economics.honest_mining_is_equilibrium(0.4, 0.0, max_lead=8)

    plt.show()





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

 .. code-block:: none

    gamma = 0.0: honest up to alpha ~ 0.331 (selfish mining pays above 0.333)
    gamma = 0.5: honest up to alpha ~ 0.250 (selfish mining pays above 0.250)




.. GENERATED FROM PYTHON SOURCE LINES 80-85

Exercise
--------
The model caps both branches at ``max_lead`` blocks. Recompute the best
response of a 40% miner with ``max_lead`` of 4, 8 and 12. Why can a larger
cap only raise it, and why does it matter more for large miners?


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

   **Total running time of the script:** (0 minutes 3.711 seconds)


.. _sphx_glr_download_api_gallery_economics_rewards_plot_04_kiayias_mining_games.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/rewards/plot_04_kiayias_mining_games.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_04_kiayias_mining_games.ipynb <plot_04_kiayias_mining_games.ipynb>`

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

      :download:`Download Python source code: plot_04_kiayias_mining_games.py <plot_04_kiayias_mining_games.py>`

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

      :download:`Download zipped: plot_04_kiayias_mining_games.zip <plot_04_kiayias_mining_games.zip>`


.. only:: html

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

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