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| max-width: 100%; | ||
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| margin: auto; | ||
| float: none!important; | ||
| } |
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| { | ||
| "cells": [ | ||
| { | ||
| "cell_type": "markdown", | ||
| "source": [ | ||
| "# The Core: A different view on explanation\n", | ||
| "\n", | ||
| "On this page another game theoretic concept *the core* is introduced and how it can be applied for explanation.\n", | ||
| "Mainly applied in the field of economics the core is a newly introduced concept in the field of explanation for AI researchers." | ||
| ], | ||
| "metadata": { | ||
| "collapsed": false | ||
| } | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "source": [ | ||
| "## Example: Writing a Paper\n", | ||
| "Let us consider three AI researchers Alice, Bob and Charlie which work together on a paper.\n", | ||
| "The paper is about a new method to explain AI models and would win the best paper award at the next conference.\n", | ||
| "Winning the best paper award would grant them a \\$500 price.\n", | ||
| "Despite their long friendship they would like to have a fair share of the prize money.\n" | ||
| ], | ||
| "metadata": { | ||
| "collapsed": false | ||
| } | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "source": [ | ||
| "As they are AI researchers they are familiar with the concept of *Shapley values*, being a fair way to distribute the prize money.\n", | ||
| "They have previously recorded their best paper award prizes:\n", | ||
| "\n", | ||
| "\n", | ||
| "| Authos | Award |\n", | ||
| "|-----------------|----------|\n", | ||
| "| | \\$ $0$ |\n", | ||
| "| Alice | \\$ $0$ |\n", | ||
| "| Bob | \\$ $0$ |\n", | ||
| "| Charlie | \\$ $0$ |\n", | ||
| "| Alice, Bob | \\$ $500$ |\n", | ||
| "| Alice, Charlie | \\$ $400$ |\n", | ||
| "| Bob, Charlie | \\$ $350$ |\n", | ||
| "| Alice, Bob, Charlie | \\$ $500$ |\n", | ||
| "\n", | ||
| "Based on these values they would like to distribute the prize money in a fair way via the Shapley values.\n", | ||
| "Running the calculations they get the following values:\n", | ||
| "\n", | ||
| "| Authos | Shapley Value |\n", | ||
| "|-----------------|---------------|\n", | ||
| "| Alice | \\$ $200$ |\n", | ||
| "| Bob | \\$ $175$ |\n", | ||
| "| Charlie | \\$ $125$ |\n", | ||
| "\n", | ||
| "An inherent assumption when computing the Shapley value is that Alice, Bob and Charlie really work together.\n", | ||
| "Yet it might be the case that Alice and Bob realize they would get a higher payoff if they would exclude Charlie, namely \\$250 each.\n", | ||
| "The concept of the core is now introduced to prevent such behavior." | ||
| ], | ||
| "metadata": { | ||
| "collapsed": false | ||
| } | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "source": [ | ||
| "## The Core\n", | ||
| "The main idea of the core is to distribute the money in such a way, that there is no incentive for anybody to leave the group.\n", | ||
| "In our example the core would be the set of all possible distributions of the prize money, where Alice, Bob and Charlie will work together.\n", | ||
| "The underlying concept capturing \"leave the group\" is *stability*.\n", | ||
| "We say a payoff distribution $\\psi(N,\\nu)$ of a game $(N,\\nu)$ is stable iff.\n", | ||
| "\n", | ||
| "$$\n", | ||
| "\\forall S\\subseteq N. \\sum_{i \\in S} \\psi(S,\\nu)_i \\geq v(S)\n", | ||
| "$$\n", | ||
| "\n", | ||
| "where $\\psi(S,\\nu)_i$ is the payoff of player $i$ in the coalition $S$.\n", | ||
| "Stability induces no need to leave the group as each player gets what he would get in a smaller\n", | ||
| "group.\n", | ||
| "The core is then defined as the set of all stable payoff distributions.\n", | ||
| "\n", | ||
| "$$\n", | ||
| " \\mathcal{C} := \\{ \\psi(N,\\nu) | \\forall S\\subseteq N. \\sum_{i \\in S} \\psi(N,\\nu)_i \\geq v(S) \\land \\sum_{i \\in N} \\psi(N,\\nu)_i = \\nu(N) \\}\n", | ||
| "$$\n", | ||
| "\n" | ||
| ], | ||
| "metadata": { | ||
| "collapsed": false | ||
| } | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "source": [ | ||
| "## The Core of Writing a Paper\n", | ||
| "Let us now compute the core for our example.\n", | ||
| "To do so we must find at least one payoff distribution, which is stable.\n", | ||
| "Let us abbreviate the authors payoffs as $a$,$b$ and $c$.\n", | ||
| "\n", | ||
| "$$\n", | ||
| " (I): a + b \\geq 500 \\\\\n", | ||
| " (II): a + c \\geq 400 \\\\\n", | ||
| " (III): b + c \\geq 350 \\\\\n", | ||
| " (IV): a + b + c = 500\n", | ||
| "$$\n", | ||
| "\n", | ||
| "From (I) and (IV) we get $c \\leq 0$ from which $c = 0$ must follow, due to $c \\geq \\nu(\\{c\\})=0$.\n", | ||
| "Now we can substitute $c=0$ into (II) and (III) to get $a \\geq 400$ and $b \\geq 350$.\n", | ||
| "Thus $a + b \\geq 750$ which is a contradiction to (IV), showing the **core is empty**.\n", | ||
| "This can be seen visually in the following plot.\n", | ||
| "\n", | ||
| ".. image:: _static/paper_game.pdf\n", | ||
| "\n", | ||
| "Notice that the line of equality (I) corresponds exactly to those payoff distributions which have $c=0$.\n", | ||
| "Due to that (I), (II) and (III) do not enclose an area, which is why the core is empty." | ||
| ], | ||
| "metadata": { | ||
| "collapsed": false | ||
| } | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "source": [ | ||
| "## The least-Core\n", | ||
| "Due to the core being empty, we introduce a related concept **the least-core**.\n", | ||
| "Firstly lets define the **e-core** to be the payoff distributions, which are stable up to a given $e$.\n", | ||
| "\n", | ||
| "$$\n", | ||
| " \\mathcal{C_e} := \\{ \\psi(N,\\nu) | \\forall S\\subseteq N. \\sum_{i \\in S} \\psi(N,\\nu)_i + e \\geq v(S) \\land \\sum_{i \\in N} \\psi(N,\\nu)_i = \\nu(N) \\}\n", | ||
| "$$\n", | ||
| "\n", | ||
| "Then the least-core is the minimal $e$ such that $\\mathcal{C_e}$ is not empty.\n", | ||
| "We can think of this subsidy $e$ as an external payment for cooperating.\n", | ||
| "In our example the professor may encourage the Alice and Bob to include Charlie in their work by giving everybody an incentive \\$$e$.\n", | ||
| "In comparison to the Shapley value the (least-)core may contain multiple payoff distributions.\n", | ||
| "Often we are interested in a single payoff distribution, for which we choose the **egalitarian-least-core**.\n", | ||
| "Given the least-core $C_{e^*}$ the egalitarian-least-core is $x \\in C_{e^*}$ with minimal $||x||_2$.\n", | ||
| "It is important to note that the least-core **always exists** whereas the core is potentially empty.\n" | ||
| ], | ||
| "metadata": { | ||
| "collapsed": false | ||
| } | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "source": [ | ||
| "## The least-Core of Writing a Paper\n", | ||
| "Computing the least-core for our example can become quite tedious as solving linear inequalities is required.\n", | ||
| "To simplify the process we use the *ExactComputer* from the *shapiq* package.\n" | ||
| ], | ||
| "metadata": { | ||
| "collapsed": false | ||
| } | ||
| }, | ||
| { | ||
| "cell_type": "code", | ||
| "execution_count": 4, | ||
| "outputs": [], | ||
| "source": [ | ||
| "import numpy as np\n", | ||
| "from shapiq.exact import ExactComputer\n", | ||
| "from shapiq.games.base import Game\n", | ||
| "\n", | ||
| "\n", | ||
| "# Define the PaperGame as described above\n", | ||
| "class PaperGame(Game):\n", | ||
| "\n", | ||
| " def __init__(self) -> None:\n", | ||
| " super().__init__(n_players=3, normalize=True, normalization_value=0)\n", | ||
| "\n", | ||
| " def value_function(self, coalitions: np.ndarray) -> np.ndarray:\n", | ||
| " coalition_values = {\n", | ||
| " (): 0,\n", | ||
| " (0,): 0,\n", | ||
| " (1,): 0,\n", | ||
| " (2,): 0,\n", | ||
| " (0, 1): 500,\n", | ||
| " (0, 2): 400,\n", | ||
| " (1, 2): 350,\n", | ||
| " (0, 1, 2): 500,\n", | ||
| " }\n", | ||
| "\n", | ||
| " values = np.array([coalition_values[tuple(np.where(x)[0])] for x in coalitions])\n", | ||
| "\n", | ||
| " return values\n", | ||
| "\n", | ||
| "\n", | ||
| "paper_game = PaperGame()\n", | ||
| "\n", | ||
| "# Initialize the ExactComputer with the PaperGame\n", | ||
| "exact_computer = ExactComputer(n_players=3, game_fun=paper_game)\n", | ||
| "# Compute the egalitarian least core abbreviated to \"ELC\"\n", | ||
| "egalitarian_least_core = exact_computer(\"ELC\")" | ||
| ], | ||
| "metadata": { | ||
| "collapsed": false, | ||
| "ExecuteTime": { | ||
| "end_time": "2024-10-04T17:04:48.123115Z", | ||
| "start_time": "2024-10-04T17:04:48.116174Z" | ||
| } | ||
| } | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "source": [ | ||
| "The egalitarian least core values can then be viewed with `.dict_values`." | ||
| ], | ||
| "metadata": { | ||
| "collapsed": false | ||
| } | ||
| }, | ||
| { | ||
| "cell_type": "code", | ||
| "execution_count": 5, | ||
| "outputs": [ | ||
| { | ||
| "data": { | ||
| "text/plain": "{(0,): 233.3333333333333, (1,): 183.33333333333337, (2,): 83.33333333333334}" | ||
| }, | ||
| "execution_count": 5, | ||
| "metadata": {}, | ||
| "output_type": "execute_result" | ||
| } | ||
| ], | ||
| "source": [ | ||
| "egalitarian_least_core.dict_values" | ||
| ], | ||
| "metadata": { | ||
| "collapsed": false, | ||
| "ExecuteTime": { | ||
| "end_time": "2024-10-04T17:04:48.128619Z", | ||
| "start_time": "2024-10-04T17:04:48.123472Z" | ||
| } | ||
| } | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "source": [ | ||
| "These values then correspond to the following payoff distribution:\n", | ||
| "\n", | ||
| "| Authos | egalitarian least-core |\n", | ||
| "|-----------------|---------------|\n", | ||
| "| Alice | \\$ $233.\\bar{3}$ |\n", | ||
| "| Bob | \\$ $183.\\bar{3}$ |\n", | ||
| "| Charlie | \\$ $83.\\bar{3}$ |\n", | ||
| "\n", | ||
| "The minimal $e$ is stored in `exact_computer._elc_stability_subsidy`." | ||
| ], | ||
| "metadata": { | ||
| "collapsed": false | ||
| } | ||
| }, | ||
| { | ||
| "cell_type": "code", | ||
| "execution_count": 6, | ||
| "outputs": [ | ||
| { | ||
| "data": { | ||
| "text/plain": "83.33333333333334" | ||
| }, | ||
| "execution_count": 6, | ||
| "metadata": {}, | ||
| "output_type": "execute_result" | ||
| } | ||
| ], | ||
| "source": [ | ||
| "exact_computer._elc_stability_subsidy" | ||
| ], | ||
| "metadata": { | ||
| "collapsed": false, | ||
| "ExecuteTime": { | ||
| "end_time": "2024-10-04T17:04:48.128991Z", | ||
| "start_time": "2024-10-04T17:04:48.128403Z" | ||
| } | ||
| } | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "source": [ | ||
| "Visualizing the egalitarian least-core in the plot we get the following:\n", | ||
| "\n", | ||
| ".. image:: _static/elc_paper_game.pdf" | ||
| ], | ||
| "metadata": { | ||
| "collapsed": false | ||
| } | ||
| } | ||
| ], | ||
| "metadata": { | ||
| "kernelspec": { | ||
| "display_name": "Python 3", | ||
| "language": "python", | ||
| "name": "python3" | ||
| }, | ||
| "language_info": { | ||
| "codemirror_mode": { | ||
| "name": "ipython", | ||
| "version": 2 | ||
| }, | ||
| "file_extension": ".py", | ||
| "mimetype": "text/x-python", | ||
| "name": "python", | ||
| "nbconvert_exporter": "python", | ||
| "pygments_lexer": "ipython2", | ||
| "version": "2.7.6" | ||
| } | ||
| }, | ||
| "nbformat": 4, | ||
| "nbformat_minor": 0 | ||
| } |