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Fix minor typos in the text book. (#228)
Also, include some solution notebooks.
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‎docs/text-book/best-responses.rst

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1. For the following games identify the best responses:
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1. For the following games identify the best responses that are strategies with
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support size 1 (i.e. strategies that just play a single action).
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1.
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‎docs/text-book/solutions/README.md

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# This contains notebooks with solutions to the exercises from the text book.

‎docs/text-book/solutions/best-responses.ipynb

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‎docs/text-book/solutions/normal-form-games.ipynb

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{
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"cells": [
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{
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"cell_type": "markdown",
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"id": "7779ca3d-99a8-4014-a001-4b4c9314e39f",
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"metadata": {},
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"source": [
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"> Obtain the full game representations $(A, B)$ for the zero sum games with row play payoff matrix given by:\n",
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"\n",
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" 1. $A =\\begin{pmatrix}1 & 3\\\\ -1 & 4\\end{pmatrix}\\qquad B =\\begin{pmatrix}-1 & -3\\\\ 1 & -4\\end{pmatrix}$"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 12,
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"id": "f365cfb4-38bb-4605-b5cd-e4476bd60978",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"Zero sum game with payoff matrices:\n",
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"\n",
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"Row player:\n",
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"[[ 1 3]\n",
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" [-1 4]]\n",
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"\n",
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"Column player:\n",
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"[[-1 -3]\n",
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" [ 1 -4]]"
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]
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},
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"execution_count": 12,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"import nashpy as nash\n",
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"import numpy as np\n",
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"\n",
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"A = np.array(((1, 3), (-1, 4)))\n",
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"game = nash.Game(A)\n",
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"game"
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]
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},
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{
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"cell_type": "markdown",
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"id": "206da237-a337-434b-b5f6-0b29ef992687",
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"metadata": {},
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"source": [
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" 2. $A =\\begin{pmatrix}1 & -2\\\\ -1 & 2\\end{pmatrix}\\qquad B =\\begin{pmatrix}-1 & 2\\\\ 1 & -2\\end{pmatrix}$"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 13,
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"id": "a8698c05-a30b-40e9-864b-081fbd31da42",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"Zero sum game with payoff matrices:\n",
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"\n",
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"Row player:\n",
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"[[ 1 -2]\n",
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" [-1 2]]\n",
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"\n",
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"Column player:\n",
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"[[-1 2]\n",
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" [ 1 -2]]"
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]
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},
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"execution_count": 13,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"A = np.array(((1, -2), (-1, 2)))\n",
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"B = np.array(((-1, 2), (1, -2)))\n",
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"game = nash.Game(A, B)\n",
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"game"
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]
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},
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{
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"cell_type": "markdown",
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"id": "07878c15-fb5d-4840-9c0d-e3fe95f1226a",
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"metadata": {},
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"source": [
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" 3. $A =\\begin{pmatrix}1 & -2 & 4\\\\ 2 & -1 & 2\\\\ 7 & -7 & 6\\end{pmatrix}\\qquad B =\\begin{pmatrix}-1 & 2 & -4\\\\ -2 & 1 & -2\\\\ -7 & 7 & -6\\end{pmatrix}$"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 14,
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"id": "fc5f7b5a-91ba-4358-b9f6-0772033fcb8c",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"Zero sum game with payoff matrices:\n",
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"\n",
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"Row player:\n",
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"[[ 1 -2 4]\n",
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" [ 2 -1 2]\n",
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" [ 7 -7 6]]\n",
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"\n",
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"Column player:\n",
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"[[-1 2 -4]\n",
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" [-2 1 -2]\n",
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" [-7 7 -6]]"
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]
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},
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"execution_count": 14,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"A = np.array(((1, -2, 4), (2, -1, 2), (7, -7, 6)))\n",
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"B = np.array(((-1, 2, -4), (-2, 1, -2), (-7, 7, -6)))\n",
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"game = nash.Game(A, B)\n",
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"game"
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]
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},
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{
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"cell_type": "markdown",
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"id": "7bafa50e-ac87-4ad8-a5ed-17a5b550066c",
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"metadata": {},
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"source": [
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"> `3.` Consider the game described as follows:\n",
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"\n",
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" > An airline loses two suitcases belonging to two different travelers. Both suitcases have the same value. An airline manager tasked to settle the claims of both travelers explains that the airline is liable for a maximum of £5 per suitcase. \n",
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"\n",
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" > To determine an honest appraised value of the suitcases, the manager separates both travelers and asks them to write down the amount of their value at no less than £2 and no larger than £5 (to the single dollar):\n",
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" > - If both write down the same number, that number as the true dollar value of both suitcases and reimburse both travelers that amount. \n",
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" > - However, if one writes down a smaller number than the other, this smaller number will be taken as the true dollar value, and both travelers will receive that amount along with a bonus/malus: £2 extra will be paid to the traveler who wrote down the lower value and a £2 deduction will be taken from the person who wrote down the higher amount. \n",
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" \n",
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" > Represent this as a Normal Form Game.\n",
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" \n",
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" \n",
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"$$\n",
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"A = \\begin{pmatrix}\n",
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"2 & 4 & 4 & 4\\\\\n",
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"0 & 3 & 5 & 5\\\\\n",
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"0 & 1 & 4 & 6\\\\\n",
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"0 & 1 & 2 & 5\n",
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"\\end{pmatrix}\n",
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"\\qquad\n",
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"B = \\begin{pmatrix}\n",
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"2 & 0 & 0 & 0\\\\\n",
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"4 & 3 & 1 & 1\\\\\n",
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"4 & 5 & 4 & 2\\\\\n",
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"4 & 5 & 6 & 5\n",
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"\\end{pmatrix}\n",
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"$$"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 15,
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"id": "f1922cd9-a1ca-40ca-b1ad-c09de6b46a9d",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"Bi matrix game with payoff matrices:\n",
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"\n",
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"Row player:\n",
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"[[2 4 4 4]\n",
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" [0 3 5 5]\n",
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" [0 1 4 6]\n",
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" [0 1 2 5]]\n",
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"\n",
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"Column player:\n",
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"[[2 0 0 0]\n",
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" [4 3 1 1]\n",
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" [4 5 4 2]\n",
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" [4 5 6 5]]"
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]
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},
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"execution_count": 15,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"A = np.array(((2, 4, 4, 4), (0, 3, 5, 5), (0, 1, 4, 6), (0, 1, 2,5)))\n",
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"B = np.array(((2, 0, 0, 0), (4, 3, 1, 1), (4, 5, 4, 2), (4, 5, 6, 5)))\n",
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"game = nash.Game(A, B)\n",
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"game"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "c8b6f40c-b1c9-4b63-b373-0c42436d8238",
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"metadata": {},
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"outputs": [],
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"source": []
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}
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],
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"metadata": {
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"kernelspec": {
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"display_name": "Python 3 (ipykernel)",
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"language": "python",
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"name": "python3"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.11.4"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 5
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}

‎docs/text-book/solutions/strategies.ipynb

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{
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"cells": [
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{
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"cell_type": "markdown",
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"id": "f11b56d3-3fc5-40c5-bde2-ff7ad5eb6207",
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"metadata": {},
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"source": [
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"3. Calculate the utilities (for both the row and column player) for the following game for the following strategy pairs:\n",
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"\n",
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" $$\n",
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" A =\n",
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" \\begin{pmatrix}\n",
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" 1 & -1\\\\\n",
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" -3 & 1\\end{pmatrix}\n",
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" \\qquad\n",
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" B =\n",
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" \\begin{pmatrix}\n",
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" -1 & 2\\\\\n",
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" 1 & -1\\end{pmatrix}\n",
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" $$\n",
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"\n",
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" 1. $\\sigma_r = (.2, .8)\\qquad\\sigma_c = (.6, .4)$\n",
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" \n",
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" $$u_r(\\sigma_r, \\sigma_c) = \\sigma_r A \\sigma_c ^T = .2\\times.6\\times 1 + .2\\times .4\\times (-1) + .8\\times.6\\times (-3) + .8\\times .4\\times 1=-1.08$$\n",
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" \n",
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" $$u_c(\\sigma_r, \\sigma_c) = \\sigma_r B \\sigma_c ^T = .2\\times.6\\times (-1) + .2\\times .4\\times 2 + .8\\times.6\\times 1 + .8\\times .4\\times (-1)=0.2 $$\n",
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" 2. $\\sigma_r = (.3, .7)\\qquad\\sigma_c = (.2, .8)$\n",
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" \n",
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" $$u_r(\\sigma_r, \\sigma_c) = -0.04$$\n",
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" \n",
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" $$u_c(\\sigma_r, \\sigma_c) = 0$$\n",
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" 3. $\\sigma_r = (.9, .1)\\qquad\\sigma_c = (.5, .5)$\n",
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" \n",
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" $$u_r(\\sigma_r, \\sigma_c) = -0.1$$\n",
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" \n",
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" $$u_c(\\sigma_r, \\sigma_c) = 0.45$$"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 5,
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"id": "dffb72f1-8d5c-48ba-bcb9-c875ea7f2731",
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"metadata": {},
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"outputs": [],
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"source": [
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"import nashpy as nash\n",
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"import numpy as np\n",
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"\n",
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"A = np.array(((1, -1), (-3, 1)))\n",
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"B = np.array(((-1, 2), (1, -1)))\n",
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"game = nash.Game(A, B)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 6,
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"id": "4b1db209-6058-46e1-85b1-df31fe5f010f",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"array([-1.08, 0.2 ])"
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]
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},
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"execution_count": 6,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"sigma_r = np.array((.2, .8))\n",
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"sigma_c = np.array((.6, .4))\n",
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"game[sigma_r, sigma_c]"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 7,
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"id": "df7c6c52-e97a-49a3-ae30-85a0bdd78499",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"array([-4.0000000e-02, 4.4408921e-18])"
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]
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},
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"execution_count": 7,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"sigma_r = np.array((.3, .7))\n",
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"sigma_c = np.array((.2, .8))\n",
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"game[sigma_r, sigma_c]"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 8,
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"id": "0fad9239-66aa-4317-8dae-800d260bb41d",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"array([-0.1 , 0.45])"
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]
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},
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"execution_count": 8,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"sigma_r = np.array((.9, .1))\n",
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"sigma_c = np.array((.5, .5))\n",
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"game[sigma_r, sigma_c]"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "fa7b3636-02db-4613-bca6-dcf0225a42ba",
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"metadata": {},
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"outputs": [],
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"source": []
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}
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],
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"metadata": {
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"kernelspec": {
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"display_name": "Python 3 (ipykernel)",
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"language": "python",
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"name": "python3"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.11.4"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 5
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}

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