ddp_ex_rust96_py.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# DiscreteDP Example: Automobile Replacement"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Daisuke Oyama**\n",
    "\n",
    "*Faculty of Economics, University of Tokyo*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We study the finite-state version of the automobile replacement problem as considered in\n",
    "Rust (1996, Section 4.2.2)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* J. Rust, \"Numerical Dynamic Programming in Economics\",\n",
    "  <i>Handbook of Computational Economics</i>, Volume 1, 619-729, 1996."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "from __future__ import division, print_function\n",
    "import numpy as np\n",
    "import itertools\n",
    "import scipy.optimize\n",
    "import matplotlib.pyplot as plt\n",
    "import pandas as pd\n",
    "from quantecon.markov import DiscreteDP"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Setup"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "lambd = 0.5  # Exponential distribution parameter\n",
    "c = 200  # (Constant) marginal cost of maintainance\n",
    "net_price = 10**5  # Replacement cost"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "n = 100  # Number of states; s = 0, ..., n-1: level of utilization of the asset\n",
    "m = 2  # Number of actions; 0: keep, 1: replace"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# Reward array\n",
    "R = np.empty((n, m))\n",
    "R[:, 0] = -c * np.arange(n)  # Costs for maintainance\n",
    "R[:, 1] = -net_price - c * 0  # Costs for replacement"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# Transition probability array\n",
    "# For each state s, s' distributes over\n",
    "# s, s+1, ..., min{s+supp_size-1, n-1} if a = 0\n",
    "# 0, 1, ..., supp_size-1 if a = 1\n",
    "# according to the (discretized and truncated) exponential distribution\n",
    "# with parameter lambd\n",
    "supp_size = 12\n",
    "probs = np.empty(supp_size)\n",
    "probs[0] = 1 - np.exp(-lambd * 0.5)\n",
    "for j in range(1, supp_size-1):\n",
    "    probs[j] = np.exp(-lambd * (j - 0.5)) - np.exp(-lambd * (j + 0.5))\n",
    "probs[supp_size-1] = 1 - np.sum(probs[:-1])\n",
    "\n",
    "Q = np.zeros((n, m, n))\n",
    "\n",
    "# a = 0\n",
    "for i in range(n-supp_size):\n",
    "    Q[i, 0, i:i+supp_size] = probs\n",
    "for k in range(supp_size):\n",
    "    Q[n-supp_size+k, 0, n-supp_size+k:] = probs[:supp_size-k]/probs[:supp_size-k].sum()\n",
    "\n",
    "# a = 1\n",
    "for i in range(n):\n",
    "    Q[i, 1, :supp_size] = probs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# Discount factor\n",
    "beta = 0.95"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Continuous-state benchmark"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let us compute the value function of the continuous-state version\n",
    "as described in equations (2.22) and (2.23) in Section 2.3."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def f(x, s):\n",
    "    return (c/(1-beta)) * \\\n",
    "        ((x-s) - (beta/(lambd*(1-beta))) * (1 - np.exp(-lambd*(1-beta)*(x-s))))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The optimal stopping boundary $\\gamma$ for the contiuous-state version, given by (2.23):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "52.86545636058919\n"
     ]
    }
   ],
   "source": [
    "gamma = scipy.optimize.brentq(lambda x: f(x, 0) - net_price, 0, 100)\n",
    "print(gamma)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The value function for the continuous-state version, given by (2.24):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def value_func_cont_time(s):\n",
    "    return -c*gamma/(1-beta) + (s < gamma) * f(gamma, s)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "v_cont = value_func_cont_time(np.arange(n))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Solving the problem with `DiscreteDP`"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Construct a `DiscreteDP` instance for the disrete-state version:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "ddp = DiscreteDP(R, Q, beta)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let us solve the decision problem by\n",
    "\n",
    "(0) value iteration,  \n",
    "(1) value iteration with span-based termination\n",
    "(equivalent to modified policy iteration with step $k = 0$),  \n",
    "(2) policy iteration,  \n",
    "(3) modified policy iteration."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Following Rust (1996), we set:\n",
    "\n",
    "* $\\varepsilon = 1164$ (for value iteration and modified policy iteration),\n",
    "* $v^0 \\equiv 0$,\n",
    "* the number of iteration for iterative policy evaluation $k = 20$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "v_init = np.zeros(ddp.num_states)\n",
    "epsilon = 1164\n",
    "methods = ['vi', 'mpi', 'pi', 'mpi']\n",
    "labels = ['Value iteration', 'Value iteration with span-based termination',\n",
    "          'Policy iteration', 'Modified policy iteration']\n",
    "results = {}\n",
    "\n",
    "for i in range(4):\n",
    "    k = 20 if labels[i] == 'Modified policy iteration' else 0\n",
    "    results[labels[i]] = \\\n",
    "        ddp.solve(method=methods[i], v_init=v_init, epsilon=epsilon, k=k)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "columns = [\n",
    "    'Iterations', 'Time (second)', r'$\\lVert v - v_{\\mathrm{pi}} \\rVert$',\n",
    "    r'$\\overline{b} - \\underline{b}$', r'$\\lVert v - T(v)\\rVert$'\n",
    "]\n",
    "df = pd.DataFrame(index=labels, columns=columns)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The numbers of iterations:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "114 \t(Value iteration)\n",
      "65 \t(Value iteration with span-based termination)\n",
      "5 \t(Policy iteration)\n",
      "6 \t(Modified policy iteration)\n"
     ]
    }
   ],
   "source": [
    "for label in labels:\n",
    "    print(results[label].num_iter, '\\t' + '(' + label + ')')\n",
    "    df[columns[0]].loc[label] = results[label].num_iter"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Policy iteration gives the optimal policy:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0\n",
      " 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1\n",
      " 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1]\n"
     ]
    }
   ],
   "source": [
    "print(results['Policy iteration'].sigma)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Takes action 1 (\"replace\") if and only if $s \\geq \\bar{\\gamma}$, where $\\bar{\\gamma}$ is equal to:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "53"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(1-results['Policy iteration'].sigma).sum()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Check that the other methods gave the correct answer:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "True\n",
      "True\n",
      "True\n"
     ]
    }
   ],
   "source": [
    "for result in results.values():\n",
    "    if result != results['Policy iteration']:\n",
    "        print(np.array_equal(result.sigma, results['Policy iteration'].sigma))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The deviations of the returned value function from the continuous-state benchmark:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1733.77910196 \t(Value iteration)\n",
      "1085.80571693 \t(Value iteration with span-based termination)\n",
      "1162.88060863 \t(Policy iteration)\n",
      "1144.47728382 \t(Modified policy iteration)\n"
     ]
    }
   ],
   "source": [
    "diffs_cont = {}\n",
    "for label in labels:\n",
    "    diffs_cont[label] = np.abs(results[label].v - v_cont).max()\n",
    "    print(diffs_cont[label], '\\t' + '(' + label + ')')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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kShOGXDkk6LBERLINJX8l/2wlNhauvx7yFzrCrmYtiCpWibdbvo1zaT6GRURy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\n38zMrM44+ZuZmdUZJ38zM7M68x85Q8FQc5kx+wAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10c94a320>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "label = 'Policy iteration'\n",
    "fig, ax = plt.subplots(figsize=(8,5))\n",
    "ax.plot(-v_cont, label='Continuous-state')\n",
    "ax.plot(-results[label].v, label=label)\n",
    "ax.set_title('Comparison of discrete vs. continuous value functions')\n",
    "ax.ticklabel_format(style='sci', axis='y', scilimits=(0,0))\n",
    "ax.set_xlabel('State')\n",
    "ax.set_ylabel(r'Value $\\times\\ (-1)$')\n",
    "plt.legend(loc=4)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the following we try to reproduce Table 14.1 in Rust (1996), p.660,\n",
    "although the precise definitions and procedures there are not very clear."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The maximum absolute differences of $v$ from that by policy iteration:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "571.657808034 \t(Value iteration)\n",
      "124.137626142 \t(Value iteration with span-based termination)\n",
      "0.0 \t(Policy iteration)\n",
      "44.2673484102 \t(Modified policy iteration)\n"
     ]
    }
   ],
   "source": [
    "for label in labels:\n",
    "    diff_pi = \\\n",
    "        np.abs(results[label].v - results['Policy iteration'].v).max()\n",
    "    print(diff_pi, '\\t' + '(' + label + ')')\n",
    "    df[columns[2]].loc[label] = diff_pi"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Compute $\\lVert v - T(v)\\rVert$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "29.0800246681 \t(Value iteration)\n",
      "27.7883386169 \t(Value iteration with span-based termination)\n",
      "2.03726813197e-10 \t(Policy iteration)\n",
      "6.59712753215 \t(Modified policy iteration)\n"
     ]
    }
   ],
   "source": [
    "for label in labels:\n",
    "    v = results[label].v\n",
    "    diff_max = \\\n",
    "        np.abs(v - ddp.bellman_operator(v)).max()\n",
    "    print(diff_max, '\\t' + '(' + label + ')')\n",
    "    df[columns[4]].loc[label] = diff_max"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Next we compute $\\overline{b} - \\underline{b}$\n",
    "for the three methods other than policy iteration, where\n",
    "$I$ is the number of iterations required to fulfill the termination condition, and\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\underline{b} &= \\frac{\\beta}{1-\\beta} \\min\\left[T(v^{I-1}) - v^{I-1}\\right], \\\\\\\\\n",
    "\\overline{b}  &= \\frac{\\beta}{1-\\beta} \\max\\left[T(v^{I-1}) - v^{I-1}\\right].\n",
    "\\end{aligned}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "26.4655743941 \t(Value iteration)\n",
      "1141.05130079 \t(Value iteration with span-based termination)\n",
      "267.494551734 \t(Modified policy iteration)\n"
     ]
    }
   ],
   "source": [
    "for i in range(4):\n",
    "    if labels[i] != 'Policy iteration':\n",
    "        k = 20 if labels[i] == 'Modified policy iteration' else 0\n",
    "        res = ddp.solve(method=methods[i], v_init=v_init, k=k,\n",
    "                        max_iter=results[labels[i]].num_iter-1)\n",
    "        diff = ddp.bellman_operator(res.v) - res.v\n",
    "        diff_span = (diff.max() - diff.min()) * ddp.beta / (1 - ddp.beta)\n",
    "        print(diff_span, '\\t' + '(' + labels[i] + ')')\n",
    "        df[columns[3]].loc[labels[i]] = diff_span"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For policy iteration, while it does not seem really relevant,\n",
    "we compute $\\overline{b} - \\underline{b}$ with the returned value of $v$\n",
    "in place of $v^{I-1}$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "4.9767550081e-09 \t(Policy iteration)\n"
     ]
    }
   ],
   "source": [
    "label = 'Policy iteration'\n",
    "v = results[label].v\n",
    "diff = ddp.bellman_operator(v) - v\n",
    "diff_span = (diff.max() - diff.min()) * ddp.beta / (1 - ddp.beta)\n",
    "print(diff_span, '\\t' + '(' + label + ')')\n",
    "df[columns[3]].loc[label] = diff_span"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Last, time each algorithm:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Value iteration\n",
      "100 loops, best of 3: 4.25 ms per loop\n",
      "Value iteration with span-based termination\n",
      "100 loops, best of 3: 4.62 ms per loop\n",
      "Policy iteration\n",
      "1000 loops, best of 3: 1.58 ms per loop\n",
      "Modified policy iteration\n",
      "1000 loops, best of 3: 1.38 ms per loop\n"
     ]
    }
   ],
   "source": [
    "for i in range(4):\n",
    "    k = 20 if labels[i] == 'Modified policy iteration' else 0\n",
    "    print(labels[i])\n",
    "    t = %timeit -o ddp.solve(method=methods[i], v_init=v_init, epsilon=epsilon, k=k)\n",
    "    df[columns[1]].loc[labels[i]] = t.best"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Iterations</th>\n",
       "      <th>Time (second)</th>\n",
       "      <th>$\\lVert v - v_{\\mathrm{pi}} \\rVert$</th>\n",
       "      <th>$\\overline{b} - \\underline{b}$</th>\n",
       "      <th>$\\lVert v - T(v)\\rVert$</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>Value iteration</th>\n",
       "      <td>114</td>\n",
       "      <td>0.00424717</td>\n",
       "      <td>571.658</td>\n",
       "      <td>26.4656</td>\n",
       "      <td>29.08</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Value iteration with span-based termination</th>\n",
       "      <td>65</td>\n",
       "      <td>0.00461827</td>\n",
       "      <td>124.138</td>\n",
       "      <td>1141.05</td>\n",
       "      <td>27.7883</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Policy iteration</th>\n",
       "      <td>5</td>\n",
       "      <td>0.00157695</td>\n",
       "      <td>0</td>\n",
       "      <td>4.97676e-09</td>\n",
       "      <td>2.03727e-10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Modified policy iteration</th>\n",
       "      <td>6</td>\n",
       "      <td>0.00137902</td>\n",
       "      <td>44.2673</td>\n",
       "      <td>267.495</td>\n",
       "      <td>6.59713</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "                                            Iterations Time (second)  \\\n",
       "Value iteration                                    114    0.00424717   \n",
       "Value iteration with span-based termination         65    0.00461827   \n",
       "Policy iteration                                     5    0.00157695   \n",
       "Modified policy iteration                            6    0.00137902   \n",
       "\n",
       "                                            $\\lVert v - v_{\\mathrm{pi}} \\rVert$  \\\n",
       "Value iteration                                                         571.658   \n",
       "Value iteration with span-based termination                             124.138   \n",
       "Policy iteration                                                              0   \n",
       "Modified policy iteration                                               44.2673   \n",
       "\n",
       "                                            $\\overline{b} - \\underline{b}$  \\\n",
       "Value iteration                                                    26.4656   \n",
       "Value iteration with span-based termination                        1141.05   \n",
       "Policy iteration                                               4.97676e-09   \n",
       "Modified policy iteration                                          267.495   \n",
       "\n",
       "                                            $\\lVert v - T(v)\\rVert$  \n",
       "Value iteration                                               29.08  \n",
       "Value iteration with span-based termination                 27.7883  \n",
       "Policy iteration                                        2.03727e-10  \n",
       "Modified policy iteration                                   6.59713  "
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Notes"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It appears that our value iteration with span-based termination is different in some details\n",
    "from the corresponding algorithm (successive approximation with error bounds) in Rust.\n",
    "In returing the value function, our algorithm returns\n",
    "$T(v^{I-1}) + (\\overline{b} + \\underline{b})/2$,\n",
    "while Rust's seems to return $v^{I-1} + (\\overline{b} + \\underline{b})/2$.\n",
    "In fact:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "i = 1\n",
    "k = 0\n",
    "res = ddp.solve(method=methods[i], v_init=v_init, k=k,\n",
    "                max_iter=results[labels[i]].num_iter-1)\n",
    "diff = ddp.bellman_operator(res.v) - res.v\n",
    "v = res.v + (diff.max() + diff.min()) * ddp.beta / (1 - ddp.beta) / 2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\lVert v - v_{\\mathrm{pi}}\\rVert$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "503.43001170479693"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.abs(v - results['Policy iteration'].v).max()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\lVert v - T(v)\\rVert$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "48.499617969617248"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.abs(v - ddp.bellman_operator(v)).max()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Compare the Table in Rust."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Convergence of trajectories"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let us plot the convergence of $v^i$ for the four algorithms;\n",
    "see also Figure 14.2 in Rust."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Value iteration"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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fRlSmVJIwKQF7qZ1BPw3Cv78/AL8UFzMxIYGevr6sHzbM9Wfos9vh3Xdhxgxr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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10dbe5fd0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "label = 'Value iteration'\n",
    "iters = [2, 20, 40, 80]\n",
    "v = np.zeros(ddp.num_states)\n",
    "\n",
    "fig, ax = plt.subplots(figsize=(8,5))\n",
    "for i in range(iters[-1]):\n",
    "    v = ddp.bellman_operator(v)\n",
    "    if i+1 in iters:\n",
    "        ax.plot(-v, label='Iteration {0}'.format(i+1))\n",
    "ax.plot(-results['Policy iteration'].v, label='Fixed Point')\n",
    "\n",
    "ax.ticklabel_format(style='sci', axis='y', scilimits=(0,0))\n",
    "ax.set_ylim(0, 2.4e5)\n",
    "ax.set_yticks([0.4e5 * i for i in range(7)])\n",
    "ax.set_title(label)\n",
    "ax.set_xlabel('State')\n",
    "ax.set_ylabel(r'Value $\\times\\ (-1)$')\n",
    "plt.legend(loc=(0.7, 0.2))\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Value iteration with span-based termination"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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4YyX+pSKyOp9VwoDmds8vs827yEsvvZTzuG/fvvTt27fc4lSqKKdTU7l73z48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\nPZ8xee1knv3Hs0y+ZjK1jN7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HJSUlRUREAgMD5bLLLpM5c+aIn5+fPPDAAxITEyNDhw4V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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10dc4d1d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "label = 'Value iteration with span-based termination'\n",
    "iters = [1, 10, 15, 20]\n",
    "v = np.zeros(ddp.num_states)\n",
    "\n",
    "fig, ax = plt.subplots(figsize=(8,5))\n",
    "for i in range(iters[-1]):\n",
    "    u = ddp.bellman_operator(v)\n",
    "    if i+1 in iters:\n",
    "        diff = u - v\n",
    "        w = u + ((diff.max() + diff.min()) / 2) * ddp.beta / (1 - ddp.beta)\n",
    "        ax.plot(-w, label='Iteration {0}'.format(i+1))\n",
    "    v = u\n",
    "ax.plot(-results['Policy iteration'].v, label='Fixed Point')\n",
    "\n",
    "ax.ticklabel_format(style='sci', axis='y', scilimits=(0,0))\n",
    "ax.set_ylim(1.0e5, 2.4e5)\n",
    "ax.set_yticks([1.0e5+0.2e5 * i for i in range(8)])\n",
    "ax.set_title(label)\n",
    "ax.set_xlabel('State')\n",
    "ax.set_ylabel(r'Value $\\times\\ (-1)$')\n",
    "plt.legend(loc=(0.7, 0.2))\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Policy iteration"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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gxniMSZOdO+HppyE2FhYudBbtMcYYkzbpqvmr6t5kR7QlfpPdEhPh44/h1lvh\n3nudFfss8RtjTPqkqeYvIjG4lvS9+i1AVTXIrVEZk4KtW6FLF/D1hd9+g6pVPR2RMcZ4pzTV/FU1\nUFWDUjgCLfGbrJaYCB98ALfdBo89BlFRlviNMSYz0j3aX0SKAVVx5vkDoKq/ujMoYy7ZtAk6d3YW\n6VmxAipV8nRExhjj/dI72r8b8CsQCbzu+jnY/WGZvC4hAd59F1q0cJr6582zxG+MMe6S3pp/b6AB\n8LuqthSRasA77g/L5GXr1zu1/eLFYdUqqFDB0xEZY0zukt7kH6eqcSKCiPip6hYRuTFLIjN5Tnw8\nvP8+DBvm1Pq7dAG5zvpVZy+epdXEVuw4sSN7gjTGmGzWoUYHRrYd6dZrpjf5HxCRosCPwFwROQns\ndWtEJk9au9ap7ZcqBX/8AeXSuG7kgLkDuLHEjczuODtrAzTGGA8p4FvA7ddM09r+IvIJ8JWqLkn2\nWgugCDBbVS+6PbI0srX9vVt8vFPLHzHCqfU/9dT1a/uXzN05l64zurKuxzqK+hfN0jiNMSYnya61\n/bcB/xOR0sC3wBRVXZjRmxoDTm3/qaegdOn01fYBTsedpuuMrnze7nNL/MYYk07p2tVPRCoCj7qO\ngsAUnC8C27ImvDTFZDV/L5OZ2v4lnad3xt/Xn8/afpYlMRpjTE6W2Zp/hrf0FZG6wDiglqr6ZjSA\nzLLk713Wr4cnn4TQUBg9On21/UtmbJ3Bi5EvsvbZtRQuUNj9QRpjTA6X2eSf3nn++UTkPhGZDMzC\n2dXvwfTeVETKicgCEdno2inwhRTK/FNE1rqOxSJSM733MTlHfDy8/Tbcfjs8/zzMnJmxxB99Jppn\nf36WCfdPsMRvjDEZlNa1/e8EHgPaAMuBr4FnVPVcBu+bALykqmtEpDCwSkTmqOqWZGV2Ac1V9bSI\n3A2MARpl8H7GgzZscJr2S5Rw5u2XL5/+a8QnxjN8+XDeWfQOLzd9mWYVm7k9TmOMySvSOuDvZeAr\noK+qnszsTVX1EHDI9fisiGwGygJbkpX5Pdkpv7veN14kIcFZk/+jj5w+/q5d09+3DzBv1zxemPUC\nFYpUYEmXJdxYwpaWMMaYzEhT8lfVVlkVgIiEAXWAZdco1g2nm8F4ic2bnb79oCBYuRIqVkz/Nfad\n3sdLkS+x6s9VDL1rKO1ubIdk5NuDMcaYK6R7Yx93cjX5TwV6q+rZVMq0BDoDTVO7zuDBgy8/joiI\nICIiwq1rAF6jAAAgAElEQVRxmrRLTIQhQ+C99+DNN+HZZ9Nf27+QcIH/Lf0fH/3+Eb0a9uKL9l9Q\nMH/BrAnYGGO8QFRUFFFRUW67XoZH+2f6xiL5gJ+BWar6cSplagHfA3er6s5Uytho/xxi+3anbz9/\nfhg3DsLD03+NWdtn8cLsF7gp5CaG3jWUSsVsNx9jjLladi3ykxXGAZuukfgr4CT+J1JL/CZnSEpy\n5uy/8Qb85z/OaH6fdM0jgT2n9vBi5ItsOLKBj+/+mDZV22RNsMYYYzyT/EXkNuBxYL2IrAYUeAWo\nCKiqjgYGAcHAp+J09MarakNPxGtSt3u3swHPxYvw229QtWr6zr/UxD/k9yG82OhFpjw0Bf98/lkT\nrDHGGMCDzf7uYs3+nqEKY8bAwIEwYAC8+CL4pnOpp8gdkfSa1YsaJWsw5K4hhBUNy5JYjTEmt/Hm\nZn/jpQ4cgG7d4NgxWLgQbropfefvP72fPpF9WHNoDcPuHsa9N9ybNYEaY4xJkSV/k2aq8MUX0K+f\n06//8svO4L60uph4kSG/DeGDpR/Qq2EvJj842Zr48wJVuHABzp51jvPnIS7Oee3Sz6QkZ6pIYqLz\nWMRpSvLxcX7mywd+fuDv7/wMCICwMOd1Y0y62f8ckyaHD0P37rBrF0RGQt266Ts/ak8Uz818jkrF\nKrGs2zIqB1fOmkBN1lKFM2fg4EH48084cuTK48QJOHnyr+PUKSfh+/hA4cLOERBwZSIvUMBJ8JcO\nHx/nPpe+ECQlOetDX7jw1xeGs2edpqe6deHWW52jSpWMrSJlTE5XrBhUqODWS1qfv7muqVOdmn6X\nLvDaa87f67Q6dPYQ/eb0Y9G+RQy9aygPVHvAFurJyeLjYe9e51ve3r2wb99fPw8ccBK+ry/nwsI4\nXLkyR8uV41hoKMdKlOBY0aKcLFyY0/7+nPbz41S+fMT4+hInwgXgQlIScUlJuO1/a1LSlV8IEhLc\ndWVjcpQHzp7lo6eeuuI16/M3WebECSfpr1oFP/4IjdKxs0JiUiKfrfyM1xe+Tte6Xdn03CYKFSiU\ndcGatEtIcKZpbNvmHFu3wo4dsHMnevAgx2+8kT21arG/ShX2V6zI/vr12R8UxEF/fw6JcCghgXhV\nQgsUICR/fkq4juL58xOcLx9V8+WjaL58FMmXj0BfX/x9fPD38cHPx4cCIvjalz9j0qVwekdTp4HV\n/E2KZs2Cp5+Ghx5y1uUPCEj7uSuiV/DszGcJLBDIp/d+yk0h6RwRaNwjMdFJ6uvXw6ZNsHEjbNxI\n0s6dHLjpJrbdcgvbq1Vje/ny7CxWjN3+/uxOTCSfjw9h/v5U8POjvJ8f5f39Ke/nR1k/P0ILFCC0\nQAGCfH2tBccYD8pszd+Sv7lCTAz07Qtz5sD48dCyZdrPPRV3ioHzBzJtyzT+e8d/6ViroyWI7BIb\nC2vXwh9/wJo1sG4dCZs3s71mTTY0bszmatXYXKYMmwsXZhtQLF8+qhYsSNWAAG4oWJDKBQtSyd+f\nSv7+FE3PKE5jjEdY8rfk7za//uosz9uypbM+f1BQ2s5TVb5a/xX95/an3Y3tePf2dylWsFiWxpqn\nxcU5iX7ZMmfXpFWrOHH0KKtbtWJNw4asCw9nfdGibBGhjJ8fNxcqRPVChbgpIIDqAQFUCwigsI2S\nN8arWfK35J9pcXHOYj1TpsDo0dC2bdrP3XZ8G8/NfI5jsccY2XYkjcqlY2CAuT5V2LPHWT5x6VJY\ntoxj+/ezsmVLVjRuzKrwcFYHBnLSx4c6hQtTt3BhahUuTM1ChahRqBCFsqCv0BjjeZb8LflnyqpV\n0KmTs1DPZ59BiRJpOy8uIY73Fr/HiOUjGNhsIL1u7UU+H6tNZlpCgtNsv2gRLF7MheXLWR0Wxm+t\nW/N7jRqsCA7mmI8P9YOCaBAYyC2BgdQrXJjwggXxsS4WY/IMS/6W/DMkPt4ZyDdiBAwdCo89lvYp\n0vN3zafHzB7ULFWToXcNpXyR8lkbbG4WHw8rVkBUFCxcyNFNm1jStCmLW7RgSXg46/z8uKFQIRoF\nBdEoKIiGgYHcGBBgid6YPM6SvyX/dNuyxantFyvmbL1btmzazjty7ggvRb7E4n2LGX7PcO678b6s\nDTQ3SkyE1ath/nz45RcObN3KwlatiGrWjMXly3Mwf34aFylC0yJFuK1IERoEBlr/vDHmbyz5W/JP\ns+Rb7775Jjz7bNpq+0maxNg/xjJwwUCerP0kgyMG25z99Ni1C+bOhblzObR6NQuaNuWXiAiiKlTg\nZIECtChWjBZFi9K8SBFqFi5s8+CNMddlyd+Sf5rs2wedOzszwiZNSvvWuxuPbKT7z91JSEpgVNtR\n1A6tnbWB5gbnzsEvv0BkJDG//EJUhQrMu/de5letSrS/PxHBwbQqWpSIokWpUaiQNeEbY9LNkr8l\n/2tShS+/dObuv/gi9O+ftr1Qzsef561f32L0H6N5PeJ1utfvjq+PjRxP1Y4dMHMmSTNn8sfRo8y5\n7z4ib7mFP4KCaFi0KHcWK8btxYpRLzDQavbGI8LCwti7d6+nwzDpVLFiRfbs2fO31y35W/JP1dGj\nTtP+tm3Obnx16qTtvLk759JjZg/ql6nP0LuGUjqwdNYG6o0SEmDJEpgxgxMLFjAnPJxZbdowu1Il\nggMCuKt4ce4KDqZ50aI23c7kCK5k4ekwTDql9u9myd+Sf4p+/hmeeQY6dnT699OyGc+lAX1L9i/h\nkzaf0KZqm6wP1JvExMCsWeiMGWxev56fWrfmp6ZNWVekCC2Cg2lTvDj3BAcTVrCgpyM15m8s+Xun\nrEr+Now4l4mJgZdegnnz4OuvoXnz65+jqkxYM4F/z/83nWp1YkOPDTag75IjR2DGDBJ+/JFFJ0/y\nY/v2/NSpEwkBAdxXsiQDixcnomhRClrt3hjjRSz55yKLF8OTT0JEhLP6a1qW5916bCvPznyWsxfP\nEtkxkjqhaewbyM2io2HaNM7NmEGkry8/tm/PzJdeIrxwYe4vWZIfSpSgVqFCtm+BMcZrWbN/LnDx\nIrz2GkyYACNHwv33p+GcxIt8sOQDhvw+hEHNB/F8w+fz9oC+Awfgu+84/dNP/BwUxPft2zO/QgUa\nFClC+1KlaFe8OOX9/T0dpTEZZs3+1xcYGMj69esJCwvzdCiXZVWzv0+mojIet2EDNGzo7Ni6Zk3a\nEv/y6OXcMvoWlh5Yyh/d/6B3o955M/H/+ScMH86pO+5gYu/e3BsSQvlBg/j69ddp17o1u5s2ZV69\nevQsW9YSvzFZrFKlSixYsACAiRMn0qxZsyy9X8uWLRk3btwVr8XExGRJ4v/kk09o0KAB/v7+dOnS\nxe3Xzwhr9vdSSUnOsrzvvAPvvw9dulx/wZ6zF88yaMEgvt74NR+1/ohHb3407zVdnzwJ33/P6e+/\nZ7q/P9+2b8+igQNpVbw4j5cqxZTixQmyFfWM8ShVzdTfpsTERHxz0DicsmXLMmjQICIjIzl//ryn\nwwGs5u+V9u2D22+HadNg+XLo2vX6iX/2jtnc/OnNnIg7wYYeG3is5mN5J/GfPw/ffEPsQw/x7eOP\n097Hhwr/+hffDx7MP++6i/3NmvFDrVr8s1QpS/zGeNiWLVvo0aMHv/32G4GBgQQHBwNw8eJF+vXr\nR8WKFSldujTPPfccFy5cAGDhwoWUL1+e//73v5QuXZouXbpw6tQp7rvvPkqWLEnx4sW57777OHjw\nIACvvvoqixYt4vnnnycoKIgXXngBAB8fH3bt2gXAmTNn6NSpEyVLlqRSpUq8/fbbl2O81DLRv39/\ngoODqVy5MrNnz071Mz3wwAO0a9fu8mfJCSz5e5FLC/bUrw+tW8PChRAefu1zjp47SsdpHXlu5nOM\nuW8MEx+YSPGA4tkTsCclJsKCBcR36cL/tW3L48eOUaZnT8a++Sbt7rmHvU2bMr12bUv4xuQw1apV\nY+TIkTRu3JiYmBhOnDgBwIABA9ixYwfr1q1jx44dREdH88Ybb1w+79ChQ5w6dYp9+/YxevRokpKS\n6NKlC/v372ffvn0EBATQs2dPAN566y2aNWvGiBEjOHPmDMOGDQO4okL0/PPPExMTw549e4iKimLS\npEmMHz/+8vvLly+nevXqHD9+nP79+9O1a9fs+PW4jf3V8xLHj0OPHrBxI8yZA3XrXru8qjJ5/WT6\nzelHx1odWd9jfd6Yvrd5MzpxIst/+40v77iDbx97jPDAQB4vV44hJUtSskABT0dojMmAMWPGsH79\neooUKQLAv//9bx5//PHLNXJfX19ef/118ufPD4Cfnx/t27e//Pjll1/m9ttvv+Y9Lg2sS0pK4ptv\nvmHdunUEBARQsWJF+vbtyxdffEHnzp0BZ+W9S/33Tz75JD179uTIkSOULFnS/R8+C3gk+YvIWKAt\ncFhVa6XwfnHgS6A04At8qKoTsjXIHGTOHKdP/5FHnHX5rzf2bO+pvXT/uTuHzh5i5j9nUr9M/ewJ\n1FOOH4evv2bPjBl8Wbkyk+69F9q0oWOFCiwpWZIqAQGejtAYr+COnsCsmFBw9OhRYmNjqV//r79l\nSUlJV4yCDwkJuZz4Ac6fP0+fPn2IjIzk1KlTqCpnz55N03iCY8eOkZCQQIUKFS6/VrFiRaKjoy8/\nDw0Nvfy4YMGCl69vyf/axgPDgUmpvP88sEZV7xGREsBWEflSVROyLcIcIDYWBgyA6dNh4kSnn/9a\nEpMSGbF8BG/++ib9mvSjb+O+5PfNf+2TvFViIsydS8wXXzD1/HkmPvwwG/79b/5RpgxflC5Nw8DA\nvDOmwRg3ySkzAa/+v1uiRAkCAgLYuHEjpUunvNz41ed8+OGHbN++nRUrVhASEsLatWupV6/e5eR/\nrb8PJUqUIH/+/Ozdu5dq1aoBsHfvXsqmdf9zL+CR5K+qi0Wk4jWKHAJquh4HAsfzWuJftcpZmrdO\nHWfBnmLFrl1+w5ENdJvRDb98fiztupQbit+QPYFmt507SRo/nl+XLWP8XXcxvUsXWhQrxgvly3Nv\n8eL4+dgwFmO8XalSpThw4ADx8fHkz58fEeHpp5+mT58+jBgxgpCQEKKjo9m4cSOtW7dO8RoxMTEU\nLFiQoKAgTpw4weDBg/92j0uD+67m4+NDhw4dGDhwIBMnTuT48eMMGTKEf/3rXxn6PImJicTHx5OY\nmEhCQgIXLlwgX758Hp2RkFP/Uo4BaojIQWAt0NvD8WSbxERn+t4998B//gNTplw78V9MvMjgqMG0\nnNiSznU688uTv+S+xB8XB1OmcOCBB3jzgw+o0rgxvf7zH2r/4x9sbdaM6fXq8WBIiCV+Y7xY8pp4\nq1atqFGjBqGhoZeb0d977z2qVKlCo0aNKFq0KK1bt2bbtm2pXq9Pnz7ExsZSokQJmjRpQps2V+5V\n0rt3b7777juKFy9Onz59/hbDsGHDCAgIIDw8nObNm9OxY8fL/f3Xi/9qb731FgEBAbz//vtMnjyZ\ngICAK2YPeILHVvhz1fx/SqXPfyAQoqp9RKQyMBeopapnUyiba1b4270bnngCChRwmvnLl792+WUH\nltF1RlcqB1fm0zafUjYo9zRJAbB5MxfHjOHnrVv5vH17fg8P59HSpelStiz1rVnfmHSxFf68U17b\n2Oc24G0AVd0pIruBasDKlAonb86JiIggIiIi6yN0I1Un2ffvDy+/DH36wLUqsecunmPQL4OYsmEK\nH9/9MY/c9EjuSYRxcTB1KjumTuXz8HAm3HUXNz7yCN0qVWJqSAgBOWjhDmOMyS5RUVFERUW57Xqe\nrPmH4dT8a6bw3ofAGVV9XURK4ST92qp6IoWyXl3zP3YMnn0Wtm2DyZOh5t9+G1eav2s+z/z8DE3K\nN2HoXUNzz5z97du5OGoUP2zfzpgHHmBdxYo8Wa4c3cqV40YbrW9MplnN3ztlVc3fI8lfRL4CIoDi\nwGHgNaAAoKo62jXCfzxQARDgXVWdksq1vDb5R0Y6q/P94x/w9tvXnsJ3Ku4U/ef0J3JnJCPbjqRN\n1TapF/YWCQnw00/s/uorRpcpw/g2bagRFET3ypW5v0QJ68M3xo0s+XunXJX83ckbk//5884Uvh9/\ndHbia9Xq2uVnbJ3BczOfo92N7XjvjvcI8kvDXr052eHDJH7+OTOXLeOztm1ZUbkyT5YrxzPly1st\n35gsYsnfO+W1Pv9ca/VqePxxp3n/elP4jp47yguzX2DlwZVMfnAyLcJaZF+g7qYKy5Zx9PPP+VyV\nkQ8+SGjz5jxXtSrTQkIoaH35xhiTbazmn00SE+F//3OOjz5y5vCnNkZPVfl6w9e8GPkiT9R6gtdb\nvk5Afi+tEV+4AN98w7Iff2REgwb83KABD5YsyXOVKlE/MNDT0RmTZ1jN3ztZs38qvCH579sHnTo5\n2/BOmgTX2i76YMxBeszswc4TOxl3/zgalm2YbXG61Z9/cmHkSL7dtInh7dtzLDSU5ypXpkuZMgTn\nz6WrDhqTg1ny906W/FOR05P/V185U/deesmZypda67aqMmHNBAbMG8CztzzLwGYD8cvnl73BusOq\nVRwaNYrPfH0Z1a4dNYsUodcNN3Bv8eL45pbpiMZ4IUv+3sn6/L3MqVPQsyf88QfMng316qVedu+p\nvTzz8zMcPXeUOU/MoU5onewL1B0SE2H6dFZ++y0f16zJzx068FipUiwID+emQnlgJ0FjTK4QGBjI\n+vXrCbtW82wuYXOpssCvvzpr8hct6qzRn1riT9IkPlvxGbeMuYUWFVuwrNsy70r8Z8+SMGwYUx99\nlKZnzvDQ009T6/HH2dWiBZ/WrGmJ3xiTZpUqVWLBggUATJw4kWbNmmXp/Vq2bMm4ceOueC0mJsbt\nif/ixYt069aNsLAwihQpQr169Zg9e7Zb75ERVvN3o4sX4bXXnNX6xoyBe+9Nveyuk7voOqMr5+PP\ns/CphdwUclP2BZpZBw5w5rPPGHfgAB8/9BCla9fmxerVaV+iBPlsbr4xJpPSsu3utSQmJnp005zk\nLm0NvGjRIsqXL8/MmTPp0KEDGzZsuGLL4Gynql59OB/B87ZsUa1fX7VtW9XDh1Mvl5iUqB///rEW\nf7+4frDkA01ITMi+IDNr7Vrd9+yz2rdPHw2OjNQOy5bpb6dOeToqY0wa5JS/lakJCwvT+fPn6+bN\nm9Xf31/z5cunhQsX1mLFiqmq6oULF7Rv375aoUIFDQ0N1R49emhcXJyqqkZFRWm5cuX0/fff19DQ\nUO3UqZOePHlS27ZtqyEhIRocHKxt27bV6OhoVVUdOHCg+vr6asGCBTUwMFB79eqlqqoiojt37lRV\n1dOnT+sTTzyhISEhGhYWpm+99dblWCdMmKBNmzbVfv36abFixTQ8PFxnzZqV5s9aq1YtnTZtWprK\npvbv5no9w7nTqmmZpAqjRkHTptCtG8yYAa5NqP5m+/HttJjQgm83fsvSrkvp16Qfvj4549tpqlRh\n3jz+6NSJx3/6idoPPURS16780bw53zRsSKMiRTwdoTEmF6lWrRojR46kcePGxMTEcOKEs6r7gAED\n2LFjB+vWrWPHjh1ER0fzxhtvXD7v0KFDnDp1in379jF69GiSkpLo0qUL+/fvZ9++fQQEBNCzZ0/A\n2WWvWbNmjBgxgjNnzjBs2DDgyp35nn/+eWJiYtizZw9RUVFMmjSJ8ePHX35/+fLlVK9enePHj9O/\nf3+6du2aps93+PBhtm/fTo0aNTL9u8qUzHxzyAkHHvw2e+SIart2qnXrqm7alHq5hMQE/XDph1r8\n/eI69Leh3lHbj4/XpClTdPajj2qrkSO17Ny5+t+dO/XkxYuejswYkwGe/FuZFpdq/qpOzbpZs2ZX\nvF+oUCHdtWvX5edLly7VSpUqqapT8/fz89OL1/j7tHr1ag0ODr78PCIiQseOHXtFmUs1/8TERC1Q\noIBu2bLl8nujRo3Sli1bXo6vatWql9+LjY1VHx8fPXytZl9VjY+P1zvuuEN79OhxzXLJpfbvRiZr\n/tbnn0GRkdCli7NYz3ffOdvwpmTrsa10nt6Z/L75+b3b71QJrpK9gaZXbCzx48fzzZIlfNCuHdqt\nG/1vuol/lCpFAevPNyZXk9czPx1XX3P/dMKjR48SGxtL/fr1L7+WlJR0qQIIQEhICPmTrSFy/vx5\n+vTpQ2RkJKdOnUJVOXv2LKrXH09w7Nixy331l1SsWJHo6OjLz0NDQy8/Lliw4OXrl0yl6VdV6dix\nI35+fgwfPjztHz6LWPJPp7g4Z13+H36AL7+Eli1TLpeYlMiQ34fw3uL3GBwxmOcaPIeP5ODkeeoU\n5z77jM+3beOj9u2p3L077998M3cFB+ee7YKNMdeUFYk7I67+m1OiRAkCAgLYuHEjpUuXTtM5H374\nIdu3b2fFihWEhISwdu1a6tWrdzn5X+vvWokSJcifPz979+6lWrVqAOzdu5eyZctm+DN17dqVY8eO\n8X//9385YjCiJf90WL8e/vlPqF4d1qyB4OCUy205toXO0zvjn8+f5U8vJ7xYePYGmh5//snxESMY\nfuIEn7ZrR/OICKbWqEGDIC/fPMgY47VKlSrFgQMHiI+PJ3/+/IgITz/9NH369GHEiBGEhIQQHR3N\nxo0bad26dYrXiImJoWDBggQFBXHixAkGDx78t3vs2rUrxXN9fHzo0KEDAwcOZOLEiRw/fpwhQ4bw\nr3/9K0Of59lnn2XLli3MmzePAqk1E2ezHFwVzTmSkuDjj53d9/r1g2++STnxJyYl8sGSD2g6rikd\na3Zkfqf5OTfx79nD/r596fPhh1Rt0YLorl1Z3KIFUxs3tsRvjMl2yWvirVq1okaNGoSGhl5uRn/v\nvfeoUqUKjRo1omjRorRu3Zpt27aler0+ffoQGxtLiRIlaNKkCW3aXLkNeu/evfnuu+8oXrw4ffr0\n+VsMw4YNIyAggPDwcJo3b07Hjh3p3LlzmuJP7tIAxDVr1lCqVCkCAwMJCgpiypQUd6nPNra873X8\n+Sd07uys2Dd5MlSunHK55LX9se3G5tykv2UL2z/5hPeLFGFaixZ0LlWKl268kbJ+XriUsDEmzWx5\nX+9ky/t6wE8/wTPPOMerr0JK+9EkJiXy0W8f8f6S93k94nV6NOiRM/v2165l7ciRvFuuHPPbtaNn\nuXJsr1KF4rbJjjHG5DmW/FMQG+s078+aBVOnwm23pVzu0kj+Ar4Fcm7f/ooVLPv8c96+4QZWtm/P\nS+HhjAkLIzCf/dMbY0xeZRngKmvWOIP66tVzHqe0hk1iUiIfL/uYdxa9k3NH8i9dyq/jx/NWrVps\neeghBlStyjcVKlAwB4wyNcYY41nW5++SlARDhsD77zs/H3885XLbj2+n8/TO+IgP4+8fT+XgVAYB\neIj++isLvviCN+rX50B4OK9Ur84TZcvaHH1j8jjr8/dOWdXnb8kfOHgQnnoKzp1z5u5XqvT3Mkma\nxIjlI3hj4RsMaj6IXrf2ylG1fY2KYs7kybzRsCHHKlbk1Ro1eKx0adtoxxgDWPL3Vpb8U5HZ5D99\nOnTvDj16wMCBkFJX+K6Tu+gyvQvxSfFMuH8CVYtXzUTE7qULFzJ78mQGN2pETLlyDLr5ZjqULo2v\nLcxjjEnGkr93suSfiowm/9hYeOklZ5neyZOhSZO/l0nSJEatHMWgXwbxctOX6dOoT47ZiEcXLmSW\nK+mfK1eO12rW5OHQUHws6RtjUmDJ3zvZVD83WrMGHnsM6tdPfVDf3lN76TqjK2cunGFR50VUD6me\n/YGmQJcsYfakSbx2663EPvwwr9WsyUOW9I0xxqRDnuoQTkqCjz6CO+90mvi//PLviV9VGfvHWG4Z\ncwutKrViadelOSLx67JlzOnRg8ZbttDvoYfod889rLvzTh4pXdoSvzHGuEFgYCB79uzxdBjZIs8k\n/z//hHvucXbgW7bM2Y3vagdjDtJ2SltGrBjB/E7zeaXZK+Tz8XDjyJo1LOjZk2Zr1tD7gQfo40r6\nHSzpG2NykUqVKrFgwQIAJk6cSLNmzbL0fi1btmTcuHFXvBYTE0NYWJjb7/XEE09QunRpihQpQuXK\nlXn77bfdfo/0yhPJ/+efnXn7t94KixZB+FVr8agqk9dNps7IOtxS+haWdVtGrVK1PBPsJZs2sfiF\nF2i5ZAnd27Xj2TZt2NC6NY+WKWOD+YwxuVpatt29lsTERDdGk3kvv/wyu3fv5vTp08yaNYvhw4cT\nGRnp2aBUNdsPYCxwGFh3nXINgHjgwWuU0dTExqo+/7xqhQqqv/6acpnDZw/rg988qDU+qaEro1em\neq1ss2OHLu/TR+8aMkTDZs/Wcbt3a3xioqejMsZ4uWv9rcwJwsLCdP78+bp582b19/fXfPnyaeHC\nhbVYsWKqqnrhwgXt27evVqhQQUNDQ7VHjx4aFxenqqpRUVFarlw5ff/99zU0NFQ7deqkJ0+e1LZt\n22pISIgGBwdr27ZtNTo6WlVVBw4cqL6+vlqwYEENDAzUXr16qaqqiOjOnTtVVfX06dP6xBNPaEhI\niIaFhelbb711OdYJEyZo06ZNtV+/flqsWDENDw/XWbNmpelzbtmyRcuVK6erVq1KU/nU/t1cr2c4\nD3uq5j8euOtaBUTEB3gPyNDXow0boGFDOHIE1q6FlFqQftj8A7VH1qZKsSqsfGYl9cvUz8it3CM6\nmvUDBvDA99/T/u67eeC++9h65510DguzufrGmDyjWrVqjBw5ksaNGxMTE8OJEycAGDBgADt27GDd\nunXs2LGD6Oho3njjjcvnHTp0iFOnTl3eRS8pKYkuXbqwf/9+9u3bR0BAAD179gTgrbfeolmzZowY\nMYIzZ84wbNgw4Mqd+Z5//nliYmLYs2cPUVFRTJo0ifHjx19+f/ny5VSvXp3jx4/Tv39/unbtes3P\n1bNnTwoVKsTNN9/MwIEDqVevntt+ZxmSmW8OmTmAilyj5g/0BnoA40hHzT8pSXX4cNUSJVTHj3ee\nX+ety2sAABKlSURBVO1E7AntOK2jVhlWRZfsW5LqN65sceSIbhs0SB97/XUtNXu2frRli55PSPBs\nTMaYXOfqv5U5zaWav6pTs27WrNkV7xcqVEh37dp1+fnSpUu1UqVKqurU/P38/PTixYupXn/16tUa\nHBx8+XlERISOHTv2ijKXav6JiYlaoEAB3bJly+X3Ro0apS1btrwcX9WqVS+/Fxsbq//f3r1HR1Xd\nCxz//oaHJpBAQhIIBEJSCAg+AyhIY0LANGlZVJfX3GoDCtfYVqnEglJMu/TqZUmvVaihKGhBRPGB\nWIGL5aE01QrWJyCpkbC4xBAumBgwiQHJ43f/mGEaIC+SmUwm+X3Wylo5Z/bZs88vA785Z++zt8Ph\n0GPHjjV5jnV1dZqbm6v9+vXTDz74oMmyZzT2d6ONV/4d8lE/ERkI3KCqk0Tk6pYeV1ICs2bB0aOw\ncycMb2Aunq0HtnLHpju4YcQN7P7Zbnr17OXBll+A8nKK/vhHHq6s5M/XXce9gwaxYuRIetuCO8YY\nX/HEeCIvzCVQUlJCVVUVY8b86+5sXV3dWc+/h4eH06PeKqUnT54kKyuLrVu3cuLECVSVysrKFo0n\nKC0tpaamhiFDhrj3RUdHU1xc7N4eMGCA+/eAgAB3/REREY3WKyIkJiZy880389JLLzFu3LiWBcAL\nOmqmWQLMr7fd7Cdy+3bnFL0ZGbB+PfTsefbrlacrmbdtHn858Bee+/FzTI6d7NEGt9ipU5Q88wyP\nFhayesoU7oyIoOCyywixpXWNMb7mhcTdGucm57CwMAIDA8nLyyMyMrJFxzz++OMUFBTw4YcfEh4e\nzp49e4iPj3cn/6a+AISFhdGjRw8KCwsZOXIkAIWFhQwaNKiNZ+ZUU1NDYGCgR+pqrY6a/McCL4vz\nrxMGpIlItapubKjwtdc+xL59cMMNkJaWRM+eSWe9/m7hu9y+4XYSoxPZ+/O99Lm4gVl9vK22lvI1\na3ji00/JSU3llrFj2RcfT+RFF7V/W4wxpgPr378/hw8fprq6mh49eiAiZGZmkpWVxdKlSwkPD6e4\nuJi8vDxSUlIarKOiooKAgACCg4MpKyvjoYceOu89Dh482OCxDoeD9PR0srOzWb16NV9//TWLFy/m\n/vvvv+BzKSkpYceOHUydOpWAgAC2b9/OunXr2L59+wXVk5ubS25u7gW/f6Pa0mfQlh9gKPBZC8qt\nopk+/2nTVEtKzu8TOVl9UudunauRv4/UDfkbmupW8Z66Oj31xhu6+O67NWLTJp3+t7/pwaoq37TF\nGNNl0cH7/GNiYtx9/qdPn9apU6dqaGiohoeHq6rqqVOn9IEHHtDY2Fjt06ePjho1SnNyclTV2ec/\nePDgs+o7cuSIJiUlae/evXXEiBG6YsUKdTgcWut6emrXrl0aFxenoaGhOmfOHFVVdTgc7tH+x48f\n14yMDA0PD9chQ4acN9r/3DEJ9Y+tr6SkRBMTEzUkJET79u2r48aN040bN7Y4Lo393Whjn79P5vYX\nkbVAEtAP5yN/DwI9XSez4pyyK4H/UdXXG6lL6+r0vK6qj498zIw3ZjAqfBRP/egpwgLDPH8izah9\n911eeOUVHpw0icv79GHhNddwWVBQu7fDGGNsbn//ZAv7NOLchX2qa6t59O+PsvSDpSxJXcItl97S\npskiWkPz8tj8zDMsGDOGPuHhLLr6ar4fGtqubTDGmPos+fsnW9inBfJL85nx5xmEBoTy6c8+ZVCw\nZwZntNjhw+zKyWF+dDRlP/gBj155JVMHDGj3Lx/GGGNMUzrFlX9tXS05/8jhkXce4ZFJj/DzsT9v\n34T7zTd8kZPDgu7d+eiKK3g4Lo7psbE2Da8xpsOwK3//ZFf+TZjy/BRO1Zzi/TveZ1josPZ749On\nOfrsszxUXMz6hATuHziQF0ePJqBbt/ZrgzHGGHOBOkXyT/leCvddex/dHO2UdFWpWL+e3+/cydLr\nr2dmfDxfjBlDqD2rb4wxxg90itv+7XkONe+9x7Pr1vGfyclM7t2b/5owgaEBAe32/sYY0xp2298/\n2Wj/RrRX8teCAjYtX878+HgGhoby2IQJxPfxwWRBxhjTCpb8/ZMl/0Z4PfmXlfHRk08yLzyc0pgY\n/vuqq0izEfzGGD9jyd8/eSv521qxjfnuO77MySFj6VKmXXMNP50yhd2pqfwwMtISvzHGtJOioiKC\ng4O98sXF4XA0OsVva61du5bU1FSP1ukNlvzPpUr566+z4De/4aphw4hNS+OL668nc8QIujssXMYY\n4w1Dhw4lMDCQ4OBggoKCCA4O5ujRowwePJjy8nKvXHQ1VWdSUpJ7bYCIiAhuuukmjh071mydt956\nK1u2bGnR+69evZqEhIQWt9eTLJvVU/PhhyyfO5cR3brxf2lp7Jk0iYfHjSPIltk1xhivEhE2b95M\neXk5FRUVlJeXn7Vsrjc0dTdBRFi2bBnl5eXs37+fEydOcO+993r8/X11J9mSP8Dhw2xdsIAr8/N5\necoUNicl8VxyMlEXX+zrlhljTJfRUDIuLCzE4XBQV1fH8ePHGTx4MJs3bwbg22+/Zfjw4bzwwgsA\nnD59mnnz5hEdHU1kZCR33XUX3333nbuuxx57jIEDBxIVFcWqVauaTbxn2tO3b19uuukm9u3bB0B5\neTkzZswgIiKCmJgYFi5c6D7m3Kt5h8PB8uXLiYuLIzQ0lNmzZwOQn5/PL37xC3bt2kVQUBCh7TwF\nfNdO/lVV/POxx0h79VV++f3vszApiR1paTaK3xhjOpAzSTokJISVK1eSmZlJSUkJWVlZxMfHk5GR\nAcD8+fM5cOAAe/fu5cCBAxQXF/Pwww8DsGXLFp544gnefvttCgoKeOutt1r8/qWlpaxfv574+HgA\nZs+eTUVFBYcOHSI3N5fnn3+eVatWndfeMzZv3szHH3/Mnj17ePXVV9m2bRsjR47k6aefZsKECVRU\nVFBWVtamGF2wtiwJ2BF+aM0ylbW1WvLii3pXdraGvfmmLt6zR79zLfNojDGdUav+r2xHQ4cO1aCg\nIA0JCdGQkBC98cYbVVX10KFDZy3Fq6p6zz336GWXXaZRUVFaVlbm3t+rVy89ePCge3vnzp0aExOj\nqqqzZs3SBQsWuF/bv39/o8vwqqomJSVpYGCghoSEaFRUlGZkZGhpaanW1tZqz549NT8/3112+fLl\nOmnSJFU9f7lfEdGdO3e6t9PT0/V3v/tdg2Ub0tjfjTYu6dvlOrNP79rF0vXreTQpiVtSUsifMIF+\nNjOfMcYgubltrkOTklp97IYNG5g0aVKz5TIzM8nJySE7O5uQkBAASkpKqKqqYsyYMe5ydXV17lv3\nR44cYezYse7XoqOjm32CICcnh1mzZp2176uvvqKmpoYhQ4acVVdxcXGj9fTv39/9e2BgIJWVlc2e\no7d1meSvRUVseuop5l5+OcMnT+adhAQu6d3b180yxpgOoy2J2yPv34LH+erq6rjzzju57bbbWLZs\nGTNnziQ2NpawsDACAwPJy8sjMjLyvOMiIyMpKipybxcWFrZqsF1YWBg9evSgsLCQkSNHuusaNOjC\nV5H15WPjnb/P/+RJ9j3+OCmvv878iRPJSUjgzbQ0S/zGGOMn6n8pWLhwIQ6Hg5UrVzJv3jymT5/u\nHjWfmZlJVlYWJSUlABQXF7Nt2zYA0tPTee655/j888+pqqpyjwW4UA6Hg/T0dLKzs6msrKSwsJDF\nixczffr0C66rf//+HD58mOrq6la1pS06b/JX5evXXmP2woUkX3IJ05KS2JuWRmorvp0ZY4zxrqau\ngs+89sknn7BkyRLWrFmDiDB//nwcDgeLFi0CYNGiRQwbNozx48fTt29fUlJS2L9/PwCpqalkZWWR\nnJxMXFwckydPbnV7nnzySQIDA4mNjeW6664jIyODmTNntqie+tvJycmMHj2aAQMGEBER0WR7PK1T\nTu9bvXs3T69dyyMJCfx7UBAPTZxo/frGmC7Npvf1Tza3fyPOSv6lpbyVk8OcYcOIDA1lSUIClwYH\n+7aBxhjTAVjy90/eSv6dY8BfTQ0Hn32WX1VU8Nm4cTxx6aVMi462OfiNMcaYBnSK5J/929+yPCGB\nueHhvBwfz8Xduvm6ScYYY0yH1SmS/5dJSexJTGSQTcdrjDHGNKtz9fkbY4xpkPX5+ydv9fl33kf9\njDHGGNMgS/7GGGNMF9Mp+vyNMcY0LdqegPJL0dHRXqnXJ33+IvInYCpwTFUvb6TMk0Aa8C1wu6ru\nbqSc9fkbY4zpUvy1z38V8IPGXhSRNOB7qjoc+BnwdHs1zJwv1wMrfZnmWZy9z2LsfRZj/+CT5K+q\nfweON1Hkx8DzrrL/APqISP8myhsvsn/M7cPi7H0WY++zGPuHjjrgbxBQVG+72LXPGGOMMW3UUZO/\nMcYYY7zEZ5P8iEg0sKmhAX8i8jTwV1V9xbWdDySq6rEGytpoP2OMMV2Ovy7sI66fhmwE7gZeEZHx\nwImGEj+07eSNMcaYrsgnyV9E1gJJQD8R+RJ4EOgJqKquUNU3ReSHInIA56N+M33RTmOMMaYz8vu5\n/Y0xxhhzYfx6wJ+IpIpIvojsF5H5vm5PZyAiUSKyQ0TyROQzEbnHtT9ERLaJyBcislVE+vi6rf5O\nRBwi8omIbHRtW4w9SET6iMg6Efnc9Xm+xmLseSKywBXfvSLyooj0tDi3jYj8SUSOicjeevsajanr\nb1Dg+qyntOQ9/Db5i4gDWIpzsqDRwC0iMtK3reoUaoBfqepoYAJwtyuuvwbeUtURwA5ggQ/b2FnM\nAf5Zb9ti7Fl/AN5U1UuAK4B8LMYe5Rq4nQlc5Rq83R24BYtzWzU0EV6DMRWRUUA6cAnOWXGXSQvm\ncfbb5A9cDRSoaqGqVgMv45wcyLSBqh49M5WyqlYCnwNROGO72lVsNXCDb1rYOYhIFPBD4Nl6uy3G\nHiIiwUCCqq4CUNUaVf0Gi7GnlQOngV4i0h0IwDkvi8W5DRqZCK+xmE4DXnZ9xg8BBTjzY5P8Ofmf\nOxHQYWwiII8SkaHAlcD7QP8zT1yo6lEgwnct6xQWA/cB9QfdWIw9JwYoFZFVrq6VFSISiMXYo1T1\nOPA48CXOpP+Nqr6FxdkbIhqJaasmxfPn5G+8SER6A68Bc1x3AM4dGWojRVtJRH6Ec1Gr3TT+uCtY\njNuiOxAP/FFV43E+NfRr7HPsUSISC9wLRAMDcd4B+CkW5/bQppj6c/IvBobU245y7TNt5Lp99xqw\nRlU3uHYfO7O+gogMAL7yVfs6gYnANBE5CLwEJIvIGuCoxdhjDgNFqvqRa3s9zi8D9jn2rLHAe6pa\npqq1wJ+Ba7E4e0NjMS0GBtcr16Jc6M/J/0NgmIhEi0hP4Cc4JwcybbcS+Keq/qHevo3A7a7fbwM2\nnHuQaRlVfUBVh6hqLM7P7Q5VnQ5swmLsEa7bo0UiEufaNRnIwz7HnvYFMF5ELnYNMpuMcxCrxbnt\nzp0Ir7GYbgR+4nrKIgYYBnzQbOX+/Jy/iKTiHNHrAP6kqot83CS/JyITgXeAz3DeVlLgAZwfpldx\nfsMsBNJV9YSv2tlZiEgiMFdVp4lIKBZjjxGRK3AOqOwBHMQ5WVg3LMYeJSL34UxKtcCnwB1AEBbn\nVqs/ER5wDOdEeG8A62ggpiKyAPgPoBpnV+22Zt/Dn5O/McYYYy6cP9/2N8YYY0wrWPI3xhhjuhhL\n/sYYY0wXY8nfGGOM6WIs+RtjjDFdjCV/Y4wxpoux5G+McRORbBHZJyJ7XHPijxOROSJycQuObVE5\nY4zv2XP+xhgARGQ8zkVaElW1xjXp0EXATmCMqpY1c/z/tqScMcb37MrfGHNGJFCqqjUAriT+bzgX\nbPmriLwNICLLROQDEflMRB507ftlA+VSRGSniHwkIq+4VtUzxnQAduVvjAFARHoBf8e5JvvbwCuq\n+o5rAaIxruVbEZG+qnpCRByucr9U1X31y4lIP+B1IFVVT4rI/cBFqvqIT07OGHOW7r5ugDGmY1DV\nb0UkHkgAkoGXXXOGw9kLjPxERDJx/v8xABgF7OPshUjGu/a/51rwpQewy/tnYYxpCUv+xhg3dd4K\nfAd4R0Q+w7l6mJuIDAXm4rzCLxeRVUBDg/wE2KaqP/Vui40xrWF9/sYYAEQkTkSG1dt1JXAIqACC\nXfuCgUqgwrW2eFq98uX1yr0PTBSR77nqDhSR4V5svjHmAtiVvzHmjN5Ajoj0AWqAA8CdwK3AFhEp\nVtXJIrIb+BwowjlG4Ixnzik3E3hJRC7CuTT0b4CCdjwfY0wjbMCfMcYY08XYbX9jjDGmi7Hkb4wx\nxnQxlvyNMcaYLsaSvzHGGNPFWPI3xhhjuhhL/sYYY0wXY8nfGGOM6WIs+RtjjDFdzP8D4GiINWmG\nBl0AAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10c98a978>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "label = 'Policy iteration'\n",
    "iters = [1, 2, 3]\n",
    "v_init = np.zeros(ddp.num_states)\n",
    "\n",
    "fig, ax = plt.subplots(figsize=(8,5))\n",
    "sigma = ddp.compute_greedy(v_init)\n",
    "for i in range(iters[-1]):\n",
    "    # Policy evaluation\n",
    "    v_sigma = ddp.evaluate_policy(sigma)\n",
    "    if i+1 in iters:\n",
    "        ax.plot(-v_sigma, label='Iteration {0}'.format(i+1))\n",
    "    # Policy improvement\n",
    "    new_sigma = ddp.compute_greedy(v_sigma)\n",
    "    sigma = new_sigma\n",
    "ax.plot(-results['Policy iteration'].v, label='Fixed Point')\n",
    "\n",
    "ax.ticklabel_format(style='sci', axis='y', scilimits=(0,0))\n",
    "ax.set_ylim(1e5, 4.2e5)\n",
    "ax.set_yticks([1e5 + 0.4e5 * i for i in range(9)])\n",
    "ax.set_title(label)\n",
    "ax.set_xlabel('State')\n",
    "ax.set_ylabel(r'Value $\\times\\ (-1)$')\n",
    "plt.legend(loc=4)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Modified policy iteration"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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KKV6+9mUeb/c4zjWdy/dmq5Bdu2DwYLP4zoQJZjkDIcTlyWh/Cf7VWl5aHjGz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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10db22048>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "label = 'Modified policy iteration'\n",
    "iters = [1, 2, 3, 4]\n",
    "v = np.zeros(ddp.num_states)\n",
    "k = 20 #- 1\n",
    "\n",
    "fig, ax = plt.subplots(figsize=(8,5))\n",
    "for i in range(iters[-1]):\n",
    "    # Policy improvement\n",
    "    sigma = ddp.compute_greedy(v)\n",
    "    u = ddp.bellman_operator(v)\n",
    "    if i == results[label].num_iter-1:\n",
    "        diff = u - v\n",
    "        break\n",
    "    # Partial policy evaluation with k=20 iterations\n",
    "    for j in range(k):\n",
    "        u = ddp.T_sigma(sigma)(u)\n",
    "    v = u\n",
    "    if i+1 in iters:\n",
    "        ax.plot(-v, label='Iteration {0}'.format(i+1))\n",
    "ax.plot(-results['Policy iteration'].v, label='Fixed Point')\n",
    "\n",
    "ax.ticklabel_format(style='sci', axis='y', scilimits=(0,0))\n",
    "ax.set_ylim(0, 2.8e5)\n",
    "ax.set_yticks([0.4e5 * i for i in range(8)])\n",
    "ax.set_title(label)\n",
    "ax.set_xlabel('State')\n",
    "ax.set_ylabel(r'Value $\\times\\ (-1)$')\n",
    "plt.legend(loc=4)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Increasing the discount factor"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let us consider the case with a discount factor closer to $1$, $\\beta = 0.9999$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "ddp.beta = 0.9999"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "v_init = np.zeros(ddp.num_states)\n",
    "epsilon = 1164\n",
    "ddp.max_iter = 10**5 * 2\n",
    "\n",
    "results_9999 = {}\n",
    "\n",
    "for i in range(4):\n",
    "    k = 20 if labels[i] == 'Modified policy iteration' else 0\n",
    "    results_9999[labels[i]] = \\\n",
    "        ddp.solve(method=methods[i], v_init=v_init, epsilon=epsilon, k=k)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "df_9999 = pd.DataFrame(index=labels, columns=columns)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The numbers of iterations:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "118900 \t(Value iteration)\n",
      "288 \t(Value iteration with span-based termination)\n",
      "7 \t(Policy iteration)\n",
      "18 \t(Modified policy iteration)\n"
     ]
    }
   ],
   "source": [
    "for label in labels:\n",
    "    print(results_9999[label].num_iter, '\\t' + '(' + label + ')')\n",
    "    df_9999[columns[0]].loc[label] = results_9999[label].num_iter"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Policy iteration gives the optimal policy:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0\n",
      " 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1\n",
      " 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1]\n"
     ]
    }
   ],
   "source": [
    "print(results_9999['Policy iteration'].sigma)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Takes action 1 (\"replace\") if and only if $s \\geq \\bar{\\gamma}$, where $\\bar{\\gamma}$ is equal to:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "43"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(1-results_9999['Policy iteration'].sigma).sum()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Check that the other methods gave the correct answer:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "True\n",
      "True\n",
      "True\n"
     ]
    }
   ],
   "source": [
    "for result in results_9999.values():\n",
    "    if result != results_9999['Policy iteration']:\n",
    "        print(np.array_equal(result.sigma, results_9999['Policy iteration'].sigma))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\lVert v - v_{\\mathrm{pi}}\\rVert$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "581.998786509 \t(Value iteration)\n",
      "4.37767255306 \t(Value iteration with span-based termination)\n",
      "0.0 \t(Policy iteration)\n",
      "4.45491330326 \t(Modified policy iteration)\n"
     ]
    }
   ],
   "source": [
    "for label in labels:\n",
    "    diff_pi = \\\n",
    "        np.abs(results_9999[label].v - results_9999['Policy iteration'].v).max()\n",
    "    print(diff_pi, '\\t' + '(' + label + ')')\n",
    "    df_9999[columns[2]].loc[label] = diff_pi"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\lVert v - T(v)\\rVert$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.0581999123096 \t(Value iteration)\n",
      "0.0558041036129 \t(Value iteration with span-based termination)\n",
      "5.96046447754e-08 \t(Policy iteration)\n",
      "0.052891805768 \t(Modified policy iteration)\n"
     ]
    }
   ],
   "source": [
    "for label in labels:\n",
    "    v = results_9999[label].v\n",
    "    diff_max = \\\n",
    "        np.abs(v - ddp.bellman_operator(v)).max()\n",
    "    print(diff_max, '\\t' + '(' + label + ')')\n",
    "    df_9999[columns[4]].loc[label] = diff_max"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\overline{b} - \\underline{b}$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.000446990132332 \t(Value iteration)\n",
      "1160.37531587 \t(Value iteration with span-based termination)\n",
      "1099.68602686 \t(Modified policy iteration)\n"
     ]
    }
   ],
   "source": [
    "for i in range(4):\n",
    "    if labels[i] != 'Policy iteration':\n",
    "        k = 20 if labels[i] == 'Modified policy iteration' else 0\n",
    "        res = ddp.solve(method=methods[i], v_init=v_init, k=k,\n",
    "                        max_iter=results_9999[labels[i]].num_iter-1)\n",
    "        diff = ddp.bellman_operator(res.v) - res.v\n",
    "        diff_span = (diff.max() - diff.min()) * ddp.beta / (1 - ddp.beta)\n",
    "        print(diff_span, '\\t' + '(' + labels[i] + ')')\n",
    "        df_9999[columns[3]].loc[labels[i]] = diff_span"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For policy iteration:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.000893980264664 \t(Policy iteration)\n"
     ]
    }
   ],
   "source": [
    "label = 'Policy iteration'\n",
    "v = results_9999[label].v\n",
    "diff = ddp.bellman_operator(v) - v\n",
    "diff_span = (diff.max() - diff.min()) * ddp.beta / (1 - ddp.beta)\n",
    "print(diff_span, '\\t' + '(' + label + ')')\n",
    "df_9999[columns[3]].loc[label] = diff_span"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Value iteration\n",
      "1 loops, best of 3: 4.27 s per loop\n",
      "Value iteration with span-based termination\n",
      "10 loops, best of 3: 19.3 ms per loop\n",
      "Policy iteration\n",
      "100 loops, best of 3: 1.99 ms per loop\n",
      "Modified policy iteration\n",
      "100 loops, best of 3: 4.32 ms per loop\n"
     ]
    }
   ],
   "source": [
    "for i in range(4):\n",
    "    k = 20 if labels[i] == 'Modified policy iteration' else 0\n",
    "    print(labels[i])\n",
    "    t = %timeit -o ddp.solve(method=methods[i], v_init=v_init, epsilon=epsilon, k=k)\n",
    "    df_9999[columns[1]].loc[labels[i]] = t.best"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Iterations</th>\n",
       "      <th>Time (second)</th>\n",
       "      <th>$\\lVert v - v_{\\mathrm{pi}} \\rVert$</th>\n",
       "      <th>$\\overline{b} - \\underline{b}$</th>\n",
       "      <th>$\\lVert v - T(v)\\rVert$</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>Value iteration</th>\n",
       "      <td>118900</td>\n",
       "      <td>4.2675</td>\n",
       "      <td>581.999</td>\n",
       "      <td>0.00044699</td>\n",
       "      <td>0.0581999</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Value iteration with span-based termination</th>\n",
       "      <td>288</td>\n",
       "      <td>0.0193439</td>\n",
       "      <td>4.37767</td>\n",
       "      <td>1160.38</td>\n",
       "      <td>0.0558041</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Policy iteration</th>\n",
       "      <td>7</td>\n",
       "      <td>0.00199425</td>\n",
       "      <td>0</td>\n",
       "      <td>0.00089398</td>\n",
       "      <td>5.96046e-08</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Modified policy iteration</th>\n",
       "      <td>18</td>\n",
       "      <td>0.00431983</td>\n",
       "      <td>4.45491</td>\n",
       "      <td>1099.69</td>\n",
       "      <td>0.0528918</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "                                            Iterations Time (second)  \\\n",
       "Value iteration                                 118900        4.2675   \n",
       "Value iteration with span-based termination        288     0.0193439   \n",
       "Policy iteration                                     7    0.00199425   \n",
       "Modified policy iteration                           18    0.00431983   \n",
       "\n",
       "                                            $\\lVert v - v_{\\mathrm{pi}} \\rVert$  \\\n",
       "Value iteration                                                         581.999   \n",
       "Value iteration with span-based termination                             4.37767   \n",
       "Policy iteration                                                              0   \n",
       "Modified policy iteration                                               4.45491   \n",
       "\n",
       "                                            $\\overline{b} - \\underline{b}$  \\\n",
       "Value iteration                                                 0.00044699   \n",
       "Value iteration with span-based termination                        1160.38   \n",
       "Policy iteration                                                0.00089398   \n",
       "Modified policy iteration                                          1099.69   \n",
       "\n",
       "                                            $\\lVert v - T(v)\\rVert$  \n",
       "Value iteration                                           0.0581999  \n",
       "Value iteration with span-based termination               0.0558041  \n",
       "Policy iteration                                        5.96046e-08  \n",
       "Modified policy iteration                                 0.0528918  "
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df_9999"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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      "text/plain": [
       "<matplotlib.figure.Figure at 0x10c94aba8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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5aOjLwMhb473K3wq4MCs+YunPu9b9jM4Y5K2Ad1TVb9N5cOr8dB6Au4SVe6HHci40ZX6K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\nAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10f14bcf8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "df_time = pd.DataFrame(index=labels)\n",
    "df_time[r'$\\beta = 0.95$'] = df[columns[1]]\n",
    "df_time[r'$\\beta = 0.9999$'] = df_9999[columns[1]]\n",
    "second_max = df_time[r'$\\beta = 0.9999$'][1:].max()\n",
    "\n",
    "plt.rcParams['axes.color_cycle'] = ['g', 'b']\n",
    "for xlim in [None, (0, second_max*1.2)]:\n",
    "    ax = df_time.ix[reversed(labels)][df_time.columns[::-1]].plot(\n",
    "        kind='barh', legend='reverse', xlim=xlim, figsize=(8,5)\n",
    "    )\n",
    "    ax.set_xlabel('Time (second)')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Darwin-13.4.0-x86_64-i386-64bit\n"
     ]
    }
   ],
   "source": [
    "import platform\n",
    "print(platform.platform())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "3.5.0 (default, Sep 23 2015, 04:41:33) \n",
      "[GCC 4.2.1 Compatible Apple LLVM 6.0 (clang-600.0.57)]\n"
     ]
    }
   ],
   "source": [
    "import sys\n",
    "print(sys.version)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.10.2\n"
     ]
    }
   ],
   "source": [
    "print(np.__version__)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
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