{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Finding the optimal policy\n",
    "\n",
    "Let's try to find the optimal policy using Poicy iteration an Value iteration in a simple grid world.\n",
    "\n",
    "The world and the optimal policy to be gound are the following:\n",
    "\n",
    "<img src=\"http://www.cs.upc.edu/~mmartin/Grid+world.jpg\" width=\"350\">\n",
    "\n",
    "Next we have the implementation of the gridworld (see previous notebook for explanations)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import random\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "\n",
    "#Define the state matrix\n",
    "state_matrix = np.zeros((3,4))\n",
    "state_matrix[0, 3] = 1\n",
    "state_matrix[1, 3] = 1\n",
    "state_matrix[1, 1] = -1\n",
    "\n",
    "#Define the reward matrix\n",
    "reward_matrix = np.full((3,4), -0.04)\n",
    "reward_matrix[0, 3] = 1\n",
    "reward_matrix[1, 3] = -1\n",
    "\n",
    "transition_matrix = np.array([[0.8, 0.1, 0.0, 0.1],\n",
    "                              [0.1, 0.8, 0.1, 0.0],\n",
    "                              [0.0, 0.1, 0.8, 0.1],\n",
    "                              [0.1, 0.0, 0.1, 0.8]])\n",
    "\n",
    "gamma = 0.999\n",
    "\n",
    "def prob_next_state(position,world_row=3,world_col=4):\n",
    "    \"Given a position in the grid-world, returns a list of possible next states\"\n",
    "    posible=[]\n",
    "    for action in range(4):\n",
    "        position_def = position\n",
    "        if(action == 0): new_position = (position[0]-1, position[1])   #UP\n",
    "        elif(action == 1): new_position = (position[0], position[1]+1) #RIGHT\n",
    "        elif(action == 2): new_position = (position[0]+1, position[1]) #DOWN\n",
    "        elif(action == 3): new_position = (position[0], position[1]-1) #LEFT\n",
    "        else: raise ValueError('The action is not included in the action space.')\n",
    "\n",
    "        #Check if the new position is a valid position\n",
    "        if (new_position[0]>=0 and new_position[0]<world_row):\n",
    "            if(new_position[1]>=0 and new_position[1]<world_col):\n",
    "                if(state_matrix[new_position[0], new_position[1]] != -1):\n",
    "                    position_def = new_position\n",
    "        posible.append(position_def)\n",
    "    return posible"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, we refactor the policy evaluation procedure using Bellman equations that we used in the previous notebook. Notice we have two optional parameters. V is when we want to continue to use a previous V valuation of states, and randomp to determine the proportion of states that will be updated by the function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "def policy_evaluation(policy_matrix, numiter, V = np.zeros((3,4)),randomp=1 ):\n",
    "    for k in range(numiter):\n",
    "        state = np.nditer(state_matrix, flags=['multi_index'])\n",
    "        #old_V = V.copy()\n",
    "        while not state.finished:\n",
    "            action = policy_matrix[state.multi_index]\n",
    "            if state[0]==0 and random.random() < randomp:\n",
    "                acum=0\n",
    "                for x in zip(prob_next_state(state.multi_index,3,4),transition_matrix[int(action),:]):\n",
    "                    acum = acum + x[1]*(reward_matrix[x[0]]+gamma*V[x[0]])\n",
    "                V[state.multi_index] = acum\n",
    "            state.iternext()\n",
    "    return V"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Auxiliar function to pretty print the policy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "def print_policy(policy_matrix2, V=np.array([]),lstates=[]):\n",
    "    policy_matrix = policy_matrix2[::-1]\n",
    "    shape = policy_matrix.shape\n",
    "    U=np.zeros(policy_matrix.shape)\n",
    "    R=np.zeros(policy_matrix.shape)\n",
    "    for row in range(shape[0]):\n",
    "        for col in range(shape[1]):\n",
    "            if(policy_matrix[row,col] == -1): \n",
    "                U[row,col]=0    \n",
    "                R[row,col]=0  \n",
    "            elif(policy_matrix[row,col] == 0): \n",
    "                U[row,col]=0    \n",
    "                R[row,col]=1 \n",
    "                #policy_string += \" ^  \"\n",
    "            elif(policy_matrix[row,col] == 1): \n",
    "                U[row,col]=1   \n",
    "                R[row,col]=0 \n",
    "                #policy_string += \" >  \"\n",
    "            elif(policy_matrix[row,col] == 2): \n",
    "                U[row,col]=0    \n",
    "                R[row,col]=-1 \n",
    "                #policy_string += \" v  \"           \n",
    "            elif(policy_matrix[row,col] == 3):\n",
    "                U[row,col]=-1   \n",
    "                R[row,col]=0                 \n",
    "                #policy_string += \" <  \"\n",
    "            elif(np.isnan(policy_matrix[row,col])): \n",
    "                U[row,col]=0    \n",
    "                R[row,col]=0  \n",
    "                #policy_string += \" #  \"\n",
    "    plt.rcParams['figure.figsize'] = (4,3)\n",
    "    if V.size==0:\n",
    "        V=np.ones(policy_matrix.shape)\n",
    "        for x in lstates:\n",
    "            V[x.multi_index]=0.7\n",
    "        plt.matshow(V,extent=[0, 4, 0, 3],vmin=0, vmax=1)\n",
    "    else:\n",
    "        plt.matshow(V,extent=[0, 4, 0, 3])\n",
    "    plt.grid()\n",
    "    X, Y = np.meshgrid(np.arange(0.5, 4.5, 1), np.arange(0.5, 3.5, 1))\n",
    "    Q = plt.quiver(X, Y,U,R)\n",
    "\n",
    "    plt.show()\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "def return_expected_action(state):\n",
    "    \"\"\"Return the expected action.\n",
    "    \"\"\"\n",
    "    lV=[]\n",
    "    for action in range(0,4):\n",
    "        acum=0\n",
    "        for x in zip(prob_next_state(state.multi_index,3,4),transition_matrix[int(action),:]):\n",
    "            acum = acum + x[1]*(reward_matrix[x[0]]+gamma*V[x[0]])\n",
    "        lV.append(acum)\n",
    "    return np.argmax(lV)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Policy iteration algorithm"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Initial policy:\n"
     ]
    },
    {
     "data": {
      "image/png": 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/UNUEgPUAlnsbi4i85OQknekADg273Qbgb89dSERWAliZvtkfmoqWvzyeH60qB3DCdArvcHwBV+1kISfFv9CF6/S8/1BdC2AtAIjIDlW91kmAoMnmsQEcX9CJyA4nyznZ1G8DMPxC4FUAjowmFBH5g5PibwcwW0Q+JiL5AG4H8JK3sYjISyNu6qvqoIh8E8CrAEIA1qlq6wgPW+tGOJ/K5rEBHF/QORqfqJ73cp2IshyP3COyEItPZCEWn8hCLD6RhVh8Igux+EQWYvGJLPT/HOJNQvqySkMAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 288x216 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Update!\n"
     ]
    },
    {
     "data": {
      "image/png": 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AbwKoBrBORKq9DkZE3nGyxl8O4LSqNqtqDMB2AI97G4uIvOTkhhpzAJy9ZrgNwP3XjyQiGwBsGB28EipH45eP50ebIgCiplN4ZS/A+ctsS5yM5KT4N7txnd7wC9WtALYCgIgcUtUaJwEyTZDnDeD8ZToROeRkPCeb+m0Arr1P81wA528nFBH5g5PiHwSwSEQqRSQHwFMA3vY2FhF5adxNfVVNiMj3ALwLIARgm6oeG+dpW90I51NBnjeA85fpHM2fqN6wu05EAccz94gsxOITWYjFJ7IQi09kIRafyEIsPpGFWHwiC/0/YLMcI9++VAUAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 288x216 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Update!\n"
     ]
    },
    {
     "data": {
      "image/png": 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hxhKRMBVyQY1JwOlXbWwCrv3iRGa2FFjad7O7ZCINXz1eHH23CmiJOkVY1kHV0iJevmJ//oDphUxUSPHPdhG2L12s3Dm3HFgOYGbbnXPXFBIgaYp52UDLl3Rmtr2Q6QpZ1W8Cak+7XQMcuZBQIhIPhRR/GzDVzKaY2TDgbuBX4cYSkTANuKrvnMuY2QPAm0AJ8Jxzbs8Af7Y8iHAxVczLBlq+pCto+cy5L31cF5Eipz33RDyk4ot4SMUX8ZCKL+IhFV/EQyq+iIdUfBEP/R//LAg6RhvJbAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 288x216 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Update!\n"
     ]
    },
    {
     "data": {
      "image/png": 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kM7Pt2SyXzaZ+A3DulQFrgaPDCSUi+SGb4r8BzDCz6WZWCtwNvBxtLBGJUsZNfXfvNbMHgNeBYuBZd9+T4W6rchEuTxXy2EDjS7qsxmfuH3m7LiIFTnvuiQRIxRcJkIovEiAVXyRAKr5IgFR8kQCp+CIB+n+LYTXv8XddBwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 288x216 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Update!\n"
     ]
    },
    {
     "data": {
      "image/png": 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Rc5x4r99GmLQ0xusHzMploVzC/6zrO/3eRd2dc2uANQBmts05d2EuA0RNnNcNtH5RZ2bbclkul139VuDYC4FPA94byVAiUhhyCX8rcK6ZnWVmpcA1wGPBjiUiQcq6q++cGzCzZcDTQAK41zn3apYfW5OP4QpUnNcNtH5Rl9P6mXO/93RdRGJOR+6JeEjhi3hI4Yt4SOGLeEjhi3hI4Yt4SOGLeOj/ATUgLeBjl26wAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 288x216 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Start with random policy\n",
    "policy_matrix = np.zeros((3,4))\n",
    "state = np.nditer(state_matrix, flags=['multi_index'])\n",
    "while not state.finished:\n",
    "    policy_matrix[state.multi_index] = np.random.choice(4, 1)\n",
    "    state.iternext()\n",
    "policy_matrix[0,3]=-1\n",
    "policy_matrix[1,3]=-1\n",
    "policy_matrix[1,1]=np.nan\n",
    "print('Initial policy:')\n",
    "print_policy(policy_matrix)\n",
    "\n",
    "\n",
    "# Enter the loop\n",
    "unchanged = False\n",
    "while not unchanged:\n",
    "    unchanged = True\n",
    "    # Policy evaluation step\n",
    "    V=policy_evaluation(policy_matrix, 100)\n",
    "    \n",
    "    # Policy improvement\n",
    "    state = np.nditer(state_matrix, flags=['multi_index'])\n",
    "    lchanged_states=[]\n",
    "    while not state.finished:\n",
    "        if state[0]==0:\n",
    "            action=return_expected_action(state)\n",
    "            if policy_matrix[state.multi_index]!=action:\n",
    "                unchanged = False\n",
    "                lchanged_states.append(state.copy())\n",
    "                policy_matrix[state.multi_index]=action\n",
    "        state.iternext()\n",
    "        \n",
    "    if not unchanged:\n",
    "        print('Update!')\n",
    "        print_policy(policy_matrix,lstates=lchanged_states)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Policy iteration without complete convergence of policy evaluation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Initial policy:\n"
     ]
    },
    {
     "data": {
      "image/png": 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+rDOzDaUsV8qu/jbg+AGPZwJvDieUiKRDKcV/BjjFzE42swrg08DD8cYSkTgNuavv7j1mdh3wKFAA7nH3l4b4sZVRhEupPI8NNL6sK2l85n7Yy3URyTmduScSIBVfJEAqvkiAVHyRAKn4IgFS8UUCpOKLBOj/Abi775rPewWAAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 288x216 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Update in iteration 1 !\n"
     ]
    },
    {
     "data": {
      "image/png": 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jgfv3AR919wd+abpFwKKBu7OApvzHLQi1wLG4Q0RI40u2Ge4+ZqiJcrmSzrmuCvgr7xbuvgxYBmBmG9z9phzmnThpHhtofElnZhtymS6XVf1m4MwrA9YDB4YTSkQKQy7FfwOYbmZTzawM+Dzwk2hjiUiUhlzVd/deM3sAeBkoBp5y961DPG1ZPsIVqDSPDTS+pMtpfENu3BOR9NGeeyIBUvFFAqTiiwRIxRcJkIovEiAVXyRAKr5IgP4fwTUrc8Bo/jsAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 288x216 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Update in iteration 2 !\n"
     ]
    },
    {
     "data": {
      "image/png": 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av4Sbk89C+YR/uOs7/cVF3d19LbAWwMyec/ez8hkgaUp53UDrl3Rm9lw+y+Wzq98KHHwh8OnA2yMZSkSKQz7hPwt80sxmmdlY4GLgt9GOJSJRGnZX3937zGwV8CiQAu5y91eG+bG1hRiuSJXyuoHWL+nyWj9z/4vDdREpcXrlnkiAFL5IgBS+SIAUvkiAFL5IgBS+SIAUvkiA/h/wBiTFJviO1wAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 288x216 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Update in iteration 3 !\n"
     ]
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 288x216 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Update in iteration 6 !\n"
     ]
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAP4AAADMCAYAAABTJB73AAAABHNCSVQICAgIfAhkiAAAAAlwSFlzAAALEgAACxIB0t1+/AAAADl0RVh0U29mdHdhcmUAbWF0cGxvdGxpYiB2ZXJzaW9uIDIuMi4wLCBodHRwOi8vbWF0cGxvdGxpYi5vcmcvFvnyVgAADr1JREFUeJzt3X1wVfWdx/H3NzcP5EEgQEAaQF0VWosjWyy1RrFStSDZsg+dru7UdqbusnaWLj6Na3e3y9ZxZmeXqS3TZcZhRLvrsjq6dlcNPhAMmqqgPBScKKKIVYJixARCbp5ubn77x40VrORewj0595zf5zWTMTEnd76Hm/c9596cc4855xARvxSFPYCIjD6FL+IhhS/iIYUv4iGFL+IhhS/iobyGb2YLzWyPme01s9vzedthM7N7zazNzFrCniUIZjbdzDaZ2W4ze9XMloc9Uz6Z2Rgze9nMdg2t30/CninfzCxhZr8xs4Zsy+YtfDNLAKuBRcB5wLVmdl6+br8A/BJYGPYQARoAbnHOfQG4CPibmN1/fcAC59wFwBxgoZldFPJM+bYc2J3Lgvnc4s8D9jrn9jnn+oEHgSV5vP1QOeeagfaw5wiKc+5959yOoc+PkvkFqg13qvxxGV1DX5YMfcTm6DUzmwYsBu7JZfl8hl8L7D/m61Zi9IvjEzM7E/hD4KVwJ8mvoV3hnUAb0Oici9P6/Ry4DRjMZeF8hm+f8f9i84jqCzOrAh4BbnTOdYY9Tz4559LOuTnANGCemc0Oe6Z8MLN6oM05tz3Xn8ln+K3A9GO+nga8l8fbl4CZWQmZ6Nc5534V9jxBcc4dBp4lPq/Z1AHfNLPfknmKvcDM/mu4H8hn+FuBc83sLDMrBa4BHsvj7UuAzMyAtcBu59xdYc+Tb2ZWY2bjhz4vB64AXg93qvxwzv3IOTfNOXcmme6anHPfGe5n8ha+c24AWAY8TeaFoYecc6/m6/bDZmYPAJuBWWbWambXhz1TntUB15HZWuwc+rg67KHyaCqwycxeIbORanTOZf2zV1yZTssV8Y+O3BPxkMIX8ZDCF/GQwhfxUNbwR3Jyg5ktzc94hSfO6wZav6jLdf1y2eKP5OSGOP/jxnndQOsXdTmtX3G2BVzm732xPblBxEc5/R1/6JTb7cA5wGrn3N99xjJLGXq0KSsrnXvGjCl5HrUwpAaKKCnO6TyISNL6RduBA610JQc/67yZ45zUATxDhzz+L/BD59wJ35Bi1swZbndzWc63GyXNLcuZP3tV2GMERusXbfO+sZ9tu3qzhn9Sr+rH8OQGES/l8qp+bE9uEPFV1hf3yJzc8B9Dz/OLyJx84+3JDSJxkMur+q+QeTcWEYkJHbkn4iGFL+IhhS/iIYUv4iGFL+IhhS/iIYUv4iGFL+IhhS/iIYUv4iGFL+IhhS/iIYUv4iGFL+IhhS/iIYUv4iGFL+IhhS/iIYUv4qHYh9+VHCSViu+Ffw4fSXMy10aImo7D6bBHiKXYh9/f75j51Xe49oaDrHvkKO0d8fpF2vNWijPm/pYf3NZGQ2OSnp54XSXmwUe7mD3/HW6/8xDPv9RDOh3fB7nRlMvbaxe0p5qSPPtiz7DLVFYYDz3axUOPdpFIQN2Xx1B/VSWLr6hk1jklmGW98Eho7ll3hL1vp4ZdprvHseb+Ttbc30n5GOPr88upv7KS+isrmTqlsO/iO37aTvcwD1Z9fY7db6bY/eZhVq4+zITqIhYtqKT+qgq+8bUKxo1NjOK0J+eppiS1U4s5/wuFd1Wpwv6tyMGvt/SycvXhnJdPp6F5Sy8vbO2lYUOSm26o5o+uqijY+B9+rIuNzcM/sB2rp9fRsKGbZ5p7eGJjN/94UzVzLxgT4ISn5hdrD9PekfteSnvHIOseOcpTTUmWLKrin26uZnptSYATjtyKf2vna3Xl/OuPFX7e3XzDeL7/F2OHXea7yw6yZXsf48YWsXBBBfVXVrLw8gomVBfu1uJj962aQk/viXdv+/ocl/1JK+0dg9ROTbD4ikrqr6pkQV055eWF/0zu5aemMzhM9/veSbHwmvcAOG9mKfVXVbD4ikq+euEYEonCfLD+2Ivrp1FUoHdB5MOfOCHBxAknDnj/gRQXzR3DnT+ayCXzyikpKexflk/73OnD30XPvdjD3/7leOqvrGTO7NKC3XM5kbNmDL+1fv7lHn52xyQWX1nJ2WcW5pb9RAr5gSny4WczvbaEn/6kJuwxAnPZxeVcdnF52GME5nvfHn5vTkamQHdERCRICl/EQwpfxEMKX8RDCl/EQwpfxEMKX8RDCl/EQwpfxEMKX8RDCl/EQwpfxEMKX8RDCl/EQwpfxEMKX8RDCl/EQwpfxEMKX8RDCl/EQwpfxENZwzez6Wa2ycx2m9mrZrZ8NAYTkeDk8vbaA8AtzrkdZnYasN3MGp1zrwU8m4gEJOsW3zn3vnNux9DnR4HdQG3Qg4lIcOxkLrFsZmcCzcBs51znp763FFgKUFMzae5D/70if1MWkK6eKVSVfxD2GIHR+kXbrbfcyrZdvVkv4ZPzlXTMrAp4BLjx09EDOOfWAGsAZs2c4ebPXnUS40ZHc8ty4rpuoPXzRU6v6ptZCZno1znnfhXsSCIStFxe1TdgLbDbOXdX8COJSNBy2eLXAdcBC8xs59DH1QHPJSIByvoc3zn3PFC41/sVkZOmI/dEPKTwRTyk8EU8pPBFPKTwRTwU+fCPdKbDHkFGaHDQ0Xl0MOwxvBT58P95ZXvYI8gIbdvVx+MbkmGP4aVIh9/63gCr7ztCe4e2+lG0vjHJ+o0KPwyRDr+hMUk6DU82dYc9ioxAQ2OSp5q6SaVyP0NU8iPy4R/7X4mO/QdS7Gzp50jnIM+/3BP2ON6JbPjJ7kGans/8wjy9SVuNqGnY+MleWoOe54+6yIa/sbmbvr5M7NpqRM+xsTc0dnMybwgjpy6y4a9vPP55vbYa0ZHsHmTTC588UO99O8Ubb6VCnMg/kQzfOUd5uXHbsvEA/PjmajJvGyBRsLOljxW3TOBL55cxsbqIu1fW8Mru/rDH8krOb71VSMyMVXfWcP/DmXcAmzO7jD9eVBXyVJKrunnl1M0r54mNSRIJ46++My7skbwTyS2+iJwahS/iIYUv4iGFL+IhhS/iIYUv4iGFX8C27ezltjsOhT2GjNBdd3cU7NmHkfw7/sf+bHEVX7+0gupx8Xz8OnBwgB2v9JFKOUpK4neA0iP3TmUgHd9DdXe29FNSXJj3W6TDr6gooqIintEDLFlYxZKF8T0wadLERNgjBOo//31K2COcUHyrEZETUvgiHlL4Ih5S+CIeUvgiHlL4Ih5S+CIeUvgiHlL4Ih5S+CIeUvgiHlL4Ih5S+CIeUvgiHlL4Ih5S+CIeUvgiHlL4Ih5S+CIeUvgiHlL4Ih5S+CIeUvgiHsoavpnda2ZtZtYyGgOJSPBy2eL/ElgY8BwiMoqyhu+cawbaR2EWERklebuElpktBZYC1NRMornltnzddEHp6plCc8vysMcITFfPFNZuXRH2GIGpHqyI9f0Ht+a0VN7Cd86tAdYAzJo5w82fvSpfN11QmluWE9d1g8z6PZR4K+wxAvPtrrn8aYzvv1zpVX0RDyl8EQ/l8ue8B4DNwCwzazWz64MfS0SClPU5vnPu2tEYRERGj3b1RTyk8EU8pPBFPKTwRTyk8EU8FPnw39mfCnsEkciJdPjOOW5ecSjsMUQiJ9Lhv/ZGP//3ZJJ3W7XVFzkZkQ6/YUM3AOs3doc8iUi0RDr89Y1JAB7fkAx5EpFoiWz4hz5Ks3l7LwCbXuimKzkY8kQi0RHZ8J94JsngUOv9/bCxWbv7IrmKbPgNjUnMMp+bZb4WkdxEMnznHJdfUsEv/qUGgHt+NpkvzxkT8lQi0RHJ8M2MH3xvHFUVmU3++LFF/PV3x4U8lUh0RDJ8ETk1Cl/EQwpfxEMKX8RDCj8kb7+b4p51R0inXdij5J1zjs432zi4aU/YowQilXI8+2J3pI8dydsFNWR46bRjy/ZeGhqTNGzo5rU3+rlx6TgSCQt7tLxI9w3Q8Zv9fLh5H4e2vE3fh1188e/jc8nFjsNpnmzqpqExydObujnSOciuTdPDHmvEIh1+eXkRU6ckGFNWmPH09zsefTpJw4YkTz6T5KOO4w8rXnN/J/c9eHTY2/iftaez4JKKIMccsVRXL23P7eXDzfto3/Eug70Dx33/9Z83sWdV07C3Ubfu+5SMLcxjMN5tTfHw4100bEjywtZe0unjv3/pNw8M+/MzaovZ2TQjwAlHLtLhf6u+im/VV4U9xgmVlhoXXlDGB20DHGwb4LnNPaSOOYP4rBklTKlJDHsbY6sK99lYcWUZY2dNpq89SX97ks7XPzju+1VnTqCorGTY27CiwnzQBvjc6cVceMEYDralOfhhmjfe+uTOSyTgS+eX/e7o0c8yOct9G6ZIhx8FZ80oYdn141l2/Xg6jw6y4blu1jcmeWJjki9+vpQH7j497BFHzMw47ZzJnHbOZP7guq/Q157k0Ja3ObR5Hx9tf5fJl57LGX8+N+wxR6y42Ljs4nIuu7iclSsm8cZb/axvTNLQ2M2vX+rhH26q5vK6wtwby0bhj6KxpxX9bi8lnXZs3dlHOu1i8zy/bEIltVfPpvbq2aT7B+jaF693R5p5dikzzy7lphuq6TicpvX9gew/VKAUfkgSCeOiuYX53DYfEqXFjPt8dPdmsqken6B6fOHuymdTuE8gRSQwCl/EQwpfxEMKX8RDCl/EQwpfxEMKX8RDCl/EQwpfxEMKX8RDCl/EQwpfxEMKX8RDCl/EQwpfxEMKX8RDCl/EQwpfxEMKX8RDCl/EQwpfxEM5hW9mC81sj5ntNbPbgx5KRIKVNXwzSwCrgUXAecC1ZnZe0IOJSHBy2eLPA/Y65/Y55/qBB4ElwY4lIkHK5YIatcD+Y75uBb7y6YXMbCmwdOjLvsRUWk59vEL0w0lAvC4Rc5x4r99GmLQ0xusHzMploVzC/6zrO/3eRd2dc2uANQBmts05d2EuA0RNnNcNtH5RZ2bbclkul139VuDYC4FPA94byVAiUhhyCX8rcK6ZnWVmpcA1wGPBjiUiQcq6q++cGzCzZcDTQAK41zn3apYfW5OP4QpUnNcNtH5Rl9P6mXO/93RdRGJOR+6JeEjhi3hI4Yt4SOGLeEjhi3hI4Yt4SOGLeOj/ATUgLeBjl26wAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 288x216 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Start with random policy\n",
    "state = np.nditer(state_matrix, flags=['multi_index'])\n",
    "while not state.finished:\n",
    "    policy_matrix[state.multi_index] = np.random.choice(4, 1)\n",
    "    state.iternext()\n",
    "policy_matrix[0,3]=-1\n",
    "policy_matrix[1,3]=-1\n",
    "policy_matrix[1,1]=np.nan    \n",
    "print('Initial policy:')\n",
    "print_policy(policy_matrix)\n",
    "\n",
    "unchanged = False                              \n",
    "for k in range(1000):              # <<------ Now we cannot stop after no change in policy\n",
    "    unchanged = True\n",
    "    # Policy evaluation\n",
    "    V=policy_evaluation(policy_matrix, 1, V)   # <<------ Only 1 swap. Reuse V. \n",
    "    \n",
    "    # Policy improvement\n",
    "    state = np.nditer(state_matrix, flags=['multi_index'])\n",
    "    lchanged_states=[]\n",
    "    while not state.finished:\n",
    "        if state[0]==0:\n",
    "            action=return_expected_action(state)\n",
    "            if policy_matrix[state.multi_index]!=action:\n",
    "                unchanged = False\n",
    "                lchanged_states.append(state.copy())\n",
    "                policy_matrix[state.multi_index]=action\n",
    "        state.iternext()\n",
    "        \n",
    "    if not unchanged:\n",
    "        print('Update in iteration',k+1,'!')\n",
    "        print_policy(policy_matrix,lstates=lchanged_states)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Policy iteration without complete sweeps of policy inprovement"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Initial policy:\n"
     ]
    },
    {
     "data": {
      "image/png": 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nvKv67j5qZtcCDwMVwFZ3fzrPr20pRLgSVc5jA40v7SKNz9x/aXNdRMqcjtwTCZCKLxIgFV8kQCq+SIBUfJEAqfgiAVLxRQL0/1K3+OayhZLRAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 288x216 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Update in iteration 1 !\n"
     ]
    },
    {
     "data": {
      "image/png": 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LU4G2pFPEpRmmrizg5Sv01w+YG2WiKMV/r7sC+m/8h/saYA2AmW1x98uiBEibQl420PKlnZltiTJdlFX9VuDkOwPWAgfOJpSI5Icoxd8MzDazi82sFPgs8ES8sUQkTllX9d190MxuBZ4BioEH3X1nlqetyUW4PFXIywZavrSLtHzm/hub6yJS4HTknkiAVHyRAKn4IgFS8UUCpOKLBEjFFwmQii8SoP8Hfn8wMB+6SQYAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 288x216 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Update in iteration 2 !\n"
     ]
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAP4AAADMCAYAAABTJB73AAAABHNCSVQICAgIfAhkiAAAAAlwSFlzAAALEgAACxIB0t1+/AAAADl0RVh0U29mdHdhcmUAbWF0cGxvdGxpYiB2ZXJzaW9uIDIuMi4wLCBodHRwOi8vbWF0cGxvdGxpYi5vcmcvFvnyVgAADpNJREFUeJzt3XtwXOV9xvHvT6uLJRksgYXwNQaKnTimOBgcsBM7IUAMaMJkmnTCNGk744zbpKQGklI6CSVt0mYSWhpPw0xGw62hDAxpLiUyF0nYQSV2wZgIR8bBNZdg2cbGWJat1X316x9SHDvY0trs2bNn3+czo0GLzp55Xq8eveecPWePuTsiEpaSuAOISP6p+CIBUvFFAqTiiwRIxRcJkIovEqCcFt/MVpjZy2a2w8xuzeW642Zm95rZPjPriDtLFMxslpmtN7NtZrbVzFbHnSmXzGySmT1nZi+Oje8f4s6Ua2aWMrNfmlnTRMvmrPhmlgLuAq4G5gPXm9n8XK2/ANwPrIg7RISGgS+7+/uAS4G/KrLXbwC43N0vBBYCK8zs0pgz5dpqYFs2C+Zyxl8M7HD3V919EHgYuC6H64+Vu7cBB+LOERV33+PuL4x9f5jRX6AZ8abKHR/VM/awbOyraM5eM7OZwLXA3dksn8vizwB2HvW4kyL6xQmJmc0BPgA8G2+S3BrbFG4H9gEt7l5M4/sucAswks3CuSy+Hef/Fc1f1FCY2WTgR8CN7n4o7jy55O4Zd18IzAQWm9mCuDPlgpk1APvcfXO2z8ll8TuBWUc9ngnszuH6JWJmVsZo6R909x/HnScq7n4Q+DnFc8xmKfAJM3ud0V3sy83sP8d7Qi6Lvwk438zOMbNy4DPAozlcv0TIzAy4B9jm7nfGnSfXzKzOzGrGvq8ErgB+HW+q3HD3v3P3me4+h9HerXP3z473nJwV392HgRuAJxk9MPSIu2/N1frjZmYPARuBeWbWaWYr486UY0uBzzE6W7SPfV0Td6gcmgasN7MtjE5SLe4+4dtexcp0Wa5IeHTmnkiAVHyRAKn4IgFS8UUCNGHxT+XiBjNblZt4haeYxwYaX9JlO75sZvxTubihmP9xi3lsoPElXVbjK51oAR99v69oL24QCVFW7+OPXXK7GfgD4C53/9vjLLOKsb82FRUVi86eNT3HUQvEsFNRlok7RWSGhksoK83qOo9EKvbx7drVSU965HjXzRzjpE7gGTvl8SfAl9z9hB9IMef8c/38xqK5IvcYf9yziJWXFN1nOBzR1rGaZQvWxB0jMsU+vsUf38nzL/ZPWPyTOqpfhBc3iAQpm6P6RXtxg0ioJjy4x+jFDf8xtp9fwujFN8Fe3CBSDLI5qr+F0U9jEZEioTP3RAKk4osESMUXCZCKLxIgFV8kQCq+SIBUfJEAqfgiAVLxRQKk4osESMUXCZCKLxIgFV8kQCq+SIBUfJEAqfgiAVLxRQKk4osESMUXCVDRF3+4b5CR4eK9AcbB7gwnc2+EfOo6WLz/7klX9MX3oRE2fPZ+fvWNx9jT+muGDvXHHSmnXn5liPcsep0v3LKPppY0fX3x3SXG3fnlrwb4xp0H+OCKnfzTd7tiyyLjy+bjtQva/udep6t957jLpCrL2Lt+O3vXb8dKjCkLplN32blMvewcqmbVYjbhjUdic/eD3ex4bWjcZXr7nMYHDtH4wCEqJxkfW1ZJw5XVNFxZzbT6aF/ivr4R1m/oo6k5zdrWXjp3Dx/52fx55dz6zf3jPv+2m8+guqo4558n1qWZMa2UC95XEXeUd0h88Q9u2cVvHt6c9fI+4hzcsovujt28tfFV3vPpi5i65NyCLf8PH+2hta0v6+X7+p2m5l6eauvjsdZevnZTLYsunBRJtq6DGf55TRePPNpzTOF/6wePHJ5wHX/zxVqqq6JIF7/bv3OAjyyt5Nu3qfg5N/vTFzH96vePu8zWbz1B90tvUlpdzpmL51C35FzOvGQOZadHU4hcum9NPX39J96HHxhwln+ykwNdI8yYluLaK6ppuKqay5dWUlkZ7UxaW5Pijtun8p2/P5MXtw7S1JymqSXNpvYBABr/9Sw+sqRy3HXUTCnO2R5gw9qZlBTo8BJf/PIplZRPOfEvV/++w0yZP43zVi6l5oLplJSm8pju3Zt+9vgv0dMb+vjrz9fQcGU1CxeUx7LlYmYsXFDBwgUVfO3mM9izd5i1rWkOdmc4b05Z3vMUilSqMLcioQiKP5FJZ53G3C8ujztGZJYvqWT5BLNqvk2rL+XzfzIl7hgyjgLdEBGRKKn4IgFS8UUCpOKLBEjFFwmQii8SIBVfJEAqvkiAVHyRAKn4IgFS8UUCpOKLBEjFFwmQii8SIBVfJEAqvkiAVHyRAKn4IgFS8UUCpOKLBEjFFwnQhMU3s1lmtt7MtpnZVjNbnY9gIhKdbD5eexj4sru/YGanAZvNrMXdX4o4m4hEZMIZ3933uPsLY98fBrYBM6IOJiLROakbapjZHOADwLPH+dkqYBVAXd1Uvpo5LwfxCk8PFbR1FO/eTk9fvcaXaF/Jaqmsi29mk4EfATe6+6Hf/7m7NwKNAPPmzvZlC9Zku+pEaetYTbGODTS+UGR1VN/Myhgt/YPu/uNoI4lI1LI5qm/APcA2d78z+kgiErVsZvylwOeAy82sfezrmohziUiEJtzHd/dngMK936+InDSduScSIBVfJEAqvkiAVHyRAKn4IgFKfPG7D2XijiCSOIkv/tfvOBB3BJHESXTxO3cPc9d93Rzo0qwvcjISXfymljSZDDy+rjfuKCKJkvjiH/1fEclOYouf7h1h3TN9ADy5vpehIY85kUhyJLb4rW29DAyMlr370AjPPNcXcyKR5Ehs8de2HLtf39SszX2RbCWy+O5OZaVxyw01ANx2cy2jHxsgItlIZPHNjDXfrGP+3HIAFi6o4F++PjXmVCLJkcjii8i7o+KLBEjFFwmQii8SIBVfJEAqvkiAVPwC9nx7P7f84/64Y8gpuvP7XaxtLcwTy07q3nmF5o+unczHPlxF7ZTi/Pu1681hXtgywNCQU1ZWfCcoffLP9/DaG0O0r5sdd5RItHcMUlZamK9bootfVVVCVVVxlh7guhWTuW7F5LhjROZAV4a9bxXvZyn84Hv1cUc4oeJtjYickIovEiAVXyRAKr5IgFR8kQCp+CIBSvTbeZJML2zp5+2uEXzsYxI3v9hPV/cIVyyrijdYQFR8ybtZ08tYvOI1ANxh8YpONqydGXOqsGhTX/KubmqKyy6edGTGr69LccnCinhDBUbFl1g0XFl95PtrrqiipKQwT20tViq+xKLhqt/tzx/9R0DyQ8WXWMyfW86cWaWUl6ODejHQwT2JhZnRcFU1//fqEJOrNf/km4ovsWm4spodrw3FHSNIKr7EZvlllSx4b3ncMYKkbSyJTXm5Ma1ec08cVHyRAKn4IgFS8UUCpOKLBEjFFwmQii8SIBVfJEATFt/M7jWzfWbWkY9AIhK9bGb8+4EVEecQkTyasPju3gYcyEMWEcmTnJ0vaWargFUAdXVTaeu4JVerLig9ffW0dayOO0ZkNL6k+0pWS+Ws+O7eCDQCzJs725ctWJOrVReUto7VFOvYQOMLhY7qiwRIxRcJUDZv5z0EbATmmVmnma2MPpaIRGnCfXx3vz4fQUQkf7SpLxIgFV8kQCq+SIBUfJEAqfgiAUp88X+zU5/LnlSDg86evcNxxwhSoovv7tx8+/64Y8gpenpjHz99PB13jCAluvgvbR/kp4+neaNTs34SNbWk+Vmzih+HRBe/qbkXgLWtvTEnkZPl7jQ1p1n/i1560iNxxwlOoou/tmV0ttCskTwvbR/k9Z3DDA5Ca5v+cOdbYou//+0MGzf3A2jWSKDfbq3B6Ca/5Fdii//YU2lGxrquWSN5ji77Y629jIx4jGnCk9jiN7WkMRv93kyzRpK8tT/Dxuf7j7x+e9/KsKl9IN5QgUlk8d2dj36oin//Vh0Ad//bWVyycFLMqSRbO3cP8cTD01ly8STOmpriuSdmcrhHu2r5lMjimxlf+LMpTK4anTJqTi/hL/50SsypJFsX/eEkrlhWdWTGX3Th6GPJn0QWX0TeHRVfJEAqvkiAVHyRAKn4MXntjSHufrCbTEbvXyfN0JDz8w29iT53JGc31JDxZTLO/27up6klTVNzLy9tH+TGVVNIpSzuaJKFroMZHl/XS1NLmifX99J9aIQX18+KO9YpS3TxKytLmFafYlJFYZZncND57yfTNDWnefypNG93HftedeMDh7jv4cPjruO/7jmbyz9UnG91nVGbor6ucN+/f6NziB/+rIem5jS/2NRPJnPszz/8iV3jPn/2jFLa182OMOGpS3TxP9UwmU81TI47xgmVlxsXX1jB3n3DvLlvmKc39jF01BXE58wuo74uNe46Tp9cvHtjP7l/WtwRxjX97FIuvnASb+7L8OZbGba/8rsXL5WCiy6oOHIuwvGcNcFrG6dEFz8Jzpldxg0ra7hhZQ2HDo/Q/HQva1vSPNaa5v3vLeeh758dd0Q5gdJSY/mSSpYvqeSO26ey/ZVB1rakaWrp5X+e7eOrN9Xy0aXJ3BpT8fPo9NNKjmylZDLOpvYBMhnXfn5CzD2vnLnnlXPTX9bSdTBD557kfmyYih+TVMq4dJGuL0iq2poUtTWFuyk/keLdgRSRE1LxRQKk4osESMUXCZCKLxIgFV8kQCq+SIBUfJEAqfgiAVLxRQKk4osESMUXCZCKLxIgFV8kQCq+SIBUfJEAqfgiAVLxRQKk4osESMUXCZCKLxKgrIpvZivM7GUz22Fmt0YdSkSiNWHxzSwF3AVcDcwHrjez+VEHE5HoZDPjLwZ2uPur7j4IPAxcF20sEYlSNjfUmAHsPOpxJ/DB31/IzFYBq8YeDqSm0fHu4xWiL00F9sedIjoaX8LNy2ahbIp/vPs7veOm7u7eCDQCmNnz7n5xNgGSppjHBhpf0pnZ89ksl82mfidw9I3AZwK7TyWUiBSGbIq/CTjfzM4xs3LgM8Cj0cYSkShNuKnv7sNmdgPwJJAC7nX3rRM8rTEX4QpUMY8NNL6ky2p85v6O3XURKXI6c08kQCq+SIBUfJEAqfgiAVLxRQKk4osESMUXCdD/A58sEf3qmFW/AAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 288x216 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Update in iteration 3 !\n"
     ]
    },
    {
     "data": {
      "image/png": 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7O/3eTd3dvQloAjCzZ919YZQASZPmsYHGl3Rm9myU5aJs6rcDR98IfDrw9omEEpHSEKX4W4GPmtksM6sAvgTcV9xYIlJMo27qu3u/ma0BHgIywM3u/uIov9ZUiHAlKs1jA40v6SKNz9x/b3ddRFJOZ+6JBEjFFwmQii8SIBVfJEAqvkiAVHyRAKn4IgH6P0idCk30uXDTAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 288x216 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Update in iteration 8 !\n"
     ]
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 288x216 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Update in iteration 9 !\n"
     ]
    },
    {
     "data": {
      "image/png": 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Rc5x4r99GmLQ0xusHzMploVzC/6zrO/3eRd2dc2uANQBmts05d2EuA0RNnNcNtH5RZ2bbclkul139VuDYC4FPA94byVAiUhhyCX8rcK6ZnWVmpcA1wGPBjiUiQcq6q++cGzCzZcDTQAK41zn3apYfW5OP4QpUnNcNtH5Rl9P6mXO/93RdRGJOR+6JeEjhi3hI4Yt4SOGLeEjhi3hI4Yt4SOGLeOj/ATUgLeBjl26wAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 288x216 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Update in iteration 14 !\n"
     ]
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 288x216 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Update in iteration 16 !\n"
     ]
    },
    {
     "data": {
      "image/png": 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Rc5x4r99GmLQ0xusHzMploVzC/6zrO/3eRd2dc2uANQBmts05d2EuA0RNnNcNtH5RZ2bbclkul139VuDYC4FPA94byVAiUhhyCX8rcK6ZnWVmpcA1wGPBjiUiQcq6q++cGzCzZcDTQAK41zn3apYfW5OP4QpUnNcNtH5Rl9P6mXO/93RdRGJOR+6JeEjhi3hI4Yt4SOGLeEjhi3hI4Yt4SOGLeOj/ATUgLeBjl26wAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 288x216 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "prob_state=1/9\n",
    "# Start with random policy\n",
    "state = np.nditer(state_matrix, flags=['multi_index'])\n",
    "while not state.finished:\n",
    "    policy_matrix[state.multi_index] = np.random.choice(4, 1)\n",
    "    state.iternext()\n",
    "policy_matrix[0,3]=-1\n",
    "policy_matrix[1,3]=-1\n",
    "policy_matrix[1,1]=np.nan    \n",
    "print('Initial policy:')\n",
    "print_policy(policy_matrix)\n",
    "\n",
    "\n",
    "unchanged = False\n",
    "for k in range(1000):\n",
    "    unchanged = True\n",
    "    # Policy evaluation\n",
    "    V=policy_evaluation(policy_matrix, 1, V)\n",
    "    \n",
    "    # Policy improvement\n",
    "    state = np.nditer(state_matrix, flags=['multi_index'])\n",
    "    \n",
    "    lchanged_states=[]\n",
    "    while not state.finished:\n",
    "        if state[0]==0 and random.random() < prob_state:   # <<---- Update only some states. \n",
    "            action=return_expected_action(state)\n",
    "            if policy_matrix[state.multi_index]!=action:\n",
    "                unchanged = False\n",
    "                lchanged_states.append(state.copy())\n",
    "                policy_matrix[state.multi_index]=action\n",
    "        state.iternext()\n",
    "        \n",
    "    if not unchanged:\n",
    "        print('Update in iteration',k+1,'!')\n",
    "        print_policy(policy_matrix,lstates=lchanged_states)    \n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Policy iteration without complete sweeps in any loop"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Initial policy:\n"
     ]
    },
    {
     "data": {
      "image/png": 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gZtvcfW4uAdKmkMcGGl/amdm2XJbLZVO/BZhy0u3JwHvnEkpEkiGX4r8EzDCzS8ysBPg88ES0sUQkSkNu6rt7n5l9GfgZkAEedvfXhvizhnyES6hCHhtofGmX0/jM/ZTddREpcDpzTyRAKr5IgFR8kQCp+CIBUvFFAqTiiwRIxRcJ0P8DJH6+pERGXiwAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 288x216 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Update in iteration 1 !\n"
     ]
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 288x216 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Update in iteration 2 !\n"
     ]
    },
    {
     "data": {
      "image/png": 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c7UqNZ/sXjxdFt44FDoSdIjgaX8zNyGehfIp/su8MOuEa0865VcAqADPb7Jybk0+AuEny2EDjizsz25zPcvlM9RuA6mPuTwL2nk4oEYmGfIr/GjDNzM4zsxLgOuCZYGOJSJCGneo753rM7BbgF0AKeNw59/Ywf7aqEOEiKsljA40v7vIanzl3wtt1EUk4Hbkn4iEVX8RDKr6Ih1R8EQ+p+CIeUvFFPKTii3jo/wHgyctDKy5knwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 288x216 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Update in iteration 3 !\n"
     ]
    },
    {
     "data": {
      "image/png": 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ZD09kZiuBlf0PO3IncPDjx4uju8YBZ6JOEZZtMG5lgseX9OcPmJnJRJkU/0KfGfSRe0w759YD6wHMbI9zbl4mAbJNkscGGl+2M7M9mUyXyaZ+LTDlvMeTgTcvJZSIxEMmxX8BuMrMpptZAXAL8FS4sUQkTENu6jvnus3sTuAZIBd43Dn3yhB/tj6IcDGV5LGBxpftMhqfOfeR3XURSTiduSfiIRVfxEMqvoiHVHwRD6n4Ih5S8UU8pOKLeOj/AT4G3qbVhStwAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 288x216 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Update in iteration 5 !\n"
     ]
    },
    {
     "data": {
      "image/png": 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Z2QJgQc/DttRwdn/8eHH09WHAyahThCfZ41sPwxYkeHzAuGwmyqb4l7pm0EX3mHbOLQeWA5jZNufcbdkEyDVJHhtofLnOzLZlM102q/p1wOgLHo8CjlxNKBGJh2yKvxW40cyuN7MC4F7gxXBjiUiY+lzVd86dM7MHgZeBFPCcc25PH09bHkS4mEry2EDjy3VZjc+cu2hzXUQSTkfuiXhIxRfxkIov4iEVX8RDKr6Ih1R8EQ+p+CIe+n+Su+OwFFHyCAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 288x216 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Update in iteration 6 !\n"
     ]
    },
    {
     "data": {
      "image/png": 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ORJ0iPBpfjpuUzUzZFP9C9ww67xnTzrlVwCoAM9vpnLspmwC5JsljA40v15nZzmzmy2ZTvw4Yf870OODw5YQSkXjIpvivAhPM7FozGwjcCTwfbiwRCVOfm/rOuU4zuw94EUgBTzrn9vTxa6uCCBdTSR4baHy5LqvxmXPn7a6LSMLpzD0RD6n4Ih5S8UU8pOKLeEjFF/GQii/iIRVfxEP/Dxfi/1jzk1iEAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 288x216 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Update in iteration 8 !\n"
     ]
    },
    {
     "data": {
      "image/png": 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SkdJQTPGfAaaa2RlmNhK4Ango3lgiEqchN/XdvdfMrgEeBXLA3e7+whB/tjKKcCUqy2MDjS/tihrfkAf3RCR7dOaeSIBUfJEAqfgiAVLxRQKk4osESMUXCZCKLxKg/wej4Q0wQt6EZQAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 288x216 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Update in iteration 10 !\n"
     ]
    },
    {
     "data": {
      "image/png": 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Dj0QbS0SilHdT3917zOwm4HEgA9zn7nvy3G1NMcKVqDSPDTS+pCtofOb+gbfrIpJyOnJPJEAqvkiAVHyRAKn4IgFS8UUCpOKLBEjFFwnQ/wOAIy8b5GhJGgAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 288x216 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Update in iteration 17 !\n"
     ]
    },
    {
     "data": {
      "image/png": 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hAyRNOa8baP2Szsw2F7JcIbv6HcDR12meBuweyVAiUhoKCX8TcKaZnW5mY4ErgUejHUtEojTsrr67D5jZNcCTQAq4x91fGebHVhVjuBJVzusGWr+kK2j9zP2PXq6LSJnTkXsiAVL4IgFS+CIBUvgiAVL4IgFS+CIBUvgiAfp/3X8xA50r8w0AAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 288x216 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Update in iteration 20 !\n"
     ]
    },
    {
     "data": {
      "image/png": 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ziwmQNuU8NtD40s7MNhezXDGr+ruAY88tnADsHkgoESkNxRR/E/AZM5tiZlXAFcDD8cYSkTgVXNV39x4zWww8BmSAu939+QJ/tjyKcCWqnMcGGl/aFTU+c//I5rqIlDkduScSIBVfJEAqvkiAVHyRAKn4IgFS8UUCpOKLBOj/AfNaFQUb+jXJAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 288x216 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Update in iteration 26 !\n"
     ]
    },
    {
     "data": {
      "image/png": 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eN9D6JZ2ZbStkuUJ29duA488MOBnYP5yhRKQ0FBL+VuBsM5tmZqOAK4CHox1LRKKUd1ff3fvNbAnwGJAC7nb3F/NcbWUxhitR5bxuoPVLuoLWz9x/7+m6iJQ5fXJPJEAKXyRACl8kQApfJEAKXyRACl8kQApfJED/D+maOcvazdnnAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 288x216 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Update in iteration 41 !\n"
     ]
    },
    {
     "data": {
      "image/png": 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Rc5x4r99GmLQ0xusHzMploVzC/6zrO/3eRd2dc2uANQBmts05d2EuA0RNnNcNtH5RZ2bbclkul139VuDYC4FPA94byVAiUhhyCX8rcK6ZnWVmpcA1wGPBjiUiQcq6q++cGzCzZcDTQAK41zn3apYfW5OP4QpUnNcNtH5Rl9P6mXO/93RdRGJOR+6JeEjhi3hI4Yt4SOGLeEjhi3hI4Yt4SOGLeOj/ATUgLeBjl26wAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 288x216 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      " Value function:\n",
      " [[ 0.84881226  0.90630541  0.95748948  0.        ]\n",
      " [ 0.797764    0.          0.69906187  0.        ]\n",
      " [ 0.7404234   0.6895106   0.64536512  0.42192624]]\n"
     ]
    }
   ],
   "source": [
    "# Start with random policy\n",
    "state = np.nditer(state_matrix, flags=['multi_index'])\n",
    "while not state.finished:\n",
    "    policy_matrix[state.multi_index] = np.random.choice(4, 1)\n",
    "    state.iternext()\n",
    "policy_matrix[0,3]=-1\n",
    "policy_matrix[1,3]=-1\n",
    "policy_matrix[1,1]=np.nan    \n",
    "print('Initial policy:')\n",
    "print_policy(policy_matrix)\n",
    "\n",
    "unchanged = False\n",
    "for k in range(10000):\n",
    "    unchanged = True\n",
    "    # Policy evaluation\n",
    "    V=policy_evaluation(policy_matrix, 1, V, randomp=1/9) # <<---- Update only some states.\n",
    "    \n",
    "    # Policy improvement\n",
    "    state = np.nditer(state_matrix, flags=['multi_index'])\n",
    "\n",
    "\n",
    "    lchanged_states=[]\n",
    "    while not state.finished:\n",
    "        if state[0]==0 and random.random() < prob_state:   # <<---- Update only some states. \n",
    "            action=return_expected_action(state)\n",
    "            if policy_matrix[state.multi_index]!=action:\n",
    "                unchanged = False\n",
    "                lchanged_states.append(state.copy())\n",
    "                policy_matrix[state.multi_index]=action\n",
    "        state.iternext()\n",
    "        \n",
    "    if not unchanged:\n",
    "        print('Update in iteration',k+1,'!')\n",
    "        print_policy(policy_matrix,lstates=lchanged_states)            \n",
    "        \n",
    "print('\\n Value function:\\n',V)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Value iteration"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 0.84881226  0.90630541  0.95748948  0.        ]\n",
      " [ 0.797764    0.          0.69906187  0.        ]\n",
      " [ 0.7404234   0.6895106   0.64536512  0.42192624]]\n"
     ]
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAP4AAADMCAYAAABTJB73AAAABHNCSVQICAgIfAhkiAAAAAlwSFlzAAALEgAACxIB0t1+/AAAADl0RVh0U29mdHdhcmUAbWF0cGxvdGxpYiB2ZXJzaW9uIDIuMi4wLCBodHRwOi8vbWF0cGxvdGxpYi5vcmcvFvnyVgAACDxJREFUeJzt3FGIXGcZxvHnye6mliZpKwlasjEJNg0NUSuVNBKvopZNK6kXWhs0eFFZECMpJNX2UqRXQhU1N4sNrRhaAq0QhFpCjUahrWliLInb0qUqLg0uNTZbBU0neb2YuYgxZk7C+fbsnPf/g4WdzZnD+232v2fmzJx1RAhALguaHgDA3CN8ICHCBxIifCAhwgcSInwgoVrDtz1m+zXbU7YfqnPfTbO91/aM7RNNz1KC7RW2D9metH3S9s6mZ6qT7ffY/q3t3/fW962mZ6qb7SHbv7P9s37b1ha+7SFJeyRtkbRO0jbb6+ra/zzwuKSxpocoqCNpV0TcKmmjpK+17P/v35I2R8RHJN0macz2xoZnqttOSZNVNqzziL9B0lREvBERZyU9JemeGvffqIg4LOl003OUEhGnIuJY7/N31P0BWt7sVPWJrn/0bo70Plrz7jXbo5LulvSjKtvXGf5ySX+54Pa0WvSDk4ntVZI+KumlZiepV++h8HFJM5IORkSb1vc9Sd+QdL7KxnWG70t8rTW/UbOwvUjS05IeiIjZpuepU0Sci4jbJI1K2mB7fdMz1cH2ZyTNRMTRqvepM/xpSSsuuD0q6c0a94/CbI+oG/2+iHim6XlKiYi3Jf1S7Tlns0nSVtt/Uvcp9mbbP7ncHeoM/4ikNbZX214o6T5JB2rcPwqybUmPSZqMiEebnqdutpfZvqH3+bWSPiXp1WanqkdEPBwRoxGxSt3ufhERX7rcfWoLPyI6knZIek7dE0P7I+JkXftvmu0nJb0gaa3tadv3Nz1TzTZJ2q7u0eJ47+Oupoeq0U2SDtl+Rd2D1MGI6PuyV1uZy3KBfHjnHpAQ4QMJET6QEOEDCfUN/2oubrA9Xs9480+b1yaxvkFXdX1VjvhXc3FDm7+5bV6bxPoGXaX1DffbILqv97X24gYgo0qv4/cuuT0q6WZJeyLim5fYZly93zbXXLPw9tEPvL/mUeeHcx1rZLjSdRADqdNZ0Or1vdtZoPPvNj1FOdPT0+pE51LXzfyXK3oDT+8tjz+V9PWI+L9/kOLmW1bGd3++pPJ+B8n5qa9q/a3fbnqMYqZefVif/tAPmh6jmMMnduqRO3/V9BjFvBTPazZO9w3/is7qt/DiBiClKmf1W3txA5BV35N76l7c8ETvef4CdS++SXtxA9AGVc7qv6LuX2MB0BK8cw9IiPCBhAgfSIjwgYQIH0iI8IGECB9IiPCBhAgfSIjwgYQIH0iI8IGECB9IiPCBhAgfSIjwgYQIH0iI8IGECB9IiPCBhAgfSIjwgYQIH0iI8IGECB9IiPCBhAgfSIjwgYQIH0iI8IGECB9IiPCBhAgfSIjwgYQIH0iI8IGECB9IiPCBhAgfSIjwgYQIH0iI8IGECB9IiPCBhAgfSIjwgYQIH0iob/i2V9g+ZHvS9knbO+diMADlDFfYpiNpV0Qcs71Y0lHbByPiD4VnA1BI3yN+RJyKiGO9z9+RNClpeenBAJTjiKi+sb1K0mFJ6yNi9qJ/G5c0LknLli29/Yc/fqS+KeeRkbPv1Vuvz/bfcEDdOHq9ziz6Z9NjFHN95zr9fepM02MUs2v3bs3GaffbrspDfUmS7UWSnpb0wMXRS1JETEiakKQP3rIyTq3cfwXjDo6b/nyv9j94sOkxirn3O1v07MYjTY9RzN1vf1z7H3y26TEaV+msvu0RdaPfFxHPlB0JQGlVzupb0mOSJiPi0fIjASityhF/k6TtkjbbPt77uKvwXAAK6vscPyJ+I6nvyQIAg4N37gEJET6QEOEDCRE+kBDhAwkRPpAQ4QMJET6QEOEDCRE+kBDhAwkRPpAQ4QMJET6QEOEDCRE+kBDhAwkRPpAQ4QMJET6QEOEDCRE+kBDhAwkRPpAQ4QMJET6QEOEDCRE+kBDhAwkRPpAQ4QMJET6QEOEDCRE+kBDhAwkRPpAQ4QMJET6QEOEDCRE+kBDhAwkRPpAQ4QMJET6QEOEDCRE+kBDhAwn1Dd/2Xtsztk/MxUAAyqtyxH9c0ljhOQDMob7hR8RhSafnYBYAc2S4rh3ZHpc0LklLly3VjX/8Yl27nleGOkv0hQNbmx6jmBvOXac7//aJpscoZklcq8/u+3zTYxTz690vVtqutvAjYkLShCStXrMqXnzf83Xtel7Z+NdP6sDio02PUczWM3foiYXtPZ3z5X99WN9/fabpMRrHWX0gIcIHEqryct6Tkl6QtNb2tO37y48FoKS+z/EjYttcDAJg7vBQH0iI8IGECB9IiPCBhAgfSIjwgYQIH0iI8IGECB9IiPCBhAgfSIjwgYQIH0iI8IGECB9IiPCBhAgfSIjwgYQIH0iI8IGECB9IiPCBhAgfSIjwgYQIH0iI8IGECB9IiPCBhAgfSIjwgYQIH0iI8IGECB9IiPCBhAgfSIjwgYQIH0iI8IGECB9IiPCBhAgfSIjwgYQIH0iI8IGECB9IiPCBhAgfSKhS+LbHbL9me8r2Q6WHAlBW3/BtD0naI2mLpHWSttleV3owAOVUOeJvkDQVEW9ExFlJT0m6p+xYAEpyRFx+A/tzksYi4iu929sl3REROy7ablzSeO/mekkn6h93Xlgq6a2mhyiI9Q22tRGxuN9GwxV25Et87X9+W0TEhKQJSbL9ckR8rMK+B06b1yaxvkFn++Uq21V5qD8tacUFt0clvXk1QwGYH6qEf0TSGturbS+UdJ+kA2XHAlBS34f6EdGxvUPSc5KGJO2NiJN97jZRx3DzVJvXJrG+QVdpfX1P7gFoH965ByRE+EBChA8kRPhAQoQPJET4QEKEDyT0HwZMxc2KknPHAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 288x216 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Value iteration\n",
    "\n",
    "V = np.zeros((3,4))\n",
    "for k in range(100):\n",
    "    unchanged = True\n",
    "\n",
    "    state = np.nditer(state_matrix, flags=['multi_index'])\n",
    "    while not state.finished:\n",
    "        if state[0]==0:\n",
    "            lV=[]\n",
    "            for action in range(0,4):\n",
    "                acum=0\n",
    "                for x in zip(prob_next_state(state.multi_index,3,4),transition_matrix[int(action),:]):\n",
    "                    acum = acum + x[1]*(reward_matrix[x[0]]+gamma*V[x[0]])\n",
    "                lV.append(acum)\n",
    "            V[state.multi_index]=max(lV)\n",
    "        state.iternext()\n",
    "        \n",
    "print(V)\n",
    "plt.matshow(V,extent=[0, 4, 0, 3])\n",
    "plt.grid()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 288x216 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "state = np.nditer(state_matrix, flags=['multi_index'])\n",
    "while not state.finished:\n",
    "    if state[0]==0:\n",
    "        lV=[]\n",
    "        for action in range(0,4):\n",
    "            acum=0\n",
    "            for x in zip(prob_next_state(state.multi_index,3,4),transition_matrix[int(action),:]):\n",
    "                acum = acum + x[1]*(reward_matrix[x[0]]+gamma*V[x[0]])\n",
    "            lV.append(acum)\n",
    "        policy_matrix[state.multi_index]=np.argmax(lV)\n",
    "    state.iternext()\n",
    "    \n",
    "print_policy(policy_matrix,V)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Try to predict optimal policy with these changes.\n",
    "\n",
    "1. Change penalty to -0,8 per step\n",
    "\n",
    "2. Try deterministic transitions\n",
    "\n",
    "3. Try the following transition matrix:\n",
    "\n",
    "transition_matrix = np.array([[0.8, 0.1, 0.0, 0.1],\n",
    "                              [0.1, 0.8, 0.1, 0.0],\n",
    "                              [0.0, 0.1, 0.8, 0.1],\n",
    "                              [0.0, 0.0, 0.5, 0.5]])\n",
    "                          \n"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
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