Switch_between_Q_and_Double_Qlearning.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "dddcde1f",
   "metadata": {
    "id": "dddcde1f"
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "a6b448f3",
   "metadata": {
    "id": "a6b448f3"
   },
   "outputs": [],
   "source": [
    "N = 3\n",
    "alpha = 0.01\n",
    "gamma = 0.99\n",
    "epochs = 200\n",
    "p = 0.8\n",
    "np.random.seed(0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "d4a62eb0",
   "metadata": {
    "id": "d4a62eb0"
   },
   "outputs": [],
   "source": [
    "def rewardnstate(a, s):\n",
    "    r, state, terminal = P[s][a]\n",
    "    if s==1:\n",
    "        r = r + np.random.uniform(-1, 1)\n",
    "    return r, state, terminal\n",
    "\n",
    "def act_functn(Q, state, state_visits, nAfes):\n",
    "    #eps = 0.2\n",
    "    #eps = 1/np.power(state_visits[state], 0.2)\n",
    "    eps = 1/np.sqrt(state_visits[state])\n",
    "    if np.random.choice([True, False], p=[1-eps, eps]):\n",
    "        action = np.argmax(Q[state, :nAfes[state]])\n",
    "    else:\n",
    "        action = np.random.choice(nAfes[state])\n",
    "    return action"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "0ZI--hBmkLnH",
   "metadata": {
    "id": "0ZI--hBmkLnH"
   },
   "outputs": [],
   "source": [
    "def train(epochs, gamma, rewardnstate, act_functn, P, seed=0, alpha=0.05):\n",
    "    def double_Qlearning(Qa, Qb,\n",
    "                         cumsum, count,\n",
    "                         count_matrix_a, maximal_value_a, \n",
    "                         maximal_value,\n",
    "                         rewardnstate, action,\n",
    "                         state, reward_list,\n",
    "                         state_visits):\n",
    "        reward, next_state, terminal = rewardnstate(action, state)\n",
    "        cumsum += reward\n",
    "        count += 1\n",
    "        alpha_a = 1/np.power(count_matrix_a[state, action], p)\n",
    "        action_star = np.argmax(Qa[next_state, :nAfes[next_state]])\n",
    "        Qa[state, action] += alpha_a*(reward + gamma*Qb[next_state, action_star] - Qa[state, action])\n",
    "        count_matrix_a[state, action] += 1\n",
    "        state_visits[state] += 1\n",
    "        maximal_value_a.append(np.max(Qa[start_state, :nAfes[start_state]]))\n",
    "        maximal_value.append(np.max(Qa[start_state, :nAfes[start_state]]))\n",
    "        reward_list.append((state, action, reward))\n",
    "        return (reward_list, maximal_value_a,\n",
    "                maximal_value, state_visits,\n",
    "                count_matrix_a, Qa,\n",
    "                next_state, terminal,\n",
    "                cumsum, count)    \n",
    "    \n",
    "    def Qlearning(Qa, cumsum,\n",
    "                  count, count_matrix_a,\n",
    "                  maximal_value_a, maximal_value, \n",
    "                  rewardnstate,\n",
    "                  action, state, reward_list,\n",
    "                  state_visits):\n",
    "        reward, next_state, terminal = rewardnstate(action, state)\n",
    "        cumsum += reward\n",
    "        count += 1\n",
    "        alpha_a = 1/np.power(count_matrix_a[state, action], p)\n",
    "        Qa[state, action] += alpha_a*(reward + gamma*np.max(Qa[next_state, :nAfes[next_state]]) - Qa[state, action])\n",
    "        count_matrix_a[state, action] += 1\n",
    "        state_visits[state] += 1\n",
    "        maximal_value_a.append(np.max(Qa[start_state, :nAfes[start_state]]))\n",
    "        maximal_value.append(np.max(Qa[start_state, :nAfes[start_state]]))\n",
    "        reward_list.append((state, action, reward))\n",
    "        return (reward_list, maximal_value_a,\n",
    "                maximal_value, state_visits,\n",
    "                count_matrix_a, Qa,\n",
    "                next_state, terminal,\n",
    "                cumsum, count)\n",
    "    \n",
    "    np.random.seed(seed)\n",
    "    nS = 3\n",
    "    nA = 8\n",
    "    gamma = 1\n",
    "    state_visits = np.ones(nS)\n",
    "    maximal_value = []\n",
    "    pref_list = []\n",
    "    pref_learning_list = [] #0: double q learning, 1: q learning \n",
    "    cumsum = 0\n",
    "    count = 0\n",
    "\n",
    "    Q = [np.random.rand(nS, nA), np.random.rand(nS, nA)]\n",
    "    Q[0][0] = 0\n",
    "    Q[1][0] = 0\n",
    "    count_matrix = [np.ones([nS, nA]), np.ones([nS, nA])]\n",
    "    maximal_value_v = [[], []]\n",
    "    nAfes = [1, 8, 2] #num_Actions_for_each_state\n",
    "    start_state = 2\n",
    "    avg_reward_per_step = []\n",
    "    return_dict = {\n",
    "            0: {0: []},\n",
    "            1: {\n",
    "                0: [], 1: [], 2: [],\n",
    "                3: [], 4: [], 5: [],\n",
    "                6: [], 7: []\n",
    "            },\n",
    "            2: {\n",
    "                0: [], 1: []\n",
    "            }\n",
    "        }\n",
    "    for i in range(epochs):\n",
    "        reward_list = []\n",
    "        state = start_state#state = np.random.choice(nS)\n",
    "        terminal = True if state==0 else False\n",
    "        choice = np.random.choice([True, False]).astype(\"int\")\n",
    "        \n",
    "        while not terminal:\n",
    "            action = act_functn(Q[choice], state, state_visits, nAfes)\n",
    "\n",
    "            # Double Q learning is preferred if \n",
    "            #  we have more than 2 previous returns for a state action pair\n",
    "            #  and the variance of return is less a threshold\n",
    "            # In case either of the above two criteria are not followed\n",
    "            # Q learning is preferred. \n",
    "            if(len(return_dict[state][action]) >= 2):\n",
    "                if(np.var(return_dict[state][action])<0.23):\n",
    "                    choice2 = np.random.choice([True, False]).astype(\"int\")\n",
    "                    \n",
    "                    (reward_list, maximal_value_v[choice2],\n",
    "                      maximal_value, state_visits,\n",
    "                      count_matrix[choice2], Qc2, next_state,\n",
    "                      terminal, cumsum, \n",
    "                     count) = double_Qlearning(Q[choice2], Q[1-choice2],\n",
    "                                              cumsum, count,\n",
    "                                              count_matrix[choice2], \n",
    "                                              maximal_value_v[choice2],\n",
    "                                               maximal_value,\n",
    "                                              rewardnstate, action,\n",
    "                                              state, reward_list,\n",
    "                                              state_visits)\n",
    "                    Q[choice2] = Qc2\n",
    "                    pref_learning_list.append(0)\n",
    "                else:\n",
    "                    \n",
    "                    (reward_list, maximal_value_v[choice],\n",
    "                    maximal_value, state_visits,\n",
    "                    count_matrix[choice], Q[choice], next_state,\n",
    "                    terminal, cumsum, count) = Qlearning(Q[choice], cumsum,\n",
    "                                                          count, count_matrix[choice],\n",
    "                                                          maximal_value_v[choice], \n",
    "                                                          maximal_value,\n",
    "                                                          rewardnstate, action, \n",
    "                                                          state, reward_list,\n",
    "                                                          state_visits)\n",
    "                    pref_learning_list.append(1)\n",
    "            else:\n",
    "                \n",
    "                (reward_list, maximal_value_v[choice],\n",
    "                 maximal_value, state_visits,\n",
    "                 count_matrix[choice], Q[choice], next_state,\n",
    "                 terminal, cumsum, count) = Qlearning(Q[choice], cumsum,\n",
    "                                                      count, count_matrix[choice],\n",
    "                                                      maximal_value_v[choice], \n",
    "                                                      maximal_value,\n",
    "                                                      rewardnstate, action, \n",
    "                                                      state, reward_list,\n",
    "                                                      state_visits)\n",
    "                pref_learning_list.append(1)\n",
    "            state = next_state\n",
    "            reward_sum = 0\n",
    "            for i in range(len(reward_list)-1, -1, -1):\n",
    "                reward_sum = gamma*reward_sum + reward_list[i][2]\n",
    "                return_dict[reward_list[i][0]][reward_list[i][1]].append(reward_sum)\n",
    "\n",
    "        pref = .0\n",
    "        pref += Q[0][2][0] - Q[0][2][1] #preference of left over right\n",
    "        pref += Q[1][2][0] - Q[1][2][1] #preference of left over right\n",
    "        pref /= float(2)\n",
    "        pref_list.append(pref)\n",
    "        avg_reward_per_step.append(cumsum/count)\n",
    "    return (avg_reward_per_step, pref_list, pref_learning_list, \n",
    "            Q[0], Q[1], maximal_value_v[0], maximal_value_v[1], \n",
    "            maximal_value, return_dict)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "f142e9e9",
   "metadata": {
    "id": "f142e9e9"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.0397106295295164\n"
     ]
    }
   ],
   "source": [
    "for i in [1]:\n",
    "    mu = i\n",
    "    P = {\n",
    "        0:{0: (0, 0, True)}, #state:{action: (reward, next state, terminal)}\n",
    "        1:{0: (mu, 0, True),\n",
    "           1: (mu, 0, True),\n",
    "           2: (mu, 0, True),\n",
    "           3: (mu, 0, True),\n",
    "           4: (mu, 0, True),\n",
    "           5: (mu, 0, True),\n",
    "           6: (mu, 0, True),\n",
    "           7: (mu, 0, True)}, \n",
    "        2:{0: (0, 1, False),\n",
    "           1: (0, 0, True)}\n",
    "    }\n",
    "    (reward_list, pref_list,\n",
    "     pref_learning_list, Qa,\n",
    "     Qb, maximal_value_a,\n",
    "     maximal_value_b, maximal_value,\n",
    "     return_dict) = train(1000, gamma, rewardnstate, act_functn, P)\n",
    "    print(pref_list[-1])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "01b974c3",
   "metadata": {},
   "source": [
    "# mu = 1, mu = -1\n",
    "# 1.0461307081651836 -0.3180550969252852  1000 epochs\n",
    "# 1.018252483260935   -0.4420569555814764  3000 epochs\n",
    "# 0.9964606531617048 -0.5036877713174344  5000 epochs\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "c7889ec8",
   "metadata": {},
   "outputs": [
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       "   0.5092972300492236,\n",
       "   0,\n",
       "   1.1620371642073726,\n",
       "   0,\n",
       "   0.05672586846015015,\n",
       "   0,\n",
       "   1.9661195072974014,\n",
       "   0,\n",
       "   1.9333033485377513,\n",
       "   0,\n",
       "   1.9781761507137074,\n",
       "   0,\n",
       "   0.5012428579339177,\n",
       "   0,\n",
       "   0.35530347460005607,\n",
       "   0,\n",
       "   0.627390108044634,\n",
       "   0,\n",
       "   1.8422533995557875,\n",
       "   0,\n",
       "   0.8603617152781082,\n",
       "   0,\n",
       "   1.6972967828866141,\n",
       "   0,\n",
       "   1.3874466773444558,\n",
       "   0,\n",
       "   0.28879563827339405,\n",
       "   0,\n",
       "   0.6428438505513294,\n",
       "   0,\n",
       "   0.3625353242956848,\n",
       "   0,\n",
       "   1.0303058942131427,\n",
       "   0,\n",
       "   0.9614827943205289,\n",
       "   0,\n",
       "   0.45228620550282694,\n",
       "   0,\n",
       "   1.1750769148781863,\n",
       "   0,\n",
       "   0.3883884152269381,\n",
       "   0,\n",
       "   1.1936390272653274,\n",
       "   0,\n",
       "   1.674652752162595,\n",
       "   0,\n",
       "   1.6287949224877807,\n",
       "   0,\n",
       "   1.725332115865506,\n",
       "   0,\n",
       "   0.22478217791290045,\n",
       "   0,\n",
       "   1.6238059968047922,\n",
       "   0,\n",
       "   1.651361642038899,\n",
       "   0,\n",
       "   0.08430200143782152,\n",
       "   ...],\n",
       "  1: [0,\n",
       "   0,\n",
       "   0,\n",
       "   0,\n",
       "   0,\n",
       "   0,\n",
       "   0,\n",
       "   0,\n",
       "   0,\n",
       "   0,\n",
       "   0,\n",
       "   0,\n",
       "   0,\n",
       "   0,\n",
       "   0,\n",
       "   0,\n",
       "   0,\n",
       "   0,\n",
       "   0,\n",
       "   0,\n",
       "   0,\n",
       "   0,\n",
       "   0,\n",
       "   0,\n",
       "   0,\n",
       "   0,\n",
       "   0,\n",
       "   0,\n",
       "   0,\n",
       "   0,\n",
       "   0,\n",
       "   0,\n",
       "   0,\n",
       "   0]}}"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "return_dict"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "SSZyZx6QdUGe",
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "SSZyZx6QdUGe",
    "outputId": "4a31ea4b-1580-4a4f-9110-c1704a6e1855"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(1897, 1897, 69, 69)"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pref_learning_array = np.array(pref_learning_list)\n",
    "maximal_value_array = np.array(maximal_value)\n",
    "qchosen = (pref_learning_array).nonzero()[0]\n",
    "qchosen_maximal_value = maximal_value_array[qchosen]\n",
    "dqchosen =(1-pref_learning_array).nonzero()[0]\n",
    "dqchosen_maximal_value = maximal_value_array[dqchosen]\n",
    "len(qchosen), len(qchosen_maximal_value), len(dqchosen), len(dqchosen_maximal_value)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "xq2VH3LEgG0h",
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 401
    },
    "id": "xq2VH3LEgG0h",
    "outputId": "8c57f2f4-6a21-4a7d-c2fa-0cc566a5c1bc"
   },
   "outputs": [
    {
     "data": {
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      "text/plain": [
       "<Figure size 1550x650 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots()\n",
    "red_dot, = ax.plot(dqchosen, dqchosen_maximal_value, \"ro\", markersize=5)\n",
    "blue_dot, = ax.plot(qchosen,qchosen_maximal_value, \"bo\",  markersize=1)\n",
    "\n",
    "ax.legend([red_dot, (red_dot, blue_dot)], [\"dq chosen\", \"q chosen\"])\n",
    "fig.set_size_inches(15.5, 6.5, forward=True)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "EaBbMCBQgpTx",
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 282
    },
    "id": "EaBbMCBQgpTx",
    "outputId": "5b4e89f3-36d3-4deb-fca3-6353ad1c599a",
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(pref_list)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "id": "65bf0b46",
   "metadata": {},
   "outputs": [],
   "source": [
    "mu = -0.1\n",
    "P = {\n",
    "    0:{0: (0, 0, True)}, #state:{action: (reward, next state, terminal)}\n",
    "    1:{0: (mu, 0, True),\n",
    "       1: (mu, 0, True),\n",
    "       2: (mu, 0, True),\n",
    "       3: (mu, 0, True),\n",
    "       4: (mu, 0, True),\n",
    "       5: (mu, 0, True),\n",
    "       6: (mu, 0, True),\n",
    "       7: (mu, 0, True)}, \n",
    "    2:{0: (0, 1, False),\n",
    "       1: (0, 0, True)}\n",
    "}\n",
    "(reward_list, pref_list,\n",
    " pref_learning_list, Qa,\n",
    " Qb, maximal_value_a,\n",
    " maximal_value_b, maximal_value,\n",
    " return_dict) = train(2000, gamma, rewardnstate, act_functn, P)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "5ca45077",
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 265
    },
    "id": "5ca45077",
    "outputId": "8c9b7f25-ec02-455f-96a2-8b9d980eaf65",
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "props = {\n",
    "    \"width\": 1,\n",
    "    \"headwidth\": 10\n",
    "}\n",
    "plt.scatter( dqchosen, dqchosen_maximal_value, s= 10, c = \"orange\")\n",
    "plt.scatter(qchosen,qchosen_maximal_value, s= 10, c = \"red\" )\n",
    "plt.plot(maximal_value)\n",
    "plt.axhline(y = max(mu, 0), color = \"r\", linestyle = \"--\", linewidth = 1)\n",
    "#plt.annotate('y = %s' % round(maximal_value[-1], 4), xy=(50000, maximal_value[-1]), xytext=(100000, 5), arrowprops = props)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "37c108a4",
   "metadata": {
    "id": "37c108a4"
   },
   "outputs": [],
   "source": [
    "def train2(epochs, gamma, rewardnstate, act_functn, P, seed=1, alpha=0.05):\n",
    "    def double_Qlearning(Qa, Qb, cumsum, count, count_matrix_a, maximal_value_a, rewardnstate, action, state, reward_list, state_visits):\n",
    "        reward, next_state, terminal = rewardnstate(action, state)\n",
    "        cumsum += reward\n",
    "        count += 1\n",
    "        alpha_a = 1/np.power(count_matrix_a[state, action], p)\n",
    "        action_star = np.argmax(Qa[next_state, :nAfes[next_state]])\n",
    "        Qa[state, action] += alpha_a*(reward + gamma*Qb[next_state, action_star] - Qa[state, action])\n",
    "        count_matrix_a[state, action] += 1\n",
    "        state_visits[state] += 1\n",
    "        maximal_value_a.append(np.max(Qa[start_state, :nAfes[start_state]]))\n",
    "        maximal_value.append(np.max(Qa[start_state, :nAfes[start_state]]))\n",
    "        reward_list.append((state, action, reward))\n",
    "        return reward_list, maximal_value_a, maximal_value, state_visits, count_matrix_a, Qa, next_state, terminal, cumsum, count\n",
    "    \n",
    "    def Qlearning(Qa, cumsum, count, count_matrix_a, maximal_value_a, rewardnstate, action, state, reward_list, state_visits):\n",
    "        reward, next_state, terminal = rewardnstate(action, state)\n",
    "        cumsum += reward\n",
    "        count += 1\n",
    "        alpha_a = 1/np.power(count_matrix_a[state, action], p)\n",
    "        Qa[state, action] += alpha_a*(reward + gamma*np.max(Qa[next_state, :nAfes[next_state]]) - Qa[state, action])\n",
    "        count_matrix_a[state, action] += 1\n",
    "        state_visits[state] += 1\n",
    "        maximal_value_a.append(np.max(Qa[start_state, :nAfes[start_state]]))\n",
    "        maximal_value.append(np.max(Qa[start_state, :nAfes[start_state]]))\n",
    "        reward_list.append((state, action, reward))\n",
    "        return reward_list, maximal_value_a, maximal_value, state_visits, count_matrix_a, Qa, next_state, terminal, cumsum, count\n",
    "    \n",
    "    np.random.seed(seed)\n",
    "    nS = 3\n",
    "    nA = 8\n",
    "    gamma = 1\n",
    "    count_matrix_a = np.ones([nS, nA])\n",
    "    count_matrix_b = np.ones([nS, nA])\n",
    "    state_visits = np.ones(nS)\n",
    "    maximal_value_a = []\n",
    "    maximal_value_b = []\n",
    "    maximal_value = []\n",
    "    pref_list = []\n",
    "    pref_learning_list = []\n",
    "    cumsum = 0\n",
    "    count = 0\n",
    "    Qa = np.random.rand(nS, nA)\n",
    "    Qb = np.random.rand(nS, nA)\n",
    "    Qa[0] = 0\n",
    "    Qb[0] = 0\n",
    "    print(\"Qa\", Qa)\n",
    "    nAfes = [1, 8, 2] #num_Actions_for_each_state\n",
    "    start_state = 2\n",
    "    return_dict = {\n",
    "            0: {0: []},\n",
    "            1: {\n",
    "                0: [],\n",
    "                1: [], \n",
    "                2: [],\n",
    "                3: [],\n",
    "                4: [],\n",
    "                5: [],\n",
    "                6: [],\n",
    "                7: []\n",
    "            },\n",
    "            2: {\n",
    "                0: [],\n",
    "                1: []\n",
    "            }\n",
    "        }\n",
    "    for i in range(epochs):\n",
    "        reward_list = []\n",
    "        state = start_state#state = np.random.choice(nS)\n",
    "        terminal = True if state==0 else False\n",
    "        while not terminal:\n",
    "            if np.random.choice([True, False]):\n",
    "                action = act_functn(Qa, state, state_visits, nAfes)\n",
    "                try:\n",
    "                    stat.variance(return_dict[state][action])\n",
    "                    x = True\n",
    "                except:\n",
    "                    x = False\n",
    "                if(x):\n",
    "                    if(stat.variance(return_dict[state][action])<0.4):\n",
    "                        (reward_list, maximal_value_a,\n",
    "                         maximal_value, state_visits,\n",
    "                         count_matrix_a, Qa,\n",
    "                         next_state, terminal,\n",
    "                         cumsum, count) = double_Qlearning(Qa, Qb,\n",
    "                                                           cumsum, count,\n",
    "                                                           count_matrix_a, maximal_value_a,\n",
    "                                                           rewardnstate, action,\n",
    "                                                           state, reward_list,\n",
    "                                                           state_visits)\n",
    "                        pref_learning_list.append(1)\n",
    "                    else:\n",
    "                        (reward_list, maximal_value_a,\n",
    "                         maximal_value, state_visits,\n",
    "                         count_matrix_a, Qa,\n",
    "                         next_state, terminal,\n",
    "                         cumsum, count) = Qlearning(Qa, cumsum,\n",
    "                                                    count, count_matrix_a,\n",
    "                                                    maximal_value_a, rewardnstate,\n",
    "                                                    action, state, reward_list,\n",
    "                                                    state_visits)\n",
    "                        pref_learning_list.append(0)\n",
    "                else:\n",
    "                    (reward_list, maximal_value_a,\n",
    "                     maximal_value, state_visits,\n",
    "                     count_matrix_a, Qa,\n",
    "                     next_state, terminal,\n",
    "                     cumsum, count) = Qlearning(Qa, cumsum,\n",
    "                                                count, count_matrix_a,\n",
    "                                                maximal_value_a, rewardnstate,\n",
    "                                                action, state,\n",
    "                                                reward_list, state_visits)\n",
    "                    pref_learning_list.append(0)\n",
    "\n",
    "            else:\n",
    "                action = act_functn(Qb, state, state_visits, nAfes)\n",
    "                try:\n",
    "                    stat.variance(return_dict[state][action])\n",
    "                    x = True\n",
    "                except:\n",
    "                    x = False\n",
    "                if(x):\n",
    "                    if(stat.variance(return_dict[state][action])<0.4):\n",
    "                        (reward_list, maximal_value_b,\n",
    "                         maximal_value, state_visits,\n",
    "                         count_matrix_b, Qb,\n",
    "                         next_state, terminal,\n",
    "                         cumsum, count) = double_Qlearning(Qb, Qa,\n",
    "                                                           cumsum, count,\n",
    "                                                           count_matrix_b, maximal_value_b,\n",
    "                                                           rewardnstate, action,\n",
    "                                                           state, reward_list,\n",
    "                                                           state_visits)\n",
    "                        pref_learning_list.append(1)\n",
    "                    else:\n",
    "                        (reward_list, maximal_value_b,\n",
    "                         maximal_value, state_visits,\n",
    "                         count_matrix_b, Qb,\n",
    "                         next_state, terminal,\n",
    "                         cumsum, count) = Qlearning(Qb, cumsum,\n",
    "                                                    count, count_matrix_b,\n",
    "                                                    maximal_value_b, rewardnstate,\n",
    "                                                    action, state,\n",
    "                                                    reward_list, state_visits)\n",
    "                        pref_learning_list.append(0)\n",
    "                else:\n",
    "                    (reward_list, maximal_value_b,\n",
    "                     maximal_value, state_visits,\n",
    "                     count_matrix_b, Qb,\n",
    "                     next_state, terminal,\n",
    "                     cumsum, count) = Qlearning(Qb, cumsum,\n",
    "                                                count, count_matrix_b,\n",
    "                                                maximal_value_b, rewardnstate,\n",
    "                                                action, state,\n",
    "                                                reward_list, state_visits)\n",
    "\n",
    "                    pref_learning_list.append(0)\n",
    "\n",
    "            state = next_state\n",
    "            reward_sum = 0\n",
    "            for i in range(len(reward_list)-1, -1, -1):\n",
    "                reward_sum = gamma*reward_sum + reward_list[i][2]\n",
    "                return_dict[reward_list[i][0]][reward_list[i][1]].append(reward_sum)\n",
    "\n",
    "        pref = .0\n",
    "        pref += Qa[2][0] - Qa[2][1] #preference of left over right\n",
    "        pref += Qb[2][0] - Qb[2][1] #preference of left over right\n",
    "        pref /= float(2)\n",
    "        pref_list.append(pref)\n",
    "        reward_list.append(cumsum/count)\n",
    "    return reward_list, pref_list, pref_learning_list, Qa, Qb, maximal_value_a, maximal_value_b, maximal_value, return_dict"
   ]
  }
 ],
 "metadata": {
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