Pricing-First_Live_Ex_Session.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "e3fc1f40",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "ea28a9b3",
   "metadata": {},
   "outputs": [],
   "source": [
    "class env:\n",
    "    def __init__(self, probs, prices):\n",
    "        self.prices = prices\n",
    "        self.probs = probs\n",
    "        \n",
    "    def round(self, arm_pulled):\n",
    "        conv = np.random.binomial(n=1, p=self.probs[arm_pulled])\n",
    "        reward = conv*self.prices[arm_pulled]\n",
    "        return reward"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "d36f261e",
   "metadata": {},
   "outputs": [],
   "source": [
    "class learner:\n",
    "    def __init__(self, n_arms):\n",
    "        self.t = 0\n",
    "        self.n_arms = n_arms\n",
    "        self.rewards = []\n",
    "        self.reward_per_arm = [[] for _ in range(n_arms)]\n",
    "        self.pulled = []\n",
    "        \n",
    "    def reset(self):\n",
    "        self.__init__(self.n_arms, self.prices, self.T)\n",
    "        \n",
    "    def act(self):\n",
    "        pass\n",
    "    \n",
    "    def update(self, arm_pulled, reward):\n",
    "        self.t += 1\n",
    "        self.rewards.append(reward)\n",
    "        self.reward_per_arm[arm_pulled].append(reward)\n",
    "        self.pulled.append(arm_pulled)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "969aa724",
   "metadata": {},
   "outputs": [],
   "source": [
    "class ucb(learner):\n",
    "    def __init__(self, n_arms, prices, T):\n",
    "        super().__init__(n_arms)\n",
    "        self.T = T\n",
    "        self.means = np.zeros(n_arms)\n",
    "        self.widths = np.array([np.inf for _ in range(n_arms)])\n",
    "        self.prices = prices\n",
    "        \n",
    "    def act(self):\n",
    "        idx = np.argmax((self.means + self.widths)*self.prices)\n",
    "        return idx\n",
    "    \n",
    "    def update(self, arm_pulled, reward):\n",
    "        reward = reward > 0\n",
    "        super().update(arm_pulled, reward)\n",
    "        self.means[arm_pulled] = np.mean(self.reward_per_arm[arm_pulled])\n",
    "        for idx in range(self.n_arms):\n",
    "            n = len(self.reward_per_arm[idx])\n",
    "            if n > 0:\n",
    "                self.widths[idx] = np.sqrt(2*np.log(self.T)/n)\n",
    "            else:\n",
    "                self.widths[idx] = np.inf"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "8bcd4e29",
   "metadata": {},
   "outputs": [],
   "source": [
    "p = [0.5, 0.4, 0.3, 0.2]\n",
    "prices = [100, 200, 800, 850]\n",
    "\n",
    "pricing_env = env(p, prices)\n",
    "T = 5000\n",
    "alg = ucb(len(p), prices, T)\n",
    "opt = np.max([a*b for a, b in zip(p, prices)])\n",
    "N_exp = 25\n",
    "\n",
    "R = []\n",
    "for _ in range(N_exp):\n",
    "    instant_regret = []\n",
    "    alg.reset()\n",
    "    for t in range(T):\n",
    "        pulled_arm = alg.act()\n",
    "        rew = pricing_env.round(pulled_arm)\n",
    "        alg.update(pulled_arm, rew)\n",
    "        instant_regret.append(opt - rew)\n",
    "    cumulative_regret = np.cumsum(instant_regret)\n",
    "    R.append(cumulative_regret)\n",
    "mean_R = np.mean(R, axis=0)\n",
    "std_dev = np.std(R, axis=0)/np.sqrt(N_exp)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "f7ef3c9b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.collections.PolyCollection at 0x2aad04e64c0>"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "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": [
    "plt.plot(mean_R)\n",
    "plt.fill_between(range(T), mean_R-std_dev, mean_R+std_dev, alpha=0.4)"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.9.7"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}