intensity.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This notebooks works like the monte_carlo notebook, except with only one price and one market situation.\n",
    "The time between adjustmensts of prices gets split into B parts and for each purchase the time slot o fit is tracked. At the end, it creates a graph. The time horizon is 1."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import random\n",
    "import csv\n",
    "import matplotlib.pyplot as plt\n",
    "import pandas as pd"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Set our start values."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "num_firm = 5                        # the number of firms which we simulate\n",
    "time_horizon = 1                    # set the time horizon which we watch to 1 \n",
    "num_market_events = 2               # the max price adjustments each firm will make\n",
    "mean_betw_customer_arrival = 2      # chose mean for how often customers appear\n",
    "num_customer_events = 4             # max 4 customers arrive in time horizon\n",
    "watched_firm = 0                   \n",
    "num_sims = 10000                    # set the number of simulations to run \n",
    "reaction_time_firm = time_horizon   # watched firm doesn't update price\n",
    "intensity_price = 7                 # the price for the watched firm\n",
    "bb = 20                             # number of times to split time horizon\n",
    "setups = [\"S1\",\"S2\",\"S3\"]           # set the names of different setups\n",
    "random.seed(32859786)               # set a random seed to always get comparable results\n",
    "#betw = between"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# calculate price depending on prices from competing firms\n",
    "def price_if_smaller_else(price, price_threshold, min_price, prices_firm, c):\n",
    "    return(\n",
    "        price \n",
    "        if min([prices_firm[c] for firm in range(num_firm) if firm!=c]) < price_threshold \n",
    "        else round(\n",
    "            min([\n",
    "                    max(min_price, prices_firm[firm] - 0.5) for firm in range(num_firm) if firm != c])\n",
    "            ,2))\n",
    "\n",
    "# define the three different setups from the paper\n",
    "s1 = {\n",
    "        '0' : lambda prices_firm, c: round(random.uniform(0,15),2),\n",
    "        '1' : lambda prices_firm, c: round(random.uniform(5,15),2),\n",
    "        '2' : lambda prices_firm, c: round(random.uniform(5,15),2),\n",
    "        '3' : lambda prices_firm, c: round(random.uniform(5,15),2),\n",
    "        '4' : lambda prices_firm, c: round(random.uniform(5,15),2)\n",
    "    }\n",
    "\n",
    "s2 = {\n",
    "        '0' : lambda prices_firm, c: price_if_smaller_else(10,5,4.9,prices_firm,c),\n",
    "        '1' : lambda prices_firm, c: price_if_smaller_else(10,5,4.9,prices_firm,c),\n",
    "        '2' : lambda prices_firm, c: price_if_smaller_else(10,5,4.9,prices_firm,c),\n",
    "        '3' : lambda prices_firm, c: price_if_smaller_else(10,5,4.9,prices_firm,c),\n",
    "        '4' : lambda prices_firm, c: price_if_smaller_else(10,5,4.9,prices_firm,c)\n",
    "    }\n",
    "\n",
    "s3 = {\n",
    "        '0' : lambda prices_firm, c: price_if_smaller_else(10,5,4.9,prices_firm,c),\n",
    "        '1' : lambda prices_firm, c: price_if_smaller_else(9,4,3.9,prices_firm,c),\n",
    "        '2' : lambda prices_firm, c: price_if_smaller_else(12,6,5.9,prices_firm,c),\n",
    "        '3' : lambda prices_firm, c: 11,\n",
    "        '4' : lambda prices_firm, c: 13\n",
    "    }"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def calc_prices(prices_firm_adjust, adjusting_firm, setup):\n",
    "    prices_firm_adjust = np.array(prices_firm_adjust)\n",
    "    #weird indexing because there are actually no price adjustments at time 0 nor on the price horizon\n",
    "    for i in range(1,len(prices_firm_adjust)-1):         \n",
    "        prices_firm_adjust[i:,adjusting_firm[i]] = setup[str(adjusting_firm[i])](prices_firm_adjust[i-1],adjusting_firm[i])\n",
    "    return prices_firm_adjust"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Calculate all the necessary information for when firms adjusts their prices and how often they adjust their prices.  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "time_betw_market_events_firm = np.random.uniform(0.8,1.2, (num_sims, num_firm, num_market_events))\n",
    "\n",
    "time_adjusts_firm = time_betw_market_events_firm\n",
    "time_adjusts_firm[:,:,0] = np.random.uniform(0, 1, (num_sims, num_firm))\n",
    "\n",
    "# calculate the time when each company adjusts their prices\n",
    "for sim_num in range(num_sims):\n",
    "    for firm in range(num_firm):\n",
    "        for i in range(1, num_market_events):\n",
    "            time_adjusts_firm[sim_num, firm, i] = (time_adjusts_firm[sim_num, firm, i-1] + \n",
    "                                                   time_betw_market_events_firm[sim_num, firm, i])   \n",
    "\n",
    "price_adjusts_before_horizon = np.array(\n",
    "    [\n",
    "        set([\n",
    "            time_adjusts_firm[sim_num].flatten()[i] \n",
    "            for i in range(len(time_adjusts_firm[sim_num].flatten())) \n",
    "            if time_adjusts_firm[sim_num].flatten()[i] < time_horizon]) \n",
    "        for sim_num in range(num_sims)])\n",
    "\n",
    "price_adjusts_before_horizon = np.array(\n",
    "    [\n",
    "        [0] + sorted(price_adjusts_before_horizon[sim_num]) + [time_horizon] \n",
    "        for sim_num in range(num_sims)])\n",
    "\n",
    "num_price_adjusts_before_horizon = np.array(\n",
    "    [\n",
    "        len(price_adjusts_before_horizon[sim_num])\n",
    "        for sim_num in range(num_sims)])\n",
    "\n",
    "adjusting_firm = np.array(\n",
    "    [[\n",
    "            min(np.where(time_adjusts_firm[sim_num]==price_adjusts_before_horizon[sim_num][i] )[0]) \n",
    "            if price_adjusts_before_horizon[sim_num][i]!= time_horizon and i!=0 \n",
    "            else -1 \n",
    "            for i in range(num_price_adjusts_before_horizon[sim_num])] \n",
    "        for sim_num in range(num_sims)])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Calculate when customers will arrive and how they will score the quality and rating of all firms."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "time_betw_customer_events = mean_betw_customer_arrival* np.random.exponential(size=(num_sims, num_customer_events)) \n",
    "\n",
    "time_customer_events = time_betw_customer_events\n",
    "\n",
    "#calculate the arrival of customers after the first one\n",
    "for sim_num in range(num_sims):\n",
    "    for customer in range(1, num_customer_events):\n",
    "        time_customer_events[sim_num, customer] = (time_customer_events[sim_num, customer-1] + \n",
    "                                                   time_betw_customer_events[sim_num, customer])\n",
    "\n",
    "rand_customer_score_qual = np.random.uniform(0,1, (num_sims, num_customer_events))\n",
    "\n",
    "rand_customer_score_rating = rand_customer_score_qual = np.random.uniform(0,0.5, (num_sims, num_customer_events))\n",
    "\n",
    "customer_score_qual_rating = np.zeros(shape=(num_sims, num_customer_events, num_firm))\n",
    "\n",
    "#calculate the score from each customer for each firm based on quality and rating of the offers\n",
    "for sim_num in range(num_sims):\n",
    "    for customer in range(num_customer_events):\n",
    "        customer_score_qual_rating[sim_num, customer, :] = (rand_customer_score_qual[sim_num, customer]*2 + \n",
    "                                                            rand_customer_score_rating[sim_num, customer]*(100-98))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Calculate the prices adjustments for each firm in the time horizon."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "prices_firm = np.array(\n",
    "    [\n",
    "        intensity_price+20.5+2*(k-2) \n",
    "        if k!=watched_firm \n",
    "        else intensity_price  \n",
    "        for k in range(num_firm)])\n",
    "\n",
    "prices_firm_adjust = np.array(\n",
    "    [[[\n",
    "                prices_firm[k]  \n",
    "                for k in range(num_firm)]\n",
    "            for _ in range(num_price_adjusts_before_horizon[s])] \n",
    "        for s in range(num_sims)])\n",
    "\n",
    "prices_firm_adjust = np.array(\n",
    "    [[\n",
    "            np.array(calc_prices(prices_firm_adjust[s], adjusting_firm[s], globals()[\"s{}\".format(setup+1)])) \n",
    "            for s in range(num_sims)]\n",
    "        for setup in range(len(setups))])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As we don't want to know what happens after or on the time horizon, we will always substract 1 when using the value num_price_adjusts_before_horizon in the following code."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# get all the customers that appear between market events, so we don't always have to iterate through each customer\n",
    "gen = np.array(\n",
    "    [[\n",
    "            list([\n",
    "                x \n",
    "                for x in range(num_customer_events) \n",
    "                if price_adjusts_before_horizon[s][i]< time_customer_events[s][x]< price_adjusts_before_horizon[s][i+1]]) \n",
    "            for i in range(num_price_adjusts_before_horizon[s] - 1)] \n",
    "        for s in range(num_sims)] )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Simulate the customer by completing the score using the current prices of each firm and then calculating a decision for each customer."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "rand_customer_score = np.array(\n",
    "    [[[[[\n",
    "                        (prices_firm_adjust[setup,s][i,k] + customer_score_qual_rating[s,c,k]) \n",
    "                        for k in range(num_firm)]\n",
    "                    for c in gen[s][i]]\n",
    "                for i in range(num_price_adjusts_before_horizon[s] - 1)] \n",
    "            for s in range(num_sims)]\n",
    "        for setup in range(len(setups))])\n",
    "\n",
    "rand_customer_decision = np.array(\n",
    "    [[[[\n",
    "                    np.argmin(rand_customer_score[setup,s][i][c]) \n",
    "                    if min(rand_customer_score[setup,s][i][c])< random.uniform(5,15) \n",
    "                    else -1 \n",
    "                    for c in range(len(gen[s][i]))]\n",
    "                for i in range(num_price_adjusts_before_horizon[s] - 1)] \n",
    "            for s in range(num_sims)]\n",
    "        for setup in range(len(setups))])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Split our time horizon in bb parts and calculate how often our watched company sells their product in each time slot."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "b = np.array(\n",
    "    [[[\n",
    "                int(round((time_customer_events[s][c]+1/bb/2)*bb))\n",
    "                for c in range(len(gen[s][i]))]\n",
    "            for i in range(num_price_adjusts_before_horizon[s] - 1)] \n",
    "        for s in range(num_sims)])\n",
    "\n",
    "At = np.array(\n",
    "    [[[\n",
    "                sum([\n",
    "                     sum([\n",
    "                          sum([\n",
    "                               1 \n",
    "                               if b[s][i][c]==t and rand_customer_decision[setup,s][i][c]==k  \n",
    "                               else 0 \n",
    "                               for c in range(len(gen[s][i]))])\n",
    "                          for i in range(num_price_adjusts_before_horizon[s] - 1)])\n",
    "                     for s in range(num_sims)])\n",
    "                for k in range(num_firm)]\n",
    "            for t in range(reaction_time_firm*bb+1)]\n",
    "        for setup in range(len(setups))])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Create our plot."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0,0.5,'E(λ$_t$)')"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = [\n",
    "        t/bb \n",
    "        for t in range(1,reaction_time_firm*bb+1)]\n",
    "\n",
    "for i in range(len(setups)):\n",
    "    plt.plot(x, [At[i,t][watched_firm]/num_sims*bb for t in range(1,len(x)+1)], label=setups[i])\n",
    "plt.legend(loc='best')\n",
    "plt.xlabel('t')\n",
    "plt.ylabel('E(λ$_t$)')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Simulated expected sales intensities of firm 1 over one period of time time for different firmetitive setups (i)-(iii), starting in market situation s0 , cf."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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M9BtIf9/+fB72OaeSThkwwJrpyW5+OFpb8MnGolsfSenZPPbffawKjWNa70AW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\n5zHh7wm8sfMNrt24ZsxfR1GqPJMkDyl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      "text/plain": [
       "<matplotlib.figure.Figure at 0x11047ba90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.show()"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "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": "3.6.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}