src/examples/SingleDummyEstimates.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "3d26ee18",
   "metadata": {},
   "source": [
    "# Single Dummy Estimates\n",
    "\n",
    "This tutorial shows the use of a neural network to estimate the number of tricks which will be taken on a given open lead in a given contract (single dummy)\n",
    "\n",
    "In this tutorial we'll access a bit lower-level APIs.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "3db7a448",
   "metadata": {},
   "outputs": [],
   "source": [
    "import os\n",
    "os.chdir('..')\n",
    "\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "import scoring\n",
    "import deck52\n",
    "\n",
    "from nn.models import Models\n",
    "from bidding import bidding\n",
    "from bidding.binary import parse_hand_f\n",
    "from bots import BotBid\n",
    "from objects import Card\n",
    "from util import hand_to_str"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "b08c7148",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "INFO:tensorflow:Restoring parameters from ../models/gib21_model/gib21-1000000\n",
      "INFO:tensorflow:Restoring parameters from ../models/gib21_info_model/gib21_info-500000\n",
      "INFO:tensorflow:Restoring parameters from ../models/lead_model_b/lead-1000000\n",
      "INFO:tensorflow:Restoring parameters from ../models/lr3_model/lr3-1000000\n",
      "INFO:tensorflow:Restoring parameters from ../models/lefty_model/lefty-1000000\n",
      "INFO:tensorflow:Restoring parameters from ../models/dummy_model/dummy-920000\n",
      "INFO:tensorflow:Restoring parameters from ../models/righty_model/righty-1000000\n",
      "INFO:tensorflow:Restoring parameters from ../models/decl_model/decl-1000000\n"
     ]
    }
   ],
   "source": [
    "models = Models.load('../models')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "b4b75d68",
   "metadata": {},
   "outputs": [],
   "source": [
    "# helper functions are defined here. it's ok to scroll over this part\n",
    "\n",
    "# this is the function which does the prediction\n",
    "# for each possible number of tricks (from 0 to 13) \n",
    "# it returns the probability that that number of tricks will be taken (single dummy)\n",
    "def single_dummy(model, hands, decl_i, strain_i, lead_card_i):\n",
    "    # an array, which is the input for the neural network, is prepared\n",
    "    X_sd = np.zeros((hands.shape[0], 32 + 5 + 4*32))\n",
    "    X_sd[:,32 + strain_i] = 1\n",
    "    X_sd[:,(32 + 5 + 0*32):(32 + 5 + 1*32)] = hands[:, (decl_i + 1) % 4]\n",
    "    X_sd[:,(32 + 5 + 1*32):(32 + 5 + 2*32)] = hands[:, (decl_i + 2) % 4]\n",
    "    X_sd[:,(32 + 5 + 2*32):(32 + 5 + 3*32)] = hands[:, (decl_i + 3) % 4]\n",
    "    X_sd[:,(32 + 5 + 3*32):] = hands[:, decl_i]\n",
    "    X_sd[:, lead_card_i] = 1\n",
    "    \n",
    "    return model(X_sd)\n",
    "\n",
    "# plotting the trick probability distribution\n",
    "def plot_tricks(tricks_softmax):\n",
    "    tricks = tricks_softmax[0]\n",
    "    plt.bar(range(14), tricks)\n",
    "    plt.title(f'E(tricks) = {tricks @ np.arange(14)}')\n",
    "    plt.xlabel('# tricks')\n",
    "    plt.ylabel('probability')\n",
    "\n",
    "def expected_score(tricks_softmax, contract, vuln):\n",
    "    tricks = tricks_softmax\n",
    "    score_by_tricks = np.array([scoring.score(contract, vuln, i) for i in range(14)])\n",
    "    return tricks @ score_by_tricks"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "fc7fa234",
   "metadata": {},
   "outputs": [],
   "source": [
    "# we take the following hand as an example\n",
    "# West plays it in 3N on a small heart lead\n",
    "# how many tricks will declarer take?\n",
    "\n",
    "hands_str = 'A8.Q752.T54.JT63 K54.T643.A92.A72 JT932.K9.Q86.854 Q76.AJ8.KJ73.KQ9'  # N E S W\n",
    "contract = '3NW'\n",
    "opening_lead = 'H2'"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "573954a6",
   "metadata": {},
   "outputs": [],
   "source": [
    "# this line just converts the string representation of the hands into a binary format\n",
    "hands = np.vstack([parse_hand_f(32)(s) for s in hands_str.split()]).reshape((-1, 4, 32))\n",
    "\n",
    "# applying the function which does the prediction\n",
    "tricks_distr = single_dummy(\n",
    "    models.sd_model.model, #Leading model\n",
    "    hands, #The hands\n",
    "    decl_i='NESW'.index(contract[-1]), #Index of declarer\n",
    "    strain_i='NSHDC'.index(contract[1]), #Index de la couleur du contrat\n",
    "    lead_card_i=deck52.card52to32(Card.from_symbol(opening_lead).code())\n",
    ")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "c6f6fdff",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0.0000000e+00, 0.0000000e+00, 8.1465862e-36, 1.7034037e-11,\n",
       "        3.6211486e-07, 1.5782302e-02, 3.2778580e-02, 7.3673598e-02,\n",
       "        1.2808882e-01, 2.0297310e-01, 2.3446704e-01, 1.8965560e-01,\n",
       "        9.5386654e-02, 2.7193969e-02]], dtype=float32)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# the result is an array representing the probability distribution of tricks\n",
    "# the first element of the array is the probability that declarer will take 0 tricks\n",
    "# the last element of the array is the probability that declarer will take 13 tricks\n",
    "tricks_distr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "a0bead58",
   "metadata": {},
   "outputs": [
    {
     "data": {
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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# we can also plot it to see it better\n",
    "plot_tricks(tricks_distr)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5ccaf6b1",
   "metadata": {},
   "source": [
    "We can see that the most likely declarer will take 10 tricks. But there is a spread, and declarer will take on average 9.57 tricks (according to the neural network)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "8923c1dc",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "308.28683088537787"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# we can also compute how much declarer is expected to score on this hand (non-vulnerable)\n",
    "expected_score(tricks_distr, contract='3N', vuln=False)[0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "ad71c875",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.7496764"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# or we can get the probability that the contract will be made (i.e declarer will take 9 or more tricks)\n",
    "np.sum(tricks_distr[:,9:])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c88dc55c",
   "metadata": {},
   "source": [
    "## Estimating on many samples at the same time\n",
    "\n",
    "A frequent use of single dummy models is to apply it on many samples to choose between possible actions (for example when making a bid or selecting the opening lead)\n",
    "\n",
    "We could just follow the steps in the above example in a loop (looping once for each sample), but it's much more efficient to make the prediction for all samples at the same time:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "6e53c7e8",
   "metadata": {},
   "outputs": [],
   "source": [
    "# we'll use a bidder bot to draw samples which are consistent with the auction\n",
    "# and give it North's cards (as North will be on lead against 3N)\n",
    "bid_bot = BotBid([False, False], 'A8.Q752.T54.JT63', models)\n",
    "\n",
    "# let's assume the following auction has occurred\n",
    "auction = [\n",
    "    'PASS', 'PASS', 'PASS', '1N', \n",
    "    'PASS', '2C', 'PASS', '2D', \n",
    "    'PASS', '3N', 'PASS', 'PASS', \n",
    "    'PASS', 'PAD_END', 'PAD_END', 'PAD_END'   # we need to put the 'PAD_END' stuff so it will look like it's North's turn to bid again\n",
    "]\n",
    "contract = bidding.get_contract(auction)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "d31e8c24",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "A8.Qxxx.Txx.JTxx Kxxx.KJ9x.QJ9x.x T9xx.T8.Kxx.KQ9x QJx.Axx.A8x.A8xx\n",
      "A8.Qxxx.Txx.JTxx Jxx.T8xx.AKQ9x.x Q9xxx.KJ9.xx.Q9x KTx.Ax.J8x.AK8xx\n",
      "A8.Qxxx.Txx.JTxx QTxx.AKJx.8xx.9x KJ9x.T9x.9xx.A8x xxx.8x.AKQJ.KQxx\n",
      "A8.Qxxx.Txx.JTxx QJ9x.JT9x.K8x.A8 Kxxx.xx.QJxx.9xx Txx.AK8.A9x.KQxx\n",
      "A8.Qxxx.Txx.JTxx xx.KT8x.KQ8x.Kxx K9xxxx.xx.Jx.Qxx QJT.AJ9.A9xx.A98\n",
      "A8.Qxxx.Txx.JTxx xxxx.AKTx.K9.9xx KQT9.9xx.J8xx.8x Jxx.J8.AQxx.AKQx\n",
      "A8.Qxxx.Txx.JTxx Jxx.KT98.K98.A9x T9xxx.Jxx.Jxxx.K KQx.Ax.AQx.Q8xxx\n",
      "A8.Qxxx.Txx.JTxx KJxx.ATxx.Jx.Qxx 9xxx.8x.K98x.A9x QTx.KJ9.AQxx.K8x\n",
      "A8.Qxxx.Txx.JTxx QJ9x.T8.Kxxx.Axx Txxxx.KJ9x.Qx.Qx Kx.Axx.AJ98.K98x\n",
      "A8.Qxxx.Txx.JTxx Kxxx.AT9x.Jxx.Q9 JT9x.J8x.KQxx.8x Qxx.Kx.A98.AKxxx\n"
     ]
    }
   ],
   "source": [
    "# we sample the hidden cards for the auction\n",
    "samples = bid_bot.sample_hands(auction)\n",
    "\n",
    "# and display a few samples just for illustration\n",
    "for i in range(10):\n",
    "    print(f'{hand_to_str(samples[i,0,:])} {hand_to_str(samples[i,1,:])} {hand_to_str(samples[i,2,:])} {hand_to_str(samples[i,3,:])}')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9a983c75",
   "metadata": {},
   "source": [
    "North's cards are always the same `A8.Qxxx.Txx.JTxx` (because it's the known hand)\n",
    "\n",
    "The rest of the cards are consistent with the Stayman sequence which occurred in the auction (West has a 1N opener without four-card majors, East has a four-card major and the strength for 3N)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "8b0b6640",
   "metadata": {},
   "outputs": [],
   "source": [
    "# now we apply the prediction on all samples (estimating how many tricks declarer will take in 3N)\n",
    "\n",
    "# we consider the following opening leads\n",
    "opening_lead_options = ['SA', 'H2', 'D4', 'CJ', 'C3']"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "6dbb241a",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "3NW\n"
     ]
    }
   ],
   "source": [
    "# for each opening opening lead option,\n",
    "# we estimate the number of tricks taken (on each sample)\n",
    "\n",
    "lead_tricks = {}\n",
    "print(contract)\n",
    "for opening_lead in opening_lead_options:\n",
    "    tricks_distr = single_dummy(\n",
    "        models.sd_model.model, \n",
    "        samples, \n",
    "        decl_i='NESW'.index(contract[-1]), \n",
    "        strain_i='NSHDC'.index(contract[1]), \n",
    "        lead_card_i=deck52.card52to32(Card.from_symbol(opening_lead).code())\n",
    "    )\n",
    "    lead_tricks[opening_lead] = tricks_distr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "6d8fd172",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{'SA': 280.6029739764299,\n",
       " 'H2': 261.3467110765077,\n",
       " 'D4': 258.5438087043963,\n",
       " 'CJ': 253.13244963947042,\n",
       " 'C3': 261.6747969372446}"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# computing declarer's expected score for each lead\n",
    "\n",
    "{lead:np.mean(expected_score(tricks, contract, False)) for lead, tricks in lead_tricks.items()}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "dbc0b203",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# to get an idea about the variance, we can compare two cards by plotting histograms of expected scores\n",
    "plt.hist(expected_score(lead_tricks['H2'], contract, False), label='H2', alpha=0.5)\n",
    "plt.hist(expected_score(lead_tricks['D4'], contract, False), label='D4', alpha=0.5)\n",
    "plt.xlabel('expected score')\n",
    "plt.ylabel('frequency')\n",
    "plt.legend();"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3.10.2 64-bit",
   "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": "3.10.2 (tags/v3.10.2:a58ebcc, Jan 17 2022, 14:12:15) [MSC v.1929 64 bit (AMD64)]"
  },
  "vscode": {
   "interpreter": {
    "hash": "db6859663c517eaff44cb590a4f83a9ffca1c2df3a6cfb3b0e93ea33231aa8bf"
   }
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}