Implementation of Donald G. Morrison and Rita D. Wheat, 1986 Misapplications Reviews: Pulling the Goalie Revisited, Interfaces, Vol. 16 No. 6, pp. 28-34 as an IPython notebook

pullinggoalie.ipynb

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      "# Pulling the goalie\n",
      "\n",
      "Writeup by Louis Luangkesorn, based on article by Morrison and Wheat, (1986) [1]\n",
      "\n",
      "    \n",
      "## Problem\n",
      "\n",
      "-   If an ice hockey team is down by one goal late in the game, should that team replace the goalie with an additional skater?\n",
      "    \n",
      "    -  In hockey, the team scoring the most goals in a game by putting the puck into the opponents net wins the game.\n",
      "    -  Season scoring\n",
      "        -  Two points for a win\n",
      "        -  One point for a tie\n",
      "        -  Zero points for a loss\n",
      "        -  Note:  Goal differential has *no* effect\n",
      "\n",
      "-  If a team pulls its goalie, it is more likely to score a goal\n",
      "-  But, the likelihood of scoring a goal for the other team increases even more!\n",
      "            \n",
      "## Model\n",
      "\n",
      "-  A team scores goals at a rate of $\\lambda$ per minute (Poisson distribution)\n",
      "-  If team *A* pulls its goalie in favor of another skater, its rate of scoring goals increases to $\\lambda_A$\n",
      "-  If team *A* pulls its goalie in favor of another skater, the *opposing* team *B* rate of scoring goals increases to $\\lambda_B$\n",
      "-  $\\lambda < \\lambda_A < \\lambda_B$ for all teams $i$\n",
      "-  The scoring process for team *A* is independent of team *B*\n",
      "-  The potential outcomes are win, tie, lose with values 2, 1, 0 respectively\n",
      "\n",
      "**When is the best time to pull a goalie?**\n",
      "\n",
      "*Thought question:  What is the immediate objective of a team that is down by one goal near the end of the game?*"
     ]
    },
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     "cell_type": "markdown",
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     "source": [
      "##  Motivation  2013 Stanley Cup Finals - Boston Bruins vs. Chicago Blackhawks\n",
      "\n",
      "###  Game 5\n",
      "\n",
      "-  Boston down 2-1\n",
      "-  Goalie Rask (BOS) pulled 3rd-19:04\n",
      "-  Bolland (CHI) scores and empty net goal 19:46 Chicago (3-1)\n",
      "\n",
      "###  Game 6\n",
      "\n",
      "-  Chicago down 2-1\n",
      "-  Goalie Crawford pulled 18:32\n",
      "-  Bickell (CHI) goal 18:44  (Crawford returns, Tied 2-2)\n",
      "-  Boland (CHI) goal 19:01  (Chicago 3-2)\n",
      "-  Rask (BOS) pulled 19:20\n",
      "-  Game ends\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "##  Analysis\n",
      "\n",
      "-  Once the goalie has been pulled by team *A*, three things can happen: \n",
      "    -  *A* can score; \n",
      "    -  *B* can score an empty net goal; \n",
      "    -  or the game ends with no goals scored by either team.\n",
      "\n",
      "Find the probabilities of each event.\n",
      "\n",
      "-  $p(0: \\lambda t)$ = The probability of zero goals being scored in a time interval of length $t$ assuming that goals scored is a Poisson process.\n",
      "-  $\\Phi_p(t)$ = The probability of obtaining a tie when your team is down by one goal with $t$ minutes remaining in the game and you pull your goalie.\n",
      "-  $\\Phi_np(t)$ = Same as above except you *do not* pull your goalie.\n",
      "\n",
      "\n",
      "-  $X_i$, = minutes until an extra man goal is scored by Team *A*, *i = 1 ... n*.\n",
      "-  $Y_i$ = minutes until an empty net goal is scored by Team *B*, *i = 1 . . . m*.\n",
      "-  $Z_i$, = minutes until the end of the game with no goals scored by *A* or *B*, *i = 1 . . . k*.\n",
      "\n",
      "-  Because the scoring model is Poisson, the likelihood that team *A* scores a goal in *X* minutes is $\\lambda_i^A e^{-\\lambda_A X}$\n",
      "-  The likelihood that team *B* scores an empty net goal in *Y* minutes is  $\\lambda_i^B e^{-\\lambda_B Y}$\n",
      "-  The likelihood that neither team *A* or *B* scores in *Z* minutes is $e^{-(\\lambda_A +\\lambda_B) Z}$\n",
      "\n",
      "$\\Phi_p(t) = [1-P_p(0;(\\lambda_A + \\lambda_B)t] \\frac{\\lambda_A}{\\lambda_A + \\lambda_B}$\n",
      "                    \n",
      "$\\Phi_{np}(t) = [1-P_p(0; 2\\lambda t)]^{1/2}$\n"
     ]
    },
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     "cell_type": "markdown",
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     "source": [
      "Note:  The real question is not who scores first, but when should the goalie be pulled given that there are *T* minutes left and you are down by one (i.e. choose $t$) so \n",
      "    \n",
      "$P(Tie, T, t) = 1-[e^{-2\\lambda(T-t)}]^{1/2} + e^{-2 \\lambda(T-t)}[1-e^{(\\lambda_A + \\lambda_B) t}]\\frac{\\lambda_A}{\\lambda_A+\\lambda_B}$\n",
      "\n",
      "Mathematically, we can take a derivitive with respect to $t$ to get an optimal value.\n",
      "\n",
      "$t^* = - \\frac{log[\\lambda(\\lambda_A-\\lambda_B)/\\lambda_A(2 \\lambda - \\lambda_A - \\lambda_B)]}{\\lambda_A + \\lambda_B}$\n",
      "\n",
      "This turns out to \n",
      "\n",
      "So, does this look right?\n",
      "\n",
      "1.  How to find estimates for $\\lambda, \\lambda_A, \\lambda_B$?\n",
      "\n"
     ]
    },
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     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "a.  Data from power plays\n",
      "    -  A player is benched (removed) due to a penalty for two minutes.  So there are now players on that team facing the standard 5 players on the other team.  \n",
      "    -  Track both when a goal is scored (by either side) or no goal scored\n",
      "    -  But this is imperfect, since the goalies are still present on both teams.  So it overestimates the chances of the team that is one person up to score.\n",
      "b.  Use data from times goalie was pulled. \n",
      "\n",
      "\n",
      "Now, given our estimates for  $\\lambda, \\lambda_A, \\lambda_B$, find the optimal time to pull a goalie.  First code the pTie and optimal $t$ functions"
     ]
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     "cell_type": "code",
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     "input": [
      "import math, random\n",
      "\n",
      "\n",
      "#lambd, lambdA, lambdB = symbols('lambd lambdA lambdB')\n",
      "lambd = 0.06\n",
      "lambdA = 0.16\n",
      "lambdB = 0.47\n",
      "\n",
      "def pTie(lambd, lambdA, lambdB, T, t):\n",
      "    firstterm = 1-math.pow((math.exp(-2*lambd*(T-t ))), 0.5)\n",
      "    secondterm = math.exp(-2 * lambd * (T-t))*(1- math.exp(-(lambdA + lambdB)*t))*(lambdA)/(lambdA + lambdB)\n",
      "    ptie =  firstterm + secondterm \n",
      "    return ptie\n",
      "\n",
      "def optimalt(lambd, lambdA, lambdB):\n",
      "    numerator = -1 * math.log((lambd * (lambdA - lambdB))/(lambdA * (2 * lambd -lambdA - lambdB)))\n",
      "    denominator = (lambdA + lambdB)\n",
      "    t = numerator/denominator\n",
      "    return t\n",
      "\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 1
    },
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     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Now, get the optimal time."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "tstar = optimalt(lambd, lambdA, lambdB)\n",
      "pstar = pTie(lambd, lambdA, lambdB, tstar, tstar)\n",
      "print(tstar)\n",
      "print(tstar, pstar)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "2.34709155754\n",
        "(2.3470915575411206, 0.19607843137254902)\n"
       ]
      }
     ],
     "prompt_number": 2
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "pt1 = pTie(lambd, lambdA, lambdB, tstar, 1.0)\n",
      "print(tstar, pt1)\n",
      "pt0 = pTie(lambd, lambdA, lambdB, tstar, 0.0)\n",
      "print(tstar, pt0)\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "(2.3470915575411206, 0.17863379736990476)\n",
        "(2.3470915575411206, 0.1313591180063588)\n"
       ]
      }
     ],
     "prompt_number": 3
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "So, if tstar is the optimal time, and the probability of getting the next goal is 0.196, what do we lose by pulling the goalie either earlier or later?\n",
      "\n",
      "Try a range of values for $t$, given $T=2.347$ and look at the probabilities."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "tvalues = [0.1 * i for i in range(40)]\n",
      "ptievalues = [pTie(lambd, lambdA, lambdB, tstar, tvalues[i]) for i in range(len(tvalues))]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 4
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import matplotlib.pylab as pyl\n",
      "ax = pyl.axes()\n",
      "ax.set_title('Probability score next goal before end of game')\n",
      "ax.set_xlabel('Time goalie pulled')\n",
      "ax.set_ylabel('Probablity')\n",
      "ax.plot(tvalues, ptievalues)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
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       "output_type": "pyout",
       "prompt_number": 5,
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        "[<matplotlib.lines.Line2D at 0x23760d0>]"
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My7p5U0sW3t7w9dfSDCVESWFzNYzk5GQAAgMDcXd3p0OHDkQ/0HXv5uZG8+bN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\nIglDCCGEWSRhCCGEMMv/AzTnRcjf/gCUAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x2260210>"
       ]
      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Now, we can see what we lose by pulling the goalie at a time other than the optimal.  Note that this says that if you do not pull the goalie when you are down by 1 with 2.45 minutes left, you only have 13% chance of tying the game."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "fractionoptiomal = [ptievalues[i]/pstar for i in range(len(tvalues))]\n",
      "fx = pyl.axes()\n",
      "fx.set_title('Fraction of optimal possibility of scoring next goal to tie game')\n",
      "fx.set_xlabel('Time goalie pulled')\n",
      "fx.set_ylabel('Fraction')\n",
      "fx.set_ylim(0, 1.05)\n",
      "fx.plot(tvalues, fractionoptiomal)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 6,
       "text": [
        "[<matplotlib.lines.Line2D at 0x25f16d0>]"
       ]
      },
      {
       "output_type": "display_data",
       "png": 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MhKbQ13ZT/TB8eV29CkycCJw9qxws79zZ3BURET1mqnVnhW+B3Lt3DwDQqVMn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\nrbLUaSpGCZC33noLy5cvx54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       "text": [
        "<matplotlib.figure.Figure at 0x2379d50>"
       ]
      }
     ],
     "prompt_number": 6
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Conclusion\n",
      "\n",
      "Traditionally, goalies were pulled if the team down by one at one minute to go.  This analysis suggests that the goalie should be pulled at around 2:40 to go.\n",
      "    \n",
      "Some limitations on the model\n",
      "\n",
      "1.  Does not recognize that different teams have different rates of scoring. [2]\n",
      "2.  The analysis ends at the next goal scored.  It does not count for the possibility that pulling the goalie enables another score. [3]\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "##  References\n",
      "\n",
      "[1]  Morrison, D. G., and R. D. Wheat. 1986. *Misapplications Reviews: Pulling the Goalie Revisited.* **Interfaces** 16 (6) (November 1): 28\u201334. doi:10.1287/inte.16.6.28. \n",
      "\n",
      "[2] Erkut, E. 1987. *Note: More on Morrison and Wheat\u2019s \u2018Pulling the Goalie Revisited\u2019.* **Interfaces** 17 (5) (September 1): 121\u2013123. doi:10.1287/inte.17.5.121.\n",
      "\n",
      "[3] Washburn, A. 1991. *Still More on Pulling the Goalie.* **Interfaces** 21 (2) (March 1): 59\u201364. doi:10.1287/inte.21.2.59."
     ]
    }
   ],
   "metadata": {}
  }
 ]
}