\n",
"\n",
"# WWW: Who Will Win?\n",
"\n",
"This is a notebook for reasoning about who will win playoff series in basketball (and could be used for other sports). The table of contents for this notebook:\n",
"\n",
"* [Model Discussion](#Models:-Introduction-and-Discussion) What model should I use to make predictions?\n",
"* [2019 NBA Playoffs](#2019-NBA-Playoffs) **Will be updated as the playoffs progress.**\n",
"* [2018 NBA Playoffs](#2018-NBA-Playoffs) An exciting season.\n",
"* [2016 NBA Playoffs](#2016-NBA-Playoffs) Ancient history; sad saga.\n",
"\n",
"---\n",
"\n",
"# Models: Introduction and Discussion\n",
"\n",
"\"It's tough to make predictions, especially [about](https://en.wikiquote.org/wiki/Yogi_Berra) [the](https://en.wikiquote.org/wiki/Niels_Bohr) future.\" That's true for the NBA basketball playoffs, where there is a wide range of opinions. Here are some models you might choose to help you make predictions:\n",
"\n",
"1. **Holistic**: I just feel that my favorite team has about a 1 in 5 chance of winning it all.\n",
"2. **Game by Game**: I think my team has a 75% chance of winning each game in the first round, then 65% for each game in the second round, but only 45% in the Conference finals, then 55% if they make it to the NBA finals. From that I'll calculate their overall chance.\n",
"3. **Point by Point**: My team has a per-game average point differential of +5.79; I'll compare that to the other teams and calculate overall chances.\n",
"4. **Play by Play**: Use [detailed statistics](https://www.basketball-reference.com/play-index/plus/shot_finder.cgi) and overhead video to [model](https://danvatterott.com/blog/2016/06/16/creating-videos-of-nba-action-with-sportsvu-data/) the game shot-by-shot, or even pass-by-pass. Not covered here.\n",
"\n",
"# Point by Point Model\n",
"\n",
"The **[Simple Rating System](https://www.sportingcharts.com/dictionary/nba/simple-rating-system-statistics.aspx) (SRS)** records the average point differential of a team over the season, with a slight adjustment for strength of schedule (see [basketball-reference.com](https://www.basketball-reference.com/leagues/NBA_2018.html)) . A great team has an SRS around 10; anything over 5 is very good.\n",
"\n",
"The Point-by-Point model says: a game is decided by a random sample from the distribution of point differentials, which is a normal (Gaussian) distribution centered around the difference of SRS scores of the two teams. So, if a team with an SRS of 7 plays an opponent with an SRS of 4, we can expect the team to win by 3, on average, but it will still lose some games. We need to know the standard deviation of the distribution to determine how often that happens; [Betlabs](https://www.betlabssports.com/blog/a-look-at-nba-team-totals/) says the standard deviation \n",
"is 10.5 points across the NBA. \n",
"The function `win_game` does the calculation of win probability given an SRS point differential, using Monte Carlo simulation:"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"%matplotlib inline\n",
"import matplotlib.pyplot as plt\n",
"from statistics import mean\n",
"from random import gauss\n",
"from collections import Counter\n",
"\n",
"def win_game(srs_diff, 𝝈=10.5, n=100000):\n",
" \"Given SRS point differential of a team against another, return game win probability.\"\n",
" return mean(gauss(srs_diff, 𝝈) > 0 for game in range(n))"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"0.61201"
]
},
"execution_count": 2,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"win_game(3)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"So, if your team has a 3-point edge over an opponent, this model predicts your team will win 61% of the time."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"\n",
"# Game by Game Model\n",
"\n",
"The next model says that a playoff series is a sequence of independent and identically distributed game results (where the probability of a single-game win could be specified using SRS, or holistically, or some other model). The idea here is to be consistent: if you believe that a team's win percentage is 60%, and you believe that games are independent, then you must believe that the team's chance of wining 4 in a row is 0.64 = 0.1296. This model ignores the fact that games aren't strictly independent, ignores the possibility of injuries, and ignores home court advantage. Why? Because these factors would change the final winning estimate by only a few percentage points, and I already have more uncertainty than that.\n",
"\n",
"The function `win_series` calculates the probability of winning a series, given the probability of winning a game:"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [],
"source": [
"def win_series(p, W=0, L=0):\n",
" \"\"\"Probability of winning best-of-7 series, given a probability p of winning a game.\n",
" The optional arguments say how many Wins and Losses the team has in the series so far.\"\"\"\n",
" return (1 if W == 4 else\n",
" 0 if L == 4 else\n",
" p * win_series(p, W + 1, L) + \n",
" (1 - p) * win_series(p, W, L + 1))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We can make a table:"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"0 point differential = 50% win game = 50% win series\n",
"1 point differential = 54% win game = 58% win series\n",
"2 point differential = 57% win game = 66% win series\n",
"3 point differential = 61% win game = 73% win series\n",
"4 point differential = 65% win game = 80% win series\n",
"5 point differential = 68% win game = 85% win series\n",
"6 point differential = 72% win game = 89% win series\n",
"7 point differential = 75% win game = 93% win series\n",
"8 point differential = 77% win game = 95% win series\n",
"9 point differential = 80% win game = 97% win series\n"
]
}
],
"source": [
"for srs_diff in range(10):\n",
" g = win_game(srs_diff)\n",
" print('{} point differential = {:4.0%} win game = {:4.0%} win series'.format(\n",
" srs_diff, g, win_series(g)))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"With a zero point differential obviously you're at 50% win percententage; with a 3 point differential you're at 61% to win a game, and 73% to win the series. This agrees very well with the \"Differential vs. Win Percentage\" [chart](http://a.espncdn.com/combiner/i?img=%2Fphoto%2F2018%2F0408%2F180408_differential.png&w=1140&cquality=40) on [this page](http://www.espn.com/nba/story/_/id/23071005/kevin-pelton-weekly-mailbag-including-nba-all-offensive-teams). \n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"\n",
"We can also do plots:"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [],
"source": [
"diff = [d/10 for d in range(101)]\n",
"game = [win_game(d) for d in diff]\n",
"series = [win_series(p) for p in game]"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [
{
"data": {
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\n",
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