2011 5th MIT Sloan Sports Analytics Conference

1. Scoring Strategies for the Underdog:
Using Risk as an Ally in Determining Optimal
Sports Strategies
Brian Skinner
bskinner@physics.umn.edu
William I. Fine Theoretical Physics Institute
University of Minnesota, Twin Cities
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2. This is not a talk about data analysis
• Every analysis begins with a question.
• Usually: “Here is some data. What can I learn from it?”
• Here: “Assume I know everything about my offense. How do
I optimize its performance?”
• Which plays do I run, and how often?
• How do I allocate shots among players?
• When (and to what extent) should I use high-risk tactics?
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3. This is not a talk about data analysis
• Every analysis begins with a question.
• Usually: “Here is some data. What can I learn from it?”
• Here: “Assume I know everything about my offense. How do
I optimize its performance?”
• Which plays do I run, and how often?
• How do I allocate shots among players?
• When (and to what extent) should I use high-risk tactics?
Goal: A simple, quantitative rule for assessing risk/reward
decisions and calculating optimal strategies.
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4. When should I take risks?
Example: A simple shooting contest – score the most
points on 100 shots
X
Shoot 2-pointers: 2PT% = 75%
X
or 3-pointers: 3PT% = 50%
Winner gets $1,600,000.
Question: Should you choose to shoot 2’s or 3’s?
Answer: It depends entirely on how good your friend is.
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5. A strategy is characterized by a
probability distribution
0.06 For a given mean
shoot 2's
shoot 3's
score, more risk implies
more variance
0.04 Which distribution should
probability density
150 8.7
you choose?
0.02
150 15
0
100 120 140 160 180 200
points scored
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6. A strategy is characterized by a
probability distribution
0.06 For a given mean
better
score, more risk implies
0.05 opponent more variance
probability density
0.04 Which distribution should
you choose?
0.03
0.02
shoot 3's Underdogs should pursue
riskier strategies.
0.01
0
100 120 140 160 180 200
points scored
6

7. A strategy is characterized by a
probability distribution
0.06 For a given mean
score, more risk implies
0.05 worse more variance
opponent
probability density
0.04 Which distribution should
shoot 2's you choose?
0.03
0.02 Underdogs should pursue
riskier strategies.
0.01
0 Favored teams should be
100 120 140 160 180 200
points scored conservative.
[Dean Oliver, 1995, http://www.rawbw.com/~deano/]
[Malcolm Gladwell, “How David Beats Goliath”,The New Yorker, 2009]
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8. What’s better: larger variance or a
higher mean?
An underdog should be
willing to sacrifice from
its mean score in order
to increase the variance
in the outcome.
Sometimes the best
strategy is the one that
leads, on average, to a
worse loss.
How can we optimize the risk/reward tradeoff?
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9. The elements of offensive strategy
A team has a set of plays it can call: i = 1, 2, 3, …
Each has point value vi and success rate pi.
“Strategy”: The number of times Ni that each play is run.
Assumption: All plays are statistically independent.
[Huizinga and Weil, “Hot Hand or Hot Head? …”, SSAC, 2009]
Consequence: The final score follows the binomial distribution.
mean: μ = Σi Ni vi pi
variance: σ2 = Σi Ni vi2 pi (1-pi)
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10. The “CLT Rule”
The Central Limit Theorem (CLT):
When many plays remain, the quantity Δ = (your score) – (opponent’s score)
is Gaussian distributed with mean μ – μopp and variance σ2 + σ2opp.
So the probability of winning depends only on the quantity
μ μopp μ = Σi Ni vi pi
Z
σ 2 σopp
2 σ2 = Σi Ni vi2 pi (1-pi)
The CLT Rule: A team’s best long-term strategy is
the one that gives the largest value of Z.
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11. Example 1: UNC vs. Duke
The recipe for an upset
UNC 2010-11: 2PT% = 0.498 Duke 2010-11: 2PT% = 0.522
3PT% = 0.317
Should UNC shoot 2’s or 3’s?
Say UNC trails by s with N poss. left
shoot
If they shoot 2’s: Z 0.048N s 3’s
2 1.99N
If they shoot 3’s: Z 0.093N s
3 2.95N
shoot
Z3 > Z2 when 2’s
s > 0.16 N
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12. Example 2: Stalling for Time
It is in the interest of the underdog to keep the game short.
underdog: p2 = 0.5
opponent: p2 = 0.55
regular possessions =
16 seconds
OR
“stalled” possessions
= 24 seconds ?
An underdog can benefit from stalling even when they
don’t have the lead.
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13. Example 3: Playing Hack-a-Shaq
A winning team protects its lead by preventing high-
variance plays.
One way: foul the opponent’s worst free throw shooter
Shaq Brian Skinner
[Getty Images/Tom Hauck]
[AP Photo/Greg Wahl-Stephens]
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14. Example 3: Playing Hack-a-Shaq
Suppose your team shoots p2 = 0.55.
Shaq’s team:
p2 = 0.5
pFT = 0.55
When should you
foul?
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15. A word about “skill curves”
In reality, a given play should not be run 100% of the time.
Play effectiveness declines with frequency of use.
This amounts to a dependence pi(Ni).
shoot 3’s
shoot
3’s
shoot
shoot 2’s
2’s
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16. Conclusions: quantifying
risk/reward decisions in sports
• When winning is unlikely, an underdog must take risk.
• “How much risk?” can be answered quantitatively.
• If different plays can be considered statistically
independent, the binomial distribution completely describes
the outcome.
• Optimal (mixed) strategies can be calculated – when many
plays remain the CLT Rule gives an excellent approximation.
• When the coach/player thinks “I know I should be doing
X, but I don’t know to what extent I should do it”, there is a
great opportunity for analytic approaches to be helpful.
Questions?
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