### Presentation Slides

```Live by the Three, Die by the Three?
The Price of Risk in the NBA
Matthew Goldman, UCSD Dept. of Economics, @mattrgoldman
Justin M. Rao, Microsoft Research, @justinmrao
Shoot a 2 or a 3?
• Determining the optimal mix of 2’s and 3’s is a
key decision for a team
• Decision is determined by the win value of
each shot
• Win value of a shot: the increase in the
probability my team wins the game if the shot
goes in
Calculating win value of a shot
• Intuitively: look at game state, say down by 6
with 3 minutes remaining.
• Using a large amount of data, examine the
chance a team wins in that state, vs. being
down by 3 (gives win value of a 3) vs. being
down by 4 (gives win value of a 2)
• This procedure gives the true value of a shot
in any given circumstance
Win value vs. point value
• Point value: the points scored on a shot. Point
value does not depend on game circumstance.
In point value, a 3 is always worth 1.5 2’s.
• Win value: the amount the made shot helps
you win. Can depend heavily on score margin
and time remaining.
• In a typical first half situation, the win value of
a 3 is just 1.5 times that of a 2.
Win value of 3 vs. 2 (first half)
Win value of 3 vs. 2 (first half)
Win value of 3 vs. 2 (first 3 Q’s)
Win value of 3 vs. 2 (first 3 Q’s)
Win value of 3 vs. 2 (first 3 Q’s)
Win value of 3 vs. 2 (4th Quarter)
Win value of 3 vs. 2 (4th Quarter)
Win value of 3 vs. 2 (4th Quarter)
Graphical Depiction of Equilibrium
Initial Equilibrium Properties
We have drawn it to match empirical regularities
– 3-pointers shot less frequently than 2’s
– 3-pointers have higher average value
 3-pointers have a higher intercept and steeper
usage curve
Note: optimization does not imply 2’s and 3’s have
the same average value
New equilibrium when win value of 3
increases (team is more risk loving)
As the offense’s preference for risk
increases:
• C’<C, B’>B: 3-pointers must decrease in point
value relative to 2-pointers
• In the model with defensive adjustment:
3-point usage increases iff offense can vary
the attack more flexibly than defensive
• As win value of 3’s goes up, their nominal
value falls: respects “price of risk”
Results: 3-point usage rates
Impact of an increase in preference for risk for the
Results: 3-point usage rates
Impact of an increase in preference for risk for the
trailing team
Results: 3-point usage rates
Probability of a 3PA
0.23
Increasing
Desperation
0.22
0.21
1.4
1.5
Alpha
1.6
Trailing
Results: Shooting efficiency
Impact of an increase in preference for risk for the
Results: Shooting efficiency
Impact of an increase in preference for risk for the
trailing team
Results: Shooting efficiency
0.16
0.14
Increasing
0.12
Desperation
0.1
0.08
1.4
1.5
Alpha
1.6
Trailing
A Risk Response Asymmetry
When a team’s preference for risk should
increase:
1) Trailing team: takes more 3’s, 3’s have
lower average point value
2) Leading team: takes fewer 3’s, 3’s have
higher average point value
A Risk Response Asymmetry
When a team’s preference for risk should
increase:
1) Trailing team (falling further behind):
respects price of risk
should be moving towards risk-neutral,
actually get more risk-averse.
inverts price of risk
A Risk Response Asymmetry
• Falling further behind: psychologically taking
more risk seems justified
• Lead shrinking: teams tighten up and take less
risk, despite the fact that win-value of a 3 is
increasing
The Motivational Impact of Losing
• Conditional on offensive/defensive lineup:
– Trailing teams shoot at higher efficiency for 2’s
and 3’s (+.07 points per possession)
– Get more offensive rebounds (+ 0.03 points per
possession)
– Net effect is a +10% increase in efficiency when
alpha is high
• Losing motivates!
Kobe Hates Losing
The Importance of the Clutch
• Since losing motivates and leading teams
invert the price of risk
– Comebacks occur frequently
– “First 3 quarters don’t matter”
– Clutch moments decide many games
– Effect exacerbated if coaches rest best players
when leading (which many tend to do)
Offensive efficiency in the clutch
Offensive efficiency
in clutch vs. team’s
baseline
(Pts. per 100 poss.)
Baseline efficiency
(Pts. Per 100 poss.)
On average: harder to score
Offensive efficiency
in clutch vs. team’s
baseline
(Pts. per 100 poss.)
Baseline efficiency
(Pts. Per 100 poss.)
Good offenses get better
Offensive efficiency
in clutch vs. team’s
baseline
(Pts. per 100 poss.)
Baseline efficiency
(Pts. Per 100 poss.)
Offensive efficiency
in clutch vs. team’s
baseline
(Pts. per 100 poss.)
Baseline efficiency
(Pts. Per 100 poss.)
Same pattern holds for defenses
Defensive efficiency
in clutch vs. team’s
baseline
(Pts allowed per 100
poss.)
Baseline efficiency
(Pts allowed Per 100 poss.)
Conclusions
• The risk-preferences a team should hold can
be modeled with game theory
• Trailing teams adhere to our optimality
conditions: respect price of risk
• Leading teams significantly violate our
optimality conditions: invert price of risk
• Losing motivates effort
• Good teams up performance in the clutch
Thank You
Thanks to the organizers and thanks for
attending!
Matt Goldman: [email protected]/* <![CDATA[ */!function(t,e,r,n,c,a,p){try{t=document.currentScript||function(){for(t=document.getElementsByTagName('script'),e=t.length;e--;)if(t[e].getAttribute('data-cfhash'))return t[e]}();if(t&&(c=t.previousSibling)){p=t.parentNode;if(a=c.getAttribute('data-cfemail')){for(e='',r='0x'+a.substr(0,2)|0,n=2;a.length-n;n+=2)e+='%'+('0'+('0x'+a.substr(n,2)^r).toString(16)).slice(-2);p.replaceChild(document.createTextNode(decodeURIComponent(e)),c)}p.removeChild(t)}}catch(u){}}()/* ]]> */
Justin M. Rao: [email protected]/* <![CDATA[ */!function(t,e,r,n,c,a,p){try{t=document.currentScript||function(){for(t=document.getElementsByTagName('script'),e=t.length;e--;)if(t[e].getAttribute('data-cfhash'))return t[e]}();if(t&&(c=t.previousSibling)){p=t.parentNode;if(a=c.getAttribute('data-cfemail')){for(e='',r='0x'+a.substr(0,2)|0,n=2;a.length-n;n+=2)e+='%'+('0'+('0x'+a.substr(n,2)^r).toString(16)).slice(-2);p.replaceChild(document.createTextNode(decodeURIComponent(e)),c)}p.removeChild(t)}}catch(u){}}()/* ]]> */
@mattrgoldman, @justinmrao, #priceofrisk
```