Live by the Three, Die by the Three?The Price of Risk in the NBA

Matthew Goldman, UCSD Dept. of Economics, @mattrgoldmanJustin 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 curveNote: 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 adjustment

• As win value of 3’s goes up, their nominal value falls: respects “price of risk”

Results: 3-point usage ratesImpact of an increase in preference for risk for the

leading team

Results: 3-point usage ratesImpact of an increase in preference for risk for the

trailing team

Results: 3-point usage rates

1.4 1.5 1.6

0.21

0.22

0.23

Alpha

Probability of a 3PA

IncreasingDesperation

TrailingLeading

Results: Shooting efficiency

Impact of an increase in preference for risk for the leading team

Results: Shooting efficiency

Impact of an increase in preference for risk for the trailing team

Results: Shooting efficiency

1.4 1.5 1.6

0.08

0.1

0.12

0.14

0.16

Alpha

Efficiency Premium of 3PA

IncreasingDesperation

TrailingLeading

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 value2) 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 risk2) Leading team (lead shrinking): when

they 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.)

Bad offenses get worse

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: mrgoldman@ucsd.eduJustin M. Rao: justin.rao@microsoft.com

@mattrgoldman, @justinmrao, #priceofrisk