Live by the Three, Die by the Three? The Price of Risk in the NBA Matthew Goldman, UCSD Dept. of Justin M. Rao, Microsoft

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 3 Q’s)

Win value of 3 vs. 2 (4 th 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’ 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 rates Impact of an increase in preference for risk for the leading team

Results: 3-point usage rates Impact of an increase in preference for risk for the trailing team

Results: 3-point usage rates

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

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 2) 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 ( 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: Justin #priceofrisk