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

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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

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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

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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.

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Win value of 3 vs. 2 (first half)

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Win value of 3 vs. 2 (first 3 Q’s)

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Win value of 3 vs. 2 (4 th Quarter)

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Graphical Depiction of Equilibrium

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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

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New equilibrium when win value of 3 increases (team is more risk loving)

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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”

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

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

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Results: 3-point usage rates

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Results: Shooting efficiency Impact of an increase in preference for risk for the leading team

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Results: Shooting efficiency Impact of an increase in preference for risk for the trailing team

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Results: Shooting efficiency

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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

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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

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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

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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!

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Kobe Hates Losing

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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)

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Offensive efficiency in the clutch Offensive efficiency in clutch vs. team’s baseline (Pts. per 100 poss.) Baseline efficiency (Pts. Per 100 poss.)

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On average: harder to score Offensive efficiency in clutch vs. team’s baseline (Pts. per 100 poss.) Baseline efficiency (Pts. Per 100 poss.)

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Good offenses get better Offensive efficiency in clutch vs. team’s baseline (Pts. per 100 poss.) Baseline efficiency (Pts. Per 100 poss.)

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Bad offenses get worse Offensive efficiency in clutch vs. team’s baseline (Pts. per 100 poss.) Baseline efficiency (Pts. Per 100 poss.)

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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.)

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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

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Thank You Thanks to the organizers and thanks for attending! Matt Goldman: Justin #priceofrisk

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