Riddler Classic

By Zach Wissner-Gross

Now that LeBron James and Anthony Davis have restored the Los Angeles Lakers to glory with their recent victory in the NBA Finals, suppose they decide to play a game of sudden-death, one-on-one basketball. They’ll flip a coin to see which of them has first possession, and whoever makes the first basket wins the game.

Both players have a 50 percent chance of making any shot they take. However, Davis is the superior rebounder and will always rebound any shot that either of them misses. Every time Davis rebounds the ball, he dribbles back to the three-point line before attempting another shot.

Before each of Davis’s shot attempts, James has a probability p of stealing the ball and regaining possession before Davis can get the shot off. What value of p makes this an evenly matched game of one-on-one, so that both players have an equal chance of winning before the coin is flipped?

My Solution

I can’t wrap my head around how to tackle this problem analytically. I’ll start by running simulations since that seems to be the easiest way to see what’s going on here.

simGame <- function (pSteal) {
  # this is the coinflip at the beginning
  possession <- sample(c("james", "davis"), 1)
  gameEnd <- F
  # each cycle starts with a player about to shoot the ball
  while (!gameEnd) {
    # james will try and steal the ball before every shot (if it's own shot this doesn't matter)
    # at the beginning bc twitter clarification
    if (sample(c(T,F), 1, prob = c(pSteal, 1-pSteal))) {
      possession <- "james"
    }
    
    # the player in possession will try and make the shot
    if(sample(c(T,F), 1)) {
      gameEnd <- T # idk if this is strictly necessary bc we're using return()
      return(possession)
    }
    
    # davis gets the rebound
    possession <- "davis"
  }
}
nSims <- 1000
df <- data.frame(pSteal = seq(0, 1, by = 0.01), meanJamesWin = 0, sdJamesWin = 0)
for (row in 1:nrow(df)) {
  p <- df[row, "pSteal"]
  winners <- c()
  for (i in 1:nSims) {
    winners <- append(winners, simGame(p))
  }
  df[row, "meanJamesWin"] <- mean(winners == "james")
  df[row, "sdJamesWin"] <- sd(winners == "james")
}
library(ggplot2)
library(ggpubr)
ggplot(df) +
  geom_point(aes(x = pSteal, y = meanJamesWin)) +
  ylab("Probability of LeBron James Winning") +
  xlab("Probability of LeBron James Stealing the Ball") +
  theme_pubr()

It’s linear — that makes our job pretty easy.

model <- lm(meanJamesWin ~ pSteal, data = df)
print(paste("The probability of either player winning becomes 0.5 when James's probability of stealing the ball is", round((0.5 - model$coefficients[1])/model$coefficients[2], 3)))
## [1] "The probability of either player winning becomes 0.5 when James's probability of stealing the ball is 0.334"

Cool, so the answer is probably one-third and working through the simulation loop gave me a tiny bit of intuition for solving this problem analytically.

The game can be split into two parts: the very first shot attempt right after the coin toss and every shot attempt after. The first shot attempt is a tiny bit different because you have the coin toss (\(\frac{1}{2}\)) in addition with the probability that James can steal the ball in the case that Davis is in possession (\(\frac{1}{2}p\)). This is then multipled by probability that whoever is in possession of the ball will end the game in sudden-death then and there (\(\frac{1}{2}\)). The probability of James winning on the first shot is:

\[(\frac{1}{2} + \frac{1}{2}p) \cdot \frac{1}{2} = \frac{1+p}{4}\]

We can now just add the probabilities of James winning on every subsequent round. The probability of the game ending on shot \(n\) is \((\frac{1}{2})^n\). We can just multiply that by \(p\) which becomes the probability that James steals the ball after Davis gets the rebound

\[\sum\limits_{n=2}^\infty p \cdot (\frac{1}{2})^n\]

which simplifies and converges to

\[\frac{p}{2}\]

Adding both terms together we get that for probability \(p\) of James stealing the ball before each shot, the chance that he wins the sudden-death game before the coin toss is

\[\frac{1 + 3p}{4}\]

Setting the expression equal to \(\frac{1}{2}\) we can solve for \(p\) and get that it equals \(\frac{1}{3}\) which matches what we found with our simulations! Here’s the math that I did to kind of figure this out — it’s probably illegible at best.

Now, we can really have some fun with this:

simGame2 <- function (pSteal, pRebound) {
  # this is the coinflip at the beginning
  possession <- sample(c("james", "davis"), 1)
  gameEnd <- F
  # each cycle starts with a player about to shoot the ball
  while (!gameEnd) {
    # james will try and steal the ball before every shot (if it's own shot this doesn't matter)
    # at the beginning bc twitter clarification
    if (sample(c(T,F), 1, prob = c(pSteal, 1-pSteal))) {
      possession <- "james"
    }
    
    # the player in possession will try and make the shot
    if(sample(c(T,F), 1)) {
      gameEnd <- T # idk if this is strictly necessary bc we're using return()
      return(possession)
    }
    
    # davis gets the rebound with pRebound
    possession <- sample(c("davis", "james"), 1, prob = c(pRebound, 1-pRebound))
  }
}
df2 <- expand.grid(pSteal = seq(0, 1, by = 0.01), 
                   pRebound = seq(0, 1, by = 0.01), 
                   jWin = 0)
nSims <- 1000
for (row in 1:nrow(df2)) {
  pSteal <- df2[row, "pSteal"]
  pRebound <- df2[row, "pRebound"]
  winners <- c()
  for (i in 1:nSims) {
    winners <- append(winners, simGame2(pSteal, pRebound))
  }
  df2[row, "jWin"] <- mean(winners == "james")
}
write.csv(df2, "2paramDF.csv")
df2 <- read.csv("heatmapDF.csv")
df2[df2$jWin < 0.52 & df2$jWin > 0.48, "jWin"] = 1

ggplot(df2, aes(pSteal, pRebound)) +
  geom_raster(aes(fill = jWin), color = "white") +
  scale_fill_gradient(low = "#FDB927", high = "#552583") +
  scale_x_continuous(breaks = seq(0, 1, by = 0.2), expand = c(0, 0)) +
  scale_y_continuous(breaks = seq(0, 1, by = 0.2), expand = c(0, 0)) +
  xlab("Prob. of James Steal") +
  ylab("Prob. of Davis Rebound") +
  coord_equal() +
  theme_pubr() +
  theme(axis.title = element_text(size = 20),
        legend.position = "none")