How You Should Keep Score in Pickup Basketball

• What if you’re playing make-it-take-it?
• Summary
• Taking action
• Other considerations
• Appendix
  • But you don’t play pickup basketball to infinity
  • Finite geometric series
  • Scoring system in pickup basketball

What if you’re playing make-it-take-it?

2s and 3s is only more balanced when you alternate possession after each score. While alternating possession is the norm in full court pickup basketball, in half court the more common play style is for the scoring team to retain possession – called “make-it-take-it.” Under these conditions⁷, calculating expected points is a little more complicated as you need to consider not just the expected points of the shot attempt but also the value associated with the chance of retaining possession (and making future shots) if you score.

Calculating the expected points for a shot is no longer just:

\(\text{(number of points)} \times \text{(shooting percentage)}\)

but:

\(\left(\text{number of points}\right) \times \left(\text{shooting percentage}\right) \\+ \left(\text{chance last shot was made}\right) \times \left(\text{number of points}\right) \times \left(\text{shooting percentage}\right) \\+ \left(\text{chance last two shots were made}\right) \times \left(\text{number of points}\right) \times \left(\text{shooting percentage}\right) \\+ \ldots \left(\text{and so on and so on}\right) \ldots\)

If we consider the expected points of an interior shot (when playing 1s and 2s) the series can be approximated as:

\[ 1 \times 0.45 + 1 \times 0.45^2 + \ldots + 1 \times 0.45^n \]

You might recognize this from calculus as a geometric series and notice that because each additional term in the series gets smaller and smaller towards 0, the total value of the sum will converge to a constant. The value of a converging infinite geometric series can be represented by the simple equation⁸:

\[ \frac{\text{(start value)}}{(1 – \text{rate})} \]

We can use this to approximate the expected points for a shot attempt (see But you don’t play pickup basketball to infinity for why this is just an approximation). If we plug in our values for a 1 pointer with 45% chance of being made:

\[ \frac{(0.45 \times 1)}{(1 – 0.45)} = 0.82 \]

and compare this to the rough expected points of a 2 pointer at 30%:

\[ \frac{(0.30 \times 2)}{(1 – 0.30)} = 0.86 \]

This shows a less than 5% difference in approximate expected points if taking an interior or long range shot when playing make-it-take⁹. This is pretty balanced, which is good news as 1s and 2s make-it-take-it is the norm for half court pickup basketball. Compare this relative balance to the expected values when playing make-it-take by 2s and 3s.

Approximate expected points for a 2 pointer in make-it-take-it:

\[ \frac{(0.45 \times 2)}{(1 – 0.45)} = 1.64 \]

vs a 3 pointer:

\[ \frac{(0.3 \times 3)}{(1 – 0.3)} = 1.29 \]

Under these conditions, an interior shot has almost 27% greater expected value than taking a long range shot (24% symmetric percent difference). In this case 2s and 3s is almost as imbalanced for interior shots as the 33% advantage we saw for distance shots when playing 1s and 2s with alternating possession. It’s good then that 2s and 3s make-it-take-it is somewhat rare.

Summary

The two conditions that lead to similar in-game expected values for interior and long range shots are make-it-take-it 1s and 2s as well as 2s and 3s with alternating possession after scores¹⁰.

# Load the gt package
library(gt)
library(dplyr)

# Function to calculate percentage difference and format the string
format_with_difference <- function(value1, value2) {
  diff <- round(abs(value2 - value1) / ((value1 + value2) / 2) * 100, 0)
  return(paste0(value1, " : ", value2, "<br>(", diff, "% difference)"))
}

# Update the data frame with formatted strings
data <- tibble(ShotType = c("1pts : 2pts", "2pts : 3pts"),
               MakeItTakeIt = c(format_with_difference(0.82, 0.86), format_with_difference(1.64, 1.29)),
               Alternating = c(format_with_difference(0.45, 0.60), format_with_difference(0.90, 0.90)))

# Create a gt table
table <- gt(data) %>%
  # Add a title and subtitle
  tab_header(
    title = "Relative differences in expected points of shot type",
    subtitle = md("Given shooting percentages of... *interior: 45% ; long range: 30%*")
  ) %>%
  # Add spanner column label with italicized text
  tab_spanner(
    label = md("**Possession after a score**"),
    columns = c(MakeItTakeIt, Alternating)
  ) %>%
  tab_spanner(
    label = md("**Points assigned**"),
    columns = c(ShotType)
  ) %>%
  # Customize column labels
  cols_label(
    ShotType = md("*interior : long-range*"),
    MakeItTakeIt = md("*make-it-take-it*"),
    Alternating = md("*alternating*")
  ) %>%
  # Add vertical lines and other styles
  tab_style(
    style = cell_borders(
      sides = "right",
      color = "lightgray",
      weight = px(1)
    ),
    locations = list(
      cells_body(columns = MakeItTakeIt),
      cells_column_labels(columns = MakeItTakeIt)
    )
  ) %>%
  tab_style(
    style = cell_borders(
      sides = "left",
      color = "lightgray",
      weight = px(2)
    ),
    locations = cells_body(columns = c(MakeItTakeIt))
  ) %>%
  tab_style(
    style = cell_borders(
      sides = "right",
      color = "lightgray",
      weight = px(2)
    ),
    locations = cells_column_labels(columns = c(ShotType))
  ) %>%
  tab_style(
    style = cell_text(
      align = "center",
      style = "italic"
    ),
    locations = cells_body(columns = c(ShotType))
  ) %>%
  tab_style(
    style = cell_text(
      color = "darkblue"
    ),
    locations = cells_title(groups = "title")
  ) %>%
  # Render the markdown in the cells
  fmt_markdown(columns = c(MakeItTakeIt, Alternating)) %>%
  tab_footnote("Symmetric absolute percent difference used as metric for % difference.") %>%
  tab_style(
    style = cell_text(size = px(10)),
    locations = cells_footnotes()
  )

# Print the table
table

Taking action

You might think that the expected values of different shots would naturally balance out. For example, if you’re playing by 1s and 2s with alternating possession (where I’ve said deep shots are more valuable) you would think players would defend long range shots more aggressively which would lead to the shooting percentage on those shots dropping relative to interior shots. In practice though, this modification does not happen as much as would be expected by fully rational players¹¹. Instead, many people anchor their play style to whatever brand of basketball they are accustomed and do not fully adjust to the scoring and possession system in place. Given these tendencies, what should you do?

1. Advocate for a balanced scoring system¹²

This means if you’re playing full court (with alternating possessions after scores), push for playing by 2s and 3s to 21 (rather than the more common 1s and 2s to 15). If you’re playing half court, you don’t need to do anything as games are typically played make-it-take by 1s and 2s, which has mostly fine incentives¹³.

2. Arbitrage

In many cases you will not be able to influence the scoring system. Under these conditions other players not fully adapting to unbalanced values of shots is an opportunity. For example, if the system advantages long range scoring, do your best to get your best shooter as many decent looks as possible and push your team to play good outside defense¹⁴.

Other considerations

The conclusion that “1s and 2s make-it-take-it” and “2s and 3s alternating possession” make for the most balanced games depends on my guesses for shooting percentages¹⁵. 45% and 30% for interior and long range are reasonable estimates for competitive pickup games but may not apply universally¹⁶.

The chart below shows, across a range of possible interior and long range shooting percentages, the relative imbalance in expected points under each of the four {possession} x {scoring} conditions¹⁷.

library(dplyr)
library(tidyr)
library(ggplot2)
library(purrr)

shooting_pcts <- crossing(
  tibble(pct_long = seq(0.10, 0.40, by = 0.01)),
  tibble(pct_short = seq(0.20, 0.60, by = 0.01))
)

shots_pts <- tibble(pts_long = c(2,3),
       pts_short = c(1, 2)
)

exp_value_alternating <- function(pts, percent) pts * percent

exp_value_makeitakeit <- function(pts, percent) (percent * pts) / (1 - percent)

total_pct_difference <- function(x, y) (x - y) / ((x + y) / 2)

data_imbalance <- crossing(shooting_pcts, shots_pts) %>% 
  mutate(pts_type = paste0(pts_short, "&", pts_long)) %>% 
  # expected value of each shot
  mutate(exp_short_alt = exp_value_alternating(pts_short, pct_short),
         exp_short_miti = exp_value_makeitakeit(pts_short, pct_short),
         exp_long_alt = exp_value_alternating(pts_long, pct_long),
         exp_long_miti = exp_value_makeitakeit(pts_long, pct_long)) %>% 
  # percentage imbalances
  mutate(alt = total_pct_difference(exp_short_alt, exp_long_alt),
         miti = total_pct_difference(exp_short_miti, exp_long_miti))

data_chart <- data_imbalance %>% 
  pivot_longer(one_of("alt", "miti"), names_to = "possession_type", values_to = "pct_imbalance") %>% 
  mutate(possession_type = ifelse(possession_type == "alt", "alternating", "make-it-take-it")) %>% 
  mutate(type_combine = paste0(pts_type, "; ", possession_type)) %>% 
  mutate(near_0 = near(pct_imbalance, 0, 0.05)) %>% 
  mutate(pct_imbalance = abs(pct_imbalance)) %>% 
  mutate(type_combine = forcats::fct_relevel(type_combine, "1&2; make-it-take-it", "1&2; alternating", "2&3; make-it-take-it", "2&3; alternating"))

data_near <- data_chart %>% 
  filter(near_0) %>% 
  mutate(near_0 = "+/- 0.05 imbalance")

data_reviewed <- data_chart %>% 
  filter(near(pct_long, 0.30), near(pct_short, 0.45)) %>% 
  mutate(label = "Standard shooting conditions:\n45% shooting from interior;\n30% shooting from long range")

# equations for perfect balance
alt_1_2 <- function(x) x / 2
alt_2_3 <- function(x) x / 1.5
miti_1_2 <- function(x) x / (2 - x)
miti_2_3 <- function(x) 2 * x / (3 - x)

perfect_balance <- list(alt_1_2, alt_2_3, miti_1_2, miti_2_3) %>% 
  map(~{tibble(pct_short = seq(0.20, 0.60, by = 0.01)) %>% mutate(pct_long = .x(pct_short))}) %>% 
  map2(list("1&2; alternating", "2&3; alternating", "1&2; make-it-take-it", "2&3; make-it-take-it"),
      ~{mutate(.x, type_combine = .y)}) %>% 
  bind_rows() %>% 
  mutate(type_combine = forcats::fct_relevel(type_combine, "1&2; make-it-take-it", "1&2; alternating", "2&3; make-it-take-it", "2&3; alternating"))


data_chart %>% 
  ggplot(aes(x = pct_short, y = pct_long))+
  geom_tile(aes(fill = pct_imbalance))+
  geom_tile(aes(fill = pct_imbalance, colour = near_0), alpha = 0, data = data_near)+
  geom_line(color = "red", data = perfect_balance)+
  geom_point(aes(shape = label), data = data_reviewed, colour = "violet")+
  scale_color_manual(values = c("red"))+
  facet_wrap(~type_combine)+
  theme_bw()+
  labs(
    title = "Imbalance of expected points by shooting percentages",
    # subtitle = "For 45%; 30%, full court 2s and 3s or make-it-take-it 1s and 2s are most balanced",
    # subtitle = "By scoring and possession rules",
    fill = "Proportion imbalance",
    x = "Interior shooting %", 
    y = "Long range shooting %",
    caption = "Symmetric absolute difference used as metric for proportion imbalance.")+
  theme(legend.key = element_rect(colour="white"),
        legend.position = "right")+
  guides(color = guide_legend(title = NULL, order = 2),
         fill = guide_legend(order = 1),
         shape = guide_legend(title = NULL))+
  scale_y_continuous(limits = c(NA, 0.4), labels = scales::percent)+
  scale_x_continuous(labels = scales::percent)+
  coord_fixed()

The greater range in color hue for make-it-take-it settings demonstrates how, as you deviate from balanced shooting percentage conditions, shot imbalance grows more quickly than when playing with alternating possession after scores. This is why the “standard shooting conditions” points in the upper right and lower left charts are pretty similar in terms of proportion imbalanced while being pretty different in terms of shooting %s distance from “perfectly balanced”.

Everything above the red “perfect balance” lines (or curves in the cases of make-it-take) can be interpreted as shooting percentages where long range shooting has an expected points advantage. Everything below them represents shooting percentages where interior shooting has an advantage. For example, in games where interior shooting percentages are still ~45% but long range shooting is a bit worse (e.g. percentages in the low 20s) 1s and 2s with alternating possession after a score can actually be more balanced than 2s and 3s. However you would be setting-up conditions for a lot of missed shots.

Sometimes it’s better to just play by all 1s.

Appendix

library(dplyr)
library(ggplot2)

finite_geometric_series <- function(start, rate, n){
  (start * (1 - rate^n)) / (1 - rate)
}

data_1s2s_finite <- bind_rows(
  tibble(shot_pts = 1,
         shot_pct = 0.45,
         points_remain = seq(2, 14, by = 1)) %>% 
    mutate(n_rounds = points_remain),
  tibble(shot_pts = 2,
         shot_pct = 0.30,
         points_remain = seq(2, 14, by = 1)) %>% 
    mutate(n_rounds = points_remain / 2)  
) %>% 
  mutate(exp_pts = finite_geometric_series(shot_pts*shot_pct, shot_pct, n_rounds)) %>% 
  mutate(shot_pts_title = ifelse(shot_pts == 1, "interior (1pts; 45%)", "long-range (2pts; 30%)")) 

data_1s2s_finite %>% 
  ggplot(aes(x = points_remain, y = exp_pts, colour = shot_pts_title))+
  geom_line()+
  scale_x_continuous(breaks = seq(2, 20, by = 2)) +  
  theme_bw()+
  ylim(0, NA) +
  labs(
    title = "Expected Value of Shots Based on Points Left in the Game",
    subtitle = "1s and 2s, make-it-take-it",
    x = "Points Left in the Game",
    y = "Expected Value",
    color = "Shot Type"
  ) +
  guides(color = guide_legend(title = NULL))

data_2s3s_finite <- bind_rows(
  tibble(shot_pts = 2,
         shot_pct = 0.45,
         points_remain = seq(3, 24, by = 1)) %>% 
    mutate(n_rounds = points_remain / 2),
  tibble(shot_pts = 3,
         shot_pct = 0.30,
         points_remain = seq(3, 24, by = 1)) %>% 
    mutate(n_rounds = points_remain / 3)  
) %>% 
  mutate(exp_pts = finite_geometric_series(shot_pts * shot_pct, shot_pct, n_rounds)) %>% 
  mutate(shot_pts_title = ifelse(shot_pts == 2, "interior (2pts; 45%)", "long-range (3pts; 30%)"))

data_2s3s_finite %>% 
  ggplot(aes(x = points_remain, y = exp_pts, colour = shot_pts_title))+
  geom_line()+
  # scale_x_continuous(breaks = seq(2, 20, by = 2)) +  
  theme_bw()+
  ylim(0, NA) +
  scale_x_continuous(breaks = seq(3, 24, by = 3)) +  
  labs(
    title = "Expected Value of Shots Based on Points Left in the Game",
    subtitle = "2s and 3s, make-it-take-it",
    x = "Points Left in the Game",
    y = "Expected Value",
    color = "Shot Type"
  ) +
  guides(color = guide_legend(title = NULL))

1. It’s not that every player has to have a 30% long range shooting percentage and 45% interior shooting percentage. It could be that some of the players on the team are far worse at shooting 3s, but they just won’t take those shots.↩︎

2. This isn’t completely true when considering make-it-take-it but as long as things don’t depart too much from the shooting percentages I’m using, most of the takeaways stay the same.↩︎

3. I’m also ignoring free throws because playing with these are exceptionally rare in pickup basketball.↩︎

4. Article on symmetric measure for percentage difference.↩︎

5. Article surveying the most common rules of pickup.↩︎

6. and finds sympathy among those without a jump shot↩︎

7. and continuing to ignore turnovers, offensive rebounds, and other complicating factors↩︎

8. Proof on Khan Academy↩︎

9. Maybe a slight edge for long range shots but pretty close (i.e. balanced).↩︎

10. The other play formats lead to imbalances that favor either long range or interior shooting.↩︎

11. Also some adjustments may be difficult to make regardless of intention.↩︎

12. Being the ethical and upstanding individual you are, you should advocate for the most balanced system available. A more cynical person might say that you should try and skew the scoring system to whichever gives your team the greatest advantage… however I think it’s more fun to just try and keep things balanced.↩︎

13. You could push for 2s and 3s with alternating possession though which is a little more balanced and has the advantage of being more robust to variations in shooting percentage which I touch-on in Other considerations.↩︎

14. Even under reasonably balanced rules you can still find opportunities where people don’t play in-line with expected value. One example, is in 1s and 2s make-it-take-it some players will be overly concerned about long range shots (and under appreciate the value of retaining possession) and defend in a way that opens themselves up to drives and high percentage interior shots. This is particularly the case if playing with less than 10 players (e.g. 3 vs 3 or 4 vs 4) where interior play is already more open and the shooting percentage on interior shots likely goes up by more than it does for long range shots.↩︎

15. There’s also the possibility that much of this post is rationalizing my own frustrations at growing up without a decent jump shot and having to always play 1s and 2s.↩︎

16. In the case of alternating possession, proportion imbalanced is linearly related to the ratio of {interior shooting %} : {long range shooting %}, e.g. in the case of 2s and 3s alternating possession, what matters is not the specific shooting percentages but how far they deviate from the ratio of 0.45 : 0.30. However when playing make-it-take-it the relationship is not linear (though within a small window of shooting percentages a linear relationship works OK as an approximation).↩︎

17. The “perfect balance” equations were found by setting the interior and long range expected points equations equal to each other and simplifying.↩︎

18. E.g. 1s and 2s make-it-take to 11, you’re down 9 – 7 with the ball… what are your chances of winning if you…↩︎

19. This allows me to put both “alternating” and “make-it-take-it” calculations on the same figures without having to clarify that the former is expected points and the latter approximate expected points.↩︎

20. In the curves I’m ignoring that some of these point values are impossible given the scoring systems.↩︎