How You Should Keep Score 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:

\(\text{(points)} \times \text{(shooting %)}\)

but:

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

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 when playing make-it-take-it (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 because 1s and 2s make-it-take-it is the norm for half court pickup basketball. Let’s compare this relative balance to the relationship between 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 shooting when playing 1s and 2s with alternating possession. So if you’re playing make-it-take-it, you actually shouldn’t do scoring by 2s and 3s¹⁰.

Summary

The two game conditions that promote similar expected values for interior and long range shots are 2s and 3s with alternating possession after scores as well as make-it-take-it by 1s and 2s¹¹.

# 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 deep shots are more valuable) players could 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 realignment does not happen as much as would be expected by fully rational players¹². 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 system¹³

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 incentives¹⁴.

2. Arbitrage

In many cases you will not be able to influence the scoring system. Other players not fully adapting to unbalanced values of shots is an opportunity. If the system advantages long range scoring, get your best shooter as many decent looks as possible and push your team to play tough exterior defense¹⁵.

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 standard shooting percentages¹⁶. 45% : 30% for interior and long range shots are reasonable estimates for competitive pickup games but may not apply universally¹⁷.

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} conditions¹⁸. See Table of imbalances in the appendix to look-up these values at any point.

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_raw <- 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(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 <- data_chart_raw %>% 
  mutate(pct_imbalance = abs(pct_imbalance))

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, expected value imbalance grows more quickly than when playing with alternating possession. Hence make-it-take-it comes with greater downside risks of incentives getting out of whack¹⁹.

The red indicates the shooting percentages for balanced incentives. 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 advantage²⁰. 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(DT)

data_table <- data_chart_raw %>%
  select(type_combine, pct_long, pct_short, pct_imbalance) %>% 
  pivot_wider(names_from = type_combine, values_from = pct_imbalance) %>% 
  # mutate(across(everything(), scales::percent, accuracy = 1)) %>% 
  rename_with(~stringr::str_replace(., "pct", "fg%"), starts_with("pct_")) %>% 
  rename_with(~paste0("diff%_", .), contains(" ")) %>% 
  dplyr::relocate(contains("short")) %>% 
  mutate(across(everything(), ~round(.x, 2)))

datatable(data_table, 
          options = list(dom = 'ltipr'), 
          filter = list(position = 'top'),
          rownames = FALSE) %>% 
  DT::formatPercentage(names(data_table), 0)

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))