The strategy of the Plinketto board

Introduction

In RedLetterMedia, they make a type of video called Best of the Worst, where they choose 3 bad movies and watch them, and decide which one of them is the best of the worst. The selection procedure uses a device called a Plinketto board.

It is a form of Galton board that is slightly longer. (It has 10 starting slots and length 19). In the Galton board, the results are normally distributed around the where it is dropped. I explore the results for the Plinketto board using simulations.

Simulations

Code

library(jsmp)
library(broom)
library(rsample)
plinketto <- function(s, width, height, row){
  if (row %% 2 == 0){
      s = s + sample(0:1, 1)
    } else {
      if (s > 1 && s < width) {
        s = s - sample(0:1, 1)
        }
      if (s == width) {
        s = s - 1
        }
    }
  row = row + 1
    
  if (row == height) {
    return(s)
  } else {
    return(plinketto(s, width, height, row))
  }
}
plotit <- function(df){
  df |> ggplot(aes(x = factor(movie_choice, levels = 1:11))) +
    geom_bar(aes(y = after_stat(count)/after_stat(count))) +
    scale_x_discrete(drop=F) +
    gg_y_percent_zero() + 
    labs(y = "", x = "movie choice")
}
set.seed(1)
nsims <- 1000

Dropping in the center

The resulting movie picks normally distributed around the drop point.

Code

tibble(movie_choice = replicate(nsims, plinketto(6, 11, 19, 0))) |> 
  plotit()

Dropping at the edge

It is less likely for the ball to end up in the corner spot. Intuitively this is caused by there being two immediate paths to all the other slots, but only one immediate paths to the corner slot.

Code

tibble(movie_choice = replicate(nsims, plinketto(1, 11, 19, 0))) |> 
  plotit()

Note that this is only the case because the final row in the Plinketto board is narrow. If the final row had been wide, the results from dropping at the edge would look like this:

Code

tibble(movie_choice = replicate(nsims, plinketto(1, 11, 20, 0))) |> 
  plotit()

Dropping at the second slot

The chance of hitting the second slot is higher if you drop at the edge, than if you drop directly above it.

Code

tibble(movie_choice = replicate(nsims, plinketto(2, 11, 19, 0))) |> 
  plotit()

Maximum chance achievable

The below plot shows the maximum chance you can get to hit a specific movie, with optimal placement.

Code

c(1:10) |> map_df(~ tibble(movie_choice = replicate(nsims, plinketto(.x, 11, 19, 0))) |> 
  janitor::tabyl(movie_choice) |> mutate(drop = .x)) |> 
  group_by(movie_choice) |> 
  summarise(max_chance = max(percent)) |> 
  ggplot(aes(x = movie_choice)) +
    geom_bar(aes(y = max_chance), stat="identity") +
    scale_x_continuous(breaks = 1:11, labels = 1:11) +
    gg_y_percent_zero() +
    labs(y = "")

Strategy

• If you are the movie placer and there is a movie you would like to avoid, put it in one of the edge slots.

• If you are both the placer and the dropper, put the movie you are interested in on slot second to the edge, and drop the ball on the edge. This gives you 32% chance to hit your movie, which is the highest achievable.

• If you are the dropper, and the movie you want is not second or third from the edge, simply drop the ball above it. This gives you at least 17.6% chance to hit your movie.