Removing white spaces in a histogram?

I came across this question here https://math.stackexchange.com/questions/2648895/why-does-fair-random-process-lead-to-unfair-result/5001301#5001301 in which someone was interested in simulating a game where two players randomly give each other money.

Using R, I tried to simulate multiple trajectories of this game to look at the distributions of different metrics:

    library(ggplot2)
    library(tidyverse)
    library(gridExtra)
    library(future)
    library(furrr)
    library(parallel)
    
    n_cores <- detectCores() - 1  
    plan(multisession, workers = n_cores)


run_money_simulation <- function(n_sims, n_exchanges, player_a_start, player_b_start) {
    set.seed(123) 
    
    sims_per_core <- ceiling(n_sims / n_cores)
    
    simulate_exchange <- function(n_exchanges, player_a_start, player_b_start) {
        person_a <- numeric(n_exchanges + 1)
        person_b <- numeric(n_exchanges + 1)
        
        person_a[1] <- player_a_start
        person_b[1] <- player_b_start
        
        for(i in 2:(n_exchanges + 1)) {
            change <- sample(c(-1, 1), 1)
            person_a[i] <- person_a[i-1] + change
            person_b[i] <- person_b[i-1] - change
        }
        
        return(list(
            final_diff = person_a[n_exchanges + 1] - person_b[n_exchanges + 1],
            max_diff = max(abs(person_a - person_b)),
            max_amount = max(c(max(person_a), max(person_b))),
            min_amount = min(c(min(person_a), min(person_b)))
        ))
    }
    
    start_time <- Sys.time()
    
    results <- future_map(1:n_sims, function(x) {
        simulate_exchange(n_exchanges, player_a_start, player_b_start)
    }, .options = furrr_options(seed = TRUE))
    
    end_time <- Sys.time()
    time_taken <- difftime(end_time, start_time, units = "secs")
 
    
    final_diffs <- sapply(results, `[[`, "final_diff")
    max_diffs <- sapply(results, `[[`, "max_diff")
    max_amounts <- sapply(results, `[[`, "max_amount")
    min_amounts <- sapply(results, `[[`, "min_amount")
    
    plot_data <- tibble(
        final_diff = final_diffs,
        max_diff = max_diffs,
        max_amount = max_amounts,
        min_amount = min_amounts
    ) %>%
        pivot_longer(everything(), 
                     names_to = "metric", 
                     values_to = "value")
    
    main_plot <- ggplot(plot_data, aes(x = value)) +
        geom_histogram(bins = 50, aes(fill = metric), color = "white", alpha = 0.7) +
        facet_wrap(~metric, scales = "free", ncol = 2) +
        scale_fill_manual(values = c("black", "red", "green4", "purple")) +
        labs(title = paste("Money Exchange Simulation Results\n",
                           "Starting amounts: A =", player_a_start, ", B =", player_b_start),
             subtitle = paste("Number of simulations:", n_sims, 
                              "| Exchanges per simulation:", n_exchanges,
                              "\nProcessed using", n_cores, "CPU cores in", 
                              round(time_taken, 2), "seconds"),
             x = "Value",
             y = "Count") +
        theme_bw() +
        theme(legend.position = "none")
    
    
    print(main_plot)
    
    
    invisible(list(
        final_diffs = final_diffs,
        max_diffs = max_diffs,
        max_amounts = max_amounts,
        min_amounts = min_amounts,
        parameters = list(
            n_sims = n_sims,
            n_exchanges = n_exchanges,
            player_a_start = player_a_start,
            player_b_start = player_b_start,
            n_cores = n_cores,
            processing_time = time_taken
        )
    ))
}

When I call the function:

run_money_simulation(100000, 100, 100, 100)

I get the following results:

I am just wondering - is there something I can do in ggplot that detects a scale break format such that the white spaces in these charts will be removed?

The problem is that you have discrete data. Histograms are a kind of density estimate, designed for continuous data that actually has a density.

Looking at the top left figure final_diff, the data being shown is always a multiple of 4. Since your bin width is slightly smaller than 4, a couple of bins completely miss the data. You could avoid the gaps by setting the bin width to 4, but that's not the best solution. The best solution is to use a display that is designed for discrete data.

The usual display for this kind of discrete data is a bar plot showing counts of each observed value. You can get that in ggplot2 using geom_bar() instead of geom_histogram. For example,

#  ... unchanged code deleted ...

  main_plot <- ggplot(plot_data, aes(x = value)) +
    geom_bar(aes(fill = metric)) +
    facet_wrap(~metric, scales = "free", ncol = 2) +
    scale_fill_manual(values = c("black", "red", "green4", "purple")) +
    labs(title = paste("Money Exchange Simulation Results\n",
                       "Starting amounts: A =", player_a_start, ", B =", player_b_start),
         subtitle = paste("Number of simulations:", n_sims, 
                          "| Exchanges per simulation:", n_exchanges,
                          "\nProcessed using", n_cores, "CPU cores in", 
                          round(time_taken, 2), "seconds"),
         x = "Value",
         y = "Count") +
    theme_bw() +
    theme(legend.position = "none")
  
##  ... more unchanged code ...

^{Created on 2024-11-21 with reprex v2.1.1}

This looks uglier than a histogram, but it is more faithful to the data. And it's less ugly in a higher resolution display, where you can see that all the bars and spaces between them are the same width.