Simulation of Gambler's Ruin (Random Walk) With R

The Gambler's Ruin problem is one of the most researched topics in the field of operational research.

Let's consider a game where a gambler is likely to win $1 with a probability of p and lose $1 with a probability of 1-p.

Now, let's consider a game where a gambler is likely to win $1 and lose $1 with a probability of 1. The player starts the game with X dollars in hand. The player plays the game until the money in his hand reaches N (N> X) or he has no money left. What is the probability that the player will reach the target value? (We know that the player will not leave the game until he reaches the N value he wants to win.)

The problem of the story above is known in literature as Gambler's Ruin or Random Walk. In this article, I will simulate this problem with R with different settings and examine how the game results change with different settings.

I will use these parameters while doing this simulation:

• The money that the player holds when he enters the game.
• The lower limit of the game.
• The upper limit of the game.
• The probability that the player earns $1 (P).
• We repeat the experiment with the above parameters (n).

Now, let's create a few scenarios from the parameters we have defined and simulate them with these scenarios.

Use Case 1

• The money that the player holds when he enters the game: $6
• The lower limit of the game: $2
• The upper limit of the game: $10
• The probability that the player earns $1: p: 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9
  • We will play the game separately for each probability value.
• We repeat the experiment with the above parameters (n): 100

Use Case 2

• The money that the player holds when he enters the game: $10
• The lower limit of the game: $1
• The upper limit of the game :$10
• The probability that the player earns $1: p: 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9
  • We will play the game separately for each probability value.
• We repeat the experiment with the above parameters (n): 100

Use Case 3

• The money that the player holds when he enters the game: $10
• The lower limit of the game is: $3
• The upper limit of the game: $20
• The probability that the player earns $1: p: 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9
  • We will play the game separately for each probability value.
• We repeat the experiment with the above parameters (n): 100

We have three basic scenarios described above. However, since we will run the same scenario again and again for different probability values (nine), each scenario will run nine times in total. At the end of the day, we will run 9*3 = 27 different scenarios.

Now, let's go to the R co:e to simulate. I will write a total of five functions to complete this simulation. These are;

• ReadInstances: With this function, I read the amount of money in the player's hand and the upper and lower limit of the game through the .txt files.

  • Sample file format (current, lower limit, upper limit): 6, 2, 10

• GamblerRun: A function that plays once with given the parameters and returns the result (win/loss).

• Simulator: The GamblerRuin function runs for as many possible scenarios as possible and records the results.

• ExportResults: The function that writes the simulation results to the file.

• Analyzer: The function that reads and predicts the simulation results written to the file and calculates the error depending on this estimation.

  • Error = |Pactual-Pestimated|/Pactual

Now, we can go to the code writing section in R.

We initialize our variables in the first place.

#Initializing Parameters
instancesFolder = "ProblemInstances"
options(warn = -1)
options(max.print = 100000000)
probabilities = c(0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9)
n = c(100)

Now, we can start writing the functions.

ReadInstances:

# Function ReadInstances(input FilesFolderPaths)
# Return: DataFrame, each record in seperate files
ReadInstances <- function(instancesFolder) {
  mydata = data.frame()
  files <- as.list(list.files(path = instancesFolder))
  for (i in 1:length(files))
  {
     line <-
          read.csv(
                   paste("ProblemInstances", .Platform$file.sep, files[i], sep = ""),
                   header = F,
                   sep = ",",
                   quote = ""
                  )
        line$UL > line$currentState && line$UL > line$LL)
    {
      mydata = rbind(mydata, line)
    }
  }
  colnames(mydata) = c('currentState', 'LL', 'UL', 'fileName')
  return(as.data.frame(mydata))
}

GamblerRuin (we play the game as a Bernoulli trial and take advantage of the rbinom function to accomplish this):

# Function GamblerRuin(inputs currentState, UL, LL, p)
# Return: string, game result win or lose
GamblerRuin <- function(currentState, UL, LL, p) { while (currentState > LL && currentState < UL)
  {
    coin = rbinom(1, 1, p)

    if (coin == 1)
    {
      currentState = currentState + 1
    }
    else
    {
      currentState = currentState - 1
    }
  }

  if (currentState == UL)
  {
    return("win")
  }
  else  
  {
    return("lose")
  }
}

Simulator:

# Function Simulator(inputs problemInstances, probabilities, n)
# Saving Scenarios and SimulationResults to Files
# Return: data frame, results of all possible scenarios (resultTable)
Simulator <- function(problemInstances, probabilities, n) {
  set.seed(0)
  resultTable = data.frame()
  scenarioID = 0
  scenarioTable = data.frame()

  for (instance in 1:nrow(problemInstances)) {
    for (prob in 1:as.integer(length(probabilities)))
    {
      for (j in 1:as.integer(length(n)))
      {
        scenarioID = scenarioID + 1
        scenrow = data.frame(problemInstances[instance,]$fileName,
                             scenarioID,
                             probabilities[prob],
                             n[j])
        scenarioTable = rbind(scenarioTable, scenrow)

        for (k in 1:n[j])
        {
          gameResult = GamblerRuin(
            problemInstances[instance, ]$currentState,
            problemInstances[instance, ]$UL,
            problemInstances[instance, ]$LL,
            probabilities[prob]
          )
          row = data.frame(problemInstances[instance, ]$fileName,
                           scenarioID,
                           probabilities[prob],
                           n[j],
                           k,
                           gameResult)
          resultTable = rbind(resultTable, row)
        }
      }
    }
    scenarioID = 0
  }


  colnames(scenarioTable) = c('instanceFileName', 'ScenarioID', 'p', 'n')
  #Save Scenario to File
  ExportResults(scenarioTable, "ScenarioFormatted.txt", 2)
  ExportResults(scenarioTable, "Scenario.txt", 1)

  colnames(resultTable) = c('instanceFileName',
                            'ScenarioID',
                            'p',
                            'n',
                            'ReplicationID',
                            'Result')


  SimulationResults <-
    subset(resultTable,
           select = c(instanceFileName, ScenarioID, ReplicationID, Result))
  #Save SimulationResults to File
  ExportResults(SimulationResults, "SimulationResultsFormatted.txt", 2)
  ExportResults(SimulationResults, "SimulationResults.txt", 1)


  return(resultTable)
}

ExportResults:

# Function ExportResults(inputs dataFrame, fileName, type)
# Saving to file (data and filename are parameters)
#type1: standart csv output, type2: formatted output
ExportResults <- function(dataFrame, fileName, type)
{
  #type1: standart csv output
  if (type == 1)
  {
    write.table(
      dataFrame,
      file = fileName,
      append = FALSE,
      quote = TRUE,
      sep = ",",
      eol = "\n",
      na = "NA",
      dec = ".",
      row.names = FALSE,
      col.names = TRUE,
      qmethod = c("escape", "double"),
      fileEncoding = ""
    )

  }
  else
    #type2: formatted output
  {
    capture.output(print(dataFrame, print.gap = 3, row.names = FALSE), file =
                     fileName)
  }
}

ReadFilesFromCsv <- function(fileName)
{
  myData = read.csv(fileName,
                    header = T,
                    sep = ",",
                    quote = "")
  return(myData)
}

Analyzer:

# Function Analyzer()
# Read recorded files and Analysis Estimation Errors
# Saving and Plotting Result of Analysis
Analyzer <- function() {
  #Read Simulation Result
  simulationResults = ReadFilesFromCsv("SimulationResults.txt")

  #converting formatted data frame
  simResults = (data.frame(
    table(
      simulationResults$X.instanceFileName,
      simulationResults$X.ScenarioID,
      simulationResults$X.Result
    )
  ))
  colnames(simResults) = c('instanceFileName', 'ScenarioID', 'Result', 'Count')

  #Seprating "win" and "lose" records to calculating p value.(using external libarary dplyr)
  library(dplyr)
  simResults_win  =  filter(simResults, Result == as.character('"win"'))
  simResults_lose =  filter(simResults, Result == as.character('"lose"'))

  #Join 2 data frame and calculate p_estimated value (using sqldf external libarary)
  library(sqldf)
  simResultsWithPestimated <-
    sqldf(
      "SELECT instanceFileName,ScenarioID,
      round((cast(simResults_win.Count as real)/(simResults_lose.Count+simResults_win.Count)),3) p_estimated
      FROM simResults_win
      JOIN simResults_lose USING(instanceFileName,ScenarioID)"
    )


  #Read Scenario Result
  scenarios = ReadFilesFromCsv("Scenario.txt")

  #changing column name
  colnames(scenarios) = c('instanceFileName', 'ScenarioID', 'p', 'n')


  #Caclulation Percentage Error using sqldf external libarary
  library(sqldf)
  estimationErrors = sqldf(
    "SELECT instanceFileName,ScenarioID,sc.p p_actual, sim.p_estimated p_estimated,
    round(abs(cast(sc.p - sim.p_estimated as real)/ sc.p),3) PercentageError
    FROM scenarios sc
    JOIN simResultsWithPestimated sim USING(instanceFileName,ScenarioID)"
  )

  #dump out staging and final data frames
  ExportResults(simResultsWithPestimated,
                "SimulationResultWithPestimated.txt",
                2)
  ExportResults(estimationErrors, "estimationErrors.txt", 2)

  #plotting estimationErrors vector
  countOfPercentageError <- table(estimationErrors$PercentageError)
  barplot(
    countOfPercentageError,
    main = "Percentage Error Distribution",
    xlab = "ErrorRates",
    ylab = "Number of Counts"
  )

}

We have completed the writing of all our functions. Now, let's write the code snippet to run the simulation and examine the results.

instances = ReadInstances(instancesFolder)
resultTable = Simulator(instances, probabilities, n)
Analyzer()
ExportResults(resultTable, 'ResultTable.txt', 2)

One of the outputs we produced since our code was worked is which scenarios work.

Another output produced is the results of the games played (only a sample set is listed).

The final output produced is the estimated values calculated for each scenario.

In the graph below, we see the distribution of the calculated error.

And that's it!