抛硬币游戏中的无限循环

Question

考虑以下掷硬币游戏：

单次游戏包括反复掷一枚公平的硬币，直到掷出的正面数和反面数之差为 4。

每次抛硬币需支付 1 美元，游戏过程中不得退出。

您在每次游戏结束时收到 10 美元。游戏的“奖金”定义为最后收到的 10 减去支付的金额。一个。模拟这个游戏，以估计许多游戏的预期赢利。湾。假设我们使用有偏见的硬币。找到使游戏公平的 P(tail) 值，这意味着预期的奖金是 0 美元。

这是我应该回答的问题，这是我的尝试

h <- function() {  
  A <- c("H", "T")  
  s <- sample(A,4, replace = T)  
  heads <- length(which(s=="H"))  
  tails <- length(which(s =="T"))  
  w <- heads - tails  
  counter <- 4  
  while (w != 4) {  
    s <- sample(A,1)  
    w <- heads - tails  
    heads <- length(which(s=="H"))  
    tails <- length(which(s =="T"))  
    counter <- counter +1  
  }  
  return(counter)  

}  
h()

但我认为这给了我一个无限循环，有人可以帮忙吗？

Answer 1

您正在根据heads 和tails 的当前值在循环的任何迭代中重新计算w。但这些值将始终为 1 和 0（或 0 和 1）。所以w 总是 -1 或 1，而不是任何其他值。

您的代码中的另一个错误是您仅在正面领先 4 时才停止。但根据规则，当反面领先 4 时，游戏也应该停止：只有绝对差异才重要。

您的代码的逻辑可以固定，但可以使用更简单的逻辑（请注意，以下代码使用不言自明的变量名称，这使得生成的代码更具可读性）：

h = function () {
    sides = c('H', 'T')
    diff = 0L
    cost = 0L
    repeat {
        cost = cost + 1L
        flip = sample(sides, 1L)
        if (flip == 'H') diff = diff + 1L
        else diff = diff - 1L
        if (abs(diff) == 4L) return(cost)
    }
}

您可以进一步简化这一点，因为硬币面的标签实际上并不重要。您所关心的只是一次抛硬币，它会返回两个结果之一。

我们可以将它实现为一个单独的函数。函数的返回值不是很重要，只要我们有一个固定的约定：它可以是c('H', 'T')，或者c(FALSE, TRUE)，或者c(0L, 1L)等。对于我们的目的，它会很方便返回-1L 或1L，以便我们的函数h 可以直接将该值添加到diff：

coin_toss = function () {
    sample(c(-1L, 1L), 1L)
}

但是有一种不同的方式来获得抛硬币：大小为 1 的 Bernoulli trial。使用伯努利试验有一个很好的特性：我们可以简单地扩展我们的函数以允许不公平（有偏见的）抛硬币。所以这里是相同的函数，但带有一个可选的bias 参数（默认情况下抛硬币是公平的）：

coin_toss = function (bias = 0.5) {
    rbinom(1L, 1L, prob = bias) * 2L - 1L
}

（rbinom(…) 返回 0L 或 1L。要将值的域转换为 c(-1L, 1L)，我们乘以 2 并减去 1。）

现在让我们更改h 以使用此功能：

h = function (bias = 0.5) {
    cost = 0L
    diff = 0L
    repeat {
        cost = cost + 1L
        diff = diff + coin_toss(bias)
        if (abs(diff) == 4L) return(cost)
    }
}

coin_toss() 是 0 或 1，但根据其值，我们要么

Answer 2

我想回答你的问题，包括 a) 和 b) 部分。我会用我的代码来节省我的时间。

这是一款很酷的游戏，其中软件模拟可能会非常有用。 游戏的基本原理是“永无止境的循环”，最终在正面和反面数量的绝对差等于 4 时结束。然后记录收益。正如康拉德鲁道夫所说，游戏是伯努利类型的。游戏模拟如下代码：

n_games <- 1000 # number of games to play
bias <- 0.5

game_payoff <- c()

for (i in seq_len(n_games)) {
  
  cost <- 0
  flip_record <- c()
  payoff <- c()
  
  repeat{
    cost <- cost + 1
    flip <- rbinom(1, 1, prob = bias)
    flip_record <- c(flip_record, flip)

    n_tails <- length(flip_record) - sum(flip_record) # number of 0s/tails
    n_heads <- sum(flip_record) # number of 1s/heads
    
    if (abs(n_tails - n_heads) == 4) {
      game_payoff <- c(game_payoff, 10 - cost) # record game payoff
      print(paste0("single game payoff: ", 10 - cost)) # print game payoff
      break
    }
  }
}

大量运行，例如在这个循环上的另一个循环，我们了解到，预期值非常接近 -6。因此，该博弈具有负期望值。它遵循以下代码：

library(ggplot2)
seed <- 122334

# simulation
n_runs <- 100
n_games <- 10000
bias <- 0.5

game_payoff <- c()
expected_value_record <- c()

for (j in seq_len(n_runs)) {
  
  for (i in seq_len(n_games)) {
    
    cost <- 0
    flip_record <- c()
    payoff <- c()
    
    repeat{
      cost <- cost + 1
      flip <- rbinom(1, 1, prob = bias)
      flip_record <- c(flip_record, flip)
      # print(flip_record)
      
      n_tails <- length(flip_record) - sum(flip_record) # number of 0s/tails
      n_heads <- sum(flip_record) # number of 1s/heads
      
      if (abs(n_tails - n_heads) == 4) {
        game_payoff <- c(game_payoff, 10 - cost) # record game payoff
        print(paste0("single game payoff: ", 10 - cost))
        break
      }
    }
  }
  expected_value_record <- c(expected_value_record, mean(game_payoff))
  game_payoff <- c()
}

# plot expected value
expected_value_record <- cbind.data.frame("run" = seq_len(length(expected_value_record)), expected_value_record)

ggplot(data = expected_value_record) +
  geom_line(aes(x = run, y = expected_value_record)) +
  scale_x_continuous(breaks = c(seq(1, max(expected_value_record$run), by = 3), max(expected_value_record$run))) +
  labs(
    title = "Coin flip experiment: expected value in each run. ", 
    caption = paste0("Number of runs: ", n_runs, ". ", "Number of games in each run: ", n_games, "."), 
    x = "Run", 
    y = "Expected value") +
  geom_hline(yintercept = mean(expected_value_record$expected_value_record), size = 1.4, color = "red") +
  annotate(
    geom = "text",
    x = 0.85 * n_runs,
    y = max(expected_value_record$expected_value_record),
    label = paste0("Mean across runs: ", mean(expected_value_record$expected_value_record)),
    color = "red") +
  theme(plot.title = element_text(hjust = 0.5), plot.caption = element_text(hjust = 0.5))

图形：

现在让我们用另一个模拟来看看问题的 b) 部分。循环被包装成一个函数，在 sapply 的帮助下，我们运行了一系列概率：

library(ggplot2)
seed <- 122334

# simulation function
coin_game <- function(n_runs, n_games, bias = 0.5){
  game_payoff <- c()
  expected_value_record <- c()
  
  for (j in seq_len(n_runs)) {
    
    for (i in seq_len(n_games)) {
      
      cost <- 0
      flip_record <- c()
      payoff <- c()
      
      repeat{
        cost <- cost + 1
        flip <- rbinom(1, 1, prob = bias)
        flip_record <- c(flip_record, flip)
        # print(flip_record)
        
        n_tails <- length(flip_record) - sum(flip_record) # number of 0s/tails
        n_heads <- sum(flip_record) # number of 1s/heads
        
        if (abs(n_tails - n_heads) == 4) {
          game_payoff <- c(game_payoff, 10 - cost) # record game payoff
          break
        }
      }
    }
    expected_value_record <- c(expected_value_record, mean(game_payoff))
    game_payoff <- c()
  }
  return(expected_value_record)
}

# run coin_game() on a vector of probabilities - introduce bias to find fair game conditions
n_runs = 1
n_games = 1000
expected_value_record <- sapply(seq(0.01, 0.99, by = 0.01), coin_game, n_runs = n_runs, n_games = n_games)

# plot expected value
expected_value_record <- cbind.data.frame("run" = seq_len(length(expected_value_record)), "bias" = c(seq(0.01, 0.99, by = 0.01)), expected_value_record)

ggplot(data = expected_value_record) +
  geom_line(aes(x = bias, y = expected_value_record)) +
  scale_x_continuous(breaks = c(seq(min(expected_value_record$bias), max(expected_value_record$bias), by = 0.1), max(expected_value_record$bias))) +
  scale_y_continuous(breaks = round(c(0, seq(min(expected_value_record$expected_value_record), max(expected_value_record$expected_value_record), length.out = 10)), digits = 4)) +
  labs(
    title = "Coin flip experiment: expected value for each probability level", 
    caption = paste0("Number of runs per probability level: ", n_runs, ". ", "Number of games in each run: ", n_games, "."), 
    x = "Probability of success in Bernoulli trial", 
    y = "Expected value") +
  geom_hline(yintercept = 0, size = 1.4, color = "red") +
  geom_text(aes(x = 0.1, y = 0, label = "Fair game", hjust = 1, vjust = -1), size = 4, color = "red") +
  theme(plot.title = element_text(hjust = 0.5), plot.caption = element_text(hjust = 0.5))

图形：

对 expected_value_record 数据帧的检查表明，当概率值在 0.32-0.33 或 0.68-0.69 范围内时，游戏是公平的。

很容易调整最后的代码以挤压更健壮的数字。