• Suppose there is a game where 1000 people flip a fair coin.
• In round 1, everyone who had tails is eliminated.
• In round 2, the remaining people flip coins again, of the remaining people, everyone who had tails is out.
• This keeps going until everyone is out.

Here is my R simulation to represent this:

# pre game set up
library(dplyr)

num_players <- 1000
results <- data.frame(player = paste0("player", 1:num_players))
status <- rep("playing", num_players)
round <- 1

# simulation
while (any(status == "playing")) {
    
    coin_flips <- ifelse(runif(num_players) > 0.5, "heads", "tails")
   
    current_round <- ifelse(status == "playing", coin_flips, "eliminated")
    
 
    status <- ifelse(status == "playing" & coin_flips == "tails", "eliminated", status)
    
   
    results <- cbind(results, current_round)
    
    colnames(results)[ncol(results)] <- paste0("round_", round)
    
    round <- round + 1
}

The results look like this:

 > head(results)
   player round_1    round_2    round_3    round_4    round_5    round_6    round_7    round_8    round_9   round_10
1 player1   heads      heads      heads      heads      heads      heads      tails eliminated eliminated eliminated
2 player2   tails eliminated eliminated eliminated eliminated eliminated eliminated eliminated eliminated eliminated
3 player3   tails eliminated eliminated eliminated eliminated eliminated eliminated eliminated eliminated eliminated
4 player4   tails eliminated eliminated eliminated eliminated eliminated eliminated eliminated eliminated eliminated
5 player5   tails eliminated eliminated eliminated eliminated eliminated eliminated eliminated eliminated eliminated
6 player6   heads      tails eliminated eliminated eliminated eliminated eliminated eliminated eliminated eliminated

I am trying to answer the following question: After the result of the simulation, can we see if this game had a unique winner? That is, was there a moment in which all other 999 players were eliminated but one player had still not been eliminated? (i.e. no ties)