We continue the topic of zero-sum games and rock paper scissors. Winnie and Lucy have now decided to embark on a million-round game of rock paper scissors. They have done their preparations very meticulously, and they are ready. Let’s start following them with the simulations of their expected game and scores.
ti <- 0
win <- 0
luc <- 0
wr <- 1/3
wp <- 1/3
ws <- 1-wr-wp
lr <- 1/3
lp <- 1/3
ls <- 1-lr-lp
for (val in 1: 1000000){
Winnie <- sample(c("Rock", "Paper", "Scissors"), 1, replace = TRUE, prob = c(wr, wp, ws))
Lucy <- sample(c("Rock", "Paper", "Scissors"), 1, replace = TRUE, prob = c(lr, lp, ls))
game_fin <- paste(Winnie = Winnie, Lucy = Lucy)
if (identical(Winnie, Lucy)) {
#print("tie")
ti <- ti + 1
} else if(Winnie == "Rock" & Lucy == "Scissors" | Winnie == "Paper" & Lucy == "Rock" | Winnie == "Scissors" & Lucy == "Paper") {
#print("Winnie Wins")
win <- win + 1
} else {
#print("Lucy Wins")
luc <- luc + 1
}
}
win * 100/ (win + luc + ti)
luc * 100/ (win + luc + ti)W: 33.3; L: 33.3. If they both follow a random strategy, giving equal weightage to each of the three options, the expected results are a third for each outcome (win, loss, tie).
After about 1000 games, Lucy notices that Winnie have a slight bias towards the rock. Since she counted hands, Lucy thinks it is around (1.2/3). Note that it did not affect the overall results, which is still at
W: 33.3
L: 33.3
Lucy sees the opportunity and adjusts her game. She increases the paper to 1.2 in 3 and starts to see the results in the next 1000 games.
W: 33.0
L: 34.0
She also reduced the proportion of scissors from her kitty and found that her winning margin increased slightly.
W: 32.7
L: 34.0
Lucy now knows that providing the paper with a higher chance (1.5/3) could fetch an even better margin, she, however, doesn’t attempt for it, suspecting Winnie would figure it out.
Lucy did not know that Winnie used to be the junior champion in her college days. Winnie was testing Lucy by giving the bait to change her from a random strategy to having a bias. Noting Lucy has changed to a more-paper-strategy, Winnie changes to a scissor biased game (1.2/3).
W: 34.0
L: 33.3
Lucy noticed it after about 1000 games. Now Lucy knows Winnie knows Lucy knows. Or strategy is getting common knowledge. She has only one way out. Go back to random. The outcome is back at 33.3% for both, irrespective of what Winnie did.
In Summary
The best strategy to win a game of rock paper scissors is that there are no strategies unless the opponent gives one. Otherwise, you stick to random choices and leave the results to randomness in the short run, or if you are on a day-long game, a likely stalemate.
