# Game scenario (expected value)

#### mathfreak33

##### New Member
You're in a game with 20 peers.

You roll a die.

Rolling a one -> 1 point & choose if you want to roll again
Rolling a two -> 2 points & choose if you want to roll again
Rolling a three -> 3 points & choose if you want to roll again
Rolling a four -> 4 points & choose if you want to roll again
Rolling a five -> Lose all points and game over!
Rolling a six -> 6 points & choose if you want to roll again

You can keep rolling until you get a five, or decide to stop rolling. If you rolled a 5, your game score is 0; if you stopped otherwise because you wanted to, your game score is the total of your points.

What is the best strategy for this game? Keep in mind that you are going up against 20 peers who are also going to be playing, so you want the highest point count.

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#### Dason

Are they playing at the same time as you (so you don't know what any of them get before you finish) or is it sequential? So player 2 knows what player 1 got, player 3 knows what players 2 and 1 got, and so on...

#### mathfreak33

##### New Member
Everyone plays in different sections of the room at the same time, so basically, you don't know what anyone else got.

#### Dason

I can't tell you if it is the best strategy but if you choose to use the strategy "keep rolling up to a maximum of N" times then you have the highest expected value at N= 5 or 6 (they give the same expected value of about 6.43.

This doesn't account for the other people playing - you might be able to use that knowledge to increase the chance of winning. You also might be able to devise a strategy that is substantially more complex than the one that I mentioned that could do better.

Here is an R script I whipped up to simulate this strategy and compare to the expected values I calculated by hand.
Code:
pts <- 1:6
pts[5] <- NA

trial <- function(n, pts){
j <- sum(sample(pts, n, replace = TRUE))
ifelse(is.na(j), 0, j)
}

sim <- function(n, N){
j <- replicate(N, trial(n, pts))
mean(j)
}

vals <- 1:7
out <- sapply(vals, sim, N = 100000)
plot(vals, out)

f <- function(n){ n * 3.2 * (5/6)^n}
cbind(out, f(vals))
Note that without specifying the distribution of strategies that the other 19 players use it will probably be difficult to find the 'optimal' strategy.

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