Probability of each team placing in each position

Question

I've been thinking about this for a while, and haven't been able to figure it out. My math teachers haven't been able to, either.

You have N teams, and each team has P players. Each player plays a (singleplayer) game, and has a W% chance of winning (each player has a different chance of winning, and this winrate is known beforehand). Let's set up an example scenario with 3 teams and the winrates of their players:

Team A: 50%, 40%, 40% Team B: 60%, 40%, 30% Team C: 90%, 50%, 20%

Whichever team gets the most wins wins the competition overall. If two teams tie for first, they both get first place and the remaining team gets third. If all three tie, they all get first place.

How do you determine, for this example, the chance for each team to get first, second, and third place? How would you determine this for any given variables in place of the ones used in this example?

Answer 1

Here is a result using simulations (in R)

w=c(0.5,0.4,0.4,0.6,0.4,0.3,0.9,0.5,0.2)
res=apply(t(w),2,function(x){
  ifelse(runif(1e5)<x,1,0)
})

res2=apply(res,1,function(x){
  c(sum(x[1:3]),sum(x[4:6]),sum(x[7:9]))
})

res3=apply(-res2,2,rank,ties.method="min")

First team:

prop.table(table(res3[1,]))

      1       2       3 
0.46692 0.29580 0.23728

Second team:

prop.table(table(res3[2,]))

      1       2       3 
0.46161 0.30760 0.23079

Third team:

prop.table(table(res3[3,]))

      1       2       3 
0.63170 0.27151 0.09679