Question

Say you are playing the General Blotto game, with 10 indistinguishable soldiers and 10 castles. You want to create a random arrangement, with each possible arrangement having equal probability.

1. In the uniform distribution, what is the probability of each arrangement?

2. If we assign each soldier independently to a random castle, show that this is not the uniform distribution on arrangements? Explain your answer. (Hint: what is the probability of putting all soldiers in the first castle?)

Answer 1

we want to know the number of non-negative integer solution of

x1 + x2 + ..x10 = 10

xi represent number of soldier in castle i , xi >= 0

number of non-negative integral solution of

x1+x2+ ...xn = r

is (n+r-1)C(n-1)

here n = 10, r = 10

hence

(10 + 10 - 1)C9

= 19C9

= 92378

hence

the probability of each arrangement = 1/92378

b)

If we assign each soldier independently to a random castle, this is not uniform distribution

It is actually multinomial distribution