# Given a set of random variables, how can I find a linear combination of these variables satisfying a constraint on the sum of their permuations?

Say I have n random variables, {X0...Xn}, n>9. I also have another set of random variables constructed from the first set, where each of these are the sum of 9 random variables in the original set, i.e. {X0+X1...+X9, X0+X1...+X10, ...}.

I can find a linear combination of random variables from the second set which satisfies some constraints; you can think of this as a game where you pick 9-variable combinations, and the solutions to the linear combination is the proportion of picks that would result in a Nash equilibrium.

Once I have that linear combination of sets, I can back out a distribution of usage of each variable from the first set; I.e. X0 is used in 80% of Nash equilibrium picks, X1 is used in 60%, etc.

My question: given the parameters of the distributions of each random variable in set 1, is it possible to find the post-Nash variable usage without first building all the permutations and computing the equilibrium distribution?

• "where each of these are the sum of 9 random variables in the original set, i.e. ${X_0+X_1...+X_9, X_0+X_1...+X_{10}, ...}$." --- those two sums have 10 and 11 terms respectively, not 9. Commented Feb 5, 2023 at 22:29