Transform -- The Game

This is a game for any plural number of players.
(No grids this time, sorry.)

There are a number of rounds in this game. The number of rounds is a multiple of the number of players. The players take turns being "the permutator", where each player is the permutator the same number of rounds.

Before starting any of the rounds, the players agree on a positive integer n. (n is the same value for all rounds in the game.)

On a round, all of the players (including the permutator) start the round by each coming up with an ordered list of n integers (positive, negative, or 0). Each player's numbers must be distinct, in that no integer occurs more than once in a particular player's list. The players each keep their lists secret from the other players for now.

Let player p's list for any particular round be {b(p,k)}, k = 1,2,3,...n.

Next, all players who are not the permutator take turns choosing terms of a list of n distinct integers(positive, negative, or zero; no integer more than once). (So, if the number of players is m, I guess to be fair, n should be a multiple of (m-1).)

Let this list be {a'(k)}, k = 1,2,3,..n.

Then, after the list is complete, the permutator forms any permutation {a(k)} of {a'(k)}.

Then everyone reveals their b-lists.

Each player p forms a sequence of n integers in this manner:

c(p,k) = sum{j=1 to n} a(n+1-j) b(p,j). k = 1,2,3...,n.

Player p's score for this round is the number of primes in {c(p,k)}.

After all the rounds are played, the players add up their scores for all the rounds. The player with the largest grand score wins.

In a variation, instead of the number of primes being the criterion for scoring, the players decide amongst themselves before each round what will be the criterion for a number in the c-list to score. Fibonacci numbers? Squares? Where each c(p,k) is coprime to c(p,k-1)?
Or maybe the choice of criterion should be totally up to the permutator (with veto power from the other players), and expressed at the beginning of each round before the b-lists are constructed.

Thanks,
Leroy Quet