Divisors/Multiples Sequence Game

Here is a game for any plural number of players.

r is a positive integer agreed upon by all players before the start of the game.
I suggest that r be congruent to 1 mod {the number of players}.

The game starts with a(1) = 1 and b(1) = 1. m and n both equal 1.

Players alternate moves.

(*)n=n+1.
(**)m=m+1.
A player on his move picks an integer a(m) that is either a divisor or a multiple of a(m-1).
a(m) must be <= r.
The player gets max(a(m),a(m-1))/min(a(m),a(m-1)) added to his score if a(m) is not among (b(1),b(2),...b(n-1)).
But the player gets
2*max(a(m),a(m-1))/min(a(m),a(m-1))
added to his score if a(m) is among (b(1),b(2),...b(n-1)).
The player continues his turn as far as he wants, but until a(m) is not amongst (b(1),b(2),...b(n-1)).
If the player is to continue his move, he goes to (**).
If a(m) is not among (b(1),b(2),...b(n-1)) and the player wants to end his move, then b(n) = a(m), and switch whose turn it is, go to (*) if n is < r.

Play until n = r, and (b(1),b(2),...b(n)) is a permutation of the integers 1 through r.
The winner is then the player with the LOWEST score.

Thanks,
Leroy Quet