Numbers To Number

This is a game for 2 players. Needed: 3 blank pieces of paper, one for each player and one common piece of paper. 2 pens/pencils, one for each player.

Let m be an integer decided on by both players. m should probably be >= 6. Players also decide who is player 1 and who is player 2 at the beginning of the game.

At the beginning of each round, that round's "binary list" is blank. (The binary list is a series of 1's, 0's and _'s written on the common piece of paper, one new binary list for each round.)

On a turn, both players secretly write down on their piece of paper any number from 1 to 2^m -1 that has yet to be written down by that player in the game (in any previous round or in the current round).

The two players' numbers are then revealed to each other.

If player 1's number is > player 2's number, then append a 1 to the right side of that round's binary list.

If player 1's number is < player 2's number, then append a 0 to the right side of that round's binary list.

If player 1's number is = player 2's number, then append a _ (underscore) to the right side of that round's binary list.

After m moves (where one move is both players moving simultaneously) have passed in the round, the round is over.
Treat the binary list as a set of binary numbers, with each _ treated as either a 0 or a 1. Convert each binary integer to decimal.
So, the binary list represents 2^(number of _'s) different integers.
Let the set of decimal integers for that round be D.

So, for instance, if the binary list (m=8) looks like this:

01_001_1,

then D for that round contains:
69, 71, 101, and 103.

Each player gets a point for every element of D that was an integer played by the player IN THAT ROUND.

After a round is over, the player crosses off all the numbers played by that player in the round (so that the player can tell the difference between numbers played in that round and numbers player in earlier rounds).

A game consists of floor((2^m -1) /m) rounds.

Add up scores from all rounds to get each player's grand score.
Highest grand score wins.

Thanks,
Leroy Quet