This is a single example of a win probability matrix, and it's clear that player 1 is the best because he is 50%+ in all matchups.

So any reasonable format should have him win more of the time, and the one that involves more games will increase that chance. This is not particularly interesting.

In fact, let's make this super mathy...

Let the win probability matrix be a random variable X, and call the max wieght row k

Let the format be a mapping f:X-->y, where y is the win probability vector.

In this example it seems we are trying to maximize y(k) subject to X=A.

But I contend this is not what we want to maximize. What we want is that the random winner be the best predictor of the best player, that is:

maximize P(A(i) >= A(j) for all rows j | z = i, where z is drawn from the distribution given by y).

By making A non-random, this example doesn't really show anything.