What is the Elo of a player who is half as good as another player? You often hear players refer to others as half as good as me, while they have the same Elo. Is there a formula to measure this? It must take in account that being half as good as Carlsen, for example, is somewhere near a 2600 GM level, while half as good as a club player, is at 1000 level.

What I think that being n times weaker than a player is.

If an n times weaker player plays the player he is n times weaker, it means that if they played an infinite match, the weaker player should get n points, while the stronger player n*x.

Any other ideas? Are there any papers written on this? If my idea were to be true, is there a formula to calculate the Elo difference?
This answer addresses one concrete portion of the OP; namely, for an arbitrary N &gt; 0, what rating difference between Player A and Player B corresponds to Player A's expected score against Player B being N times the expected score for Player B? I'll speak in terms of the USCF's Elo rating system, since I'm most familiar with it.

If Player A's expected score is N times that of Player B when they face each other, then Player A's expected score is N / (N + 1) while Player B's expected score is 1 / (N + 1) (since the two expected scores must sum to 1). Let A be Player A's rating and let B be Player B's rating. According to the USCF rating system, these ratings are intended to indicate that Player A's expected score is then

1 / (1 + 10^((B - A) / 400))


Thus if we solve the equation

1 / (1 + 10^((B - A) / 400)) = N / (N + 1)


for A - B, we will have an answer as to what rating difference corresponds to Player A's expected score being N times greater than Player B's when they match up. What we get is that

A - B = -400 * ln(1 / N) / ln(10)


So we have the following approximate rating differences corresponding to certain values of N:

N     A - B
-     -----
1     0
1.5   70
2     120
3     191
4     241
5     280