Say there are two athletes competing at jumping. Both will always jump anywhere between 0m and 1m. For both athletes there the probability distributions p1(L) and p2(L) that determine the likelihood of each of them jumping a certain distance. The athletes will each jump once, and the one who jumps farther wins. The probability of each of them winning can be written in discrete case as $$ P^1_{win} = \sum_i \sum_j p^1_i p^2_j \delta(i>j)$$ $$ P^2_{win} = \sum_i \sum_j p^1_i p^2_j \delta(j>i)$$ and in continuous case as $$ P^1_{win} = \int_x \int_y p^1(x) p^2(y) \delta(x>y)dxdy$$ $$ P^2_{win} = \int_x \int_y p^1(x) p^2(y) \delta(y>x)dxdy$$ Here $\delta$ is the identifier function, which is 1 if its argument is true and 0 if false. Also known as Heaviside step function.
Thus the better athlete can be determined by comparing $P^1_{win}$ and $P^2_{win}$.
Now the questions:
- Conjecture 1: All possible athletes can be ordered in such a way, that each athlete would have a non-negative winning probability against all those below, and non-positive winning probability against all those above. True or False?
- Conjecture 2: If conjecture 1 is true, it should be possible to assign a number to each jumper that could be used equivalently to perform the above ordering. Further, it should be possible to compute this number only from the distance probability distribution of each individual jumper. True or False? If True, give an example of such a number.
I have high suspicion that this is a well-studied question. I just don't know what to google for...
I have realised that the first conjecture is False, thus the 2nd conjecture is irrelevant.
Imagine 3 different competitors with the same expected value: $$P^A[1] = 1.0$$ $$P^B[0] = 1.0 - 1.0 /1.9$$ $$P^B[1.9] = 1.0 / 1.9$$ $$P^C[0] = 1.0 - 1.0 /2.1$$ $$P^C[2.1] = 1.0 / 2.1$$
It can be easily verified that according to the above rules, A is more likely to win against C, C against B, and B against A. This is a loop, and thus A,B and C can not be sorted by increasing performance.
This game (a very simplified version of your favourite TCG) shows very interesting behaviour - each curve is good against something and bad against something else.