Comparing / Averaging two very different numbers

Question

I have two examples called a and b. Each example has two scores:

a: 0.5 and 0.001

b: 0.8 and 0.0001

If the higher score wins, then b wins when looking at the first set (0.8 > 0.5). But I also want to take the second numbers into consideration. Just adding them up doesn't help because 0.80001 > 0.5001. But if the second number is significantly larger in example a then I want a to win regardless of the first set of number.

Is there any techniques of how these numbers can be combined?

The numbers are probabilities and the probabilities of the second set can get very small.

Answer 1

You could use a weighted average. That is, multiply the second value with some fixed value before adding them. Then it is up to you to decide exactly how important the second value is, which you choose by chosing the value you multiply with. You could also use a geometric mean (check wikipedia).

Answer 2

So a few things to consider I guess, but an example to start you off

$X_{i}(a) = \dfrac{P_i(a)}{P_i(b)+P_i(a)}$ since for your cases above leads to player a has scores $X(a) = \begin{pmatrix} 38.5\\ 90.9\end{pmatrix}$ whilst player b will have $X(b) = \begin{pmatrix} 61.5\\ 9.1\end{pmatrix}$ .