You have a yellow button and a blue button. The yellow button saves 10 lives. The blue button saves 1 life. However, both buttons have an independent, unknown probability of working. Your objective is to save as many lives as you can. Which button do you hit?
Is the decision arbitrary as the probability of the buttons working is completely unknown or is it mathematically logical to choose one over the other?
From a mathematical viewpoint, the decision is arbitrary. However, most people would probably be more comfortable with the yellow button.
The probability of success $X_i$ for each button $i=Y,B$ is unknown. If you assume that $X_i$ is i.i.d., for instance, the yellow button is preferable. OF course, this or other similar assumptions, while consistent with the given information, are arbitrary.
Another approach to argue for the yellow button is to assume that the best criterion is minimise the maximum regret. This is allegedly more compelling, but it offers you no guarantee that you saving as many lives as possible (the criterion says only that, in the worst case, you could not have saved more).