mulur

So CoffeeScript meets C++?

Ok, my solution is needlessly complicated because unbounded numbers between -Infinity and Infinity has a mean of 0, so if we just pick 0 and say "if hand1 < 0 then next is bigger, otherwise smaller" strategy already gives 75% accuracy long term.

pyk's solution above works 66.6% even for the first try...

Oh that's interesting. Thank you for the clarification.

Hmm I'm not exactly good with maths (or puzzles for that matter) but here is a blind stab:

By break even, I think you mean that the game is played multiple times, as long as necessary. If you are not following a martingale like strategy (edit: you can't anyways, the bets are fixed) and respond randomly (or with full faith that this is your lucky day, doesn't matter really), you are expected to break even with 50% chance.

How can we improve and make better than 50%?

'A' chooses two distinct integers each time. They can choose among infinite number of integers. For the first guess, I can't do better than 50% chance but if I start to keep history of the numbers that were picked it feels to me like I can do better than chance if I assume A chooses their distinct integers uniformly random by expecting a uniform distribution. I don't have much ideas about how to judge uniformness in an unbounded set of integers though but maybe something like: if first hand is less than the average of all previous numbers, I'd say "other number is higher", otherwise I'd say "other number is lower". I feel like this would improve my chances to more than 50% in the very long term (at infinite plays perhaps?) but am not able to prove it.

If A is not choosing their numbers randomly with a true RNG then I'd play many games at 50% chance then run their choices through a neural network to extract patterns then I'd do better than random for sure.

>This is incredibly wrong. The guarantee is based 100% on facts. This is literally the only thing it's based on.

I don't really follow you here. You mean that the guarantee is about safety? I mean back then our life expectancy was lower. The fact that humanity survived with old methods doesn't mean what we did was safe(r) in general. It just means that enough people survived so that humanity didn't die off as a whole. That is a very weak guarantee.