## It’s your move

I’m thinking of two numbers between 0 and 1. Your goal is to guess a number which falls in between my two numbers. Each guess costs you \$1, and if you guess correctly you win the reciprocal of the length of my range (ie if I am thinking of 0.2 and 0.4, a correct guess wins you \$5). At any time you may request that I choose of a new pair of numbers, and of course I will pick a new pair of numbers whenever you win.

What’s your strategy? Under certain conditions, this game is fair. How might you be able to have a positive expectation?

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1. Martin Gillette

The best strategy would be to pick the midpoint, 0.5, every time. So long as the numbers are chosen randomly each time that’s your best bet.

2. I can only think of directions in regards to the “Fairness” part of the challenge.
Well, if , c, the mean of your two numbers [a, b], has a small variance from the centre, than I’ll go with the previous comment – and always pick 0.5.
If such a variance is imposed (or viewed), then you actually need to always go with [0, 1]. Giving me a single \$ each time – and costing me the same. Hence, E=0.
So imposing small Var on your mean value, is a recipe for a “Fair” game.
For that matter, if YOU have to meet such a constraint, then it hardly matters around which point are your two #s revolving.
If all I know, is the above constraint, I am taking a binary-search to find your mean using say a kind of Win-Stay-Lose-Shift on the values I’m raffling.
For large enough N (considering the constraint is still on, and has remained the same throughout) I will eventually find your mean, and will always gamble on it.
This is only fair in the large N sense. Asymptotically fair. I still lose money during “excavation”.
Again, since for large enough N I’m statistically winning – your best bet is to lower my income from every single hit – again choosing 0,1 as your two #s.
So having a small variance from a (secretly) pre-chosen, fixed point, is a recipe for us both playing nice – Keeping our Dollars (or Shekels) in our pockets. Besides, we’ve got 8-legged Paul for such coin-flips