Another interesting proposition.

[thylacine]

Two players Writer (W) and Picker (P) play a game.

W antes 2 dubloons, P antes 1 dubloon to make a pot of 3 dubloons (no more betting).

W has 3 identical slips of paper. W writes a distinct number on each piece of paper. (Any real numbers are allowed.) These 3 slips of paper are then mixed up in a hat. Two of them are randomly selected and read out loud.

P must try to guess whether the 3rd and unseen number is the Biggest, Middle, or Smallest of the 3 numbers. If correct then P wins the pot, otherwise W wins it.

If both try to play well, then what can P's expected win be?

Note that if P simply guesses at random, then his EV is zero (i.e. the antes are `fair') and W can do nothing to make P's EV less than this.

But can P have positive EV regardless of what W does, and if so how big an EV can P be sure of?

You need to describe how W selects sets of 3 numbers, and how P guesses based on the two seen numbers. You may assume that P knows W's strategy.

and if you do all that ...

In general W antes N-1, P antes 1, W chooses N numbers, P sees N-1 of them, and must guess where the unknown number ranks among the list of N numbers.

[bigpooch]

It is quite clear that P has a +EV: this is just a
generalization of the two envelope paradox: use a more
intricate switching type function. Normally, picking the
last (unseen) number as in the middle all the time will
yield an EV = 0 but using a modified switching function
based on the maximum when thinking of picking the last
number as higher and similarly for the opposite direction
(when the probability functions say to pick last as min
and max choose middle).

Of course, one can generalize this for more than 3 numbers;
again, pick max or min accordingly depending on the
switching functions. I didn't work out technical details
but the idea seems quite clear. On the other hand, more
precise analysis would be required for maximizing EV. Did
you happen to find an answer for this maximization?

[thylacine]

N=2 here is same as N=2 in `An interesting proposition' and both are `envelope switching', I guess. For bigger N these problems differ.

I was not sure what you were saying in your post. This stuff seems interesting and fairly deep. It's fun to think about but I've got other things I should be doing. But I'm curious to know what is known about these kinds of things. Your other post saying `the standard answer is...' suggested to me that this was a well explored topic. But when I start thinking about W's possible probability distributions of N-tuples of numbers, and P's possible responses (in either version of these problems), I realise this could be deep stuff.