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.