Accept or pass the die
Players A and B alternate, A first. The player to move rolls a fair 21-sided die with faces . They either accept, which ends the game with the opponent paying them the face value (a negative face means they pay), or pass, and then the other player rolls under the same rules. There is no limit on the number of turns.
Both play optimally. Find A's expected payoff.
Fractions, decimals with at least 3 significant digits, and expressions such as sqrt(2)/2, e^-1 or pi/4.
If you pass, your opponent faces exactly the game you were facing. If that game is worth to the player about to roll, what is passing worth to you?
, where is the face. Guess which faces are accepted, solve, and check the guess.
h
s Shown after a correct answer. Revealing it first marks the problem as seen, not solved.
Idea. The game looks the same from the point of view of whoever is about to roll. If it is worth to that player, passing hands the opponent a game worth to them, so to you. You accept a face exactly when .
Step 1: the equation. The game favours the player who rolls, so and you accept every face that is not too negative.
Step 2: guess and solve. Suppose you accept faces and above and pass on and below (9 faces). Then so and .
Step 3: check. With , passing is worth . Accepting is better than , and passing beats accepting . So the guess is consistent, and A's expected payoff is .
The trap. Refusing every negative face. Accepting a loss of 1 is right here, because passing is worth to you: the opponent gets the favourable position.