Two tetrahedral dice until a tie
Each turn you roll a red and a blue fair four-sided die, faces 1 to 4. If the blue die is higher, you are paid blue minus red dollars; otherwise you are paid nothing. If the two dice show the same number, the game ends after that turn; otherwise you play another turn.
Find the fair value of the game.
Fractions, decimals with at least 3 significant digits, and expressions such as sqrt(2)/2, e^-1 or pi/4.
Compute the expected payment of one turn and the expected number of turns.
The turn that ends the game pays nothing, since its dice are equal.
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s Shown after a correct answer. Revealing it first marks the problem as seen, not solved.
Idea. Every turn is an independent copy of the same roll. The payment per turn has a fixed mean, the number of turns is geometric, and the two multiply.
Step 1: one turn. Out of 16 outcomes, blue beats red by 1 in 3 of them, by 2 in 2, by 3 in 1. So a turn pays on average
Step 2: the number of turns. A turn ends the game with probability , so there are 4 turns on average.
Step 3. Whether the game continues depends only on the dice already rolled, so Wald's identity applies:
Check by first step. With the value, , which again gives .