Doubles wipe you out
You play a game with two fair dice. Your winnings start at 0. Before each roll you may stop and keep your winnings. If you roll:
- if the two dice show different numbers, their sum is added to your winnings;
- if they show the same number (a double), the game ends and you lose everything.
You want to maximise your expected final winnings. The best strategy is to keep rolling while your winnings are below some level and to stop once they reach it. Find that level: the smallest amount of winnings at which stopping is optimal. (If stopping and rolling are equally good, count it as stopping.)
Fractions, decimals with at least 3 significant digits, and expressions such as sqrt(2)/2, e^-1 or pi/4.
Compare stopping now with rolling exactly once more and then stopping.
Given that a roll is not a double, what is the expected sum of the two dice?
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s Shown after a correct answer. Revealing it first marks the problem as seen, not solved.
Idea. Compare stopping now with rolling once more and then stopping. Here that one-step comparison gives the optimal rule, because once it says stop it keeps saying stop.
Step 1: one more roll. A double has probability . All 36 outcomes have sums adding up to and the six doubles to , so the 30 non-doubles add up to and a non-double roll adds on average. With winnings , rolling once and then stopping gives on average
Step 2: the threshold. This beats stopping at when At the two are equal.
Step 3: why the one-step rule is optimal. If , every roll that does not end the game raises , so you stay in the region where one more roll does not pay. When the region where stopping wins the one-step comparison cannot be left, the rule “stop as soon as one more roll is not profitable” is optimal. Below 35, rolling once beats stopping, so stopping is not optimal there.
So you roll while your winnings are below 35 and stop as soon as they reach 35 or more. Played this way from 0, the game is worth about .
A relative. In the game Pig, one die is rolled, a 1 wipes the turn total and the other faces add to it. The same comparison, , gives the familiar rule “hold at 20”.