The 1933 movie "The Devil's Brother" (also known under the title "Fra Diavolo") takes place in the Northern Italy of the early 18th century. Stan Laurel and Oliver Hardy play the fierce robbers Stanlio and Ollio. They manage to steal 100 gold coins from the rich Lord Rocburg, and then discuss at length how to share the loot. Finally they decide to play the following game.
- In every step, Stanlio picks a handful of gold coins from the loot.
Then Ollio decides, whether this handful should go to himself or to Stanlio. - Once all coins have been assigned, the game ends.
- Once one of them has received 9 handfuls, the game also ends. In this case the other player (who has received at most 8 handfuls up to that moment) receives all the remaining gold coins.
Question: What is the highest number of gold coins that Stanlio can guarantee for himself?
(As usual, we assume that both players use optimal strategies.)
Answer
Ok, I'll have a stab. The highest number Stanlio can guarantee is:
46 coins
Optimal strategy:
An optimal strategy for Ollio is to claim a handful that has 6 or more coins, and to pass on handfuls of 5 or fewer coins.
So ...
Stanlio knows this, and so knows that producing a handful with less than 5 coins will result in a bigger pot for Ollio left after 9 rounds. Producing a handful with more than 6 coins just reduces the pot left for himself after 9 rounds.
Which means ...
If Stanlio produces 5 coins at a time, Ollio will let him keep them. This means after 9 rounds Stanlio will have 45 coins and Ollio can claim the remaining 55 coins. If Stanlio produces 6 coins at a time then Ollio will claim them. Meaning that after 9 rounds Ollio will have 54 coins and Stanlio can keep the remaining 46.
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