All Drunk

So, whats the solution to running out of interesting puzzles? Stop posting. I guess the other solution is to post non-interesting puzzles, or possibly interesting non-puzzles, but that seems like so much more work than not posting at all. Anyway, I guess one should post the solution to the drinking game puzzle from last time.

First of all, it is obvious that Alice can get a 1/2-1/2 split with Bob if she wants, so the only question is if she can do any better than that. Bob being forced to drink from the bottle at least once is a disadvantage for him, if not for that rule it is clear that the optimal strategy is 1/2-1/2 in general. Since Bob being forced to drink from the bottle is a disadvantage, we can assume he will try to get rid of that, specifically if the bottle ever has less poison than the cup, Bob will immediately drink from the bottle and the remaining bottles will get split 1/2-1/2.

As soon as Bob has drank from the bottle, the remaining bottles get split 1/2-1/2. If we get to the last bottle and Bob has not yet drank from the bottle, Alice will do a 1-0 split, forcing Bob to drink all of the poison in the bottle. Thus, if there are two bottles left, let us assume Alice does an x-y split (x>y and x is the amount in the bottle). If Bob choses x, he will have free choice on the next one and in the end Bob will have drank x+1/2 and Alice will have drank y+1/2. If Bob choses y, Alice will be able to force him to drink all of the last bottle and so Bob will drink y+1 and Alice will drink x. Bob will see this coming and choose whichever one is lower, so Alice does best to make them equal.

We have x+1/2=y+1 and x+y=1, so we get x=3/4 y=1/4, so if there are two bottles left and Bob must still drink from a bottle once, Alice splits 3/4-1/4 and forces Bob to drink 5/4. Obviously if there are two bottles left and Bob does not have to drink from a bottle, then Bob drinks half of each of the remaining bottles for a total of 1.

Now, if there are three bottles left and Bob must drink from the bottle once (the original problem), then Alice splits the first bottle x-y (x>y and x the bottle) and Bob can select either x+1 or y+5/4. Setting those equal and using x+y=1 we get x=5/8 y=3/8. So Alice divides up the first bottle 5/8-3/8 and forces Bob to drink 13/8 in total, whereas Alice only drinks 11/8.

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