In this sharing game theory puzzle, 3 friends take turns taking from a jar of 1000 candies to share. For example, the 1st friend could take 500 candies, then the 2nd friend could take 400, and the 3rd friend would take the remaining 100.

No one wants to be seen as greedy, but no one wants to end up with the fewest candies either. As such, their goals are (in order of preference):

- Do not end up with the most candies, nor the fewest candies (a tie for most or fewest also fails this condition)
- End up with as many candies as possible

All of them are logical, rational, know each other’s goals, but cannot communicate before or during sharing. How many candies should each friend end up with?

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1st friend will have 334, 2nd friend will have 666, and 3rd friend will have 0.

Intuitively, if the 1st friend takes a lot of candies, the 2nd friend can “win” (satisfy goal 1) by taking 1 less than the 1st friend (or all remaining candies if that is not possible). It turns out this cutoff is at 335: if the 1st friend takes 335 or more, the 2nd friend will always take 1 fewer, guaranteeing a win.

For example, if 1st friend takes 335, 2nd would take 334 to win, as 3rd can only take the remaining 331 (because 3rd can no longer satisfy goal 1, they maximize goal 2).

But if the 1st friend takes 334, 2nd is stuck with no way to win:

- If 2nd takes 332 or fewer, 3rd wins by taking 333.
- If 2nd takes 333 or more, 3rd cannot win, so 3rd would take all remaining candies and no one wins.

Thus 2nd has no way to satisfy goal 1, so 2nd would take all remaining 666 candies. This means the 1st friend wins with 334 candies, 2nd friend will have 666, and 3rd friend will have 0.

Since this satisfies goal 1 for the 1st friend, there is no reason for the 1st friend to consider any smaller number (which would be worse for goal 2).

It’s not said that there cannot be leftover candy.

Each could take 333 candies, which would satisfy the 1st condition as well as the second for all of them, if all of them could satisfy all goals why wouldn’t they?

Regarding solution, why wouldn’t the second friend take 333 candies, if he does this maximizes the amount of candy he gets, while also making sure he does not receive the most nor the least candies.

Leftover candy is technically allowed.

Each taking 333 candies is not a stable equilibrium – the 1st friend would take 334 instead, because this still ensures they satisfy the 1st goal but also maximizes the 2nd goal. There isn’t any goal of helping others achieve their goals.

As for the 2nd friend taking 333, the 3rd friend would take 333 as well, meaning both of them would end up with the least candies. Unable to fulfill the 1st goal, the 2nd friend would just maximize their candies instead, taking 666. But it seems the wording could be ambiguous, I’ll re-word it to make clear that “least” and “most” should include ties for least and most. Thanks!