• #### Secret Santa variants

From leflynn@21:1/5 to All on Sun Dec 11 16:20:01 2022
You have been invited to a Secret Santa pick-and-swap event with nine other Rec.Puzzlers. It will use the rules in one of these four games.

1. Progressive swapping, all presents are in the game, unknown values.
The first version of the game is very simple. There are ten unwrapped gifts with presents of unknown but different values, and all participants have the same value system. They have been wrapped to appear identical. The ten participants are randomly
assigned turns.
1.a. The first participant selects a package and opens it, revealing it contents to everyone.
1.b. The second participant has a choice. They can take what the first participant opened or open and reveal a new gift. If they take the first participant’s gift, then the first participant opens and reveals a new one.
1.c. The third participant has a choice. They can take either of the two revealed gifts or open and reveal a new gift. If they take a gift that participant can either open a new gift or take the other remaining participants gift. If they take a gift the
third participant only option is to open and reveal another gift. That is, gifts can only be taken once per round and the round ends when some chooses (or is forced) to open a new gift.
1.d. The play for the fourth through tenth participants proceeds with the increased number of options for taking someone’s gift. When the last gift is open, the game is over and everyone keeps what they have.
What is the optimal strategy for each player?

2. Limited swapping (one per turn), all presents are in the game, unknown values.
This is the same as the previous case but there is only one taking per turn. That is, the new participant can either take any revealed gift or open a new one. If they take a gift from someone, that participant’s only choice is to open a new gift.
What is the optimal strategy for each player?

3. Progressive swapping, presents are “protected” after two swaps, unknown values.
This is the same as the first case but when a present has been taken twice, it is out of the game and whoever took it the second time will get to keep it at the end of the game.
What is the optimal strategy for each player?

4. Limited swapping (one per turn), presents are “protected” after two swaps, unknown values.
This is a combination of 2 and 3.
What is the optimal strategy for each player?

Other variations can be made with the values of the presents known, more or fewer than two swaps before presents are protected, or some set maximum number of swaps greater than one per turn.

L. Flynn

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