Each Christmas my wife attends a party where a bunch of suburban erstwhile party-girls get together and A) drink and B) exchange ornaments. Looking for any excuse to get invited to hang out with a bunch of drunk soccer-moms, every year I express sincere scientific interest in their peculiar mechanism of matching porcelain trinket to plastered Patricia. Alas I am denied access to their data.
So theory will have to do. Here is the game they play. Each dame brings with her an ornament wrapped in a box. The ornaments are placed on a table and the ladies are randomly ordered. The first mover steps to the table, selects an ornament and unboxes it. The next in line has a choice. She can steal the ornament held by her predecessor or she can select a new box and open it. If she steals, then #1 opens another box from the table. This concludes round 2.
Lady #N has a similar choice. She can steal any of the ornaments currently held by Ladies 1 through N-1 or open a new box. Anyone whose ornament is stolen can steal another ornament (she cannot take back the one just taken from her) or return to the table. Round N ends when someone chooses to take a new box rather than steal.
The game continues until all of the boxes have been taken from the table. There is one special rule: if someone steals the same ornament on 3 different occasions (because it has been stolen from her in the interim) then she keeps that ornament and leaves the market (to devote her full attention to the eggnogg.)
Theoretical questions:
- Does this mechanism produce a Pareto efficient allocation?
- Since this is a perfect-information game (with chance moves) it can be solved by backward induction. What is the optimal strategy?
- How can this possibly be more fun than quarters?
Cheap Talk 
12 comments
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December 13, 2009 at 10:37 pm
Alex
To answer question 3, it’s more fun when played with gag gifts.
For question 2, wouldn’t you need to know the preferences of the people at the party to solve the game and determine the optimal strategy for each player? This may be especially difficult since people might not even think about their own preferences until they see the gifts that have been revealed.
For question 1 if you allow players to trade gifts at the end (which presumably nobody would have a problem with), then yes.
December 14, 2009 at 8:10 am
Susan Simpson
1. Nope, although allowing trading at the end of the game might help. It’d happen if A got a Santa ornament, and B likes it but takes her chances with an unopened present and get a reindeer, which A likes but she does not.
When I played this game as a teenager, with various social groups I was a part of, the (adult) organizers made the mistake of letting everyone open a new present and then choose who to trade with to “upgrade.” I assume they did this to save time, but all it resulted in was people trying to leave others stuck with the most useless gift — i.e., the guy who shaves his head ends up with a set of combs, or the kid allergic to peanuts gets the pile of reece’s pieces.
December 14, 2009 at 1:02 pm
The Wife
Definitely not Pareto efficient. I had to return one of the ornaments that I got. Thank goodness there was a gift receipt with it! 🙂
December 14, 2009 at 1:44 pm
Alex
Pareto efficiency doesn’t require everyone to be happy, though. It requires that the outcome be one where you can’t make anybody better off without making somebody else worse off. I still contend that if you’re allowed to trade at the end then it is Pareto efficient. Even if the kid who gets the Reese’s Pieces is allergic to peanuts, the game is still Pareto efficient if the Reese’s Pieces are less desirable to everyone else than the gift they already have. And if that is not the case then the kid can trade his Reese’s Pieces for something else.
December 14, 2009 at 1:53 pm
chug
Who cares about Pareto efficiency – I like hanging out with drunk soccer moms! And soccer dads, too!
And loved “matching porcelain trinket to plastered Patricia”
Looked for a synonym for ornament that starts with “p”, and the closest I came is pride or popcorn:
matching porcelain popcorn to plastered Patricia
December 14, 2009 at 8:24 pm
Daniel Reeves
This is normally referred to as a White Elephant Gift Exchange.
My sister and I have been hashing out the version of this that we plan to use for our family gift exchange:
http://etherpad.com/welephant
December 15, 2009 at 10:53 am
devilyouknow
I know this game as Yankee Swap. Thick on the ground in New England anyway. There are rule variations, most commonly the concluding swap. Google should turn up the variations.
December 15, 2009 at 11:08 am
Alicia
The main issue in being totally Pareto efficent has to do with limit of 3 steals of any one ornament/present. With unlimited Steals of ornaments/presents you would reach a pareto efficent outcome.
However, with only 3 steals you keep the game moving so that not everyone immediatly swaps for the best option ( example the “you have died of dysentry” tee shirt at last weekends yankee swap or the ipod in the office christmas special)
December 15, 2009 at 11:03 pm
Noah Yetter
Perhaps this is hopelessly naive, but how can you label this “a perfect information game” when all the relevant information is wrapped up in either boxes (payoffs) or the participants’ heads (preferences)? If the ornaments weren’t wrapped, and everyone ranked the ornaments in the same order, THEN you could solve it.
December 16, 2009 at 12:57 pm
Lyle_s
I looked this up earlier and, if I understood it correctly, perfect information means that every player has witnessed what has happened in that game up to any point in time in the game. If it can be solved through backwards induction, then we’re starting essentially at the end of the game and so all but one gift has been opened.
I don’t know how to solve this stuff but my family does the Yankee Swap (as my New England relatives call it) / White Elephant (Midwesterner relatives) so I want to know the answer. If you saw all of the other players pick and steal, you’d probably have a pretty good idea of what the general consensus good gifts were and who had particular attractions to a specific gift. If the group is large enough, some presents might be out of the mix (although not how Alicia stated. If I read the rules correctly, one person must steal the same gift 3 times for it to drop out of the game, not just that the present can only be stolen 3 times overall.)
My goal is to end up with the present I like the most that is still available. If I don’t like any of the presents I’ve seen, I take the last box. Worst case scenario is it’s another present I don’t like. If the one I like has just been revealed to the players or has never been stolen by anyone else during the game, I grab it immediately and I’m one up on the stealer tally. Hopefully, if someone grabs it from me, I can induce some sort of gift stealing whirlpool that allows it to either end up with me un-stolen or allow me to steal it 3 times.
What if I’m after a hotly-traded commodity, though? Unless I can see a path of preferences that will get that thing to land in my lap and stay there, I probably don’t have much chance of landing it in the end. But, if there’s a clear top 3 gifts out there and I’m the last picker, I know I can land one of them just by picking the best gift out there and letting the churn begin because it should end up in a triangle pattern of exchange that I’ve inserted myself into. Eventually, one of the other two will bow out with the best gift (as they’ve likely stolen it at least once before I did) and then another person and gift come into the triangle. I guess, even if the web of stealers expanded beyond 3 it wouldn’t affect my ability to get one of the top gifts because it creates more chances for me to steal the gift I’m interested in without someone else dropping out of the game with that gift.
So, if I’m the last guy picking, I’m taking the thing I like the most, unless they all stink. I would probably apply the same strategy if I was anywhere close to the end, knowing that I’ll at least build up some history with the gifts I’m interested in.
Now, who can tell me the real answer? I don’t want to get stuck with a ball and gag this year!
December 16, 2009 at 1:11 pm
jeff
yes i was assuming preferences were known. but a perfect information game does not require that the contents of the boxes are known. the hidden gifts just means we have to add moves by nature (which resolve the uncertainty about the contents of the box.)
December 16, 2009 at 2:03 pm
devilyouknow
If the game concludes with a non-recourse swap, I don’t see how it could be Pareto efficient. Come to think of it, every round ends with a non-recourse swap (i.e.–a non-recourse unwrapping). Every round in which the unwrapper is other than the last permissable player compounds the likelihood that the outcome is not Pareto efficient.