The Game Theory of Yankee Swap

A deep dive into the most mid holiday party game

It’s December, and that means it’s the season for dominating dumb holiday games.

Sure, there’s a lot of serious gaming going on right now. The WSOP is having its annual winter series in the Bahamas. Ditto the World Poker Tour, which is in Vegas right now for its championship. And every credible home card game in the country is running its annual club championships sometime in the next week; my Oh Hell club has its championship on Wednesday, and my home game poker tournament series championship is Thursday.

So did I do any preparation for those events over the weekend? Nope. I spent Sunday afternoon watching the pathetic Giants and working out the formal strategy for my family’s annual Christmas-night Yankee Swap.

The World of Gloriously Dumb Games

For those unfamiliar, Yankee Swap (or White Elephant, or Dirty Santa, or just a grab bag) is a popular gift-giving game played at lots of holiday gatherings. Everyone brings a wrapped present and puts it on the table. You then draw numbers to set a pick order. When it’s your turn to pick, you can either take a gift from the middle and unwrap it, or steal any gift that has already been unwrapped. If your gift gets stolen, you have the same options. Rinse and repeat until everyone has a gift.

Yankee Swap is a dumb game. And I mean that affectionately, like how Philip Seymour Hoffman’s Lester Bangs describes rock n’ roll.

Dumb games are not stupid games. Stupid games—like Tic-Tac-Toe or War—aren’t worth playing except to kill time. Dumb games, played by people with the right balance of too-much-competitiveness, little or no self-consciousness, and laugh-while-you-play absurdity, can be utterly transcendent. A frantic energy in an all-consuming artificial world, as you chase a victory that means literally nothing and everything. All while you can’t stop laughing.

Musical Chairs is the classic of the genre. Pit has stood the test of time as one of the best commercially-sold dumb games, but Hungry Hungry Hippos is best-in-class (though Drone Home is great too). The dumb card game Gabes is the single one I’m most nostalgic for, having played it endlessly for an entire summer with my neighborhood friends when we were 14.

But the greatest of the dumb games is Talking Rule Spoons—and don’t worry, you’re going to get 3,000 words on that later this week. You really haven’t lived until you’ve seen your quiet, unassuming neighbor dive across your kitchen counter in her Christmas sweater to try to rip a spoon out of a child’s hand, all while everyone—including the neighbor and the child—is losing it laughing.

Seconds before the beginning of a 13-player Talking Rule Spoons war at our Christmas party earlier this month.

As far as dumb games go, Yankee Swap is decidedly mid. And that’s being generous. It’s not frantically-paced like so many of the great dumb games. Too many normies play it who are too shy about stealing a great gift, and who would never do it purely for the spite and the laughs. You can get stuck with a bummer gift that no one will ever steal and basically be out of the game. It takes a bit too long. And the stakes are a little bit too high; you are playing for actual prizes rather than just the bragging rights of crushing your friends and family at a dumb game.

The Useless Game Theory of Yankee Swap

My family is very into games, both serious and dumb. We also love deconstructing the strategies, especially for the dumb games. I mean, who doesn’t get into hour-long arguments on Christmas evening about proper Hungry Hungry Hippos tactics?

Naturally, our Yankee Swap is pretty cutthroat. In the gloriously dumb way. No one thinks twice about ripping the best gift out of grandma’s hands. Everyone cheers raucously when someone does it. People angle strategically for the best gifts, and generally are thoughtful about strategy. Lots of hooting and hollering.

If you look around the internet, you can get a lot of advice on how to “win” at Yankee Swap, but the problem is that the advice is almost always too generic. Yankee Swap strategy is very sensitive to (1) the rules you are playing under; (2) the distribution of value of the prizes; (3) the relative preferences of you and your competitors; and (4) the degree to which other players will be ruthless EV maximizers and/or make plays out of spite.

So while it’s fun to look at the game theory of a Yankee Swap, it’s not actually all that helpful beyond setting up the basic principles of strategic play, which aren’t that hard to derive without game theory.¹

But we’re still going to (briefly) do it. Here’s the plan: first, I’m going to lay out the rules for my family’s Yankee Swap. Then we’ll walk through the game theory of it under constraints that make the game solvable, in order to understand some of the dynamics. Then we’ll relax the assumptions and talk about strategy for the real-world of my mom’s living room on Christmas night.

One Decent Way to Play Yankee Swap

Here are the Yankee Swap rules my family uses:

1. Everyone puts a wrapped gift on the table, so the number of gifts equal the number of players. This is also the number of rounds that will take place. So ten people, ten gifts, ten rounds.

2. Each player draws a number out of a hat. That is the pick order. Each number corresponds to a round.

3. The person with number 1 goes first and picks a gift to unwrap. Anytime a gift is unwrapped, that is the end of the round.

4. The person with number 2 now starts the second round. They may either steal the gift from player 1, or unwrap a gift. If they unwrap a gift, round two ends and player 3 begins a new round. if they steal a gift, the player who had the gift stolen may unwrap a gift and end the round, or may steal another gift.

5. No gift may be stolen twice in the same round. Once a gift is unwrapped and a new round begins, all gifts are again eligible to be stolen.

6. The game ends when the last gift is unwrapped. If you are playing with 10 people, this will be at the end of the 10th round. When the last gift is unwrapped, everyone gets the gift they currently hold.

These are not the only reasonable rules for Yankee Swap, but I think they are the best. The main thing to avoid is gift-stealing cycles. You can’t have Alice and Bob and Chris just stealing the same gift in a circle until someone becomes shamed and gives up. That’s stupid. Some people cap the total number of times a gift can be stolen, but that artificially limits strategy and takes gifts out of play early. The above rules mean every gift is in play every round until the game ends, but limits the endless steals of the same gift within rounds. It’s a solid system.

Of course, the steal rules also form the basis of all Yankee Swap strategy.

Solving My Family’s Yankee Swap, Under Constraints

We have to add two constraints in order to make this game solvable.

1. The prizes must all be fully known; and

2. All players must have identical preference orderings.

The easiest way to do this is model the prizes as money. So, for example if you have five players, make the gifts $5, $4, $3, $2, $1.

Now we can start looking at some games. First, the trivial game with two players, Alice and Bob. We have gifts of $2 and $1. In the random draw, Alice has drawn round 1 and Bob has drawn round 2.

Alice goes first and opens a gift. Round 1 ends. Now Bob can steal the gift or open the unwrapped one. The strategy is trivial—Bob steals if Alice has opened $2, but opens the unwrapped gift if Alice has opened $1. Since Alice cannot steal the $2 back—remember, gifts can only be stolen once per round—she is forced to unwrap the $1, which ends the game.

So the expected value of this game for Alice is $1 and for Bob is $2. The strategic insight is obvious but important: if you go last under these rules, you will always get the best gift. If the best gift is already unwrapped, you will steal it and the game will end before anyone can steal it back; if the best gift is not unwrapped, you will unwrap it and the game will end. Going last guarantees the best prize.

Now consider the three player game, the first non-trivial one. Alice, Bob, and Chris have drawn numbers 1, 2, and 3, respectively. There are wrapped prizes of $1, $2, and $3. What are their expected values for playing the game?

Well, we know Chris is always going to get the $3. He goes last, so he will always steal it or unwrap it. His expected value for the game is therefore $3. What about the other two? This can easily be worked out by common-sense logic:

Alice goes first. There are three scenarios, which each happen 1/3 of the time:

1. If Alice unwraps the $1, no one will ever steal it from her. Bob is going to unwrap, since all the wrapped gifts are better than $1. So she is stuck with $1 and Bob and Chris and essentially playing the 2-player game with $2 and $3 prizes. Whatever Bob draws, Chris will end up with the $3, either by stealing or unwrapping. [Result: Chris $3, Bob $2, Alice $1]

2. If Alice unwraps the $3, Bob will steal if from her, and Alice will then be forced to unwrap. If she unwraps the $1, Chris will steal the $3 from Bob, and Bob will unwrap the $2, ending the game. If she unwraps the $2, Chris will steal the $3 from Bob, Bob will steal the $2 from Alice, an Alice will unwrap the $1, ending the game. [Result: Chris $3, Bob $2, Alice $1]

3. If Alice unwraps the $2, Bob has the first real choice of the game. He can either steal the $2, or he can unwrap, getting either $1 or $3.

   1. If Bob steals the $2, Alice unwraps getting either $1 or $3. If she unwraps $1, Chris unwraps the $3 and ends the game. If she unwraps the $3, then Chris steal the $3, and Alice gets to steal the $2 from Bob, who then has to unwrap $1 and end the game.

   2. If Bob unwraps, he either gets $1 or $3. If he unwraps $3, Chris steals it, Bob then steals the $2 from Alice, and Alice unwraps $1 and the game ends. If Bob unwraps $1, Chris unwraps $3 and the game ends.

   This means Bob stealing or unwrapping is equivalent. In either case, he gets $2 half the time and $1 half the time. [Result: Chris $3, Bob $2 half the time and $1 half the time, Alice $2 half the time and $1 half the time]

Now this is interesting. If Alice unwraps $1 on her first turn, she is obviously stuck with it. If she unwraps $3 on her first turn, she also ends up with $1. But if she unwraps $2 on her first turn, somehow she sometimes gets to keep it. What is going on here?

There are two strategic insights: first, if you get to pick second in the last round, you get the second best prize. Everyone knows that Chris is going to start the last round by getting the $3. So if you can get the $3 in round before that, you know that Chris will steal if from you in the last round, and then you can get the $2. That’s why it does Alice no good to unwrap the $3. Bob can just steal if from her, knowing that Chris will steal it from him in the last round, and that he will then be able to either steal or unwrap the $2.

Second insight: if Chris unwraps to start the last round, Alice and Bob get stuck with whatever gifts they are currently holding. That is, there is no chance for Bob to steal, since Chris unwrapping the $3 ends the game.

And this is why Alice does better when she draws the $2; it sets up two parallel situations, both of which can help her: if Bob steals her $2, she has a chance to unwrap the $3 and therefore get the second pick in the last round after Chris steals it from her; or, if Bob doesn’t steal, he has a chance to draw the $1, and Chris will end the game by unwrapping the $3.

Here’s a table showing the outcomes and expected values of the 3-person game:

But again, the specific outcomes are less important than some of the strategic implications:

• If you go last in this game, you always get the best prize;

• If you hold the best prize going into the last round, you get the second best prize;

• If you unwrap the worst prize, you will always be stuck with it;

• If you unwrap a great prize early, it doesn’t help you much.

I won’t bother going through the details, because it gets too complicated to do by common sense, but here are the outcome and EV tables for the 4-player game:

Let’s consider the strategy for Chris. He gets to go first in round 3. He knows that Dan is going to get the $4 gift in round 4. So why is Chris’ EV less than $3? Why can’t he just grab the $4 gift in round 3, ensuring that Dan will take it from him, and he can then steal or open the $3 gift in the last round?

Because the $4 is only available to steal half the time. The other half of the time, when it is Chris’ turn to go at the start of round 3, the $4 gift is still wrapped. And unless the two unwrapped gifts Bob and Alice have are the $1 and $2 gifts (leaving the $3 and $4 gifts wrapped), it cannot help him to unwrap. He is always no worse off (if the wrapped gifts are $2 and $4) or better off (if the wrapped gifts are $4 and $1) stealing the $3 gift from Alice or Bob.

When he does steal the $3 gift, he hopes that the $4 gift remains unwrapped, so Dan will unwrap it and he can keep the $3. When Alice or Bob unwraps the $4, they now hold the best prize going into the last round, and will be able to steal the $3 from Chris, which will leave him to steal the $2.

One strategic upshot of this is that stealing is usually as good or better than unwrapping. Unwrapping great gifts doesn’t let you keep them, and unwrapping the worst gifts leaves you stuck with them.

Notice also how important the random draw for pick order is under these rules. You usually end up with the prize that corresponds to your pick. Any strategic advantage you can get from smart play will be within the constraint that your pick number controls a lot of your destiny.

Relaxing the Constraints: Basic Real-World Strategy

When you get to the real world Yankee Swap at my family’s house later this month, you run into three realities that complicate the basic game theory:

• No one knows what the gifts are, or how good they are. This applies both globally and during the game. The broad dictate in our Yankee Swap is to put in something worth $20-$50, but that’s a very loose constraint. And you do get about 1/4 of the entries being complete gag gifts. So the ex ante distribution of value isn’t known. But neither is the value of the remaining unwrapped gifts at any point in play; in the toy games above, when there is one gift left, all the players know its exact value. In my mom’s living room, no one knows whether it’s a sweet set of cutting boards or a gag DVD box set from the 90s. The game plays very differently if the distribution of prizes is 20% great, 50% ok, and 30% gag garbage than if it’s 10%, 10%, and 80% across the same dimensions. You can make estimates about your relatives, but you can’t know for sure.

• Everyone has different preference orderings for the gifts. In the toy games, we make the preferences uniform among the players by converting everything to money. But there’s no way to solve for an equilibrium with Alice, Bob, and Chris if we are looking at a toaster, a phone case, and sweater, unless we know their preferences. And even if we can correctly guess their ordinal preferences, we still don’t know the relative utilities.

Luckily, both of these realities imply strategic actions that you can use to improve the chances you get gifts higher up on your preference list than you should based simply on the random draw that assigns you your round number.

The key strategic insight:

• The maximizing strategy is the one that will allow you to go earliest in the last round, since that allows you to both see all the gifts and get a gift that can never again be stolen from you. Obviously, whoever has the last number is going to go first in that round, and they are going to get whatever gift that want, except for the one remaining wrapped gift, which they cannot risk opening, as it may be a gag gift. If you have the last number, congrats, you won by sheer luck. If you do not have the last number, your basic strategy revolves around figuring out how to pick second in the last round, which comes down to figuring out how to get the gift that the person with the last number wants most. Essentially, you want them to steal from you first in the last round.

• The good news is that you know who is going to pick last right from the start! If it’s Aunt Mary, then you know you should be aiming to be holding the gift that is Mary’s first preference as you head into the last round. So long as Mary steals that from you, you will get your top pick, excluding the remaining wrapped gift and excluding the gift Mary just stole from you.

• The bad news is twofold: first, even if you know Mary’s preferences perfectly, you won’t know what Mary’s preferred gift is until the round immediately beforehand, when you will see all but one of the gifts Mary can choose from. Second, even if you do spot something early that you know Mary will put top value, you can’t just grab it early. Someone else might take it from you. Your safest path is to acquire the gift that Mary wants in the round immediately before she picks, such that no one else can take it from you.

If you’ve ever thought about something recursively, you can probably see where this is going. The clear winning strategy under my family’s Yankee Swap rules is to continually steal the gift that the person who goes first in the next round wants the most. That way, you will always go second in each round, and can continue to steal the most desired gift of the next person to go, so long as it’s not the same most-desired gift as the person who just stole it from you.

Indeed, a “perfect game” in our Yankee swap would be something like having the 6th pick, stealing the visible gift the person who picks 7th likes best, then stealing the visible gift the person who picks 8th likes best, and so on, until you steal the gift the last person likes best, they steal it from you on their turn, and then you pick whatever you want from the rest of the gifts.

Someday, god willing, I will do this.

There are complications.

One was mentioned above: if two people in a row have the same top preference, you will not be able to get it, because it will just have stolen from you.

But the most important complication is that if anyone else catches on to what you are doing or figures out the strategy, they can start doing it and preempting you. The person with pick 8 doesn’t have to steal from you the gift you know they like best, they can steal the gift they know the person with pick 9 likes best! Then they get to second next round…unless the person with pick 10 also catches on! This strategy is not in equilibrium. It’s exploitative, and relies on other people making honest picks instead of strategic picks.

Setting up a Yankee Swap, Loudonville NY circa Christmas 2018

And that gets to the general heart of the strategy. You need to be stealing things that other people prefer, rather than things that you want. Your whole goal is to stay in the game and maximize how early you get to pick in the last round. You can only do this by holding a popular item. You should absolutely steal something that you don’t really like but you know everyone else loves.

And for the love of god, you really need to avoid unwrapping. There’s a lot of advice about Yankee Swap along the lines of calculating average values and such, but the bottom line is that you need to be very risk averse with unwrapping, because there’s a huge negative asymmetry in unwrapping a bad gift.

If you unwrap a gag gift, you are stuck with it. No one is ever stealing from you. And if you unwrap an ok gift, you are going to be way down the steal chain in any given round, which means you may be forced to unwrap again, rather than steal one of the opened gag gift and lock yourself into garbage. And every time you are forced to unwrap, you risk locking yourself into a gift no one else wants.

So set aside your preferences and aim to always be holding a gift that is popular and will likely be stolen early in the round. That will allow you to steal another popular gift and keep you early in the order in subsequent rounds. That advice alone will do you well. Steal good stuff, even if you don’t like it.

To really maximize your EV, take into account the preferences of the early pickers in the future rounds, both the person holding the number that goes next and the people likely to be stolen from early on. When it is your turn to steal, you need to consider the preferences of the next person in the pick order, and then the preferences of the person they are likely to steal from first. Those are your targets.

Unwrapping Strategy

What if you have one of the very few first numbers? You are almost certainly going to have to unwrap at some point, because it’s going to be your turn and every visible item will either already have been stolen that round, or will be a useless gag gift that takes you out of the game.

There’s obviously no substitute for being a complete luckbox and landing one (or more) killer early unwraps that you are forced to take. That can also draw some great resentment from the crowd.

But it’s important to remember that unwrapping isn’t random. There’s not a great signal from the size of the gift or whether its a box or bag, but the wrapping paper can give you a clue as to who brought the gift, and that is incredibly valuable information.² As much as knowing peoples’ picking preferences can help you make steal decisions, knowing their Yankee Swap purchasing decisions might be worth more.

First and foremost, if you can identify someone who would never put a gag gift into the mix, that’s your go to if you are forced to unwrap. Guarantees you something of value. After that, you are looking for the big spenders. This is why it’s hard to grab stuff my sister put in; she has a really polarized Yankee Swap Buying Range. Her good stuff is easily going to be the best stuff in there, and will set you up nicely for strategic play. But she also goes gag maybe 25% of the time. ³

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Of course, all of this assumes that none of my relatives read this article and take it to heart. Not an unreasonable assumption. But then again, so is the assumption that your quiet neighbor won’t launch herself across the table for a spoon.

Enjoy all the dumb games this month!

1. There’s no equilibrium for pretty much any real-world Yankee Swap, and people are making so many strategic mistakes anyway that you’d want to play an exploitative strategy even if there was a GTO one available.

2. It’s not cheating per se to watch who brings what gift and puts it on the table, but I think it’s outside the sprit of the game. It’s like a soft angleshoot in poker.

3. The poker players will also enjoy that there are blockers in Yankee Swap, in the form of the gift you wrapped. Someone can be drawing dead from the middle, but only you know it.