Short Announcement: This is basically my last post for the year. I might write a few “best of the year posts.” Also In January, I plan to mostly write Monday puzzles and Tuesday game theory (so no Weds – Fri posts) in order to work on other projects. This was a great year for the blog–thanks, and hope you have a happy New Year!

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While researching Christmas-related game theory topics, I stumbled into a stunning example in the book Game Theory and Applications, Volume 8.

The setting is a short-story from the American author O. Henry published in 1906. In The Gift of the Magi, a young husband and wife couple want to give each other the best Christmas gifts. The husband wishes to buy a set of combs for his wife’s lovely, long hair, and the wife wishes to buy a watch chain for her husband’s pocket watch. But neither has enough money to buy the gift for the other person.

How should this game play out, and how does true love affect the outcome?

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"All will be well if you use your mind for your decisions, and mind only your decisions." Since 2007, I have devoted my life to sharing the joy of game theory and mathematics. MindYourDecisions now has over 1,000 free articles with no ads thanks to community support! Help out and get early access to posts with a pledge on Patreon. .

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Selfish lovers

Let’s take a step back and consider a different situation. Imagine two lovers who are in a short-term relationship and could be described as “selfish”: neither cares about creating a long-term relationship and each hopes to profit in the short-term. How might a lover view the optional act of gift-giving?

Well, each person is likely to think strategically. That is, each person is going to consider the best thing to do in response to what the other person might do.

So a strategic, selfish lover might think as follows. “I have to decide whether to spend time and money for a gift. Now if my significant other does not give me a gift, I’m clearly better off not wasting my money in giving a gift. On the other hand, if my significant other does give me a gift…well, that’s good for me, and I am still better off not giving a gift!”

Now both lovers are likely to think this way, and so neither ends up giving a gift. This equilibrium would not suit most people, and it in fact would scare people that their gifts might not be reciprocated.

One objection is that the lovers were selfish, and they only cared about their personal gain. Relationship advisors often point out that a loving marriage cannot be selfish, it has to be self-less, with each partner really caring about the other person.

So this brings us to the next point: how does the game play out under the assumption of perfect love?

True love does not conquer all!

Let’s return to the O. Henry story. The wife Della wishes to buy a watch chain for her husband Jim. And Jim wishes to buy combs for his wife.

Each wishes to buy a nice present, but neither has enough money. In fact, the only thing of value Della has is her hair, and the only thing of value Jim has is his pocket watch. In the story, each realizes that the item of value can be pawned in order to purchase the gift for the spouse. How might each view the game?

If Della and Jim were selfish, then the payoffs of the game would be as follows.

–Both withhold (W) gifts. Each person retains the personal item of value, and maintains the status quo of 0. –One withholds (W), another gives (G). The person who gave the gift has sacrificed and gets nothing in return, for a payout of -2. The person who withholds the gift has received something of value and lost nothing, netting a payout of 1. –Both give (G) gifts. They both sacrifice their items of value, and so the wife ends up with combs but she has sold her hair, and the husband ends up with a watch chain but he has sold his watch. They each get a payout of -1.

Here is the game matrix.

If Della and Jim play this game, each can reason like the selfish lovers example that withholding a gift is the best strategy. In fact, not giving a gift is a dominant strategy, and they end up in the best outcome.

But you see, there is a hitch! The payoffs in the game are presented in terms of a selfish couple. In fact, we already explained that Della and Jim are in true love. They care so much about each other that their utility is not based on personal well-being. They in fact value their utility according to how the other person feels.

Mathematically, if we write the utility as the ordered pair (player 1, player 2) = (a, b), then the true love funciton is a mapping that switches the payoffs:

TrueLove(a, b) = (b, a)

In other words, Jim’s payoff is how well Della feels, and Della’s payoff is how well Jim feels.

How does the game play out now? We will re-write the matrix after applying the TrueLove transformation to the payoffs.

How does the game play out now? Look at the payoffs closely–this is the Prisoner’s dilemma! Della and Jim both find it is a dominant strategy to give a gift. And this is precisely what happens in the O. Henry story: Della sells her hair to buy a watch chain, and Jim sells his watch to buy combs for his wife. They both sacrifice and end up in the worst outcome!

The O. Henry story concludes with a moral that the couple realized their mutual love for each other, and that was a gift more important than the unwise sacrifices they made. This is pretty much the statement that gifts are an act of costly signaling.

Nevertheless, the fact that the couple preferred the equilibrium should not obscure the fact that that ended up making a bad choice, trapped by the wrong incentives. So I prefer to remember the moral presented in the game theory text: “True Love gives no protection from the Prisoners’ Dilemma.”