Textbook game theory prescribes players try to maximize selfish payouts. This assumption can accurately depict business negotiations, sports strategy, war games, a wide range of animal behavior, and even the emergence of cooperation. But, critics point out, the dismal science has it wrong. Humans are often altruistic and game theory overlooks the power of true love. Or does it?

Perfect love only exists in a perfect world. Lovey-dovey couples on Valentines Day proclaim they care so much about each other. But the gifts couples exchange reveal a secret self-interest, which is why people primarily give chocolates, flowers, and go out to dinner. That is, people who say they are not self-interested unknowingly corroborate the model.

But perhaps it is a coincidence? Might people who are truly in love end up in similar outcomes as people who are self-interested? This is a question we can analyze using game theory, only we have to modify what each person is trying to maximize. We can then analyze the Nash equilibrium of each game and see how each people would act.

As we’ll show, true love is not as powerful as songs and poems would have you believe. People in true love would encounter many of the same dilemmas that are studied in textbook game theory. Let’s see why.

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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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The TrueLove Utility Mapping

Let’s imagine that a standard game has players 1 and 2 getting the payoffs a and b. The standard model has that each person cares only about their individual payoff. That is, player 1 cares only about getting a, and player 2 cares only about getting b. Neither cares about what the other gets; and each is trying to make their own number as large as possible.

We can model TrueLove as a situation where a player’s happiness depends solely on the other person. In other words, the TrueLove function is a mapping that switches the payoffs, as follows:

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

Now player 1 cares only about the amount b, which is what is awarded to player 2, and player 2 cares only about the amount a that is given to player 1.

How does the TrueLove mapping affect the Nash equilibrium of common games?

(Credit: I read about the TrueLove mapping in Game Theory and Applications, Volume 8).

Staying On Budget

You and your partner both want to treat each other to expensive gifts. However, your household cannot afford if you both get expensive gifts for each other, but it would be okay if only one of you splurges. You both agree in principle to a spending limit, but there is nothing holding either of you from secretly splurging on a nice gift.

Let’s consider the situation as a game. Each of you has a choice of staying on budget or splurging. If you both stay on budget, then you both feel fiscally responsible and earn a payout of 2 points each. If only one person splurges, then the gift receiver is ecstatic at the lavish present and gets 3 points, while the person who splurged is okay with the boring gift (which was within budget) and gets 0 points. If you both splurge, then you are both very happy with the gifts, but you both feel guilty the budget is broken, leading to a payout of 1. That is, you are both less happy than if you had stayed on budget, which is overall more important than having nice gifts that you cannot afford.

How does this game play out? We will analyze it using my free online game theory solver. One player is called Rose, who chooses a row of the matrix, and the other player is Colin, who chooses a column of the matrix.

Let’s denote the choice of splurging as Up for Rose and Left for Colin, and the choice of staying within budget as Down for Rose and Right for Colin. (We’ll do a similar re-labeling of strategies for the rest of the games in this post). Then we can input the payouts for the various combinations, and the solver will calculate the Nash equilibria in pure and mixed strategies.

There is a single Nash equilibrium of Rose choosing Down and Colin choosing Right, which corresponds to both people being responsible and staying in budget. Each person gets a payoff of 2.

In fact, it is a dominant strategy for each person to stay within the budget. The logic is simple: if your partner splurges, then you should stay on budget–to get the expensive gift while the household still stays within budget; and if your partner stays within the budget, then you should also stay within the spending limit to be responsible.

A game with this payout structure is more generally known as a game of deadlock. The thing is, this game is probably not even mentioned in any of the texts that I consider the best game theory books. Why not?

Well, the game is just boring! Each player has a dominant strategy to cooperate. There is simply no dilemma because both people want to do the right thing and they do. There is nothing to analyze and so game theory does not dwell on studying these types of games.

But let’s consider the game from a different perspective. How might TrueLove change the game? Now each person cares so much about the other person being happy that each person’s payoff is the point total of the other person. This means the payoffs in the game are transposed. What happens now? Let’s use the game theory solver to find out.

There is a unique Nash equilibrium of Up for Rose and Left for Colin. That is, both people splurge on the gifts! They both end up with a payout of 1, and they are both worse off than if they had just followed the agreement to stay within budget to get a payout of 2! What has happened is that each person cares so much about the other person that each person would rather splurge to make the other person happy even with the risk of blowing the budget. In other words, each person has a dominant strategy to splurge, and therefore the household budget gets broken for sure.

This is a situation that many couples experience, and it’s precisely each person’s love for the other person that ruins the situation for both of them. Does the game look familiar to you? The new game is a prisoner’s dilemma! That is, the TrueLove mapping has transformed a boring game of pure coordination into a tricky dilemma where each person is tempted not to cooperate.

This game alone demonstrates that true love cannot conquer game theory, as the TrueLove mapping makes otherwise self-interested rational people do crazy things to mutual detriment.

Furthermore, if even one person was selfish and not splurged on the other, then the household would at least have stayed within budget. So there is a case to be made that selfishness can be mutually beneficial.

Let’s continue to analyze how the TrueLove mapping influences the outcome in other commonly analyzed games.

Battle of Sexes

In the very first episode of I Love Lucy, Fred and Ethel are celebrating their 18th wedding anniversary. Ethel wants to celebrate by going out to a nightclub. Fred wants to go out and watch a fight. While they disagree about it, both would prefer to be with the other than go out alone. Suppose each gets 2 from the preferred choice, 1 from the partner’s preferred choice, and 0 from going out without the partner.

We will re-label the choices so fight = Up/Left and nightclub = Down/Right. Here is the game matrix and the Nash equilibria, as calculated by my free online game theory solver.

There are 3 Nash equilibria. They can both go to one choice or the other, or they can play a mixed strategy between the choices. (In I Love Lucy, they ultimately both go to the Copa Cabana nightclub).

How does TrueLove change the game? Now each person prefers to go to the partner’s choice. Fred would rather go to the nightclub, and Ethel would rather go to the fight. The payoff entries in each cell become transposed.

Surprisingly, the game has almost exactly the same equilibria!

Now instead of arguing for what each person wants to do the game is transformed into each fighting to do what the other person wants! Once again, TrueLove does not solve the dilemma! And if one person was selfish, then the couple could amicably go to that person’s choice without disagreement.

Stag Hunt

Two hunters can either seek out a stag or a hare. If they both go for the stag, they can capture it and each gets a payout of 3. But if only one person goes for the stag, then that person cannot capture it and goes home with no food (0). Anyone that hunts a hare can get it for sure, and the smaller prey means a smaller payout of 1.

Here is the game inputted into the solver, with a re-labeling of strategies as Up/Left = stag and Down/Right = hare.

Again there are 3 possible Nash equilibria. Both can hunt the stag, both can hunt the hare, or they can play a mixed strategy. The game is a model of cooperation: if both can agree to hunt the stag, they are both better off. But each is tempted to go alone and guarantee some food with the hare.

How does TrueLove affect the game? Here’s the game after the TrueLove mapping.

Only the mixed strategy equilibrium is eliminated. Now it seems reasonable that they would cooperate since cooperating is a weakly dominant strategy. So TrueLove would seem to improve the outcome if we could have empathy for the other side. Nonetheless, it does not rule out getting stuck in a trap where each person would still selfishly go for the hare–and this is assuming each party has true love for the other person!

Game of Chicken

Two teenagers are driving towards each other in a test of guts. If both swerve, then everyone mostly forgets (6 to each). If only one swerves out of the way, that person is a chicken (2) and the other person wins everyone’s respect (7). But if neither swerves, then both end up severely injured (0 to each).

In the standard game, each person is hoping the other swerves out of the way. Here is the game in the solver, with the re-labeling of strategies as Up/Left = don’t swerve and Down/Right = swerve.

There are 3 Nash equilibria: either only one person swerves out of the way, or they play a mixed strategy.

The TrueLove transformation has the following effect on the payouts.

Now each person has a dominant strategy of swerving, and there is a unique equilibrium of both parties being chicken. This is the first of the games where true love influences the outcome in an undeniably positive way.

The interesting part is this: if one person is playing according to true love, then that person is going to swerve. So it would be a best response of someone who is selfish not to swerve! So you can see how the dynamic might play out in a relationship where one person truly cares about the other person and the other party is happy to play selfish.

Prisoner’s Dilemma

Two prisoners are held on a crime without physical evidence. They are separated and an officer wants to get a confession. If each person stands pat, then they will both serve a minimal 1 year in jail. If only one person confesses, that person goes free, and the other person gets 3 years in jail as a penalty. If they both confess, they will pay the price and serve 2 years each.

If they could both trust the other person, they would stand pat and only serve 1 year each. Suppose we label the strategies as Up/Left = stand pat and Down/Right = confess.

Each player is better off confessing no matter what the other person does, and so they end up where both players confess and serve 2 years each.

The story seems like a tragedy of selfishness: if they could only do what’s best for the group, instead of what’s best for the individual, they could end up in a mutually beneficial outcome. But how would true love change the game?

Now true love does conquer the game, and not confessing is a dominant strategy.

This would seem to be a win for true love. However, that is not entirely the case! While true love can win in games of prisoner’s dilemmas, it will also transform some games of pure cooperation into prisoner’s dilemmas, as explained in the first example about staying in budget for gift-giving.

(Another example is O. Henry’s tale about two lovers who are deciding whether to make an individual sacrifice to get a gift for the other. The ironic ending, as I explained in this post, is an example of how true love actually ruins the obvious choice.)

So while true love solves actual prisoner’s dilemmas, it also creates prisoner’s dilemmas where none existed before. So it’s not obviously clear that true love is a net benefit.

Matching Pennies

Two players show “heads” or “tails.” Player 1 gets a 1 point if they match; player 2 gets 1 point if they do not match.

Clearly the game is about randomizing the choices so the opponent cannot outguess what you are doing. Let’s analyze the game, with a re-labeling of strategies as Up/Left = heads and Down/Right = tails.

Matching pennies is an example of a game that has no pure strategy Nash equilibrium and a unique mixed strategy.

How would true love change the game?

The equilibrium is exactly the same, which makes sense because now player 1 wins if they do not match and player 2 wins if they do match. So the two players have exchanged roles and therefore the game is exactly the same.

In Summary: True Love is Not Much Different!

As shown above, TrueLove cannot overcome the inherent conflict in these games. In the battle of the sexes and the stag hunt game, TrueLove does not significantly change the equilibrium outcomes. In matching pennies, true love does absolutely nothing.

The prisoner’s dilemma and game of chicken are more interesting. The TrueLove mapping does materially improve the outcome. But notice: this is only if both players follow it! Should one player know the other is playing by true love, then a selfish player could easily capitalize and prey on this behavior, and so TrueLove is not an evolutionarily stable solution.

Furthermore, true love happens to create some prisoner’s dilemmas in games which were previously games of cooperation, as in the example of staying in budget. TrueLove makes otherwise self-interested rational people pursue actions for mutual destruction. So we cannot categorically say that true love conquers all. In fact, true love exchanges one problem for another, exchanging textbook prisoners’ dilemmas for textbook games of pure cooperation.

The conclusion is that true love cannot overcome game theory, no matter what people who have not studied game theory claim.

The truth about Valentine’s Day is that true love is not much different from self-interest. For the first time in American history, there are more single adults than there are married adults. But somehow I don’t think Hallmark is going to print either message any time soon.