Wednesday, October 21, 2009

The Economics of Love (as told by T-Rex)

In today's Dinosaur Comics, T-Rex ponders the economics of romance. Specifically, he discusses the notion that there is an extremely high probability that someone exists in the world that is a better match for you than a mate you may currently be involved with. One of the reasons for this is that people tend to (but do not always) end up with those whom they have met through close proximity, i.e. friends they might have known since childhood, classmates, co-workers, people they might have met at a bar or party, etc. The problem is that if one is trying to maximize a happiness utility equation, then proximity serves to impose a pretty significant constraint on the optimization problem.

Were it an unconstrained equation, the maximum happiness would likely be greater. That is, there is a greater change of you finding an even better match--and hence deriving a greater level of satisfaction--if you were completely unrestrained. If you didn't have a job, responsibilities and such, and you could spend your entire life in pursuit of true love. And you would have a greater change of finding it.

At a more micro level, suppose your entire realm of existence constituted a particular bar filled with people. Further, let's say you spotted someone very attractive heading into the VIP area. You tried to go after that person, but the bouncer would not let you in under any circumstances. That is a constraint that's preventing you from meeting someone who very may well be the love or your life. Really, the greater area and time you can cover, the more of a chance you have of meeting this person.

Then, however, T-Rex considers another issue. He claims that if you've been with someone for long enough such that you've developed a considerable amount of history (and that this person makes you very happy), then even if you were to meet someone who you were more compatible with, it would not rationally make sense to trade up. According to Rex:

Let's say you're happy with Person A (95% perfect) and you meet Person B who is 99% perfect. It doesn't make sense for you to leave Person A for B if you've been with them for years! You lose out on all your shared history, and that's like a times two multiplier! [...] Romance with a times two multiplier!

The thing is that this argument depends on a little more than how long you have already been in a relationship. In fact, it makes some implicit assumptions. The first thing it assumes is that love/satisfaction is an constantly increasing function of time. Moreover, it seems to be an exponentially increasing function in this case. What T-Rex seems to be saying here is that as time goes by, you build a history and develop shared experiences with your significant other. These experiences give you pleasure as they occur, but moreover they continue to give you pleasure after they occur so long as you can share and relive them with that person. In a way, you're compounding these events. If you suddenly left this person, then you would no longer have someone to share these experiences with and hence they would lose some meaning. Consider an inside joke with your girlfriend or boyfriend. This joke would obviously not bring you the same satisfaction if you retold it to someone else. I believe this is why T-Rex refers to the cost of leaving Person A as "losing out on all your shared history."

Of course, depending on your view of romance, this might work differently. For instance, the "love/time" relationship might be more in shape of a logistic curve.

reprinted from wikipedia.org
First, the curve increases slowly (likely to reflect the start-up costs of any relationship: anxiety, learning new things about them, etc.), then increases by a larger amount (to reflect the comfort stage of a relationship) and then it levels off (when you start getting bored). With this curve, it is entirely possible that an individual reach a level in his relationship where new and shared experiences bring little to no additional pleasure. It is even possible for curves to start to decrease! It is at this point that people consider leaving to find new, potentially greater experiences.

For simplicity, let us assume that the love curve is increasing and exponential (i.e. increasing returns to scale). Then, you would intuitively assume that you would only want to leave Person A for Person B if your love function with B increased at a higher rate than with A. However, this is really only true if you have enough time left in your life span in order to realize these benefits.

This brings us to another crucial element of what T-Rex is saying. Since there is a cost associated with breaking up with Person A, then in the short-run you would lose out. However, since satisfaction is increasing with time, then it would make sense in the long-run to leave Person A given that you do have the time to compensate for the loss you experience. Consider the following graph that reflect this:

In this graph, Person A is the red line, Person B is the blue line, and Person C is the green line. Suppose that you meet Person B relatively early in your life span. As you can see, it is possible that in the long run, Person B's function would cross Person A's function (at age x) and attain a greater happiness in the long-run. However, suppose you meet person C. Despite the fact that person C's function is increasing more rapidly than Person A's and B's, the fact that you meet person C so late in the game does not leave you enough time to hit the same level as A or B (before your unfortunate death at the hands of the end of the graph).

Therefore, if we make the assumptions that we did in this argument, then T-Rex is certainly right about having an incentive to stay given that you do not have the time to reach that level of satisfaction with anyone else. However, if there is enough time, then just the opposite could be true.

Ah, the good ol' romance multiplier. Incidentally, I'm a relatively young guy. I keep trying to make this argument to people I break up with, and they only end up getting mad at me. What's the deal?

Update: This also varies based on whether you're trying to simply attain a higher level of happiness in the long-run or whether you're trying to achieve an overall greater total happiness throughout your life. If the latter, then before breaking up, you would want to make sure that you have enough time in your life to both attain a higher level of happiness than you would with Person A and sustain that level of happiness for a while until death.

39 comments:

Loren said...

In your last example, would you definitely leave person A for person B? While it may be true that after age x, you will be happier with B than with A, you will be less happy until that point. Does 15 years of being happier with B at the end of your life outweigh 30 years of being happier with A in the middle of your life?

I feel like a more accurate measure would be if you took the integral of the two curves from the point you meet B until death and then choose the larger one, so as to maximize total happiness.

ShadowBanker said...

Loren - See the update at the bottom.

Anonymous said...

"Integrate your Love Curve."

Is it me, or did I just say something dirty?

Dang, I knew a Math Degree would be good for something!!

Mike LeGower said...

Here's a potential complication: these are all romantic happiness curves, ceteris paribus. If you are married to A, you will also incur significant psychic costs from the break-up, both short-term (just the crappiness of breaking up) and long term (kids? visitation rights?). This is not to mention your likely financial costs of divorce. So if we're assuming that we're just trying to maximize romantic happiness, as long as romantic happiness isn't dependent on income or other types of happiness, I think this is solid.

repsilat said...

I may be misunderstanding a little here, but...

I'm pretty sure things change when you take "breaking up" into account. Specifically, if you're with Person A when you meet Person B and change mates your happiness integral doesn't reset to zero - the new blue line begins where the old red line leaves off, not at the x-axis.

By the graph shown, the rate of happiness increase from Person B is always greater than that from Person A (steeper line) meaning that regardless of how long you're going to live B is *always* worth the trade up (barring some extra cost-of-break-up.) "Catching up" would only make sense if the blue line was initially shallower than the red, but achieved a greater "top speed".

ShadowBanker said...

Anonymous - Oh so many math jokes can go along with this. A friend of mine actually attempted to write a math hip hop once, though I forget how it went, I remember it started "Concavity changes / I know my domain, you better check what your range is."

Mike - You're absolutely right. And the cost assumptions I make are admittedly lax.

repsilat - That's a good point, though the reason I had the overall happiness go down is because I do think there is a cost associated with breaking up. That is, T-Rex seems to think that a relationship involves some combination of compatibility and shared experiences. Once you break up with A, you no longer have that shared history, so you can't jump jump onto B's curve and start from the same level of happiness. Because you lose those experiences (and endure whatever other costs of breakup), your current level of happiness would go down if you leave for Person B.

Readly said...

I'd definitely go for greater total happiness. The area under the curves is where it's at (pun intended)!

Adam said...

Great story, but it may not be sufficient to explain the phenomena. The lovers here can only make decisions based on knowledge they have, and the actual slope of the romantic curve in a relationship isn't known in the beginning.

The actual romantic happiness slope isn't a perfect curve, but a number that jumps up and down wildly (though it trends upward, like the S&P or something).

So if you're trying to decide whether to switch partners, you also have to take probability into account. How good is the data I'm using to estimate the curve? Is it representative of what our actual shared moments will be?

Calatar said...

Perhaps the love equation of happiness would include some sinusoidal behavior to account for the ups and downs of a relationship. Keeping the same upward trend, and perhaps with an increasing period and a decreasing amplitude, to account for general acceptance of each other resulting in a more constant happiness.

Relationships probably start with cosine-like behavior, not starting from zero. We are often extraordinarily happy at the beginning of a relationship. People tend to exit relationships during the down cycle of the sinusoid, the local minima if you will. Particularly when another person's initial relationship happiness value looks promising (initial maximum on relationship 2 greater than one of the minima on relationship 1).

If you grant that every relationship has different constants, it explains why some relationships approach zero happiness, but start off well (due to a high initial value). This also explains why people make bad decisions regarding relationships, by relatively measuring temporary happiness benefits.

Anonymous said...

It seems like black widows would be the ones maximizing their satisfaction in this model, as long as they keep the transitions smooth.

seriousopinions said...

I suggest moving the line in Graph 1 to the right such that the negatives are not used. It is a mostly arbitrary move, but the graph moving into negatives suggest something else entirely.

If you have spent zero time with somebody, you have a non-relationship. How that can go any lower (how time can go into negatives, or how anything can be less than nothing) is beyond me.

On a scale of happiness increase, obviously negatives do not belong anywhere. The negatives are better represented by positives on a different spectrum, perhaps time spent away from someone rather than with them. Depending on the person, this can also increase happiness. In such a case it would be likely that spending time with the same person would decrease happiness.

If I were to interpret Graph 1 as a depiction of opposites, then you are asking me to see 0.0 X, 0.5 Y as the surrogate origin (co-existing with an original origin,for some reason), and the negative numbers as meta-representative of negative emotions. It is only in this way that I might make concessions.

Nevertheless, these graphs are adequate in representing a small set of fixed perceptions and their potential outcomes. In seriousness, if you have to look at a graph to decide whether you want to stay with Person A or go with Person B, they might as well be equal in romance since apparently your heart can't decide.

If I may make another suggestion, another post titled "The Economics of Relationships" would be neat, and the subject content would be the actions done in order to please the other(s) and the return in investment thereof (e.g. flowers and a dinner date in return for intercourse of some kind).

Sirrah said...

The question is, would you ever know the difference?

http://xkcd.com/584/

ShadowBanker said...

Adam - You're absolutely right, but I'm assuming certainty in my analysis. If we were to assume that the individual in question was uncertain of the rewards of the new relationship, this would be an exercise in expected utilities. Maybe a future post is in order.

Calatar - The problem is it's hard to generalize people's happiness during relationships. I was going for simplicity here. Indeed some might start happy then decline then rise again. For others, beginning stages could be rather awkward and it isn't until the comfort zone that true happiness enters.

Seriousopinions - Please keep making suggestions! We love hearing them. I think that's a great idea for a post.

Sirrah - Love that xkcd comic!

Anonymous said...

Perhaps to take into account the ups and downs of a relationship one should use geometric brownian motion instead of simple smooth curves (although I suppose in order to plot it you would have to take the mean and end up with a smooth curve anyway or you could do a monte carlo type effort and end up with a specific path)

Gotta keep things interesting

ShadowBanker said...

That would be a great way to describe the randomness of relationships. We should write a paper on the economics of romance. Want to join me, anonymous?

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By the graph shown, the rate of happiness increase from Person B is always greater than that from Person A (steeper line) meaning that regardless of how long you're going to live B is *always* worth the trade up (barring some extra cost-of-break-up.) "Catching up" would only make sense if the blue line was initially shallower than the red, but achieved a greater "top speed".

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