market

Marriage and matching

pnrj mathematical models, personal , , , , , , ,

Oct 10 JDN 2459498

When this post goes live, I will be married. We already had a long engagement, but it was made even longer by the pandemic: We originally planned to be married in October 2020, but then rescheduled for October 2021. Back then, we naively thought that the pandemic would be under control by now and we could have a wedding without COVID testing and masks. As it turns out, all we really accomplished was having a wedding where everyone is vaccinated—and the venue still required testing and masks. Still, it should at least be safer than it was last year, because everyone is vaccinated.

Since marriage is on my mind, I thought I would at least say a few things about the behavioral economics of marriage.

Now when I say the “economics of marriage” you likely have in mind things like tax laws that advantage (or disadvantage) marriage at different incomes, or the efficiency gains from living together that allow you to save money relative to each having your own place. That isn’t what I’m interested in.

What I want to talk about today is something a bit less economic, but more directly about marriage: the matching process by which one finds a spouse.

Economists would refer to marriage as a matching market. Unlike a conventional market where you can buy and sell arbitrary quantities, marriage is (usually; polygamy notwithstanding) a one-to-one arrangement. And unlike even the job market (which is also a one-to-one matching market), marriage usually doesn’t involve direct monetary payments (though in cultures with dowries it arguably does).

The usual model of a matching market has two separate pools: Employers and employees, for example. Typical heteronormative analyses of marriage have done likewise, separating men and women into different pools. But it turns out that sometimes men marry men and women marry women.

So what happens to our matching theory if we allow the pools to overlap?

I think the most sensible way to do it, actually, is to have only one pool: people who want to get married. Then, the way we capture the fact that most—but not all—men only want to marry women, and most—but not all—women only want to marry men is through the utililty function: Heterosexuals are simply those for whom a same-sex match would have very low utility. This would actually mean modeling marriage as a form of the stable roommates problem. (Oh my god, they were roommates!)

The stable roommates problem actually turns out to be harder than the conventional (heteronormative) stable marriage problem; in fact, while the hetero marriage problem (as I’ll henceforth call it) guarantees at least one stable matching for any preference ordering, the queer marriage problem can fail to have any stable solutions. While the hetero marriage problem ensures that everyone will eventually be matched to someone (if the number of men is equal to the number of women), sadly, the queer marriage problem can result in some people being forever rejected and forever alone. (There. Now you can blame the gays for ruining something: We ruined marriage matching.)

The queer marriage problem is actually more general than the hetero marriage problem: The hetero marriage problem is just the queer marriage problem with a particular utility function that assigns everyone strictly gendered preferences.

The best known algorithm for the queer marriage problem is an extension of the standard Gale-Shapley algorithm for the hetero marriage problem, with the same O(n^2) complexity in theory but a considerably more complicated implementation in practice. Honestly, while I can clearly grok the standard algorithm well enough to explain it to someone, I’m not sure I completely follow this one.

Then again, maybe preference orderings aren’t such a great approach after all. There has been a movement in economics toward what is called ordinal utility, where we speak only of preference orderings: You can like A more than B, but there’s no way to say how much more. But I for one am much more inclined toward cardinal utility, where differences have magnitudes: I like Coke more than Pepsi, and I like getting massaged more than being stabbed—and the difference between Coke and Pepsi is a lot smaller than the difference between getting massaged and being stabbed. (Many economists make much of the notion that even cardinal utility is “equivalent up to an affine transformation”, but I’ve got some news for you: So are temperature and time. All you are really doing by making an “affine transformation” is assigning a starting point and a unit of measurement. Temperature has a sensible absolute zero to use as a starting point, you say? Well, so does utility—not existing. )

With cardinal utility, I can offer you a very simple naive algorithm for finding an optimal match: Just try out every possible set of matchings and pick the one that has the highest total utility.

There are up to n!/((n/2)! 2^n) possible matchings to check, so this could take a long time—but it should work. I’m sure there’s a more efficient algorithm out there, but I don’t have the mental energy to figure it out at the moment. It might still be NP-hard, but I doubt it’s that hard.

Moreover, even once we find a utility-maximizing matching, that doesn’t guarantee a stable matching: Some people might still prefer to change even if it would end up reducing total utility.

Here’s a simple set of preferences for which that becomes an issue. In this table, the row is the person making the evaluation, and the columns are how much utility they assign to a match with each person. The total utility of a match is just the sum of utility from the two partners. The utility of “matching with yourself” is the utility of not being matched at all.


ABCD
A0321
B2031
C3201
D3210

Since everyone prefers every other person to not being matched at all (likely not true in real life!), the optimal matchings will always match everyone with someone. Thus, there are actually only 3 matchings to compare:

AB, CD: (3+2)+(1+1) = 7

AC, BD: (2+3)+(1+2) = 8

AD, BC: (1+3)+(3+2) = 9

The optimal matching, in utilitarian terms, is to match A with D and B with C. This yields total utility of 9.

But that’s not stable, because A prefers C over D, and C prefers A over B. So A and C would choose to pair up instead.

In fact, this set of preferences yields no stable matching at all. For anyone who is partnered with D, another member will rate them highest, and D’s partner will prefer that person over D (because D is everyone’s last choice).

There is always a nonempty set of utility-maximizing matchings. (There must be at least one, and could in principle have as many as there are possible matchings.) This actually just follows from the well-ordering property of the real numbers: Any finite set of reals has a maximum.

As this counterexample shows, there isn’t always a stable matching.

So here are a couple of interesting theoretical questions that this gives rise to:
1. If there is a stable matching, must it be in the set of utility-maximizing matchings?

2. If there is a stable matching, must all utility-maximizing matchings be stable?

Question 1 asks whether being stable implies being utility-maximizing.
Question 2 asks whether being utility-maximizing implies being stable—conditional on there being at least one stable possibility.

So, what is the answer to these questions? I don’t know! I’m actually not sure anyone does! We may have stumbled onto cutting-edge research!

I found a paper showing that these properties do not hold when you are doing the hetero marriage problem and you use multiplicative utility for matchings, but this is the queer marriage problem, and moreover I think multiplicative utility is the wrong approach. It doesn’t make sense to me to say that a marriage where one person is extremely happy and the other is indifferent to leaving is equivalent to a marriage where both partners are indifferent to leaving, but that’s what you’d get if you multiply 1*0 = 0. And if you allow negative utility from matchings (i.e. some people would prefer to remain single than to be in a particular match—which seems sensible enough, right?), since -1*-1 = 1, multiplicative utility yields the incredibly perverse result that two people who despise each other constitute a great match. Additive utility solves both problems: 1+0 = 1 and -1+-1 = -2, so, as we would hope, like + indifferent = like, and hate + hate = even more hate.

There is something to be said for the idea that two people who kind of like each other is better than one person ecstatic and the other miserable, but (1) that’s actually debatable, isn’t it? And (2) I think that would be better captured by somehow penalizing inequality in matches, not by using multiplicative utility.

Of course, I haven’t done a really thorough literature search, so other papers may exist. Nor have I spent a lot of time just trying to puzzle through this problem myself. Perhaps I should; this is sort of my job, after all. But even if I had the spare energy to invest heavily in research at the moment (which I sadly do not), I’ve been warned many times that pure theory papers are hard to publish, and I have enough trouble getting published as it is… so perhaps not.

My intuition is telling me that 2 is probably true but 1 is probably false. That is, I would guess that the set of stable matchings, when it’s not empty, is actually larger than the set of utility-maximizing matchings.

I think where I’m getting that intuition is from the properties of Pareto-efficient allocations: Any utility-maximizing allocation is necessarily Pareto-efficient, but many Pareto-efficient allocations are not utility-maximizing. A stable matching is sort of a strengthening of the notion of a Pareto-efficient allocation (though the problem of finding a Pareto-efficient matching for the general queer marriage problem has been solved).

But it is interesting to note that while a Pareto-efficient allocation must exist (typically there are many, but there must be at least one, because it’s impossible to have a cycle of Pareto improvements as long as preferences are transitive), it’s entirely possible to have no stable matchings at all.

The “market for love” is a bad metaphor

pnrj core principles, critique of neoclassical economics , , , , , , , , , , , , , , , , ,

Feb 14 JDN 2458529

Valentine’s Day was this past week, so let’s talk a bit about love.

Economists would never be accused of being excessively romantic. To most neoclassical economists, just about everything is a market transaction. Love is no exception.

There are all sorts of articles and books and an even larger number of research papers going back multiple decades and continuing all the way through until today using the metaphor of the “marriage market”.

In a few places, marriage does actually function something like a market: In China, there are places where your parents will hire brokers and matchmakers to select a spouse for you. But even this isn’t really a market for love or marriage. It’s a market for matchmaking services. The high-tech version of this is dating sites like OkCupid.
And of course sex work actually occurs on markets; there is buying and selling of services at monetary prices. There is of course a great deal worth saying on that subject, but it’s not my topic for today.

But in general, love is really nothing like a market. First of all, there is no price. This alone should be sufficient reason to say that we’re not actually dealing with a market. The whole mechanism that makes a market a market is the use of prices to achieve equilibrium between supply and demand.

A price doesn’t necessarily have to be monetary; you can barter apples for bananas, or trade in one used video game for another, and we can still legitimately call that a market transaction with a price.

But love isn’t like that either. If your relationship with someone is so transactional that you’re actually keeping a ledger of each thing they do for you and each thing you do for them so that you could compute a price for services, that isn’t love. It’s not even friendship. If you really care about someone, you set such calculations aside. You view their interests and yours as in some sense shared, aligned toward common goals. You stop thinking in terms of “me” and “you” and start thinking in terms of “us”. You don’t think “I’ll scratch your back if you scratch mine.” You think “We’re scratching each other’s backs today.”

This is of course not to say that love never involves conflict. On the contrary, love always involves conflict. Successful relationships aren’t those where conflict never happens, they are those where conflict is effectively and responsibly resolved. Your interests and your loved ones’ are never completely aligned; there will always be some residual disagreement. But the key is to realize that your interests are still mostly aligned; those small vectors of disagreement should be outweighed by the much larger vector of your relationship.

And of course, there can come a time when that is no longer the case. Obviously, there is domestic abuse, which should absolutely be a deal-breaker for anyone. But there are other reasons why you may find that a relationship ultimately isn’t working, that your interests just aren’t as aligned as you thought they were. Eventually those disagreement vectors just get too large to cancel out. This is painful, but unavoidable. But if you reach the point where you are keeping track of actions on a ledger, that relationship is already dead. Sooner or later, someone is going to have to pull the plug.

Very little of what I’ve said in the preceding paragraphs is likely to be controversial. Why, then, would economists think that it makes sense to treat love as a market?

I think this comes down to a motte and bailey doctrine. A more detailed explanation can be found at that link, but the basic idea of a motte and bailey is this: You have a core set of propositions that is highly defensible but not that interesting (the “motte”), and a broader set of propositions that are very interesting, but not as defensible (the “bailey”). The terms are related to a medieval defensive strategy, in which there was a small, heavily fortified tower called a motte, surrounded by fertile, useful land, the bailey. The bailey is where you actually want to live, but it’s hard to defend; so if the need arises, you can pull everyone back into the motte to fight off attacks. But nobody wants to live in the motte; it’s just a cramped stone tower. There’s nothing to eat or enjoy there.

The motte comprised of ideas that almost everyone agrees with. The bailey is the real point of contention, the thing you are trying to argue for—which, by construction, other people must not already agree with.

Here are some examples, which I have intentionally chosen from groups I agree with:

Feminism can be a motte and bailey doctrine. The motte is “women are people”; the bailey is abortion rights, affirmative consent and equal pay legislation.

Rationalism can be a motte and bailey doctrine. The motte is “rationality is good”; the bailey is atheism, transhumanism, and Bayesian statistics.

Anti-fascism can be a motte and bailey doctrine. The motte is “fascists are bad”; the bailey is black bloc Antifa and punching Nazis.

Even democracy can be a motte and bailey doctrine. The motte is “people should vote for their leaders”; my personal bailey is abolition of the Electoral College, a younger voting age, and range voting.

Using a motte and bailey doctrine does not necessarily make you wrong. But it’s something to be careful about, because as a strategy it can be disingenuous. Even if you think that the propositions in the bailey all follow logically from the propositions in the motte, the people you’re talking to may not think so, and in fact you could simply be wrong. At the very least, you should be taking the time to explain how one follows from the other; and really, you should consider whether the connection is actually as tight as you thought, or if perhaps one can believe that rationality is good without being Bayesian or believe that women are people without supporting abortion rights.

I think when economists describe love or marriage as a “market”, they are applying a motte and bailey doctrine. They may actually be doing something even worse than that, by equivocating on the meaning of “market”. But even if any given economist uses the word “market” totally consistently, the fact that different economists of the same broad political alignment use the word differently adds up to a motte and bailey doctrine.

The doctrine is this: “There have always been markets.”

The motte is something like this: “Humans have always engaged in interaction for mutual benefit.”

This is undeniably true. In fact, it’s not even uninteresting. As mottes go, it’s a pretty nice one; it’s worth spending some time there. In the endless quest for an elusive “human nature”, I think you could do worse than to focus on our universal tendency to engage in interaction for mutual benefit. (Don’t other species do it too? Yes, but that’s just it—they are precisely the ones that seem most human.)

And if you want to define any mutually-beneficial interaction as a “market trade”, I guess it’s your right to do that. I think this is foolish and confusing, but legislating language has always been a fool’s errand.

But of course the more standard meaning of the word “market” implies buyers and sellers exchanging goods and services for monetary prices. You can extend it a little to include bartering, various forms of financial intermediation, and the like; but basically you’re still buying and selling.

That makes this the bailey: “Humans have always engaged in buying and selling of goods and services at prices.”

And that, dear readers, is ahistorical nonsense. We’ve only been using money for a few thousand years, and it wasn’t until the Industrial Revolution that we actually started getting the majority of our goods and services via market trades. Economists like to tell a story where bartering preceded the invention of money, but there’s basically no evidence of that. Bartering seems to be what people do when they know how money works but don’t have any money to work with.

Before there was money, there were fundamentally different modes of interaction: Sharing, ritual, debts of honor, common property, and, yes, love.

These were not markets. They perhaps shared some very broad features of markets—such as the interaction for mutual benefit—but they lacked the defining attributes that make a market a market.

Why is this important? Because this doctrine is used to transform more and more of our lives into actual markets, on the grounds that they were already “markets”, and we’re just using “more efficient” kinds of markets. But in fact what’s happening is we are trading one fundamental mode of human interaction for another: Where we used to rely upon norms or trust or mutual affection, we instead rely upon buying and selling at prices.

In some cases, this actually is a good thing: Markets can be very powerful, and are often our best tool when we really need something done. In particular, it’s clear at this point that norms and trust are not sufficient to protect us against climate change. All the “Reduce, Reuse, Recycle” PSAs in the world won’t do as much as a carbon tax. When millions of lives are at stake, we can’t trust people to do the right thing; we need to twist their arms however we can.

But markets are in some sense a brute-force last-resort solution; they commodify and alienate (Marx wasn’t wrong about that), and despite our greatly elevated standard of living, the alienation and competitive pressure of markets seem to be keeping most of us from really achieving happiness.

This is why it’s extremely dangerous to talk about a “market for love”. Love is perhaps the last bastion of our lives that has not been commodified into a true market, and if it goes, we’ll have nothing left. If sexual relationships built on mutual affection were to disappear in favor of apps that will summon a prostitute or a sex robot at the push of a button, I would count that as a great loss for human civilization. (How we should regulate prostitution or sex robots are a different question, which I said I’d leave aside for this post.) A “market for love” is in fact a world with no love at all.