Saturday, August 27, 2011

Sevenths are awesome

Is it common knowledge that sevenths are awesome?  In my experience, no.

If your 6th grade math teacher didn't drill it into your brain, you may not know off the top of your head that 1/7th is .142857 repeating.  The cool thing is that if you did know that off the top of your head, you could also know the decimal equivalents of 2/7, 3/7, ... , 6/7 without any additional memorization.

2/7=.285714 repeating
3/7=.428571 repeating

hey...those are the same string of 6 numbers just shifted over by a few places! It continues:

4/7=.571428 repeating
5/7=.714285 repeating
6/7=.857142 repeating

And there you have it.  Somehow they're all just the same infinite string of numbers starting at slightly different positions.  Like many math facts, if you carry this around your whole life, you will find many excuses to use it, and you will at some point even have a great nerd moment where someone else actually wants to know 3000/7 and you can tell them 428.571428 repeating before they have even located their calculator.  Mathemagical.

Perhaps The Answer to the Ultimate Question of Life, The Universe, and Everything is more like 6/7 than 6*7.  In any case, why do sevenths work like this and what deeper math is at play here?  Feel free to offer any thoughts you may have.

Added 8/30: Apparently these are called cyclic numbers!  Thank you, internet.

Friday, August 19, 2011

Usury 3: Me, myself, and I(ntertemporal tradeoffs?)

In the original usury post I said that we could explain a pattern of [splurging up front, massive debt thereafter] with time preferences that strongly favor the present.  Here are a couple possibilities.

1. For starters, maybe people's preferences -- while consistent over time -- are just really impatient.  They care a lot more about the beer and chips today than their increased poverty a year from today.  There's certainly nothing wrong with this in theory.  A person's goal isn't to maximize their bank account in the long run; rather, they care about their own rate of time preference, and it could be making them better off overall to borrow at 15% today in order to consume that much more in the present.

While possible, some find it unconvincing.  Also, yawn.  It is actually written into my economonomics contract that consistency gets a verbal and written yawn.  Do you prefer option 2?  At the very least, it's actually interesting...

2. Maybe people's preferences display "time inconsistency."  Their "true" preferences are rather patient, but in the moment they cannot resist the beer and chips even though it's "really" not good for them overall.  If they could commit in advance to not borrowing so much from the credit card company, they would.

Time inconsistency is often considered irrational, but let me take a few minutes to make a relatively deep comment about the nature of preferences.

First of all, the idea that you are actually a single entity over time is at best a simplification, and at worst an actively detrimental illusion.  "Multiple selves" is not just a way of looking at things...it's reality, right?  At the level of reality, you are not the same person tomorrow!  There is some transition rule that turns your constituent atoms of today into your atoms of tomorrow, and why should they necessarily have to agree with each other?  And even if you were physically constant over time, different instances of you wouldn't have the same bodily experience from the time-dated consumption of a bag of chips (yesterday-you enjoys yesterday-chips more than today-you enjoys yesterday-chips), and different versions of yourself could easily have different preferences over when to eat chips.  Would you expect a hundred clones in a room to agree over who gets the chips?  More likely they'd all want the chips for themselves; they have the "same" preferences but over different identities, which is to say they don't really have the same preferences at all, over states of the world.

That is the default.  We can try to model things with consistency, and we can look for reasons why consistency might come to the surface, but it's important to keep in mind that consistency is not the privileged default state of the world.  It is special case if it turns out that different versions of yourself all agree with each other.

There are about 8 directions we could go from here.  We could describe some salient theoretical possibilities for the shape of our time preferences.  We could talk about empirical evidence for the actual shape of our time preferences in various settings.  Or we could go a level deeper and talk about where the preferences came from in the first place, and what sorts of preferences were likely to have arisen.  We could talk about the powerful (but not all-powerful) forces pushing our preferences in the direction of consistency, and reasons we might systematically deviate from consistency.

But right now I just want to make a simple remark. As soon as there is any disagreement whatsoever between different instances of "your" preferences, what's to say which is the "true" or "right" set of preferences?  Six years in advance you want to abstain from the beer and chips.  In six years, you don't.  What does it mean to say the advanced preferences are the "real" ones?  In some sense there are simply many agents with different preferences in conflict with each other, doing what they can to get their way.  Now perhaps most agents agree that You-2017 should not go for the chips, while You-2017 disagrees, in which case are we really making a comment about social optimality when we say You-2017 is in "error"?  And if social optimality is on the table, by all means, please tell me: what weights are we using?  How is it related to the discounting that's already taking place?  It is so very nice if preferences start out consistent, because then everyone agrees on everything...but as soon as there's disagreement, there's a big discussion to be had. (More to say, another time).

I'm happy to abstract away from this, and a single, consistent agent over time is a great fit in many situations.  But at the level of reality, I don't exactly like to think of time inconsistency as irrational. Irrationality presupposes a correct set of preferences which are being violated, but there is no correct set of preferences here, only a bunch of preferences in disagreement.

(The discussion of why a particular set of preferences may jump out and seem to be "correct"...is also for another time).

Sunday, August 14, 2011

Usury 2: Relativity?

In the previous usury post, I said that one way we could explain a pattern of [splurging up front, massive debt thereafter] is if the same consumption bundles are not offered over time.  What if people strongly prefer today's bundles to tomorrow's?  Of course, the meat of the explanation is in giving you a compelling reason to believe this may be the case.

So, here's a reason.  Suppose there are relative effects together with concavity.  That is:
  1. Utility depends on what your peers are doing.  When other people are eating better food, you get higher utility from better food yourself.
  2.  The losses from being worse off than your peers are more severe than the gains from being better off than them.  When you smell the steaks your neighbors are eating, you really want steak too.  When your neighbors aren't eating steaks, you would still like a steak but not nearly as much as when it was being dangled in front of you.
If these hold, then people care about their position relative to the pack, and in particular it's better to stick with the pack than to spend some time below and some time above it.  When your friends are splurging backed by 15% interest credit card debt, you feel like you need to as well.  When you crash later, at least they will be crashing too.  So, we can get stuck in an equilibrium where, given that everyone's taking on massive debt, it's optimal for each individual to take on massive debt.

So when I say that different consumption bundles are offered over time, I don't mean that the physical goods are any different over time.  I'm imagining social effects as part of the consumption bundle, and that's what's changing.

*

A couple of comments.  First, I like this explanation but I do think of it as a partial explanation.  It doesn't explain why this equilibrium was selected in the first place, but it goes a long way to explaining the extent of people's willingness to do what other people are doing.  At the very least it serves to magnify tendencies that may have their roots in other explanations.

For example, perhaps credit card companies pulled one over on people, maybe they managed to actually trick them.  I'm  not going to reject this explanation per se, but a takeaway from today is that there can be much less of this than you would require if it were the whole story.  We don't need everyone to be deluded, to end up in an equilibrium where everyone is taking bad deals.

Second, I think the above is a nice story for understanding status quo effects more generally.  Especially when it's costly to gather the info necessary to find the best course of action, "do what everyone else is doing" is a particularly cheap and safe shortcut.  If they're right, great.  If they're wrong, at least you all go down together.

I'm going to be loose here, but note that without the concavity and relative effects, people would observe the crowd's opinion, and then perhaps do better by putting even a slight amount of their own effort into further information-gathering.  (With everyone doing this, the crowd could actually get pretty smart).  But here, you have to put a serious amount of effort forward in order to get enough additional info to make deviating from the crowd worthwhile at all.  (So collective ignorance is supported as an equilibrium).

I expect that:

  • Most people model most of their behavior off the status quo among their peer group, while selecting a few areas to focus on much more intently, as opposed to spreading their attention across all areas.
  • Status quo sunscreen usage doesn't necessarily reflect the available information together with how much people care about health risks per se, ignoring social effects.  Maybe we're just in a bad equilibrium.

Friday, August 12, 2011

Does usury exist?

Dan Lemire writes:
What is usury? Lending money knowing that it will make people poorer... 
Economists who expect people to be rational won’t believe in usury. Surely, people borrow money to be better off. Yet the average credit card debt per household is over $15,000 in the USA (and it grows all the time, of course). Does anyone believe that these $15,000 are invested in highly profitable ventures? (They would need to be highly profitable since credit cards often charge over 15% for loans.) 
...can anyone convince me that credit card companies produce wealth by charging exorbitant interests to people who use the money to buy beer and chips? 
First, as a pure matter of tone, I'm going to interpret "economists who expect people to be rational" not as "economists, who expect people to be rational," but rather as "the subset of economists who expect real people to actually be rational."  And by way of preterition, I'm not going to make any comment about the difference between a useful model and actually believing something literally about the real world, or about the variously constraining definitions of rationality and whether the aforementioned subset is even nonempty under the definition that noneconomists typically have in mind when they talk about economic rationality.  Instead, we will charge ahead and leave the question of whether economists are delusional fools to another day.

Now, accepting this definition of usury, the first step is to be careful about what it means to make someone "poorer."  Dan is taking "poorer" to mean something like "lower present value of wealth with respect to whatever interest rate," so that borrowing at 15% makes you poorer unless you're taking that borrowed money and investing it in another project with an even higher return.  But if that's poorer, then forget borrowing at 15% --- consuming makes you poorer!  It's fine to use "poorer" in this way, but then you don't get to equate "poorer" with "worse off."  A rational person's allegiance is to making himself better off, not maximizing his bank account!  (Rational people consume, of course).

By contrast, economists might take "poorer" to directly mean "worse off," i.e. having access to less desirable consumption bundles.  But then borrowing at 15% doesn't necessarily make you poorer.  It gives you access to fewer bundles tomorrow, but more today.  If you want the extra consumption today more than the foregone consumption tomorrow, you are not poorer for borrowing to enable that consumption pattern.

Upcoming, a few ways to complete the story. Perhaps -- for a special reason I will soon make clear -- the same consumption bundles are not offered over time, in which case people could simply prefer the consumption bundles offered today to the ones offered tomorrow.  Or perhaps the bundles are constant over time but there is something going on with their time preferences that makes them prefer to consume a lot up front.

Friday, August 5, 2011

Tuesday, August 2, 2011

Relationship advice for economists: Don't do it with models.

Economics has plenty to say about relationships, both in terms of individual behavior and how it aggregates. Today I want to frame the individual's problem as a peculiar sort of search problem.

For today, take as given that each individual has a bunch of characteristics, which make different people better or worse matches for each other. For instance, you may prefer someone who likes football as much as you, or someone who shares a lot of your beliefs.

The objective, loosely, is to spend as long as possible with as good a match as possible. But there is incomplete information. You don't know anyone else's vector of characteristics, and you have to meet and get to know them to find out. Some characteristics are uncovered quickly (e.g. physical appearance), others take weeks, months, or years to discover ("just how much can I trust this person?").

So you set about to find a good match, repeatedly sampling from a pool of candidates. You're dating someone, all the while learning more about them and their compatibility with you. On any day you can either opt to continue the relationship another day, or reject your partner and move onto the next possibility. In this simple example, the expected value of rejecting/starting anew is constant, so as usual the problem gives rise to a rejection threshold: if you should ever learn enough bad information that the expected value of continuing the relationship drops below the expected value of moving on, you move on.

Now here's a twist. All through your life, you continue to meet people and learn things about them, whether in a relationship or not. But if you are in a relationship, there's a cap on how much you're allowed to learn about anyone else. Conversely, there's a cap (not necessarily the same) on how much you are able to learn about someone else (in terms of relationship compatibility) without being in a relationship with them.

Together, these say that to find out whether someone will be a good enough match for you, you will eventually have to enter into a relationship, and once in a relationship, your outside search behavior is critically limited. Also, we are now imagining a model where the expected value of moving on is not constant, but rather depends on your current best candidate outside of the relationship, which is determined by what you've learned about them. (You can think of this as the best candidate who is willing to be in a relationship with you).

Excuse me while I make up some terminology. Say that a relationship is t-stable if in period t, neither party prefers to match with their best candidate outside of the relationship. That is, a relationship continues to the next period precisely if it is t-stable. Furthermore, say that a relationship is t-superstable if it would be t-stable for any possible best outside candidate. That is, given each partner's period-t expected value of the relationship, there's no possible outside option that could entice them away from their current relationship.

(The t is there to remind us that this really is a dynamic system that unfolds over time; superstability is a function of the information you currently know.  You could have a really great first date and be superstable in the first period of the relationship, but lose it later on as more characteristics become known.  And you may never know whether your relationship will last into old age; people change in ways they can't entirely predict. Perhaps it takes being 50 to know that you'll be compatible as 50-year-olds.   Keep it in mind, but for the rest of the post I'm just going to drop the t.)

And let me be clear that superstability does not mean you're matched with your "soul mate," your best possible match in the universe.  It means that out-of-relationship signals are simply not informative enough to compete with all the great things you know for sure about the person you're with.

To be sure, whether we should expect to find superstable matches depends on the particular social rules that govern the sort of search people are allowed/able to do inside and outside of what society has defined as a "relationship."  But I think the answer is generally yes.  Compatibility depends on so many important things that are impossible to observe outside of a relationship; attractiveness and surface personality and so forth are ultimately not a large enough part of the quality of a match. After enough searching, you may find yourself in a particular relationship and discover that it's a really great match, way better than you expected beforehand, way better than you could ever expect beforehand, from anyone. And then you will be done.

I know this is a long post, but I don't like to split these things up into disjoint pieces.  Read on if you want to see what else falls out of the model, now that we have it.

Observations, questions, extensions



Which sorts of characteristics are really important for match quality?  I haven't said anything about how characteristics get converted into match quality, but presumably the function is not simple.  It takes the two vectors of characteristics and combines them in a potentially complicated way.  In reality, perhaps we observe the match quality directly, but not the function that produced it from the characteristics.  We can theorize about the relationship between characteristics and quality, but how good should we expect these theories to be?

If we know that certain characteristics lead to higher match quality, we can put a lot of effort into surrounding ourselves with people who have those characteristics.  We can apply to schools or jobs that attract like-minded people, and let the process do a lot of screening for us.  On the other hand, if we don't actually have a good idea of where match quality comes from, then our attempts to screen for preliminary characteristics are largely a waste of effort.  If we mostly need to be in a relationship to find out the quality of the relationship, then we should try to have more relationships, not "better" ones.


*

I glossed over behavior out of a relationship above, but the model actually doesn't rule out it being optimal to spend time out of any relationship, even if there are currently people willing to match with you. That comes from keeping the caps separate, as I said:
If you are in a relationship, there's a cap on how much you're allowed to learn about anyone else. Conversely, there's a cap (not necessarily the same, presumably higher) on how much you are able to learn about someone else without being in a relationship with them.
By becoming close friends with someone while single, perhaps you can get relatively far into a relationship while still being allowed to search over other people. Once you are in a relationship, there may be a hard bound on how much time you can spend with other people you might be interested in. But out of a relationship, it may be that you can get pretty close to multiple people, without "dating" and thereby activating the restriction on how you spend your time outside of that particular relationship. Not necessarily a bad idea. Yes, dating can irrevocably destroy a friendship and all the time that went into it. But on the flip side, the time you spend in an ultimately failed relationship is more costly per day, since it more seriously restricts your search behavior. So the more you know about someone prior to starting a relationship with them, the less you have to figure out when time is being billed at a higher rate.

Another reason to stay (or become) single that's not captured by this model is this: when you're single, you are significantly more likely to discover who else is interested in you. Here we're assuming this is just known, but clearly it's private information in real life...info which people are reluctant to give up, especially if you're spoken for.


*

In a match, quality along easily observable characteristics versus less observable characteristics says something about how the match is likely to have been initiated. For example, if a really high-quality person is matched with someone who's low-quality along the most visible dimensions, it is more likely that they knew each other for long enough to uncover some of the less visible dimensions before dating. If you feel your most easily-observable characteristics don't adequately represent who you really are, you may have much better luck cultivating friendships than asking people out on the fly.

*

And if you do ask someone out on the fly, their expected unobservable quality could even be decreasing in their attractiveness, conditional on them saying yes. Lower quality people are more willing to say yes, so if they're willing and attractive, it could mean that they are deficient in less observable categories. Similarly, if you know your friend is really high-quality, and her boyfriend seems "not good enough" when you first meet him, then you might actually conclude that he's better than expected along the deeper dimensions you can't observe in one interaction. Better, perhaps, than you'd expect a really handsome guy to be.  (But I don't know how far it makes sense to push this line of counterintuitive reasoning.  Maybe your friend is just deluded.)

Empirically, we may find that relationship length is decreasing in attractiveness of the other party, over the range of time during which a lot of hidden characteristics are discovered. Hmm. (Perhaps economists shouldn't do it with models after all).

*
Note that these effects are dampened along dimensions where there is widespread disagreement about "quality."  It could be interesting to think through how more and less observable characteristics break down into matters of personal taste versus widespread agreement over quality. 

*

Finally, a note on this search model vis-a-vis the matching literature. The Gale-Shapley Deferred Acceptance algorithm is beautiful and simple, but it assumes complete information. How might we adapt the notion of a stable matching (across all of society) to our environment?

Well, we could say a t-stable matching is a matching where every individual relationship is t-stable. But it seems to lose its edge, no? As soon as we introduce incomplete information and a cost of searching (here it takes time), the space of t-stable matchings gets very large, and there's no reason for any such "stable" matching to survive for any length of time. On the other hand, a "t-superstable matching" is much closer to the Gale-Shapley notion of "stable matching." But in general there's no reason to expect a t-superstable matching to exist, so we might want to think about conditions that would ensure it.  I suspect they would be far too restrictive to be realistic though.  Where would we take this?  On the one hand, incomplete information is nice in that it potentially supports a lot of matchings that we see in the real world, which would not survive under complete information.  But on the other hand, that also means the theory isn't going to give any sharp predictions.  And the matchings are unlikely to be stable over time, because there is constantly new information flowing, which changes people's decisions.  I suppose superstability could potentially be a decent solution concept for the subset of people who are married, but this isn't without its problems either.

Put a bunch of people in a room. Give each person one observable characteristic, and one characteristic that can only be discovered by spending a period with them. What happens?

Bayesian approaches to matching have hardly been touched.