David R. Henderson

Asymmetric Loss Functions

Some Data to Ponder... Kling, Krugman, and Krugman on...

One of the most important things I learned from my Economics professors at UCLA was at a cocktail party at the late Sherwin Rosen's place in Rochester, New York. Labor economist Finis Welch was visiting from UCLA and I had left UCLA to be an assistant professor at the Rochester B-School the previous summer. How we got to this topic doesn't matter (although, believe it or not, I remember) but Welch used the term "asymmetric loss function" and said that if you didn't have one, you would miss half your flights.

I got the point immediately. The asymmetry is that a mistake in one direction (say, too much) has a much lower cost than a mistake in the other direction (in this case, too little.) Back to flights. If the loss to you of being 30 minutes too early for a flight is the same as the loss to you of being 30 minutes too late, that is, if you had a symmetric loss function, you would be late roughly as many times as you would be early. Of course, we're not. And the reason we're not is that the cost of being 30 minutes too late is typically a large multiple of the cost of being 30 minutes too early.

Once I had that concept, I started seeing all kinds of decisions through that lens. Should I pack a raincoat just in case it rains? Typically, the cost of having the excess weight is small relative to the cost of not having the raincoat if it does rain. So yes, I should. If I have a sore throat and want some lozenges to carry me through the day, should I carry 4 in my pocket (the number I expect to use) or 8? The cost of not having enough is substantially larger than the cost of the extra little pocket space and the slightly degraded wrappers on the unused lozenges. So I should carry 8.

Of course, I've ignored probabilities here. If I'm packing for a trip and just taking a carry-on, I might not take the raincoat if the weather forecast for where I'm going is less than a 10% probability of rain. So there are some implicit probabilities lurking in the background.

So let's have a fun game. (I think it's fun, at least). Give some actual examples from your life where you have an asymmetric loss function and the asymmetry is huge.

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CATEGORIES: Cost-benefit Analysis

COMMENTS (27 to date)
Paul Ralley writes:

Looking both ways before crossing the road?

Brian Bergfeld writes:

Government's response to the possibility of climate change.

Brandon Berg writes:


Usems writes:

Accepting a slightly sub-optimal job offer when none other is present. The asymmetry increases the closer the offer is to the optimal one.

Mo writes:

My sense is that the general public has an ALF - probably somewhat cognitively biased - when it comes to nuclear power.

Should we build more nuclear power? If yes this brings a small risk of a huge disaster. This potential loss is seen as many multiples of the (typically unseen) losses from not using nuclear.

As with many things, the "seen" loss (a meltdown), combined with over-estimated probabilities, looms much larger in the public's mind than the "unseen" losses (i.e. higher energy prices, cancers from coal emissions).

Charlie Deist writes:

The recent recession seems to be strong evidence that the Fed should have a more asymmetric loss function with respect to inflation. Even minor disinflation (falling NGDP) has much higher costs than mild above-target inflation, at least if Scott Sumner is right (and I think he is).

Constant writes:

Having more than enough versus having too little gas in my car to get me to my next destination, with no gas station between me and the destination.

Having more air available than I need to breathe versus having too little. And other biological necessities.

Having more than enough points to graduate versus having not enough.

In the other direction (less is much better, later is much better):

My laptop is a computer small enough to be portable. Anything to do with portability probably involves a size or weight asymmetry.

If I am pulled over and my breath tested, much better to have less alcohol in my blood than the legal limit rather than more.

If you're watching a movie on TV, much better to shut off the TV after the movie is over, as opposed to before.

Clavin writes:

I recently returned from Afghanistan and the ALF most prevalent was "how much ammunition to carry?"

The odds of a firefight in Kabul were low (but not zero) and the weight of the ammunition was also not trivial. In addition, there was a significant possibility of long-term musculo-skeletal injuries along with the short-term loss of mobility. However, these factors did not outweigh the least favorite sound in combat - the sound of your rifle going "click" instead of "bang".

So the ALF was:
Too much ammo - tough to sprint and future injury
Not enough ammo - dead

This thinking process extends to all other carried gear - water, sleeping bag, knife, batteries, etc. Military life has many instances of non-monetary economic decisions.

BTW, this blog has a following in the military acquisition community - please keep up the high quality discourse.

Thomas writes:

Isn't the classic example the college guy with a condom in his wallet?

Jonathan writes:

In the Army, we use the phrase "better to have it and not need it than need it and not have it" to describe asymmetric loss functions. My best example is my aid bag. I pack it with all manner of specialty equipment for very specific injuries. My loss if I don't use it is the extra weight; if I don't bring it the loss could be a human life. It is hard to resist the impulse to try to pack an entire ICU into your aid bag.

mike writes:

Gun ownership.

Lambda writes:

1) Like Brandon Berg said: seatbelt.

2) Airport

3) Not much else. I'm pretty reckless I guess.

My general "philosophy" on this stuff - no, "practice" is the right word - is to minimize hassles on a moment-by-moment basis.

So I'd go like this with the lozenges - I'd take four, even if I might need eight, because four is less; and if I was out too long and needed more than four, I'd just go home, minimizing THAT hassle.

It's pretty irrational, but, hey, I'm WINNING!

John Hall writes:

Continuing this example, I once missed a flight while at school and I decided that the loss of admitting to people that I missed my flight was greater than the additional cost of a more expensive ticket that would allow me to not need to admit to others than I had missed my flight.

Shayne Cook writes:

Perhaps you could forward this to the NFL and the NFL Players Union.

Matt writes:

I think my entire morning routine is based on this concept. Eat breakfast, drink coffee, go to bathroom, double check to make sure I have everything (wallet, keys, cell phone) before I leave the house; these are all things I make sure to do before I leave the house because missing any of them can screw up my entire day.

Jeremy, Alabama writes:

An ALF might be between an incandescent light bulb vs an energy-efficient florescent one that blows mercury everywhere if you break it. Unfortunately, the govt appears to have made that decision for you.

Perhaps there should be a follow-up post on how the govt has intervened in questions that really ought to be ALFs.

Philo writes:

The information embodied in a non-symmetrical loss function could equally well be embodied in a symmetrical one, by adjusting the scales of measurement. (A logician would prefer the term 'non-symmetrical' to 'asymmetrical.)

Cassie writes:

In my family, it seems my temper is less of a loss than the time it takes my husband and kids to properly prepare for situations. Hmmm....

Faze writes:

Pascal's Wager?

rpl writes:

There's an old saying amongst pilots that captures this principle. You tend to hear it whenever the weather is marginal: "Better to be down here wishing you were up there, than up there wishing you were down here."

Really, though, there are so many examples of asymmetric loss functions (e.g., anything you buy insurance for) that it would be more interesting to ask, where are loss functions not asymmetric?

rpl writes:

Right after I hit submit I thought of another interesting variation on the question, which is, to what extent to people overreact to asymmetric loss functions. To use David's example, the cost of missing a flight is much higher than the cost of twiddling your thumbs at the airport, but it's not infinitely higher. I once knew a guy who argued that if you don't miss a flight once a year or so, you're wasting too much time at the airport. However, I suspect most of us don't think that way; instead, we build enough slack into our schedules that we never miss a flight (I know I never have). Ultimately, that costs us more than if cut it finer and accepted the occasional missed flight.

Luke writes:

re: rpl
Symmetric loss functions:
Which candidate to vote for in a two-party system?

Tracy W writes:

Jonathan - clearly you don't subscribe to the theory of a first aid kit for hiking by a doctor I know (who will remain anonymous):
"All a first aid kit needs is band-aids, aspirins and a pistol. The pistol is for those medical problems that can't be fixed with the first two items."
(On thinking about it, it's not entirely clear if you're meant to carry a bullet as well or just bash the patient's head-in with the pistol).

I find hiking interesting because while the losses remain the same, the costs of carrying protection against asymmetries is a lot higher. I do pack a bit more in my first aid kid than the doctor would recommend, but the last trip I was on that ran into bad weather I used everything in my pack except for what was in the first aid kit and the emergency food supplies, which I feel quite good about.

Anyway, another asymmetric loss function - buying a computer that can handle the most demanding tasks you want of it, rather than the average.

Guy in the Veal Calf Office writes:

Speaking your mind to loved ones.

James Oswald writes:

Retail inventories.

dullgeek writes:

I've noticed is that once you tell someone about ALFs, they'll tend to see every loss function as an ALF, and will become over cautious for a good number of things that don't have the ALF curve.

In fact, I suspect that a huge portion of people's tendency to prefer the precautionary principle is what distinguishes them from economists, who ask about the opportunity cost of a potential decision. E.g. most people see ALFs where they don't exist. I wonder if this is why many struggle with economics which suggests that we examin the opportunity cost before making a decision.

And climate change is a great example of this. An economist asks, "What is the cost of prevention?" The general public relates much more strongly to the precationary principle. The general public automatically assumes that climate change is an ALF.

So to me, a more interesting question than the one proposed is the opposite: how often do people assume a decision has an ALF w/o examining it? I speculate that the answer is most of the time.

Warren Gibson writes:

The phrase "cost function" makes me itch because it implies that values are measurable. They're not, they're only ranked. All we can say for sure is that if you choose to carry an umbrella you prefer to bear the cost of carrying it over the likely cost of not carrying it.

Of course you can say "I strongly prefer to carry an umbrella" but I say that has no meaning for economics.


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