expected utility

Bundling the stakes to recalibrate ourselves

PNRJ cognitive science, heuristics and biases, personal , , , , , , , , , ,

Mar 31 JDN 2460402

In a previous post I reflected on how our minds evolved for an environment of immediate return: An immediate threat with high chance of success and life-or-death stakes. But the world we live in is one of delayed return: delayed consequences with low chance of success and minimal stakes.

We evolved for a world where you need to either jump that ravine right now or you’ll die; but we live in a world where you’ll submit a hundred job applications before finally getting a good offer.

Thus, our anxiety system is miscalibrated for our modern world, and this miscalibration causes us to have deep, chronic anxiety which is pathological, instead of brief, intense anxiety that would protect us from harm.

I had an idea for how we might try to jury-rig this system and recalibrate ourselves:

Bundle the stakes.

Consider job applications.

The obvious way to think about it is to consider each application, and decide whether it’s worth the effort.

Any particular job application in today’s market probably costs you 30 minutes, but you won’t hear back for 2 weeks, and you have maybe a 2% chance of success. But if you fail, all you lost was that 30 minutes. This is the exact opposite of what our brains evolved to handle.

So now suppose if you think of it in terms of sending 100 job applications.

That will cost you 30 times 100 minutes = 50 hours. You still won’t hear back for weeks, but you’ve spent weeks, so that won’t feel as strange. And your chances of success after 100 applications are something like 1-(0.98)^100 = 87%.

Even losing 50 hours over a few weeks is not the disaster that falling down a ravine is. But it still feels a lot more reasonable to be anxious about that than to be anxious about losing 30 minutes.

More importantly, we have radically changed the chances of success.

Each individual application will almost certainly fail, but all 100 together will probably succeed.

If we were optimally rational, these two methods would lead to the same outcomes, by a rather deep mathematical law, the linearity of expectation:
E[nX] = n E[X]

Thus, the expected utility of doing something n times is precisely n times the expected utility of doing it once (all other things equal); and so, it doesn’t matter which way you look at it.

But of course we aren’t perfectly rational. We don’t actually respond to the expected utility. It’s still not entirely clear how we do assess probability in our minds (prospect theory seems to be onto something, but it’s computationally harder than rational probability, which means it makes absolutely no sense to evolve it).

If instead we are trying to match up our decisions with a much simpler heuristic that evolved for things like jumping over ravines, our representation of probability may be very simple indeed, something like “definitely”, “probably”, “maybe”, “probably not”, “definitely not”. (This is essentially my categorical prospect theory, which, like the stochastic overload model, is a half-baked theory that I haven’t published and at this point probably never will.)

2% chance of success is solidly “probably not” (or maybe something even stronger, like “almost definitely not”). Then, outcomes that are in that category are presumably weighted pretty low, because they generally don’t happen. Unless they are really good or really bad, it’s probably safest to ignore them—and in this case, they are neither.

But 87% chance of success is a clear “probably”; and outcomes in that category deserve our attention, even if their stakes aren’t especially high. And in fact, by bundling them, we have even made the stakes a bit higher—likely making the outcome a bit more salient.

The goal is to change “this will never work” to “this is going to work”.

For an individual application, there’s really no way to do that (without self-delusion); maybe you can make the odds a little better than 2%, but you surely can’t make them so high they deserve to go all the way up to “probably”. (At best you might manage a “maybe”, if you’ve got the right contacts or something.)

But for the whole set of 100 applications, this is in fact the correct assessment. It will probably work. And if 100 doesn’t, 150 might; if 150 doesn’t, 200 might. At no point do you need to delude yourself into over-estimating the odds, because the actual odds are in your favor.

This isn’t perfect, though.

There’s a glaring problem with this technique that I still can’t resolve: It feels overwhelming.

Doing one job application is really not that big a deal. It accomplishes very little, but also costs very little.

Doing 100 job applications is an enormous undertaking that will take up most of your time for multiple weeks.

So if you are feeling demotivated, asking you to bundle the stakes is asking you to take on a huge, overwhelming task that surely feels utterly beyond you.

Also, when it comes to this particular example, I even managed to do 100 job applications and still get a pretty bad outcome: My only offer was Edinburgh, and I ended up being miserable there. I have reason to believe that these were exceptional circumstances (due to COVID), but it has still been hard to shake the feeling of helplessness I learned from that ordeal.

Maybe there’s some additional reframing that can help here. If so, I haven’t found it yet.

But maybe stakes bundling can help you, or someone out there, even if it can’t help me.

Against Self-Delusion

PNRJ cognitive science, core principles, personal , , , , , , , , , , , , , ,

Mar 10 JDN 2460381

Is there a healthy amount of self-delusion? Would we be better off convincing ourselves that the world is better than it really is, in order to be happy?


A lot of people seem to think so.

I most recently encountered this attitude in Kathryn Schulz’s book Being Wrong (I liked the TED talk much better, in part because it didn’t have this), but there are plenty of other examples.

You’ll even find advocates for this attitude in the scientific literature, particularly when talking about the Lake Wobegon Effect, optimism bias, and depressive realism.

Fortunately, the psychology community seems to be turning away from this, perhaps because of mounting empirical evidence that “depressive realism” isn’t a robust effect. When I searched today, it was easier to find pop psych articles against self-delusion than in favor of it. (I strongly suspect that would not have been true about 10 years ago.)

I have come up with a very simple, powerful argument against self-delusion:

If you’re allowed to delude yourself, why not just believe everything is perfect?

If you can paint your targets after shooting, why not always paint a bullseye?

The notion seems to be that deluding yourself will help you achieve your goals. But if you’re going to delude yourself, why bother achieving goals? You could just pretend to achieve goals. You could just convince yourself that you have achieved goals. Wouldn’t that be so much easier?

The idea seems to be, for instance, to get an aspiring writer to actually finish the novel and submit it to the publisher. But why shouldn’t she simply imagine she has already done so? Why not simply believe she’s already a bestselling author?

If there’s something wrong with deluding yourself into thinking you’re a bestselling author, why isn’t that exact same thing wrong with deluding yourself into thinking you’re a better writer than you are?

Once you have opened this Pandora’s Box of lies, it’s not clear how you can ever close it again. Why shouldn’t you just stop working, stop eating, stop doing anything at all, but convince yourself that your life is wonderful and die in a state of bliss?

Granted, this is not generally what people who favor (so-called) “healthy self-delusion” advocate. But it’s difficult to see any principled reason why they should reject it. Once you give up on tying your beliefs to reality, it’s difficult to see why you shouldn’t just say that anything goes.

Why are some deviations from reality okay, but not others? Is it because they are small? Small changes in belief can still have big consequences: Believe a car is ten meters behind where it really is, and it may just run you over.

The general approach of “healthy self-delusion” seems to be that it’s all right to believe that you are smarter, prettier, healthier, wiser, and more competent than you actually are, because that will make you more confident and therefore more successful.

Well, first of all, it’s worth pointing out that some people obviously go way too far in that direction and become narcissists. But okay, let’s say we find a way to avoid that. (It’s unclear exactly how, since, again, by construction, we aren’t tying ourselves to reality.)

In practice, the people who most often get this sort of advice are people who currently lack self-confidence, who doubt their own abilities—people who suffer from Impostor Syndrome. And for people like that (and I count myself among them), a certain amount of greater self-confidence would surely be a good thing.

The idea seems to be that deluding yourself to increase your confidence will get you to face challenges and take risks you otherwise wouldn’t have, and that this will yield good outcomes.

But there’s a glaring hole in this argument:

If you have to delude yourself in order to take a risk, you shouldn’t take that risk.

Risk-taking is not an unalloyed good. Russian Roulette is certainly risky, but it’s not a good career path.

There are in fact a lot of risks you simply shouldn’t take, because they aren’t worth it.

The right risks to take are the ones for which the expected benefit outweighs the expected cost: The one with the highest expected utility. (That sounds simple, and in principle it is; but in practice, it can be extraordinarily difficult to determine.)

In other words, the right risks to take are the ones that are rational. The ones that a correct view of the world will instruct you to take.

That aspiring novelist, then, should write the book and submit it to publishers—if she’s actually any good at writing. If she’s actually terrible, then never submitting the book is the correct decision; she should spend more time honing her craft before she tries to finish it—or maybe even give up on it and do something else with her life.

What she needs, therefore, is not a confident assessment of her abilities, but an accurate one. She needs to believe that she is competent if and only if she actually is competent.

But I can also see how self-delusion can seem like good advice—and even work for some people.

If you start from an excessively negative view of yourself or the world, then giving yourself a more positive view will likely cause you to accomplish more things. If you’re constantly telling yourself that you are worthless and hopeless, then convincing yourself that you’re better than you thought is absolutely what you need to do. (Because it’s true.)

I can even see how convincing yourself that you are the best is useful—even though, by construction, most people aren’t. When you live in a hyper-competitive society like ours, where we are constantly told that winning is everything, losers are worthless, and second place is as bad as losing, it may help you get by to tell yourself that you really are the best, that you really can win. (Even weirder: “Winning isn’t everything; it’s the only thing.” Uh, that’s just… obviously false? Like, what is this even intended to mean that “Winning is everything” didn’t already say better?)

But that’s clearly not the right answer. You’re solving one problem by adding another. You shouldn’t believe you are the best; you should recognize that you don’t have to be. Second place is not as bad as losing—and neither is fifth, or tenth, or fiftieth place. The 100th-most successful author in the world still makes millions writing. The 1,000th-best musician does regular concert tours. The 10,000th-best accountant has a steady job. Even the 100,000th-best trucker can make a decent living. (Well, at least until the robots replace him.)

Honestly, it’d be great if our whole society would please get this memo. It’s no problem that “only a minority of schools play sport to a high level”—indeed, that’s literally inevitable. It’s also not clear that “60% of students read below grade level” is a problem, when “grade level” seems to be largely defined by averages. (Literacy is great and all, but what’s your objective standard for “what a sixth grader should be able to read”?)

We can’t all be the best. We can’t all even be above-average.

That’s okay. Below-average does not mean inadequate.

That’s the message we need to be sending:

You don’t have to be the best in order to succeed.

You don’t have to be perfect in order to be good enough.

You don’t even have to be above-average.

This doesn’t require believing anything that isn’t true. It doesn’t require overestimating your abilities or your chances. In fact, it asks you to believe something that is more true than “You have to be the best” or “Winning is everything”.

If what you want to do is actually worth doing, an accurate assessment will tell you that. And if an accurate assessment tells you not to do it, then you shouldn’t do it. So you have no reason at all to strive for anything other than accurate beliefs.

With this in mind, the fact that the empirical evidence for “depressive realism” is shockingly weak is not only unsurprising; it’s almost irrelevant. You can’t have evidence against being rational. If deluded people succeed more, that means something is very, very wrong; and the solution is clearly not to make more people deluded.

Of course, it’s worth pointing out that the evidence is shockingly weak: Depressed people show different biases, not less bias. And in fact they seem to be more overconfident in the following sense: They are more certain that what they predict will happen is what will actually happen.

So while most people think they will succeed when they will probably fail, depressed people are certain they will fail when in fact they could succeed. Both beliefs are inaccurate, but the depressed one is in an important sense more inaccurate: It tells you to give up, which is the wrong thing to do.

“Healthy self-delusion” ultimately amounts to trying to get you to do the right thing for the wrong reasons. But why? Do the right thing for the right reasons! If it’s really the right thing, it should have the right reasons!

Is Singularitarianism a religion?

PNRJ ethics, futurism , , , , , , ,

 

Nov 17 JDN 2458805

I said in last week’s post that Pascal’s Mugging provides some deep insights into both Singularitarianism and religion. In particular, it explains why Singularitarianism seems so much like a religion.

This has been previously remarked, of course. I think Eric Steinhart makes the best case for Singularitarianism as a religion:

I think singularitarianism is a new religious movement. I might add that I think Clifford Geertz had a pretty nice (though very abstract) definition of religion. And I think singularitarianism fits Geertz’s definition (but that’s for another time).

My main interest is this: if singularitarianism is a new religious movement, then what should we make of it? Will it mainly be a good thing? A kind of enlightenment religion? It might be an excellent alternative to old-fashioned Abrahamic religion. Or would it degenerate into the well-known tragic pattern of coercive authority? Time will tell; but I think it’s worth thinking about this in much more detail.

To be clear: Singularitarianism is probably not a religion. It is certainly not a cult, as it has been even worse accused; the behaviors it prescribes are largely normative, pro-social behaviors, and therefore it would at worst be a mainstream religion. Really, if every religion only inspired people to do things like donate to famine relief and work on AI research (as opposed to, say, beheading gay people), I wouldn’t have much of a problem with religion.

In fact, Singularitarianism has one vital advantage over religion: Evidence. While the evidence in favor of it is not overwhelming, there is enough evidential support to lend plausibility to at least a broad concept of Singularitarianism: Technology will continue rapidly advancing, achieving accomplishments currently only in our wildest imaginings; artificial intelligence surpassing human intelligence will arise, sooner than many people think; human beings will change ourselves into something new and broadly superior; these posthumans will go on to colonize the galaxy and build a grander civilization than we can imagine. I don’t know that these things are true, but I hope they are, and I think it’s at least reasonably likely. All I’m really doing is extrapolating based on what human civilization has done so far and what we are currently trying to do now. Of course, we could well blow ourselves up before then, or regress to a lower level of technology, or be wiped out by some external force. But there’s at least a decent chance that we will continue to thrive for another million years to come.

But yes, Singularitarianism does in many ways resemble a religion: It offers a rich, emotionally fulfilling ontology combined with ethical prescriptions that require particular behaviors. It promises us a chance at immortality. It inspires us to work toward something much larger than ourselves. More importantly, it makes us special—we are among the unique few (millions?) who have the power to influence the direction of human and posthuman civilization for a million years. The stronger forms of Singularitarianism even have a flavor of apocalypse: When the AI comes, sooner than you think, it will immediately reshape everything at effectively infinite speed, so that from one year—or even one moment—to the next, our whole civilization will be changed. (These forms of Singularitarianism are substantially less plausible than the broader concept I outlined above.)

It’s this sense of specialness that Pascal’s Mugging provides some insight into. When it is suggested that we are so special, we should be inherently skeptical, not least because it feels good to hear that. (As Less Wrong would put it, we need to avoid a Happy Death Spiral.) Human beings like to feel special; we want to feel special. Our brains are configured to seek out evidence that we are special and reject evidence that we are not. This is true even to the point of absurdity: One cannot be mathematically coherent without admitting that the compliment “You’re one in a million.” is equivalent to the statement “There are seven thousand people as good or better than you.”—and yet, the latter seems much worse, because it does not make us sound special.

Indeed, the connection between Pascal’s Mugging and Pascal’s Wager is quite deep: Each argument takes a tiny probability and multiplies it by a huge impact in order to get a large expected utility. This often seems to be the way that religions defend themselves: Well, yes, the probability is small; but can you take the chance? Can you afford to take that bet if it’s really your immortal soul on the line?

And Singularitarianism has a similar case to make, even aside from the paradox of Pascal’s Mugging itself. The chief argument for why we should be focusing all of our time and energy on existential risk is that the potential payoff is just so huge that even a tiny probability of making a difference is enough to make it the only thing that matters. We should be especially suspicious of that; anything that says it is the only thing that matters is to be doubted with utmost care. The really dangerous religion has always been the fanatical kind that says it is the only thing that matters. That’s the kind of religion that makes you crash airliners into buildings.

I think some people may well have become Singularitarians because it made them feel special. It is exhilarating to be one of these lone few—and in the scheme of things, even a few million is a small fraction of all past and future humanity—with the power to effect some shift, however small, in the probability of a far grander, far brighter future.

Yet, in fact this is very likely the circumstance in which we are. We could have been born in the Neolithic, struggling to survive, utterly unaware of what would come a few millennia hence; we could have been born in the posthuman era, one of a trillion other artist/gamer/philosophers living in a world where all the hard work that needed to be done is already done. In the long S-curve of human development, we could have been born in the flat part on the left or the flat part on the right—and by all probability, we should have been; most people were. But instead we happened to be born in that tiny middle slice, where the curve slopes upward at its fastest. I suppose somebody had to be, and it might as well be us.

Sigmoid curve labeled

A priori, we should doubt that we were born so special. And when forming our beliefs, we should compensate for the fact that we want to believe we are special. But we do in fact have evidence, lots of evidence. We live in a time of astonishing scientific and technological progress.

My lifetime has included the progression from Deep Thought first beating David Levy to the creation of a computer one millimeter across that runs on a few nanowatts and nevertheless has ten times as much computing power as the 80-pound computer that ran the Saturn V. (The human brain runs on about 100 watts, and has a processing power of about 1 petaflop, so we can say that our energy efficiency is about 10 TFLOPS/W. The M3 runs on about 10 nanowatts and has a processing power of about 0.1 megaflops, so its energy efficiency is also about 10 TFLOPS/W. We did it! We finally made a computer as energy-efficient as the human brain! But we have still not matched the brain in terms of space-efficiency: The volume of the human brain is about 1000 cm^3, so our space efficiency is about 1 TFLOPS/cm^3. The volume of the M3 is about 1 mm^3, so its space efficiency is only about 100 MFLOPS/cm^3. The brain still wins by a factor of 10,000.)

My mother saw us go from the first jet airliners to landing on the Moon to the International Space Station and robots on Mars. She grew up before the polio vaccine and is still alive to see the first 3D-printed human heart. When I was a child, smartphones didn’t even exist; now more people have smartphones than have toilets. I may yet live to see the first human beings set foot on Mars. The pace of change is utterly staggering.

Without a doubt, this is sufficient evidence to believe that we, as a civilization, are living in a very special time. The real question is: Are we, as individuals, special enough to make a difference? And if we are, what weight of responsibility does this put upon us?

If you are reading this, odds are the answer to the first question is yes: You are definitely literate, and most likely educated, probably middle- or upper-middle-class in a First World country. Countries are something I can track, and I do get some readers from non-First-World countries; and of course I don’t observe your education or socioeconomic status. But at an educated guess, this is surely my primary reading demographic. Even if you don’t have the faintest idea what I’m talking about when I use Bayesian logic or calculus, you’re already quite exceptional. (And if you do? All the more so.)

That means the second question must apply: What do we owe these future generations who may come to exist if we play our cards right? What can we, as individuals, hope to do to bring about this brighter future?

The Singularitarian community will generally tell you that the best thing to do with your time is to work on AI research, or, failing that, the best thing to do with your money is to give it to people working on artificial intelligence research. I’m not going to tell you not to work on AI research or donate to AI research, as I do think it is among the most important things humanity needs to be doing right now, but I’m also not going to tell you that it is the one single thing you must be doing.

You should almost certainly be donating somewhere, but I’m not so sure it should be to AI research. Maybe it should be famine relief, or malaria prevention, or medical research, or human rights, or environmental sustainability. If you’re in the United States (as I know most of you are), the best thing to do with your money may well be to support political campaigns, because US political, economic, and military hegemony means that as goes America, so goes the world. Stop and think for a moment how different the prospects of global warming might have been—how many millions of lives might have been saved!—if Al Gore had become President in 2001. For lack of a few million dollars in Tampa twenty years ago, Miami may be gone in fifty. If you’re not sure which cause is most important, just pick one; or better yet, donate to a diversified portfolio of charities and political campaigns. Diversified investment isn’t just about monetary return.

And you should think carefully about what you’re doing with the rest of your life. This can be hard to do; we can easily get so caught up in just getting through the day, getting through the week, just getting by, that we lose sight of having a broader mission in life. Of course, I don’t know what your situation is; it’s possible things really are so desperate for you that you have no choice but to keep your head down and muddle through. But you should also consider the possibility that this is not the case: You may not be as desperate as you feel. You may have more options than you know. Most “starving artists” don’t actually starve. More people regret staying in their dead-end jobs than regret quitting to follow their dreams. I guess if you stay in a high-paying job in order to earn to give, that might really be ethically optimal; but I doubt it will make you happy. And in fact some of the most important fields are constrained by a lack of good people doing good work, and not by a simple lack of funding.

I see this especially in economics: As a field, economics is really not focused on the right kind of questions. There’s far too much prestige for incrementally adjusting some overcomplicated unfalsifiable mess of macroeconomic algebra, and not nearly enough for trying to figure out how to mitigate global warming, how to turn back the tide of rising wealth inequality, or what happens to human society once robots take all the middle-class jobs. Good work is being done in devising measures to fight poverty directly, but not in devising means to undermine the authoritarian regimes that are responsible for maintaining poverty. Formal mathematical sophistication is prized, and deep thought about hard questions is eschewed. We are carefully arranging the pebbles on our sandcastle in front of the oncoming tidal wave. I won’t tell you that it’s easy to change this—it certainly hasn’t been easy for me—but I have to imagine it’d be easier with more of us trying rather than with fewer. Nobody needs to donate money to economics departments, but we definitely do need better economists running those departments.

You should ask yourself what it is that you are really good at, what you—you yourself, not anyone else—might do to make a mark on the world. This is not an easy question: I have not quite answered for myself whether I would make more difference as an academic researcher, a policy analyst, a nonfiction author, or even a science fiction author. (If you scoff at the latter: Who would have any concept of AI, space colonization, or transhumanism, if not for science fiction authors? The people who most tilted the dial of human civilization toward this brighter future may well be Clarke, Roddenberry, and Asimov.) It is not impossible to be some combination or even all of these, but the more I try to take on the more difficult my life becomes.

Your own path will look different than mine, different, indeed, than anyone else’s. But you must choose it wisely. For we are very special individuals, living in a very special time.

How do people think about probability?

PNRJ cognitive science, critique of neoclassical economics , , , , , , , , , , , ,

Nov 27, JDN 2457690

(This topic was chosen by vote of my Patreons.)

In neoclassical theory, it is assumed (explicitly or implicitly) that human beings judge probability in something like the optimal Bayesian way: We assign prior probabilities to events, and then when confronted with evidence we infer using the observed data to update our prior probabilities to posterior probabilities. Then, when we have to make decisions, we maximize our expected utility subject to our posterior probabilities.

This, of course, is nothing like how human beings actually think. Even very intelligent, rational, numerate people only engage in a vague approximation of this behavior, and only when dealing with major decisions likely to affect the course of their lives. (Yes, I literally decide which universities to attend based upon formal expected utility models. Thus far, I’ve never been dissatisfied with a decision made that way.) No one decides what to eat for lunch or what to do this weekend based on formal expected utility models—or at least I hope they don’t, because that point the computational cost far exceeds the expected benefit.

So how do human beings actually think about probability? Well, a good place to start is to look at ways in which we systematically deviate from expected utility theory.

A classic example is the Allais paradox. See if it applies to you.

In game A, you get $1 million dollars, guaranteed.
In game B, you have a 10% chance of getting $5 million, an 89% chance of getting $1 million, but now you have a 1% chance of getting nothing.

Which do you prefer, game A or game B?

In game C, you have an 11% chance of getting $1 million, and an 89% chance of getting nothing.

In game D, you have a 10% chance of getting $5 million, and a 90% chance of getting nothing.

Which do you prefer, game C or game D?

I have to think about it for a little bit and do some calculations, and it’s still very hard because it depends crucially on my projected lifetime income (which could easily exceed $3 million with a PhD, especially in economics) and the precise form of my marginal utility (I think I have constant relative risk aversion, but I’m not sure what parameter to use precisely), but in general I think I want to choose game A and game C, but I actually feel really ambivalent, because it’s not hard to find plausible parameters for my utility where I should go for the gamble.

But if you’re like most people, you choose game A and game D.

There is no coherent expected utility by which you would do this.

Why? Either a 10% chance of $5 million instead of $1 million is worth risking a 1% chance of nothing, or it isn’t. If it is, you should play B and D. If it’s not, you should play A and C. I can’t tell you for sure whether it is worth it—I can’t even fully decide for myself—but it either is or it isn’t.

Yet most people have a strong intuition that they should take game A but game D. Why? What does this say about how we judge probability?
The leading theory in behavioral economics right now is cumulative prospect theory, developed by the great Kahneman and Tversky, who essentially founded the field of behavioral economics. It’s quite intimidating to try to go up against them—which is probably why we should force ourselves to do it. Fear of challenging the favorite theories of the great scientists before us is how science stagnates.

I wrote about it more in a previous post, but as a brief review, cumulative prospect theory says that instead of judging based on a well-defined utility function, we instead consider gains and losses as fundamentally different sorts of thing, and in three specific ways:

First, we are loss-averse; we feel a loss about twice as intensely as a gain of the same amount.

Second, we are risk-averse for gains, but risk-seeking for losses; we assume that gaining twice as much isn’t actually twice as good (which is almost certainly true), but we also assume that losing twice as much isn’t actually twice as bad (which is almost certainly false and indeed contradictory with the previous).

Third, we judge probabilities as more important when they are close to certainty. We make a large distinction between a 0% probability and a 0.0000001% probability, but almost no distinction at all between a 41% probability and a 43% probability.

That last part is what I want to focus on for today. In Kahneman’s model, this is a continuous, monotonoic function that maps 0 to 0 and 1 to 1, but systematically overestimates probabilities below but near 1/2 and systematically underestimates probabilities above but near 1/2.

It looks something like this, where red is true probability and blue is subjective probability:

cumulative_prospect
I don’t believe this is actually how humans think, for two reasons:

  1. It’s too hard. Humans are astonishingly innumerate creatures, given the enormous processing power of our brains. It’s true that we have some intuitive capacity for “solving” very complex equations, but that’s almost all within our motor system—we can “solve a differential equation” when we catch a ball, but we have no idea how we’re doing it. But probability judgments are often made consciously, especially in experiments like the Allais paradox; and the conscious brain is terrible at math. It’s actually really amazing how bad we are at math. Any model of normal human judgment should assume from the start that we will not do complicated math at any point in the process. Maybe you can hypothesize that we do so subconsciously, but you’d better have a good reason for assuming that.
  2. There is no reason to do this. Why in the world would any kind of optimization system function this way? You start with perfectly good probabilities, and then instead of using them, you subject them to some bizarre, unmotivated transformation that makes them less accurate and costs computing power? You may as well hit yourself in the head with a brick.

So, why might it look like we are doing this? Well, my proposal, admittedly still rather half-baked, is that human beings don’t assign probabilities numerically at all; we assign them categorically.

You may call this, for lack of a better term, categorical prospect theory.

My theory is that people don’t actually have in their head “there is an 11% chance of rain today” (unless they specifically heard that from a weather report this morning); they have in their head “it’s fairly unlikely that it will rain today”.

That is, we assign some small number of discrete categories of probability, and fit things into them. I’m not sure what exactly the categories are, and part of what makes my job difficult here is that they may be fuzzy-edged and vary from person to person, but roughly speaking, I think they correspond to the sort of things psychologists usually put on Likert scales in surveys: Impossible, almost impossible, very unlikely, unlikely, fairly unlikely, roughly even odds, fairly likely, likely, very likely, almost certain, certain. If I’m putting numbers on these probability categories, they go something like this: 0, 0.001, 0.01, 0.10, 0.20, 0.50, 0.8, 0.9, 0.99, 0.999, 1.

Notice that this would preserve the same basic effect as cumulative prospect theory: You care a lot more about differences in probability when they are near 0 or 1, because those are much more likely to actually shift your category. Indeed, as written, you wouldn’t care about a shift from 0.4 to 0.6 at all, despite caring a great deal about a shift from 0.001 to 0.01.

How does this solve the above problems?

  1. It’s easy. Not only don’t you compute a probability and then recompute it for no reason; you never even have to compute it precisely. Just get it within some vague error bounds and that will tell you what box it goes in. Instead of computing an approximation to a continuous function, you just slot things into a small number of discrete boxes, a dozen at the most.
  2. That explains why we would do it: It’s easy. Our brains need to conserve their capacity, and they did especially in our ancestral environment when we struggled to survive. Rather than having to iterate your approximation to arbitrary precision, you just get within 0.1 or so and call it a day. That saves time and computing power, which saves energy, which could save your life.

What new problems have I introduced?

  1. It’s very hard to know exactly where people’s categories are, if they vary between individuals or even between situations, and whether they are fuzzy-edged.
  2. If you take the model I just gave literally, even quite large probability changes will have absolutely no effect as long as they remain within a category such as “roughly even odds”.

With regard to 2, I think Kahneman may himself be able to save me, with his dual process theory concept of System 1 and System 2. What I’m really asserting is that System 1, the fast, intuitive judgment system, operates on these categories. System 2, on the other hand, the careful, rational thought system, can actually make use of proper numerical probabilities; it’s just very costly to boot up System 2 in the first place, much less ensure that it actually gets the right answer.

How might we test this? Well, I think that people are more likely to use System 1 when any of the following are true:

  1. They are under harsh time-pressure
  2. The decision isn’t very important
  3. The intuitive judgment is fast and obvious

And conversely they are likely to use System 2 when the following are true:

  1. They have plenty of time to think
  2. The decision is very important
  3. The intuitive judgment is difficult or unclear

So, it should be possible to arrange an experiment varying these parameters, such that in one treatment people almost always use System 1, and in another they almost always use System 2. And then, my prediction is that in the System 1 treatment, people will in fact not change their behavior at all when you change the probability from 15% to 25% (fairly unlikely) or 40% to 60% (roughly even odds).

To be clear, you can’t just present people with this choice between game E and game F:

Game E: You get a 60% chance of $50, and a 40% chance of nothing.

Game F: You get a 40% chance of $50, and a 60% chance of nothing.

People will obviously choose game E. If you can directly compare the numbers and one game is strictly better in every way, I think even without much effort people will be able to choose correctly.

Instead, what I’m saying is that if you make the following offers to two completely different sets of people, you will observe little difference in their choices, even though under expected utility theory you should.
Group I receives a choice between game E and game G:

Game E: You get a 60% chance of $50, and a 40% chance of nothing.

Game G: You get a 100% chance of $20.

Group II receives a choice between game F and game G:

Game F: You get a 40% chance of $50, and a 60% chance of nothing.

Game G: You get a 100% chance of $20.

Under two very plausible assumptions about marginal utility of wealth, I can fix what the rational judgment should be in each game.

The first assumption is that marginal utility of wealth is decreasing, so people are risk-averse (at least for gains, which these are). The second assumption is that most people’s lifetime income is at least two orders of magnitude higher than $50.

By the first assumption, group II should choose game G. The expected income is precisely the same, and being even ever so slightly risk-averse should make you go for the guaranteed $20.

By the second assumption, group I should choose game E. Yes, there is some risk, but because $50 should not be a huge sum to you, your risk aversion should be small and the higher expected income of $30 should sway you.

But I predict that most people will choose game G in both cases, and (within statistical error) the same proportion will choose F as chose E—thus showing that the difference between a 40% chance and a 60% chance was in fact negligible to their intuitive judgments.

However, this doesn’t actually disprove Kahneman’s theory; perhaps that part of the subjective probability function is just that flat. For that, I need to set up an experiment where I show discontinuity. I need to find the edge of a category and get people to switch categories sharply. Next week I’ll talk about how we might pull that off.

What do we mean by “risk”?

PNRJ critique of neoclassical economics , , , , , , , , , ,

JDN 2457118 EDT 20:50.

In an earlier post I talked about how, empirically, expected utility theory can’t explain the fact that we buy both insurance and lottery tickets, and how, normatively it really doesn’t make a lot of sense to buy lottery tickets precisely because of what expected utility theory says about them.

But today I’d like to talk about one of the major problems with expected utility theory, which I consider one of the major unexplored frontiers of economics: Expected utility theory treats all kinds of risk exactly the same.

In reality there are three kinds of risk: The first is what I’ll call classical risk, which is like the game of roulette; the odds are well-defined and known in advance, and you can play the game a large number of times and average out the results. This is where expected utility theory really shines; if you are dealing with classical risk, expected utility is obviously the way to go and Von Neumann and Morgenstern quite literally proved mathematically that anything else is irrational.

The second is uncertainty, a distinction which was most famously expounded by Frank Knight, an economist at the University of Chicago. (Chicago is a funny place; on the one hand they are a haven for the madness that is Austrian economics; on the other hand they have led the charge in behavioral and cognitive economics. Knight was a perfect fit, because he was a little of both.) Uncertainty is risk under ill-defined or unknown probabilities, where there is no way to play the game twice. Most real-world “risk” is actually uncertainty: Will the People’s Republic of China collapse in the 21st century? How many deaths will global warming cause? Will human beings ever colonize Mars? Is P = NP? None of those questions have known answers, but nor can we clearly assign probabilities either; Either P = NP or not, as a mathematical theorem (or, like the continuum hypothesis, it’s independent of ZFC, the most bizarre possibility of all), and it’s not as if someone is rolling dice to decide how many people global warming will kill. You can think of this in terms of “possible worlds”, though actually most modal theorists would tell you that we can’t even say that P=NP is possible (nor can we say it isn’t possible!) because, as a necessary statement, it can only be possible if it is actually true; this follows from the S5 axiom of modal logic, and you know what, even I am already bored with that sentence. Clearly there is some sense in which P=NP is possible, and if that’s not what modal logic says then so much the worse for modal logic. I am not a modal realist (not to be confused with a moral realist, which I am); I don’t think that possible worlds are real things out there somewhere. I think possibility is ultimately a statement about ignorance, and since we don’t know that P=NP is false then I contend that it is possible that it is true. Put another way, it would not be obviously irrational to place a bet that P=NP will be proved true by 2100; but if we can’t even say that it is possible, how can that be?

Anyway, that’s the mess that uncertainty puts us in, and almost everything is made of uncertainty. Expected utility theory basically falls apart under uncertainty; it doesn’t even know how to give an answer, let alone one that is correct. In reality what we usually end up doing is waving our hands and trying to assign a probability anyway—because we simply don’t know what else to do.

The third one is not one that’s usually talked about, yet I think it’s quite important; I will call it one-shot risk. The probabilities are known or at least reasonably well approximated, but you only get to play the game once. You can also generalize to few-shot risk, where you can play a small number of times, where “small” is defined relative to the probabilities involved; this is a little vaguer, but basically what I have in mind is that even though you can play more than once, you can’t play enough times to realistically expect the rarest outcomes to occur. Expected utility theory almost works on one-shot and few-shot risk, but you have to be very careful about taking it literally.

I think an example make things clearer: Playing the lottery is a few-shot risk. You can play the lottery multiple times, yes; potentially hundreds of times in fact. But hundreds of times is nothing compared to the 1 in 400 million chance you have of actually winning. You know that probability; it can be computed exactly from the rules of the game. But nonetheless expected utility theory runs into some serious problems here.

If we were playing a classical risk game, expected utility would obviously be right. So for example if you know that you will live one billion years, and you are offered the chance to play a game (somehow compensating for the mind-boggling levels of inflation, economic growth, transhuman transcendence, and/or total extinction that will occur during that vast expanse of time) in which at each year you can either have a guaranteed $40,000 of inflation-adjusted income or a 99.999,999,75% chance of $39,999 of inflation-adjusted income and a 0.000,000,25% chance of $100 million in inflation-adjusted income—which will disappear at the end of the year, along with everything you bought with it, so that each year you start afresh. Should you take the second option? Absolutely not, and expected utility theory explains why; that one or two years where you’ll experience 8 QALY per year isn’t worth dropping from 4.602056 QALY per year to 4.602049 QALY per year for the other nine hundred and ninety-eight million years. (Can you even fathom how long that is? From here, one billion years is all the way back to the Mesoproterozoic Era, which we think is when single-celled organisms first began to reproduce sexually. The gain is to be Mitt Romney for a year or two; the loss is the value of a dollar each year over and over again for the entire time that has elapsed since the existence of gamete meiosis.) I think it goes without saying that this whole situation is almost unimaginably bizarre. Yet that is implicitly what we’re assuming when we use expected utility theory to assess whether you should buy lottery tickets.

The real situation is more like this: There’s one world you can end up in, and almost certainly will, in which you buy lottery tickets every year and end up with an income of $39,999 instead of $40,000. There is another world, so unlikely as to be barely worth considering, yet not totally impossible, in which you get $100 million and you are completely set for life and able to live however you want for the rest of your life. Averaging over those two worlds is a really weird thing to do; what do we even mean by doing that? You don’t experience one world 0.000,000,25% as much as the other (whereas in the billion-year scenario, that is exactly what you do); you only experience one world or the other.

In fact, it’s worse than this, because if a classical risk game is such that you can play it as many times as you want as quickly as you want, we don’t even need expected utility theory—expected money theory will do. If you can play a game where you have a 50% chance of winning $200,000 and a 50% chance of losing $50,000, which you can play up to once an hour for the next 48 hours, and you will be extended any credit necessary to cover any losses, you’d be insane not to play; your 99.9% confidence level of wealth at the end of the two days is from $850,000 to $6,180,000. While you may lose money for awhile, it is vanishingly unlikely that you will end up losing more than you gain.

Yet if you are offered the chance to play this game only once, you probably should not take it, and the reason why then comes back to expected utility. If you have good access to credit you might consider it, because going $50,000 into debt is bad but not unbearably so (I did, going to college) and gaining $200,000 might actually be enough better to justify the risk. Then the effect can be averaged over your lifetime; let’s say you make $50,000 per year over 40 years. Losing $50,000 means making your average income $48,750, while gaining $200,000 means making your average income $55,000; so your QALY per year go from a guaranteed 4.70 to a 50% chance of 4.69 and a 50% chance of 4.74; that raises your expected utility from 4.70 to 4.715.

But if you don’t have good access to credit and your income for this year is $50,000, then losing $50,000 means losing everything you have and living in poverty or even starving to death. The benefits of raising your income by $200,000 this year aren’t nearly great enough to take that chance. Your expected utility goes from 4.70 to a 50% chance of 5.30 and a 50% chance of zero.

So expected utility theory only seems to properly apply if we can play the game enough times that the improbable events are likely to happen a few times, but not so many times that we can be sure our money will approach the average. And that’s assuming we know the odds and we aren’t just stuck with uncertainty.

Unfortunately, I don’t have a good alternative; so far expected utility theory may actually be the best we have. But it remains deeply unsatisfying, and I like to think we’ll one day come up with something better.