Thursday, March 16, 2006

Non-zero and being Risk-Averse

As an applied physics major in college, one of the things that I found difficult was solving many of the more complex equations that you would come across in mechanics (Orbital equations), E&M (Maxwell's equations), quantum mechanics (Schrodinger's) and the like. This is not to say that I didn't like or wasn't any good at math. But when solving such equations, you invariably needed to reduce some number of terms to zero because they were negligible relative to the other terms, and I just never could wrap my head around that. (As an example, take a look at equations 4.53-4.55 here)

At what point does it really become negligible? A hundred to one? A thousand? I mean, it's non-zero, right? That means it's gotta have some kind of effect, or so I thought.

In what may appear to be an unrelated note, Monkeyboy linked to this cool article a few days (weeks?) ago. Even though the article is about risk, I particularly like this quote:

Advice to people wishing to become smarter: Get in the habit of assuming that everything is more complex than you imagine.

Anyway, it makes some interesting points on how to evaluate risk -- beyond just straight probabilities and expected ROI -- using examples of the lottery, insurance and... Russian Roulette. The author suggests that, as a thought experiment, you might consider how much you value your own life, in very tangible, quantitative terms:
Suppose you have the option to play Russian Roulette, in return for which you will receive a fee of x. The gun has one million chambers, one of which holds a bullet. If you get the bullet, you die. Otherwise you collect the fee. What is the minimum value for x that will induce you to play? Would you play if x were one million dollars?
The author says he'd do it for a million. (the discussion follows with another step or two.) Me, I couldn't find a workable number for x. I'm sure part of that stems from a lesson I learned a number of years ago that can be summarized in a fortune that read: "an ounce of gold cannot buy an ounce of time." (Let's just say bonus plans can be evil. In essence, for a carrot, THEY OWN YOU. Ok, lesson learned.) But, I think a larger aspect is that even when you're talking about a million to one odds, IT'S STILL ONE -- a finite number. Non-zero, baby. And you know what? Shit happens. Shit happens to me. And I guess maybe I value life too much. Perhaps under other circumstances, my response might differ. But given my present circumstances, I value life, my life, too much. Or at least I value it more than money.

Tristram Shandy said...

Interesting point, but of course, setting terms to zero is something one most often does in physics to see if it works. The curl of the electric field should be zero in a vacuum, so you run the numbers in Maxwell’s Equations and you get… Light! It’s a bit like looking for your keys under the lampost, but if you don’t, you end up with a whole bunch of insoluble problems and physics becomes rather depressing. Especially now in the time of computational science, analytic solutions are only vague approximations of many phenomena that are currently being explored with greater physical precision (albeit with different choices of approximation) by computers.

As far as risk goes, you don’t get paid a million dollars to get in your car and drive somewhere, but what are the chances you’ll be killed in an auto accident? Not to sound too much like an advertising campaign, but life takes risks. They’re unavoidable. Just as the curl of E ain’t quite zero in air (as opposed to a vacuum), the probability of most events never vanishes entirely.

Speaking of which, we just opened our show on “Cosmic Collisions” at the Museum. The Educators Guide even has a section on risk as it relates to potential impacts.

A thought-provoking post.

Mark Dominus said...

Well, you say that, but I bet you don't really mean it. Let me give you an example. Let us suppose you own a bicycle helmet. When you bought your bicycle helmet, did you buy the very top of the line, the one that offered the very best protection to your precious, irreplaceable head? Or did you opt to strike a balance between safety and cost, accepting a higher risk of injury and death in return for more money in your pocket?

And if you *don't* own a bicycle helmet, why the heck not? Let's not hear any nonsense about not owning a bicycle! What if you trip and fall walking down the streat and crack your head open? Hey, the odds might be a million to one, but it's still one, a finite number, not zero, as you said!

So why don't you go around wearing a top-of-the-line bicycle helmet everywhere?

tallasiandude said...

Well, the snarky answer would be that the bicycle helmet doesn't fit over the twelve inches of padding that I normally choose to encase my body within on my day-to-day travels. And it makes it hard to get into a car, too.

But yes, all hyperbole aside, you both have a point, and it all gets back to utility. The bottom line: I don't really NEED ten million dollars, but I do need to drive my car (to get to the office once in a while so they'll keep paying me, which then translates directly into food and shelter). At least, I need to drive more so than not.

Still, the Russian Roulette exercise somehow breaks down for me -- something about how it scales (or doesn't scale) -- although I'm having difficultly putting my finger on exactly what it is that doesn't jive.

Perhaps it has something to do with perception and perceived risk, which seems to somehow recursively feed back into the ideas of utility and costs. I'm afraid of bullets; I'm ok with asphalt. (There's also some component of control in the car and bicycle examples. The RR game is pure numbers. But that's probably just feeding into the perception equation.)

Perhaps my answer doesn't really put a value on my life, it just says that right now, when I stop and think about it, life is good. My needs are met, and then some. Why be greedy, y'know? (That may just be utility at it's boundary conditions, I suppose. Or the whole problem with trying to wrap your brain around very large numbers.)

Or maybe it's just an indication that I have a hard time with the idea of assigning value to life in terms of dollars and cents, or really any kind of concrete measurement system. That whole everyone-finds-their-own-path kind of thing.

Mathematically, I'd just call it orthogonal.

degustibus@gmail.com said...

You should leave aside statistics in considering Russian Roulette. You have no desire to kill yourself. If you put a gun to your head and pull the trigger then you are taking action which really has only two possible outcomes, one is immediate death by your own hand. Even people who don't particularly fear death resist suicide (Socrates comes to mind). It's the difference between a fighter pilot who volunteers for an exceedingly dangerous mission and the kamikaze. Yes, life is full or risk and no one here gets out alive, but that doesn't mean one should be willing to kill oneself for some money. Granted, people start smoking knowing the statistics so I'm sure plenty would take the gamble for 1,000,000. People risk their lives just to sneak across the desert in order to pick strawberries or steal cars from Home Depot all the time not far from where I sit.

yes1 said...
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