Editore"s Note
Tilting at Windmills

Email Newsletter icon, E-mail Newsletter icon, Email List icon, E-mail List icon Sign up for Free News & Updates

January 27, 2006
By: Kevin Drum

RISKY BUSINESS....Virginia Postrel has an interesting column in the New York Times today. At least, it's interesting for people like me who are fascinated by research into how people evaluate risk and uncertainty. First, there's a brief test:

  1. A bat and a ball cost $1.10 in total. The bat costs $1 more than the ball. How much does the ball cost?

  2. If it takes five machines five minutes to make five widgets, how long would it take 100 machines to make 100 widgets?

  3. In a lake, there is a patch of lily pads. Every day, the patch doubles in size. If it takes 48 days for the patch to cover the entire lake, how long would it take for the patch to cover half the lake?

What makes this interesting is that it's not really a math/logic test. That is to say, it is a math/logic test, but Shane Frederick of MIT says it's more than that. It's an indicator of your tolerance for risk: high scorers tend to prefer risky gambles more than low scorers, even when the gambles aren't especially favorable. On the other hand, there's also this:

For instance, 80 percent of high-scoring men would pick a 15 percent chance of $1 million over a sure $500, compared with only 38 percent of high-scoring women, 40 percent of low-scoring men and 25 percent of low-scoring women.

Unless that's a misprint, I just have to wonder what kind of moron would take $500 over a 15% chance of a million bucks? That's crazy unless you're dead broke and a goon with a baseball bat is coming after you with your kneecaps in his sights.

Anyway, read the whole thing. Interesting stuff. Quiz answers are at the end.

Kevin Drum 1:59 PM Permalink | Trackbacks | Comments (193)

Bookmark and Share
 
Comments

Whatever. Math (or, math/logic) is hard. If you're into NYT columns, I thought Krugman today was more interesting than this men are from Mars/women are from Venus malarkey.

Posted by: Chocolate Thunder on January 27, 2006 at 2:03 PM | PERMALINK

Would you take $1 million sure thing over a 15% shot at $100 million?

Posted by: Ron Byers on January 27, 2006 at 2:04 PM | PERMALINK

Those are common IQ-test-type questions, although the last one usually is used in conjuction with describing the growth of bacteria. You'd think that the tendency to gamble would decrease with the tendency to think things through. Go figure.

Posted by: tbrosz on January 27, 2006 at 2:07 PM | PERMALINK

Now you're talking Ron! That's a good question.


I like these posts, Kevin. The trolls stay home and the geeks come out to play.

Execpt that there's a sex angle, which always tends to spoil the fun. Have to ask my wife that one.

Posted by: David in NY on January 27, 2006 at 2:09 PM | PERMALINK

Unless that's a misprint, I just have to wonder what kind of moron would take $500 over a 15% chance of a million bucks?

Well, I would think that would be the "morons" who understand risk much better than you. $500.00 is a much better return for nothing than an 85% chance at nothing.

Keep buying those Power Ball tickets Kevin.

Posted by: Jeff II on January 27, 2006 at 2:09 PM | PERMALINK

Ron B., a very good question that illustrates how we satisfice, or choose an outcome that's good enough.

Posted by: David W. on January 27, 2006 at 2:11 PM | PERMALINK

I think how one is doing economically and age would have something to do with willingness to take risk.

I'd pay $500 for a 15% chance at a million dollars (but I'd take the sure million in Ron Byers scenario).

Posted by: Frank J. on January 27, 2006 at 2:11 PM | PERMALINK

tbroz,
The successful take risks and can figure what risks are worth taking.

Posted by: Frank J. on January 27, 2006 at 2:13 PM | PERMALINK

Jeff II,
Apparently, people who do poorly at logic tests think the way you just described.

Posted by: Frank J. on January 27, 2006 at 2:14 PM | PERMALINK

It's cute the way Ms. Postrel insists several times that "there's no right answer", while making clear from her other comments that smart people are expected to do the calculation and maximize the expected value, subject to an economist-approved discount rate.

So someone who would rather have $3400 this month than $3800 next month isn't necessarily wrong because of the large discount rate this implies. About half the country is living from paycheck to paycheck and is worried about how they're going to get next month's rent. People also have experience that makes them interpret the problem differently than stated: $3400 cash in hand is a sure thing; a promise for $3800 next month might be broken.

Posted by: Joe Buck on January 27, 2006 at 2:14 PM | PERMALINK

Would you take $1 million sure thing over a 15% shot at $100 million?

That gets to the heart of the risk analysis. For most of us $1 million will make a big difference, so it'd be a bad risk to pass that up for a shot at $100 million. But for a lot of middle-class people, a $500 sure thing versus 15% shot at $1 million is pretty straightforward: $500 will have a minor impact on one's life so why not risk going for the $1 million. As for Kevin's point, someone who's destitute may see $500 as having a huge impact and worth taking over the risk of trying for $1 million.


Posted by: puppethead on January 27, 2006 at 2:15 PM | PERMALINK

Unless that's a misprint, I just have to wonder what kind of moron would take $500 over a 15% chance of a million bucks?

A poor college student might make this choice. And you might make the same choice if the amount were $5000 (which might be of equivalent value to you as the $500 is to the college student).

Posted by: Jennifer Susse on January 27, 2006 at 2:16 PM | PERMALINK

"80 percent of high-scoring men would pick a 15 percent chance of $1 million over a sure $500."

Perhaps high-scoring men also have higher incomes, so that $500 wouldn't mean that much to them. Kevin's reaction to people who turn this bet down being a case in point.

Posted by: AB on January 27, 2006 at 2:16 PM | PERMALINK

In the interest of accuracy in the media, shouldn't the answer for the price of the ball be 10 cents.

Posted by: Chaz on January 27, 2006 at 2:19 PM | PERMALINK

I think Frank J.'s exactly right. I'd imagine the high scorers probably are probably in a higher income bracket (intelligence generally predicts income) than the low scorers and so $500 wouldn't make a huge difference to them, yet $1M would. For the low scorers in lower income brackets $500 might mean being able to make the rent this month.

In Ron Byer's scenario for a middle class or upper middle class person the $1M would be enough to hugely change their life (retire). Given the same choice to someone worth $50M and I would guess they might choose the 15% chance at $100M.

These statistics are meaningless unless they are controlled for income levels.

Posted by: Adventuregeek on January 27, 2006 at 2:21 PM | PERMALINK

How's the great American novel coming Frank?

Posted by: cq on January 27, 2006 at 2:23 PM | PERMALINK

Well, I was correct on the answers - Now, should I bet the 3 or the 4 horse in the first at Santa Anita today, or should I wheel for a Superfecta?

And I see the "Cal Pundit spammers" are alive and well today - 4 in a row. The Schaife foster home and day care center does such wonders with their little cretins.

Posted by: thethirdPaul on January 27, 2006 at 2:24 PM | PERMALINK

Unless that's a misprint, I just have to wonder what kind of moron would take $500 over a 15% chance of a million bucks?

Careful, your OC background is showing.

While I could certainly use $150,000, I definitely need $500.

Posted by: jerry on January 27, 2006 at 2:26 PM | PERMALINK

Two points:

I would take a 15% chance at one million dollars over a certain five-hundred dollars, and a certain one-million dollars over a 15% chance at one-hundred million.

The first reason is simply one of utility: Another $500 is nice, but it won't really change my life in a noticeable way. Just one more early mortgage payment. But a million dollars, even after taxes it finishes the mortgage, pays for the kids to go to college, and leaves some change for the future. Major life expenses cleared, and life significantly simplified.

But, $100 million is just icing on the cake. After all the other stuff is cleared up, you have gobs of cash left for living out dreams of consumption and philanthropy. It's nice, but it's not worth an 85% chance of losing the ability to radically simplify your life.

The second, more subtle point, is that the expected value heuristic is better when the game is repeated. (The heuristic is to just calculate .15 * $100,000,000 = $15,000,000 > $1,000,000 so take the chance for one hundred million). If you play the same wager once a week for the next thirty years, the law of large numbers kicks in and you will almost certainly have about 15 times as much money as those who play it safe. But if you get to play the game only once, you will probably get zilch. So risk taking is worth it only if you can take lots of risks.

Posted by: Nate on January 27, 2006 at 2:27 PM | PERMALINK

Thanks Kevin. I have been trying to write a chatty essay for the math phobic on how to solve word problems. The conversion to rate per machine is hard for people to get their minds around until they see it done.

I think the extra (and unstated) variable in the final question has to do with how much you need the money, or how much money you have ever had. In other words, it depends on how magnificent that sum of $500 actually looks to you. I could probably scrounge $500 somewhere if I desperately needed it, but the chance of winning $150,000 would be a true windfall. I would chance the 15 percent, because there isn't much I have available to me that could earn me an extra 150k. Somebody who has a car loan due the day after tomorrow will be more enticed by the sure thing.

In other words, it is less of a risk for somebody with rational expectations of high income to go for the jackpot. There will be other jackpots down the line.

There are, of course, other variables that we are being too polite to mention. People who can solve these problems have above average math talent, which means that statistically, they are more likely to be educated, to have higher income, and to have higher expectations of higher income. For them, the grand slam is not only a possibility, it is an expectation. You didn't give any data on the age breakdown of who took the test and how the scores broke down, but that would be another variable.

And last but not least, there is of course the unhappy fact that lots of people don't have a real good feel for what 15 percent of anything actually is. To the people who treat it as a mysterious question, the retreat to hard reality is easy.

Posted by: Bob G on January 27, 2006 at 2:27 PM | PERMALINK

All right, the first question is designed to get the incorrect reply of 10 cents, the last question is designed to get the incorrect response of 24 days, what incorrect answer is the second question driving at?

Posted by: hank on January 27, 2006 at 2:28 PM | PERMALINK

I just have to wonder what kind of moron would take $500 over a 15% chance of a million bucks? That's crazy unless you're dead broke and a goon with a baseball bat is coming after you with your kneecaps in his sights.

So, as joebuck and several commenters have already implied, it's not really a test of risk tolerance, either ... it's a test of economic insecurity.

Posted by: Swopa on January 27, 2006 at 2:29 PM | PERMALINK

cq,
I have a sure thing to do first before I invest more time in a risk.

Posted by: Frank J. on January 27, 2006 at 2:29 PM | PERMALINK

"I just have to wonder what kind of moron would take $500 over a 15% chance of a million bucks?"

Anyone who ever studied probability would.

The expected value of a sure $500 is just that, $500. The expected value of the million-dollar wager is (0.85 chance of nothing) + (0.15 chance of $1,000,000) = $150,000.

So you take the 15% shot, every time, and it ain't close. It's too bad people think this result is counter-intuitive. It's actually one of the founding principles of all statistics.

Posted by: mmy on January 27, 2006 at 2:30 PM | PERMALINK

My spouse notes that most women probably would take the sure money because they aren't sure about their financial strait.

That is, most women make less money and have less financial ndependence than men.

Posted by: Crissa on January 27, 2006 at 2:32 PM | PERMALINK

I suppose this is where we quote Brewster's Millions, right?

Posted by: Al on January 27, 2006 at 2:32 PM | PERMALINK

hank,
100 minutes would be my guess for the most common incorrect answer to number 2 (follows the pattern).

Posted by: Frank J. on January 27, 2006 at 2:32 PM | PERMALINK

In the real world, any promise of future money needs to be heavily discounted. What if the person offering you the deal doesn't really have $1 million? What happens if multiple people turn up to claim the prize -- does it go to court? What if you lose the lottery ticket?

Who is this person that's offering me a 15% chance at a million dollars? Those offers don't grow on trees. Why offer this to me -- doesn't this person have friends? The whole thing sounds like a scam. I know just what's going to happen: Next week, the 15% chance at $1 million will turn into a 20% chance at $5 million, which will suddenly morph into a 30% chance at $2 million, and then I will "win" and be able to "claim my prize" by mailing $15K in "taxes" to a P.O. box.

If I take the $500 cash, I get the money up front. No strings, no unknowns, no need to put faith in anyone.

Naturally, people who have lots of skill with story problems are more likely to interpret the question as a story problem, and will "correctly" take the lottery ticket over the cash. Nigerian scammers and lawyers don't exist in the happy world of story problems.

Posted by: Mike B on January 27, 2006 at 2:32 PM | PERMALINK

I am the statistical counter-example. I scored 100%, and I can tell you I am one of the most risk averse people I know, both with money and with more life and death matters (like skiing really fast). Of course, I am female, and that in and of itself skews the conclusions. On the other hand, I do fit the patience profile.
But what about accountants? Surely they would do well as a group on these tests, and surely they are risk averse as well?

Posted by: lisainVan on January 27, 2006 at 2:32 PM | PERMALINK

Oops, misread Kevin's post. Strong cough medicine will do that to you. He was actually saying the morons are the people who would take the $500, and I agree with him. If kids aren't learning this kind of math in junior high, something is seriously wrong.

Posted by: mmy on January 27, 2006 at 2:34 PM | PERMALINK

$500.00 is a much better return for nothing than an 85% chance at nothing

bah.

$500 would do essentially nothing for my overall financial situation. $1M would do a lot.

Posted by: cleek on January 27, 2006 at 2:36 PM | PERMALINK

In the interest of accuracy in the media, shouldn't the answer for the price of the ball be 10 cents.

No. The answer is derived from "x + (x + 1) = 1.10" or "2x + 1 = 1.10", where "x" is the price of the ball. Why 2x? Because the bat is one dollar more than the ball, which means the price of the bat has to include the amount of the ball plus the $1.

Posted by: puppethead on January 27, 2006 at 2:38 PM | PERMALINK

Since a few commenters seem to have missed my point, I was specifically saying that if you're really poor or desperate, the $500 sure thing would make sense. For most of us, even those of us who don't make much money, $500 is simply not enough that it's worth taking it when the alternative has an expected payout 300 times higher.

But yes, if you're really poor or desperate, then the $500 sure thing might make sense.

And of course, Ron is correct to ask how things change if you make it a sure $1 million instead of $500. Given the diminishing utility of money, there are a lot more people who would sensibly take that vs. a 15% chance of a billion dollars.

Posted by: Kevin Drum on January 27, 2006 at 2:39 PM | PERMALINK

"So you take the 15% shot, every time, and it ain't close. It's too bad people think this result is counter-intuitive. It's actually one of the founding principles of all statistics."

Expected utility = expected value is "one of the founding principles of all statistics"? Hmm.

Now, I'd totally take even a 15% shot at $100M over a sure thing of $1M, but my utility function is unusually close to linear pretty far out there.

Posted by: gundryggia on January 27, 2006 at 2:39 PM | PERMALINK

In the interest of accuracy in the media, shouldn't the answer for the price of the ball be 10 cents.

Posted by: Chaz on January 27, 2006 at 2:19 PM | PERMALINK

Only if the medium is Fox News, and W had decreed it so.

Posted by: Tony on January 27, 2006 at 2:39 PM | PERMALINK

So, as joebuck and several commenters have already implied, it's not really a test of risk tolerance, either ... it's a test of economic insecurity. Posted by: Swopa

Yes and no.

As mmy explained much more eloquently than me, in grossest terms, $500.00 is a sure thing. A 15% chance at $1M is really long odds.

However, if someone is already wealthy, it's correct that $500.00 perhaps means nothing to them. However, as someone firmly in the upper-middle income bracket, $500.00 pays for 4 days of skiing and lodging at Vail, and I can fly to Denver for free (long story). $500.00 buys a new set of boards and bindings this Spring when the sales start. $500.00 buys 100' of hedging I'll be putting in this spring. $500.00 pays for a roundtrip ticket to Kauai this summer. $500.00 in a Roth IRA will be several thousand dollars by the time I retire. So $500.00 is nothing to sneeze at.

Posted by: Jeff II on January 27, 2006 at 2:41 PM | PERMALINK

You need to mark up that "sure million" by a bucket or nine. A teardown on my block goes for more than that. A sure house, when I live in a nice rent-controlled apartment, isn't worth much to me and sure as hell doesn't mean retirement. Make it a sure $10M and I'll take it over a 15% chance at $100M, even though the expected value is much, much less. Otherwise, I'll take the free lottery ticket over the free house I don't really need.

Posted by: wcw on January 27, 2006 at 2:41 PM | PERMALINK

ROFL.

99% +-eps of Americans can't even add fractions, let alone think usefully about the concept of *expectation*.

And in other news, dog bites man.

Posted by: cdj on January 27, 2006 at 2:44 PM | PERMALINK

The article says the answers are:


5cents
5widgets
47 days


I don't get the first two answers.

Puppethead's answer doesn't help.

Posted by: Ace Franze on January 27, 2006 at 2:44 PM | PERMALINK

The problem with facing it at a 15% chance of $1,000,000 being valued at $150,000 is the assumption of repetitive risk.

If there is no repeition, it is worth precisely $0 to 85% and $150,000 to 15%.

Hence money now being more worthwhile.

Still, just because you can do the math has nothing on whether you will take actual risks. Risking $500 for $1,000,000 isn't an actual risk. If there's x number in a pool, there's no risk at all, and the pool is really small.

If 1/4 of people walking into a room get a TV, and you walk in with three people, you're gonna get a TV.

That's not a risk.

Posted by: Crissa on January 27, 2006 at 2:46 PM | PERMALINK

It's an indicator of your tolerance for risk: high scorers tend to prefer risky gambles more than low scorers, even when the gambles aren't especially favorable.

On the examples given, it seems to indicate "understanding" of risk. "... Even when the gambles aren't especially favorable" seems to mean "favorable but not especially so." If that is true, then the people who score high on the calculations can more easily recognize when the gambles are favorable. Again, that would be "understanding" of risk rather than "tolerance" of risk.

Posted by: contentious on January 27, 2006 at 2:48 PM | PERMALINK

It's an indicator of your tolerance for risk: high scorers tend to prefer risky gambles more than low scorers, even when the gambles aren't especially favorable.

First, there is a big difference between saying what you would hypothetically do, than making an actual choice with actual consequences.

Second, the hypotheticals don't appear to measure what people like as much as indicate what they think the "right" answer is.

Third, both scores on the quiz and answers to the risk assesments could reflect an abstract approach versus concrete. A concrete thinker may be more likely to take the "obvious" and incorrect answer as well as more likely to take concrete reward now over potential. Concrete thinkers and abstract thinkers approach things differently and there are smart and dumb of each.

Fourth, those who dont get the questions right may have less confidence of their ability to evaluate risk, thereby making them more likely to choose the safer option, even if they like risk when they understand it more or less.

(I have a relatively low tolerance for risk, and got 3 out of 3)

Posted by: Catch 22 on January 27, 2006 at 2:48 PM | PERMALINK

"A 15% chance at $1M is really long odds."

So is there ANY sum of money you'd pay for a 15% shot at $1M? $20? $50? $250?

Posted by: gundryggia on January 27, 2006 at 2:48 PM | PERMALINK

Humans do risk reward assessments all the time. Do I stay with this good steady job or do I risk it all to start my own business? Do I sell my business for a million dollars or wait for the 10 million it might be worth in a couple of years? Do I hunt buffalo or do I pick berries this morning? I am not sure but I think prudent risk taking is necessary to achieve any success in life. For our ancestors prudent risk taking was probably necessary for survival. Anyway sure things sure are boring. Most people are willing to take some risk. The only way to determine if a risk is worthwhile is to develop an deep understanding and appreciation of the facts surrounding the risk. Dispite what mathematitians might argue risk assessment is not a linear task. It should not be surprising that more intelligent people are willing to take greater risks. The true sign of intelligence is knowing just what risks to take.

Posted by: Ron Byers on January 27, 2006 at 2:49 PM | PERMALINK

OK, so the bat is $1.05, which is $1 more than .05, and $1.05 + .05= $1.10.

Posted by: Ace Franze on January 27, 2006 at 2:49 PM | PERMALINK

How to express this without math equations? If you buy the ball for 10 cents, how much does the bat cost? You know it's going to cost $1 more than the ball, so it must cost $1.10. That means you owe a total of $1.20, which is not the answer. But if the ball costs five cents, the bat will cost $1.05 and the total is $1.10, which is correct. The trick is that you must include the price of the ball in the price of the bat.

The second question is really the least-tricky, I think. If five machines make five widgets in five days, how long does it take each machine to make its own widget? The answer is five minutes. This means one machine makes one widget in five minutes, two machines make two widgets in five minutes, etc. So 100 machines will take the same five minutes to make 100 widgets. The key to this question is how long a single machine takes to do its work.

Posted by: puppethead on January 27, 2006 at 2:54 PM | PERMALINK

I don't get the first two answers.

5 cents:

if A costs a dollar more than B, and A + B = 1.10
then B + (B + 1.0) = 1.1.0
2B + 1.0 = 1.10
2B = .1
B = .1 / 2
B = .05

5widgets
first you have to assume each machine is independent of the others. if you do that, you know each machine takes 5 minutes to make a widget, no matter how many other machines there are.

Posted by: cleek on January 27, 2006 at 2:55 PM | PERMALINK

Put it this way: What number of people would have to pay $500 for the chance at a $1,000,000 to pay for the payouts?

There's many ways to look at risk: Utility, Expected value, etc...

Posted by: Crissa on January 27, 2006 at 2:55 PM | PERMALINK

However, it could also mean it takes five machines to make one widget...

Also, one hundred machines is twenty sets of five machines; twenty times five widgets is one hundred.

Posted by: Crissa on January 27, 2006 at 2:58 PM | PERMALINK

Unless that's a misprint, I just have to wonder what kind of moron would take $500 over a 15% chance of a million bucks? That's crazy unless you're dead broke and a goon with a baseball bat is coming after you with your kneecaps in his sights.

You're still missing it Kevin, but this is the situation much of America (that doesn't live behind the orange curtain) lives in. Unless by goons with baseball bats you are referring to Lieberman, Reid, Biden and their wonderful Bankruptcy Bill gift to the credit card industry.

Posted by: jerry on January 27, 2006 at 2:59 PM | PERMALINK

Nate: Excellent post.

Posted by: S Ra on January 27, 2006 at 3:00 PM | PERMALINK

So is there ANY sum of money you'd pay for a 15% shot at $1M? $20? $50? $250? Posted by: gundryggia

That's wasn't the equation. I'm not taking any risk nor "paying" anything taking the $500.00 because I'm not losing anything in exchange. That, as the expression goes, is free money.

However, to answer your question, the typical minimum lotto wager of $1.00 is certainly worth a 15% chance at any sum above the arbitrary figure of, say, $500.00 because the odds given beat any state lottery by a factor of several thousand.

Posted by: Jeff II on January 27, 2006 at 3:01 PM | PERMALINK

If you don't trust the person offering the $1 Million to run a fair bet (How do you know they won't somehow always miss the 15% chance to pay you.) than the $500 bet is safer.

Posted by: MobiusKlein on January 27, 2006 at 3:02 PM | PERMALINK

That's a pretty big lake by the way.

Posted by: Sam Shen on January 27, 2006 at 3:02 PM | PERMALINK

Nonsesne. People that can figure out those problmes are anmost assuredly better at figuring out actual odds and making a logical decision.

As for pinking the "risky" gamble (the 15% chance), that's an expected payout of $150K, an easy pick over the sure $500. Hardly risky to me, absent unstated externalities such as some pressing "necessity" for the $500 immediately (such as having to pay off a guard to escape a death sentence in some totalitarian country, or a need to buy food at once to keep from certain starvation).

I'd note that it is primarily poor and uneducated people that waste money on state lotteries, with expected payoffs of some 50 cents on the dollar. They do so based on poor risk analysis, but near as I can tell, the lure of a megamillions payoff is, in their mind, worth "more" than the actual monetary value; the idea of being a millionaire is appealing to them, and the lottery is the only chance they have of getting it, no matter how long the odds, and no matter that financially it's a "losing bet".

Cheers,


Posted by: Arne Langsetmo on January 27, 2006 at 3:02 PM | PERMALINK

For everyone, the emotion of money will cause you to have a personal answer. $500 isn't a lot of money (compared to $1,000,000), but to some it could make all the difference.

When you think of the question as a pure math problem you can go either direction:
1) make it so low, that people would surely pick the risk [guaranteed 5 cents vs. 15% chance at $100]

or 2) make it so large that no one will take the risk [guaranteed $500 trillion vs. 15% chance at $1 quintillion]

The math is the same, but the emotion has changed. Though, I sure would like to see the $1,000,000,000,000,000.00 check.

Posted by: rusrus on January 27, 2006 at 3:04 PM | PERMALINK

That's a pretty big lake by the way

yeah that's what i was thinking - or reallllly small lily pads.

2^48 is 281,474,976,710,656. two hundred trillion times the original size.

Posted by: cleek on January 27, 2006 at 3:05 PM | PERMALINK

I don't get the first two answers.

To solve these difference problems:
-- Take the difference out of the pot and set it aside
-- Split what's left in the pot (10 cents)
-- Then give all of the difference to the larger

In the second case, just look at what has changed. You have 20 times as many machines, but you have to make twenty times as many widgets, so they offset.

Posted by: Mornington Crescent on January 27, 2006 at 3:05 PM | PERMALINK

Bingo. Makes more sense if you remember $500 dollars ain't to Bill Gates what it is to Mary who needs to make rent.

Or, as the economists call it, "the declining marginal utility of money." Ton Byers nailed it--it's not that you don't understand the question, it's that the people who wrote the test didn't understand the anser.

Posted by: theorajones on January 27, 2006 at 3:07 PM | PERMALINK

"So is there ANY sum of money you'd pay for a 15% shot at $1M? $20? $50? $250? Posted by: gundryggia

That's wasn't the equation. I'm not taking any risk nor "paying" anything taking the $500.00 because I'm not losing anything in exchange. That, as the expression goes, is free money."

No, that is the question. If you take the 15% chance at $1M bet, you're paying $500 (the sure thing if you decline the bet) for a shot at the risky payout.

Posted by: gundryggia on January 27, 2006 at 3:08 PM | PERMALINK

I test does actually measure cognitive ability (and time preference). The thing is, women who do well are risk averse whereas men who do well are risk seeking.

People who score low are impatient and don't gamble. They want to use what they can get now and don't invest.

The study can be read here. (Via Marginal Revolution.)

Posted by: aaron on January 27, 2006 at 3:08 PM | PERMALINK

Hey!

What's with that slam at the end against my alma mater, Toledo U!? Leave the put-downs to those who've been there and lived to tell the tale!

(full disclosure: Engineering grad. got 3 out of 3)

Posted by: U of T grad on January 27, 2006 at 3:09 PM | PERMALINK

Or, to answer the author's question:

"Would You Take the Bird in the Hand, or a 75% Chance at the Two in the Bush?"

Depends how hungry I am.

Posted by: theorajones on January 27, 2006 at 3:09 PM | PERMALINK

Someone with low income who needs $400 for car repairs might want to just take the money.

They need the $400; it means something to them. But probably have no idea what to do with a million.

Posted by: aaron on January 27, 2006 at 3:15 PM | PERMALINK

No, that is the question. If you take the 15% chance at $1M bet, you're paying $500 (the sure thing if you decline the bet) for a shot at the risky payout. Posted by: gundryggia

No. You'd be paying it (apparently) because I'm not that dumb.

My 500 birds in the hand versus your 1M birds in the bush the next county over, and the bridge is out.

Posted by: Jeff II on January 27, 2006 at 3:17 PM | PERMALINK

I hate these things, they make me feel dumb. And I want the sure 500$, call me a idiot but 15% is no good.

Posted by: Joseph on January 27, 2006 at 3:18 PM | PERMALINK

We always preferred a "Bird" in the hand.

Posted by: Celtic Fan on January 27, 2006 at 3:22 PM | PERMALINK

Kevin, I bet five will get you ten you're wrong.

Posted by: Thinker on January 27, 2006 at 3:23 PM | PERMALINK

There's been some fairly definitive work done on risk attitude and on decisions under uncertainty.

Wealth DOES enter into the evaluation of uncertain prospects, and so $500 for certain can be seen as more attractive by some people than by others. If you have no money at all, for example, $500 could be very attractive indeed compared to an uncertain prospect of ANY amount of money. Conversely, as many posts upthread have indicated, if $500 won't make that much difference to you, but $1 million will, then you might be willing to take the risk.

Precise, spare formulations of this phenomenon have been constructed and are easy to populate and test.

One common misconception, though, is that an expected value calculation gives the answer. Expectation assumes risk-neutrality, which typically is only true for entities when they are considering amounts that are negligible as a fraction of their wealth. A 15% shot at $1 million is equivalent to a certain $150,000 only for very wealthy individuals and corporations. To all but a very few individuals, a certain $150,000 is preferable to a 15% chance at $1 million; this is because individuals are by and large risk-averse.

Posted by: bleh on January 27, 2006 at 3:23 PM | PERMALINK

It's amazing how many people on this thread are contorting the situation into some rationalization that taking the $500 is the correct choice. It clearly is not. I think a pretty strong argument can be made that the poor wouldn't have their lives changed with $500 anymore than the well off would.

And the "avoid eviction" argument has a whiff of the "approving torture in a ticking bomb" scenario.

Of course, if a poor person had an option of $500 vs. 15% chance of 1MM...he could just treat the option like a pollution credit and sell it for thousands, which it would EASILY bring. Then everyone's happy.

Posted by: michael on January 27, 2006 at 3:26 PM | PERMALINK

I have no doubt that many people spend more money on the lottery than they can really afford, and this is not a good thing.

However, it's completely simple-minded to just say "the expected payout is terrible, so it's dumb to ever play". For those without a problem (the majority, I would guess) gambling is a form of entertainment. Is it worth 1$ or 5$ to walk around for a few days with the faint buzz of "oooh, what if...." in your mind?

I don't know really. But it doesn't seem like any more obvious a waste of money than most of the other myriad things we "waste" money on for our own entertainment.

How much utility did you really get from dropping 300$ to upgrade your 17" monitor to the nice 19 incher you're looking at now?

Posted by: skippy on January 27, 2006 at 3:27 PM | PERMALINK

"And last but not least, there is of course the unhappy fact that lots of people don't have a real good feel for what 15 percent of anything actually is. To the people who treat it as a mysterious question, the retreat to hard reality is easy.

Posted by: Bob G"

This is a great subject, Kevin. Decision theory is hugely important in medicine. For example, surgeons have actually been asked to extrapolate the probablity of a breast cancer on biopsy from a 15% risk mammogram result. A majority will interpret a much higher risk of a cancer.

A clever demonstration was done at Dartmouth by a lecturer in decision theory. He flipped 5 coins, then laid them out in a line and asked a roomful of graduate students to estimate the order of heads and tails from left to right. Then, without looking at the estimates which had all been passed to the aisle and tabulated by another lecturer, he predicted that the sequence would be HTHHT in 76% of the papers. It was about 70%. His explanation was that human estimates of randomness are predictable. The student assumes that the sequence will be Heads, Tails, Heads, then realizes that the sequence will be HTHTH. He/she then changes to a HTHH to try to achieve "randomness."

Another was the choice heuristic. A student headed for the library is stopped by a friend and asked to a party. The probability is not evenly divided between library and party as a significant number will go home. They will avoid a choice. This was demonstrated in the New England Journal which showed in a paper last year that offering a choice of NSAIDS (like Motrin) to patients (with knee osteoarthritis) reduces the number who actually take one. Used car salesmen figured that out years ago and never show more than one car.

Intermountain Health Systems introduced a computer-based decision system in writing orders for respirators in ICU about 10 years ago and discovered that it reduced mortality by half.

Another reason why I support single payer in spite of otherwise libertarian politics.

Posted by: Mike K on January 27, 2006 at 3:28 PM | PERMALINK

If you take the 15% chance at $1M bet, you're paying $500 (the sure thing if you decline the bet) for a shot at the risky payout

i'm not "paying" it, since i never had it to begin with. if i miss the $1M, i walk away with not a penny less than i had to start with.

Posted by: cleek on January 27, 2006 at 3:29 PM | PERMALINK

Jeff II - Of course you are paying something for the $500 - the right to a lottery with a 15% chance of a payoff! Opportunity cost ...

Crissa - You are correct, for 85% the payoff is nothing, but for the other 15% the payoff is $1,000,000, not $150,000. That doesn't mean there aren't risk-averse people who will refuse to make the trade, but those people take risks with worse odds every day.

Posted by: Matt on January 27, 2006 at 3:30 PM | PERMALINK

Cleek - look up the term "opportunity cost".

Posted by: cdj on January 27, 2006 at 3:31 PM | PERMALINK

And this sort of thing should apply to real world economics as well.

A rich man could (if such an opportunity were available) easily buy a 15% chance at $1 million from a poor man for $500. Even better the rich man can buy 100 such chances for $50,000. Giving him a 99.999991% chance of tripling his money or better.

Posted by: jefff on January 27, 2006 at 3:34 PM | PERMALINK

lol - so many foolish $500 people...

Take two groups of 100 people. Each member of group A takes the "bird in the hand" option - the $500. Each member of Group B "swings for the fences" and takes their 15% shots at the million.

Add up the total earnings for each group.

Which group do you think will be ahead?

Now vary the number of members of the groups (both have 200, both have 50, and so on).

Sheesh.

Posted by: cdj on January 27, 2006 at 3:35 PM | PERMALINK

Even with the goon coming after you with the baseball bat, you might say "Joey, instead of the $500, how about I split this 15% million-dollar gamble with you? 50-50? 40-60? 20-80?"

Posted by: Joey the Bat on January 27, 2006 at 3:36 PM | PERMALINK

If 100 people took the bet and pooled their winnings:

Sure $500: $50,000 divided up 100 ways = $500 each.

15% chance at $1m: 15 people will likely get the prize, $15 million divided up 100 ways = $150,000 each.

The reason to use 100 people and not one is to illustrate the likely payoff risk vs. reward for one person. Even if each person doesn't approach payoff that way, the organization offering the proze certainly would.

Posted by: phobos deimos on January 27, 2006 at 3:38 PM | PERMALINK

Mike K.,
I would have guessed all heads; as likely as any other sequence.

The novel Cryptonomicon has an interesting part about humans trying to achieve randomness and the predictability of human nature being exploited to break that "randomness."

Math and logic are fun!

Posted by: Frank J. on January 27, 2006 at 3:39 PM | PERMALINK

"If you take the 15% chance at $1M bet, you're paying $500 (the sure thing if you decline the bet) for a shot at the risky payout

i'm not "paying" it, since i never had it to begin with."

You were granted a sure $500 when you got the chance to play the game (lucky you). The 500/1M choice is identical to this: I hand you $500 and then give you the option to trade in your $500 for a 15% chance at $1M.

I thought that breakdown might clarify the bet for people, but apparently not...

Posted by: gundryggia on January 27, 2006 at 3:39 PM | PERMALINK

I thought that breakdown might clarify the bet for people, but apparently not

look at my bank statement before and after the "deal". tell me if i've paid anyone anything.

i'm not arguing economics, i'm arguing semantics.

Posted by: cleek on January 27, 2006 at 3:44 PM | PERMALINK

Phobos -

My reason for using 100, then varying it to 200 and then 50 was to point out that the expected value (per capita) is independent of the number of people in the groups. In particular, it applies to the case where each group contains just one person.

Apparently that got lost in the shuffle...

Oh well - I guess this thread illustrates the point of Drum's post - lol

Posted by: cdj on January 27, 2006 at 3:45 PM | PERMALINK

Maybe I'm stupid here, but I'm not so sure those questions tested one's ability to understand the quesstion. For instance, it is possible to get all those right and not understand statistics. Was there a test that showed a correlation between these questions and an understanding of statistics?

Posted by: theorajones on January 27, 2006 at 3:45 PM | PERMALINK

gundryggia,

So your saying that we'd be freerolling for a 15% chance at 1MM? That someone gave us a $500 coupon for the freeroll first?

After this explanation, not taking the 15% shot is an absolutely mindboggling choice.

Posted by: michael on January 27, 2006 at 3:45 PM | PERMALINK

Joseph, You want a sure $500? I'll pay you the $500 if you take the chance and give my your winnings, win or lose.

I bet y'all my lunch that even the most destitute could find a player, relative, or usurous lender who would give even better deals.

Posted by: Joey the Bat on January 27, 2006 at 3:48 PM | PERMALINK


gundryggia -- Yes, taking the 15% bet and not accepting the free $500 is called "opportunity cost".

(I even forgot that in my previous post)

Posted by: phobos deimos on January 27, 2006 at 3:49 PM | PERMALINK

theorajones -

(a) probability is not statistics. Roughly speaking, they are converses (picture arrows pointing in opposing directions) of each other.

(b) questions 1 & 2 require only thinking ability. question 3 arguably requires only the same, but some previous contact ("feel for") geometric/exponential growth is useful.

(c) a justified answer to the $500/$1 million question requires the most basic probability knowledge possible.

All imo.

Posted by: cdj on January 27, 2006 at 3:49 PM | PERMALINK

cdj - whoops, I was posting mine before yours showed up. No comment intended on yours.

Posted by: phobos deimos on January 27, 2006 at 3:51 PM | PERMALINK

Maximizing expected log wealth is the natural decision rule for problems like this. In which case you should prefer a 15% chance at $1000000 to a sure $500 iff your initial wealth exceeds $191.45. Similarly you should prefer a 15% chance at $100000000 to a sure $1000000 iff your initial wealth exceeds $1002506.45.

Posted by: James B. Shearer on January 27, 2006 at 3:56 PM | PERMALINK

I got the first question wrong. d'oh.

Kevin, if you're desperate for money, the sure $500 may make more sense than the 85% chance of not getting $1,000,000.

Say you're flat broke and about to be thrown out of your apartment and onto the street, with all of your belongings. Consider: 100% chance of another month indoors or 85% chance of no money and being homeless? Not in southern California, but a northerly state in winter. Some people would take the $500 and keep on trying to find a job (or a second job). Not necessarily moronic.

Posted by: Librul on January 27, 2006 at 3:57 PM | PERMALINK

Regarding the discussion of how a "sure thing" choice compares to a riskier payout of a bigger sum, why is it that nearly everyone here can understand and accept that the utility of this money has an impact on the decision and yet if you took the exact same sort of utility argument into the realm of tax code fairness, we'd be stuck in another partisan shoutfest?

I also think the $500 may have been a typo only because it's too far off from the expected payout of the gamble.

But the previous poster has it exactly right that "expected payout" is great for homo economicus but if you ratchet up the "sure thing" choice I guarantee that a very large percentage of people will be humming an old Steve Miller song (yeah, "Take the Money and Run") long before we get to $150,000.

[if you were ever sucked into watching "Who Wants To Be a Millionaire" don't you think you've seen that pattern, i.e. taking the sure thing over higher expected payouts?]


Point is that like it or not, if you took the 15% gamble for the $1 million over a sure $50K, and lost I think the chances are good that you would be kicking yourself over losing $50K in 2 minutes, not saying "oh well, the expected payout was higher and I started out with nothing".

IOW, yes I think you would at least partially internalize this as a "loss". Can't we all come up with examples of this in our 'real lives'?

So $500 vs. 15% at $1 million maybe a moronic choice, but would $50K or $100K still be so "moronic" for those of you expected payout literalists? Explain that one to the spouse. Oh well, the expected payout was $150,000 after all so why should I have taken the $100,000?

:^)

Posted by: JR on January 27, 2006 at 3:59 PM | PERMALINK

phobos -

It's all good - nothing wrong with confirmation of known results...

It might be worth noting that this faux debate about the value of the $500 vs .15 chance at $1 million parallels the faux debate about evolution vs ID.

In particular, there is no debate within the scientific community on the matter. The only debate is from people who don't understand the science.

The fact is simple - except under unstated and strained hypotheses, a 15% shot at $1 million is worth more that a 100% shot at $500. About $149,500 more, on average, in fact.

"But, but, but, what if The Professional was waiting in the wings, to snipe anyone who won the $1 million! It's not worth nothing if you're DEAD, is it! Yah!"

LOLOL

Posted by: cdj on January 27, 2006 at 4:01 PM | PERMALINK

Answer:

I take the 15% shot at the $1 million, find an investment banker, and sell them the bet for ~$120,000.

Posted by: Urinated State of America on January 27, 2006 at 4:08 PM | PERMALINK

Point is that like it or not, if you took the 15% gamble for the $1 million over a sure $50K, and lost I think the chances are good that you would be kicking yourself over losing $50K in 2 minutes, not saying "oh well, the expected payout was higher and I started out with nothing".

well, if you put it that way. yeah, leaving $50K on the table is a kick in the pants.

so there's obviously a value under which the sure thing isn't worth any emotional investment. for me, that's over $500.

Posted by: cleek on January 27, 2006 at 4:10 PM | PERMALINK

This exercise is counter-intuitive. I've got a pretty good understanding of probability and I don't think that makes me more willing to take risks.

Besides, money means different things to different people. For one thing, you can't take it with you.

Posted by: Tripp on January 27, 2006 at 4:15 PM | PERMALINK

cdj: Good point.

There may be people who would take the $500 over the 15% chance of $1M, but those here defending them aren't convincing me that they aren't morons.

The defenders probably did their own math wrong and are rationalizing that they aren't bad people They can't tell that a 15% chance at 1M$ isn't obscenely better than a 50-50 chance of $1000 or a 0.05% chance at $100000.


Posted by: Joey the Bat on January 27, 2006 at 4:21 PM | PERMALINK

What if someone gives you five hundred bucks, than offers you a one-time-only 15% chance at a million, for the low, (really low) price of five hundred bucks?

Same exact situation, only you're actually paying money...maybe there's something to this whole opportunity cost idea after all.

Posted by: Boronx on January 27, 2006 at 4:21 PM | PERMALINK

Interesting thread, Kevin. I do a fair amount of risk assessment in my work, so I am somewhat tuned into this topic. The points you raise are valid ones...

To tie this topic into our favorite subject, George W. Bush, I have been consistently amazed at how poorly this president and this Administration assess and manage risk. They are abject failures when it comes to these things!

For example, by spending $500 million on radiation detectors in American ports, or less than we spend in two weeks in Iraq, we could have a 90% probability of detecting significant quantities of enriched uranium, plutonium or other highly radioactive substances being brought into critical American seaports. Same thing applies to deploying x-ray or "sniffer" devices in airport luggage screening areas. Instead, we piss $200 billion down our legs in Iraq to accomplish absolutely nothing, while Osama bin Laden makes audiotapes mocking us.

I could give many more examples, such as fully funding the Nunn-Luger Act, spending $1 billion on preventative maintenance for the New Orleans levees, instead of $10 billion to repair them after they breach, strengthening cockpit doors pre-9-11, like the Gore Commission recommended in 1996 and on and on and on....

The Republicans are piss poor risk assessors and risk managers and are making us all less safe every second they hold the reins of power!!!

Posted by: Stephen Kriz on January 27, 2006 at 4:31 PM | PERMALINK

This study shows that men do better on famous brainteasers than women. Dr. Frederick interprets this as men are smarter than women, but an alternative explanation is that men are more familiar with old, famous brainteasers. The study was published in Journal of Economic Perspectives, presumably because no psychology journal would publish a study witih such an obvious confound.
http://debfrisch.com/archives/2006/01/boyz_r_smarter_1.html

Posted by: DF on January 27, 2006 at 4:34 PM | PERMALINK

Stephen Kriz:

The republicans are good assessors and risk managers, however they are optimizing their commissions on war contracting, reconstruction work, etc. Somebody is making money off of the 200B$ Individually, they are making good deals for their contributors, if not their constituents.

Posted by: Joey the Bat on January 27, 2006 at 4:36 PM | PERMALINK

For anybody interested in the human perception of risk (especially as it relates to traffic safety), here's an excellent book:
"Risk" by John Adams
http://www.amazon.co.uk/exec/obidos/ASIN/1857280687/

Posted by: TG on January 27, 2006 at 4:38 PM | PERMALINK

cdj-

But regardless of math, since the example used money, you do have to adjust for socio-economic status. As Ron pointed out above, how many people would take the sure $1million, even with the 15% promise of $100m?

If the example involved paperclips, you'd probably get the more statistically correct answer.

You can apply the same logic to why some people stay in stable, low paying jobs instead of taking riskier, but potentially higher paying jobs. Statistics rarely factor into these decisions, or rather get outweighed by real-world factors like their savings, whether they have a family, healthcare needs, etc.

Posted by: tinfoil on January 27, 2006 at 4:39 PM | PERMALINK

Who here would give $200,000 for a 15% chance at $400 million dollars? (Individually, not by pooling money or otherwise changing the numbers.)

I'd presume that anyone who thought it foolish for a poor person to take $500 over a 15% chance at a million would do this, right?

Or how about this bet: I'll roll a die. If a six comes up, I will give you 20,000 times the entire value of your assets. If any other number comes up, you have to give me everything you own.
Mathematically speaking, you'd be a fool not to take me up on it, right?

Posted by: steve burnap on January 27, 2006 at 4:41 PM | PERMALINK
Unless that's a misprint, I just have to wonder what kind of moron would take $500 over a 15% chance of a million bucks?

Oh, sure, if its a frequently repeating proposition, in the long run, your better off taking the chance at a million.

But if things like that don't come up often (which they don't), taking the $500 makes a lot of sense; specifically, if it doesn't repeat frequently enough that the present value of the $1 million based on the mean time you expect to receive it in the future is greater than $500 (the net present value of the $500-right-now-no-risk), then, well, you're better off with the $500.

(Fiddling with Excel, with a 5% annual discount rate, you want to go for the $1 million if you get comparable opportunties frequently enough that your expected time to get a payoff is less than about 156 years -- which itself kind of presumes that you expect to live a little longer than most people.)

Lots of people trying to analyze this kind of thing pull out elementary probability, look at the mean payout, and say "you should go with the chance."

That answer is wrong. What you should ask yourself is, given the frequency with which events like this present themselves, which approach to them, if carried out throughout my life, is most likely to leave me better off. In this case, I would suspect that the frequency of opportunities to trade $500 for a 15% shot at $1,000,000 is low enough that at any reasonable discount rate, most people end up better off if they reject such opportunities consistently than if they take them.

And that's even before we get into things like whether or not the utility of $1,000,000 is actually 2,000 times the utility of $500, since really its the "net present utility" that matters; using financial values is just a convenient (easy-to-calculate) but often misleading proxy, particularly, it ignores the fairly well-established observation that money, like most commodities, has decreasing marginal utility.

Posted by: cmdicely on January 27, 2006 at 4:44 PM | PERMALINK

tinfoil -

"But regardless of math,"

I stopped reading your remark right there.

Posted by: cdj on January 27, 2006 at 4:55 PM | PERMALINK

The bat and ball question proves only that algebra is false. Either whatever they are thinking of can't be expressed in language at all or the chances that ball costs 5 cents are equal to zero.

Posted by: afigbee on January 27, 2006 at 4:58 PM | PERMALINK

These are more gullibility questions than anything else.

1. The ball could cost $1 and the bat $2, and the total price could be $1.1 if you have a coupon of some kind.

2. What if increasing the number of widget makers isn't linear? What if you need to spend 2 years building a big enough widget factory?

3. What if the size of the pond changes every day?

It's yet another reminder that both libertarians and liberals can't think outside the box.

-- HuffAndBlow

Posted by: TLB on January 27, 2006 at 5:00 PM | PERMALINK

cmdicely -

LOL - you rock!

And don't forget to consider arbitrage opportunities based on the "contest" being run in different countries, with varying exchange rates!

LOLOL

Posted by: cdj on January 27, 2006 at 5:01 PM | PERMALINK

Oh, sure, if its a frequently repeating proposition, in the long run, your better off taking the chance at a million.Posted by: cmdicely

There is no "long run." It's a one shot deal. The offer matters only, as has been stated many times upthread, if $500.00 means a lot to you in the short run or if it has no significant impact on your financial situation. If you are in the latter category, then a 15% shot a $1M means more.

look at the mean payout, and say "you should go with the chance."

There is no "mean payout." $500.00 is the median payout bracketed by nothing or $1M.

And that's even before we get into things like whether or not the utility of $1,000,000 is actually 2,000 times the utility of $500, since really its the "net present utility"

That statement applies only to meth addicts or others of impaired mental capacities. $1M, even if invested conservatively, nets a "utliltiy" many times greater than 2,000 times the net utility of $500.00.

Posted by: Jeff II on January 27, 2006 at 5:05 PM | PERMALINK

"Mike K.,
I would have guessed all heads; as likely as any other sequence.

The novel Cryptonomicon has an interesting part about humans trying to achieve randomness and the predictability of human nature being exploited to break that "randomness."

The interesting part was how uniform the pattern of trying to create randomness was. Of course, there was no "right answer." The 76% figure has been replicated in many groups and the Dartmouth example (that I witnessed) was in a pretty intelligent group.

There are whole bunches of other examples of behavior patterns. One comment above mentions that the choice of $500 or a 15% chance at a million does not cost anything. Nothing is lost if the person chooses the 15% and loses. However, there is the "sunk cost" heuristic in which people do act as though the choice has costs.

An example often used is losing a theater ticket. Having lost the ticket, what will the loser pay to replace it ?

The way that utilities are often determined is called "the standard gamble." Kevin's example of $500 or 15% chance is an example of that. You can keep raising the payout until the choices are even between the money and the 15% chance of a million. That amount is the "utility" of a 15% chance on a million (or on getting cancer if you use it in a reverse choice).

Posted by: Mike K on January 27, 2006 at 5:06 PM | PERMALINK

Steve Burnap: Nope, you presume wrongly.

As many people have pointed out above, it isn't just math/probability that makes the difference, it is the perceived utility.

As to your examples, there probably aren't many reading here that can afford to risk $200K, myself included, and I still think it would be foolish for the poor person or anybody to take the sure $500 over the 15% chance of a million.

On your second example, if my net worth was in the $500 range, i would feel foolish to not take your bet. If I had significant assets, I could understand not losing them.

But that wasn't the initial question. The other problems all had clear answers that are easy to see if you understand a little math. I believe it was not a typo that people have a hard time understanding risk and probabilities. To a (fabled) risk-neutral person a $500 sure thing would be equivalent to a 50-50 chance at $1000 or a 0.05% chance of $1,000,000. A 15% chance of $1000000 is obscenely (300times) better than the $500 sure thing, far outweighing any normal understanding of equivalence. Unless there is some mental defect, perverse choice, or ticking-bomb thing going on to justify the $500 answer, it is innumeracy.

Posted by: Joey the Bat on January 27, 2006 at 5:09 PM | PERMALINK

Jeff II -

There's no mean payout? Wow! I had no idea we were dealing with a Cauchy distribution, or somesuch. Thanks for the heads up!

Posted by: cdj on January 27, 2006 at 5:12 PM | PERMALINK

hey, I understand perfectly the logic of the delayed payment giving a high discount rate. Logically, the 15% chance of the million makes more sense. And yes, I got all three right.

But then, I am an unemployed graduate student trying to figure out how to pay my rent on tuesday and living on the same spaghetti that I've been eating all week waiting for some checks to come in. That $500 means I eat for a month and can buy a textbook I desperately need. of course I'll take it.

Posted by: northzax on January 27, 2006 at 5:12 PM | PERMALINK

This reminds me of a WSJ article, back during the high point of the dot com bubble, that pointed out quite clearly that almost all of the money invested in the dot com bubble companies _before_ their IPOs, at pennies per share, was put in by "venture capitalists" who expected that only one in ten of their investments, the industry average, would actually make money for them.

I guess they call them 'founder shares' because when the stock founders, they don't lose much?

That is, they put in the early money, other people after the IPOs and later stock sales ran the paper value of the stock way up believing they could pick which of the ten percent were worth owning -- ran the whole market way up -- til it went way down.

A later article pointed out that indeed, the VCs got back their anticipated profits -- one in ten of the companies, roughly, had made significant money for them.

The rest? Well, remember, the VCs had paid pennies for their stock before the IPOs. When it became junk, if they hadn't dumped it, they'd only lost pennies.

The VCs weren't gambling in the same way the ordinary stock market buyers were, since they'd bought their shares before the companies went public at a tiny fraction of the cost the public later paid, per share.

The bubble was those who either gambled or thought they had a sure thing, after the IPOs.


I still haven't figured out who was taking more of a chance, but I suspect the VCs risked less, given their, um, economies of scale.

Posted by: me on January 27, 2006 at 5:21 PM | PERMALINK

what kind of moron would take $500 over a 15% chance of a million bucks?

With this oh-so-respectful observation, Kevin reveals himself yet again as someone who has never wanted for money.

Posted by: Nell on January 27, 2006 at 5:31 PM | PERMALINK

Wow! I had no idea we were dealing with a Cauchy distribution, or somesuch. Thanks for the heads up!
Posted by: cdj

Hardly. A Cauchy distribution is a distorted bell curve. A three point distribution describe by the bargin here would be graphed at zero on the left, blip up in the center at $500.00, and then take off like a rocket on the right side. I think.

(Yes. I had to look this up. Cauchy distribution. I'm just sure.)

Posted by: Jeff II on January 27, 2006 at 5:32 PM | PERMALINK

cmdiceley: ???

The present value of a 15% chance of $1000000 is $150000. Three hundred times the $500 sure thing.

Even if it only comes up once during your long life, a one-in-seven chance at $1000000 is far more valuable than $500.

The choice that was posed is not $500 every week versus a once in a lifetime one-in-seven chance at $1000000, it is a once in a lifetime opportunity to choose the $500 or the million-dollar gamble.

If it is an honest gamble, take it, or if you don't like the risk sell the risk for $600 to one of the lots of people who think it a good deal.

Posted by: Tony the Bat on January 27, 2006 at 5:32 PM | PERMALINK
There is no "long run." It's a one shot deal.

See, that's the exact mistake people make. You can't analyze decision-making in terms of one-shot deals in isolation, and applying probability and looking at the mean result and deciding on that basis. That is a reasonable way of looking at expected utility of things that you can choose to do as much as you like, essentially at will. Its not a good way of assessing decision-making in general, and its particularly bad for infrequently repeated (and, a fortiori, actual "one time") opportunities.

A good case can be made that a "decision" is right, from the perspective of utility maximization, if you are more likely than not, over your lifetime, to experience more utility from acting in accord with the rule which generates than you would from any other rule applying to the same kind of situations.

With this example, you are most likely to end up worse off if it is a one-time kind of opportunity. Which means you probably shouldn't take it.

The offer matters only, as has been stated many times upthread, if $500.00 means a lot to you in the short run or if it has no significant impact on your financial situation.

I read what has been said upthread. I'm doing this thing called "disagreeing" with the kind of analysis that undergirds that argument (though I agree to the extent that decreasing marginal utility with increasing wealth is an important factor, here.)

There is no "mean payout."

Yes, there is. The mean payout from taking the gamble is the chance of getting any payout (0.15) times the amount of the payout ($1,000,000), or $150,000. As has been stated, correctly, many times upthread.

What's wrong is looking at this $150,000 vs. the $500 mean (and fixed) payout of taking the other option, and saying therefore that the choice to gamble is worth $149,500 more than the choice not to gamble. That analysis represents the limit case of infinite repetition (i.e., if you are able to repeat that decision again and again of times, the values should converge to $500 times the number of repetitions for not gambling, and $149,500 times the repetitions for gambling.)

However, real world decision-making, in order to maximize utility, needs to take into account the frequency with which opportunities with a given payoff distribution occur, and the value of present versus future results, and consider whether a given decision model is more likely than not to be optimal in terms of utility produced over the lifetime of the actor.

Events with unusual, infrequently experienced payoff matrices -- like the $500 vs. 15% of $1,000,000 scheme -- cannot be expected to converge over the lifetime of a real actor, and so using just the basic statistics and comparing means is a bad way to assess even the financial value of the decision.

Posted by: cmdicely on January 27, 2006 at 5:36 PM | PERMALINK

Need your experience & opinion: Online political news survey ($10 reward possible)

Hello, all.
My name is Daekyung Kim, a Ph.D. student studying journalism and mass communication in Southern Illinois University at Carbondale. I am emailing to ask you to do me a favor. I am now working on my dissertation whose topic is about online news and want to gather information about how and why Internet users are using news Web sites for political information based on an online survey.

This online survey will approximately take 15-20 minutes to complete. After collecting the data, I will draw to pick up 50 respondents among those who complete this survey and each will be given $ 10 gift card. Your experience will be very useful in understanding how politically interested online users are using online news and the consequent effects on traditional news.

Would you please spend some time to fill out this survey ?
http://www.surveymonkey.com/s.asp?u=594061481532
(please click on the address, OR if not working, copy it into the URL address)

You can withdraw the survey at any time you want. All responses will be kept confidential and only be used for academic purposes. This survey has been reviewed and approved by the SIUC Human Subject Committee. So, there are no questions that may identify personal information.

Thank you very much in advance for participating.

Posted by: danny on January 27, 2006 at 5:38 PM | PERMALINK

To state the ball and bat problem without algrebra,

The bat brings in a value = to or > than 1.00, but no higher than 1.10.

If the ball cost .01, the bat costs 1.09, so that would be 1.08 more than the ball.

If the ball cost .02, the bat costs 1.08, so that would be 1.06 more than the ball.

If the ball cost .03, the bat costs 1.07, so that would be 1.04 more than the ball.

If the ball cost .04, the bat costs 1.06, so that would be 1.02 more than the ball.

If the ball cost .05, the bat costs 1.05, so that would be 1.00 more than the ball.

So that's why.

Posted by: cld on January 27, 2006 at 5:40 PM | PERMALINK

Wow, three correct answers and the three horse ran 2nd in the lst race at Santa Anita. However, Purrfect Vixen, did bail me out in the 2nd, even though, she was chalk.

Better start studying the 8th and get out race.
Thank you, Al Gore, for allowing me to watch and bet online.

Posted by: thethirdPaul on January 27, 2006 at 5:45 PM | PERMALINK

Like Contentious, I am also puzzled by the description of high scorers as risk takers. The high scorers seem to be better risk assessors, able to think through the problem before them and make better judgements. Choosing $3800 next month over $3400 now is hardly risky: a return of almost 12% in one month puts me in the realm of credit card companies, and certainly don't take risks.

Posted by: dpl on January 27, 2006 at 5:51 PM | PERMALINK

Kevin's suspicions are probably correct- there is a typo. $500 doesn't mean a lot in today's society, and most people are not getting kicked into the street if they don't take the sure thing. Unless it is a matter of life and death, or the equivalent of broken knee caps, it is foolish not the take the 15% chance at a million dollars. However, I would not be too surprised to learn that such a large number of low quiz scorers actually are unable to evaluate the proposed gamble. Having read this thread with the number of people who are trying to defend taking the $500, I suspect a lot them were unable to answer any of the 3 quiz question correctly without looking at the answers.

Posted by: Yancey Ward on January 27, 2006 at 6:01 PM | PERMALINK

To state the ball and bat problem without algrebra,

Why bother, when the algebra is so simple? My kids spend hours doing simple math problems for school homework, because the school teaches fuzzy-friendly math instead of cold hard algebra. The fact that so many reasonably intelligent people here can't easily get all three problems correct is a sad reflection on our school systems.

As for the million dollar question, the "correct" answer involves more than just knowing math. A more interesting analysis would involve comparing exactly how much of a sure thing a person would be willing to give up to win million dollars to that person's economic status. I would not be certain that poor people in general are necessarily more willing to take the sure $500 in this case.

Posted by: Nemo on January 27, 2006 at 6:07 PM | PERMALINK

Jeff II -

My reference to the Cauchy distribution was facetious - it's the most well-known distro that has no mean.

I was making fun of your comment.

Posted by: cdj on January 27, 2006 at 6:10 PM | PERMALINK

What's wrong is looking at this $150,000 vs. the $500 mean . . .

What's wrong with looking it at this way is that $500.00 is the median payout of three numbers because there are only there possible outcomes or variables. There is no mean, and you don't have the chance of "winning" 15% of the $1M. Accepting the offer you get nothing, $500.00 or $1M.

Posted by: Jeff II on January 27, 2006 at 6:10 PM | PERMALINK

Sorry if somebody already demonstrated this, but it's quite simple. If you are offered this choice 100 times and take the $500 every time you end up with $50,000. If you are offered it 100 times and choose the $1,000,000 every time you end up with $15,000,000. Only the very desparate or the very stupid would choose the $500.

Posted by: Sisyphus on January 27, 2006 at 6:10 PM | PERMALINK
Even if it only comes up once during your long life, a one-in-seven chance at $1000000 is far more valuable than $500.

Its not a question of how often the exact offer comes up, its a question of how often similar offers come up.

What I am simply saying is any decision rule under which a person is more likely than not to end up worse off over the course of their life than if they followed another rule is, pretty much by definition, suboptimal.

For kinds of choices that are frequently repeated, results will most likely converge over a persons lifetime to the statistical mean result, making it somewhat reasonable to treat that result as the value of the gamble (though one has to be cautious there, too; the gamblers ruin problem means that risk still needs to be discounted.)

But gambles of types that are particularly infrequent call for a different analysis; if for any definable class of gambles, you operate under a rule which makes you more likely than not to end up behind over your lifetime, then you've got a bad decision rule, even if the mean result -- i.e., where the results would converge with infinite repetition -- is much better with your rule.

Of course, often, in practice, there are ways to change the kind of decision faced, particularly to make it act like it is more frequent and more likely to converge. For instance, if this deal was offered to each of, say, 30 people (with independent draws), if they socialized the benefit by agreeing to share the returns, it makes a lot of sense, since each of then reasonably expects to be ahead vs. them all taking the flat $500.

Posted by: cmdicely on January 27, 2006 at 6:12 PM | PERMALINK

Sisyphus,

So what gamble did you take that left you with that damn rock?

Posted by: Yancey Ward on January 27, 2006 at 6:13 PM | PERMALINK
What's wrong with looking it at this way is that $500.00 is the median payout of three numbers because there are only there possible outcomes or variables.

Um, no.

There is one possible outcome of the decision not to gamble ($500), which is both the mean and median of that choice.

There are two possible outcomes of the decision to gamble ($0 and $1,000,000). Because of the 85/15 distribution, $0 is the median result of that decision, $150,000 is the mean. Even though there is no chance to get the $150,000.

You can't combine the two, because they are the results of different decisions, not part of the distribution representing the same circumstance.

There is no mean,

Yes, there is, for either decision. For the decision to gamble, the mean is not a value that can actually occur itself, but that's not uncommon for means.

and you don't have the chance of "winning" 15% of the $1M.

That's true, but not particularly relevant.

Accepting the offer you get nothing, $500.00 or $1M.

No, if you accept the offer to gamble, you cannot get the $500. You either get nothing or $1 million.

Posted by: cmdicely on January 27, 2006 at 6:18 PM | PERMALINK

"Having read this thread with the number of people who are trying to defend taking the $500, I suspect a lot them were unable to answer any of the 3 quiz question correctly without looking at the answers."

The other issue is that utilities differ for different people and it skews the distribution by implying that IQ and risk taking are always correlated. Maybe it's true but there are personality types whose IQ and risky behavior are inversely correlated. One of them ran me off the road the other day.

The sex differences were noted but there are probably age differences too. This comes out very distinctly in medical applications.

The economic factor also applies. Survey designers are always (or should be) warned that people with low socio-economic status will try to answer a question guessing the preferred answer of the person asking. The designer should always ask important questions several times in a survey and slant the apparent desired result both ways so that the bias is neutralized.

One way to do this is to complete the standard gamble by moving the two variables around. Increase the $500 to 550, 600, etc. Then shift the 15% to 16% or 20%. When you get to even choices, you've established the utility for THAT PERSON. Then try it with others until you have a pattern. What we've seen here is an example of the method. Not results of a study. Although it is fun to do the quiz. In a small series of three MDs, only one got all three correct.

Posted by: Mike K on January 27, 2006 at 6:19 PM | PERMALINK

Jeff II -

Wow. You keep saying there is no mean.

It's just false, no matter how many times you say it.

Just because there's no 3.5 on a die does NOT mean that the mean of a die throw isn't 3.5.

No matter how many times you say to the contrary.

Posted by: cdj on January 27, 2006 at 6:20 PM | PERMALINK
Sorry if somebody already demonstrated this, but it's quite simple. If you are offered this choice 100 times and take the $500 every time you end up with $50,000. If you are offered it 100 times and choose the $1,000,000 every time you end up with $15,000,000.

Yes, which is why if you expect to get 100 chances -- or even, say, 5 -- like this over the course of your life, you should certainly play every single time you can.

That's not the scenario presented.

Posted by: cmdicely on January 27, 2006 at 6:23 PM | PERMALINK

Jeff II -

Ok fine, I'll try to be helpful.

Maybe it would be helpful for you to think of "the mean" in terms of "what one should charge to let someone play the game".

In the simpler case of throwing a die, where the payoff is the number rolled, imagine you're the carnival scheister, trying to get people to play this game. The price you should charge to break even would be 3.5 - and that's the case even though 3.5 is not in the space of outcomes for the game. That latter fact is simply irrelevant to where the break-even point for payoffs is.

Posted by: cdj on January 27, 2006 at 6:23 PM | PERMALINK

And maybe it's also helpful for you to notice, since you keep talking about the median, and it being one of the outcomes, that:

THE MEDIAN NEED NOT BE ONE OF THE OUTCOMES EITHER.

In particular, the case where there are an even number of samples is one where the median will not in general be one of the outcomes.

Posted by: cdj on January 27, 2006 at 6:28 PM | PERMALINK

Joey at bat: if your entire net worth is $500, and you lose it, what are you going to eat next month?

You are, at the one hand, demanding that we consider this a pure question of numbers and on the other, admitting that you "can't afford to risk $200,000". That's an admission that it's more than just a question of probability, but also a question of what your position in life is.

It is, for example, fashionable to look down on someone who spends a dollar on the lottery because "in the long run", it's not a money earner. But people who do so ignore the very real fact that for a person with absolutely no prospects of financial advancement, the slight chance of a complete life-changing event outweighs the loss of a dollar. A fifty-year-old unskilled laborer making $19k a year likely has no chance of getting out of poverty, and therefore may well hold a miniscule chance of doing that more valuable than a dollar.

For the record, I got every one of those questions right, without looking at the answer. Thing is, this isn't just about math.

I personally find this thing suspect and would love to see data on the socioeconomic status of the people in the study. Choosing $3800 next month over $3400 now is hardly risky, but presumes that you can afford to forgo $3400 this month. My own suspicion is that the people who answered correctly tended to be people like me, a white-collar engineer type who was raised on stupid little logic puzzles like that. Those sorts of people tend to have more money.

It seems like a lot of people look upon these questions as ones of financial management, ignoring the fact that at a very real level, it isn't just about maximizing dollars, but about making sure you have enough to eat, clothes to wear, a place to live, etc.

A question I wish they'd ask: You can have a salary of $50,000 a year, or a salary of $10,000 a year plus a randomly chosen bonus between $0 and $100,000 a year. Which would you choose? The math gives a very clear answer, but I suspect it's one many wouldn't choose. Stability of income has a value.

Posted by: steve burnap on January 27, 2006 at 6:30 PM | PERMALINK

Let's get Kevin on Deal or No Deal, just for snicks.

Posted by: Howie Mandell on January 27, 2006 at 6:31 PM | PERMALINK
In particular, the case where there are an even number of samples is one where the median will not in general be one of the outcomes.

To be pedantic, the case where, for some integer n, the number of samples is 2n and the nth sample is not equal to the n+1th sample is one where the median will not be one of the outcomes.

OTOH, if you have an even number of results like 1,2,2,3, the median will be one of the outcomes.

Posted by: cmdicely on January 27, 2006 at 6:35 PM | PERMALINK

and you don't have the chance of "winning" 15% of the $1M.

That's true, but not particularly relevant. Posted by: cmdicely

That's what I said. But you and many others have stated that $150,000 is the "average" payout when there is no average payout based on the 15% chance of winning $1M.

Again, with this offer, there are only three possible outcomes. 1) You take the $500.00. Decide not to take the $500.00 and 2) get nothing or 3) or "win" the $1M. I count that as three possible outcomes. But then again, as you and others insist that $150,000 is the average payout or even that $350,000(?) is the average payout, perhaps I'm not current with the "new" math.

There is no average because this is a single iteration with a single person. It's not a best of ten or, as you people seem to be thinking, 3 iterations that result in all three possible outcomes. Only then could you achieve an average payout of $333,333.333

Posted by: Jeff II on January 27, 2006 at 6:35 PM | PERMALINK

cmdicely -

You didn't actually disagree with what I said.

In particular, I didn't say that an even number of samples *couldn't* have the median as one of its outcomes... Just that it won't in general.

The Great Pedant Uprising Of 2006!

:)

Posted by: cdj on January 27, 2006 at 6:38 PM | PERMALINK

"Why bother, when the algebra is so simple?"

Somebody asked. And, in fact, that's actually how I figured it out. I can't remember one word of algebra. I don't know why it's even taught.

If your kids do math for fun, they're math-brained kids. Do they spend five or six hours a day lifting weights or jogging? Those are someone else's kids.

For most people algebra is worthless. Like grammar.


Posted by: cld on January 27, 2006 at 6:42 PM | PERMALINK

cld -

Thanks very much for the clearest possible statement of why America is where America is, and the direction it's heading.

Your comment should be turned into a public service announcement and run on tv.

Thanks again!

Posted by: cdj on January 27, 2006 at 6:45 PM | PERMALINK
But you and many others have stated that $150,000 is the "average" payout

I'm pretty sure I've used "mean" consistently rather than "average", though the two usually mean the same ("average" can also mean "median" and even sometimes "mode", so I prefer to avoid it in discussions where the difference between those is important; in really annoyingly mathematical discussions, its sometimes important to distinguish the "arithmetic mean" -- what we are talking about -- from other possible means, but that isn't an issue, so far, here.) But, yes, it is the (arithmetic) mean payout of the gamble.

when there is no average payout based on the 15% chance of winning $1M.

There is an mean payout. The mean payout is $150,000. There is no chance of getting exactly the mean, but that's irrelevant to the existence of the mean.

Again, with this offer, there are only three possible outcomes. 1) You take the $500.00. Decide not to take the $500.00 and 2) get nothing or 3) or "win" the $1M. I count that as three possible outcomes.

Unless the decision to take the money is random, it is wrong to group the two possible decisions in one distribution. Instead, you have two choices, each of which has its own result distribution. The first choice is to keep the $500. The distribution here is obvious -- you have a 100% chance of getting $500, the mean result is $500, the median is $500. The second choice is to gamble, and have an 85% chance of getting 0, and a 15% chance of getting $1,000,000. The mean there is $150,000. The median is $0.

But then again, as you and others insist that $150,000 is the average payout or even that $350,000(?) is the average payout, perhaps I'm not current with the "new" math.

There is nothing new about this. It might help you to review what this Wikipedia article, as well as some of the linked references.

Posted by: cmdicely on January 27, 2006 at 6:48 PM | PERMALINK
You didn't actually disagree with what I said.

That was a pedantic elaboration on, rather than a pedantic correction to, what you said.

Posted by: cmdicely on January 27, 2006 at 6:49 PM | PERMALINK

cdj,

Glad to help.

Only twice in my life since high school have I run into questions that could have been answered with algebra, they were both trivial, and the answers were soon forthcoming --without algebra.

That teaching it might strengthen your limp math muscle is undeniable, but teaching it so it can be remembered past the last test seems impossible.

Algebra has applicability to the real life of perhaps .01% of the population, and most of those people have such anomolous brains they border on being retarded in anything except math.

It's like gym, except the gym-brained are equally retarded throughout their brains.

Posted by: cld on January 27, 2006 at 7:18 PM | PERMALINK

That was a pedantic elaboration on, rather than a pedantic correction to, what you said.
Posted by: cmdicely

Since when is pedant synonymous with full of shit and/or taking out your ass?

Posted by: Jeff II on January 27, 2006 at 7:19 PM | PERMALINK

The $3400/$3800 was not presented as a question on risk tolerance, it was a question on patience.

They are also not discussing only the results of the few research anecdotes presented but findings from the entire body of research, probably hundereds of experiments.

I think they also did say that high scorers are better risk asessors and acted closer to expected utility than low scorers.

Posted by: jefff on January 27, 2006 at 7:21 PM | PERMALINK

"Would you take $1 million sure thing over a 15% shot at $100 million?"
Posted by: Ron Byers on January 27, 2006 at 2:04 PM | PERMALINK


If you live in a very poor part of the world would you take a sure $1 or a 15% shot at $100?

If you were an idiot would you take a $50,000 bribe and a 10% chance at a 25 year jail sentence or take no bribe and no jail time? It all depends upon where you're from and what your frame of reference is. If you're Republican it all changes and you take the bribe to fit in.

Posted by: MarkH on January 27, 2006 at 7:46 PM | PERMALINK

Algebra has applicability to the real life of perhaps .01% of the population, . . .Posted by: cld

Peggy Sue figured that out long ago. Most of us traverse fairly complicated worlds with nothing but cipherin'. Or you can blame it all on HP who has made it possible to do enormously complicated equations with calculators.

Posted by: Jeff II on January 27, 2006 at 7:52 PM | PERMALINK

"All right, the first question is designed to get the incorrect reply of 10 cents, the last question is designed to get the incorrect response of 24 days, what incorrect answer is the second question driving at?"
Posted by: hank on January 27, 2006 at 2:28 PM | PERMALINK


When it says "the patch" it's vague. From the start it takes 24 days. From the 47th day it takes only 1 more to cover the whole lake.

But, of course, if Jack Bauer is tending the lake it takes a whole t.v. season at one hour per episode. :-)

Posted by: MarkH on January 27, 2006 at 7:53 PM | PERMALINK

"The article says the answers are:
5cents
5widgets
47 days
I don't get the first two answers.
Puppethead's answer doesn't help."
Posted by: Ace Franze on January 27, 2006 at 2:44 PM | PERMALINK


1.00 + .05 + .05 = 1.10

The second one is tougher.

5 widgets isn't right as the question asks how "long" it takes to produce 100 widgets, not how "many".

It depends upon whether the widgets are passed from one machine to another, as in an assembly line sequencing or if each machine produces a widget by itself. If each machine produces a widget, then 100 widgets take 100 minutes * 5 (time it takes to produce 1 [ or 5 ] ). But, if they're passed from machine to machine it's a much more complicated calculation and I won't hazard a guess.

The lily pad doubles from day to day and covers the pond by day 47. My joke above said 24 because that was the only way I could include the Jack Bauer joke. From day 47 to 48 it doubles again and in so doing covers the other half of the lake. So, how long it takes to cover half the lake always depends upon where you start in terms of the pad's size.

Posted by: MarkH on January 27, 2006 at 8:02 PM | PERMALINK

Kevin,

A bird in the hand is worth two in the bush. I think Benjamin Franklin said that. Why do you hate our founding fathers? ;-)

kgb

Posted by: kgb on January 27, 2006 at 8:03 PM | PERMALINK

Shoot, somehow my math brain is working, but it isn't coming out my fingertips right. The machines produce 100 widgets in 5 minutes, just as 5 produce them in 5 minutes or 6 would produce them in 5 minutes or 7 would produce them in 5 minutes, etc. And, yes the lily pad is going to get very big unless Jack Bauer can torture it into telling him who planted the bomb, er it.

The problem isn't the answers. We've all got plenty of answers. We just don't understand the questions.

Posted by: MarkH on January 27, 2006 at 8:11 PM | PERMALINK

So my question is, if $500 is obviously too low for most of us to give up decent odds of a million, what amount of sure money is your tipping point? I think mine is $10K, maybe somewhat less (I have a pretty low income but plenty of savings which figures in to my not feeling too desperate).

And I got all the questions right, and I would describe myself as being pretty darn risk averse. And I am a chick.

Posted by: J.B. on January 27, 2006 at 8:41 PM | PERMALINK

I guess you've never been poor. Alot of people would take the 500, better a bird in hand than in bush.

Posted by: emel on January 27, 2006 at 8:43 PM | PERMALINK

The way to understand the widget question is to change "machine" to "woman", change "five minutes" to "nine months" and change "widgets" to "babies".


And yes, the question is poorly worded. It should say "five identical machines", not "five machines".

Posted by: steven burnap on January 27, 2006 at 9:17 PM | PERMALINK

Simple. Take the $500! Then you can buy 500 bats from the people who think they cost $1 and sell them to the people who think they cost $1.10. Relieve them of their valueless balls since their hands are now full of bats. Sell the balls to the guys who gave up the bats who think the balls are worth 0.10.

Keep the $500. Hire a hall with the extra $100. Start juggling classes for 1000 people who have either two bats or two balls for $20 per hour. Brand and franchise the enterprise and have an IPO. 15% of 1,000,000? I wouldn't get out of bed for it.

Posted by: phil on January 27, 2006 at 9:52 PM | PERMALINK

This has been a fascinating discussion. For the record, I got questions 2&3 right, but #1 went right past me, until puppethead explained it.

As for the money questions, I understand the mathematics of probability behind them, but I think it's not that simple in the real world. I think one's answer to money questions depends crucially on one's own financial circumstances.

Consider the following three situations:
1. 5 cents sure thing vs. 15% chance at $100
2. $500 sure thing vs. 15% chance at $1 million
3. $50,000 sure thing vs. 15% chance at $100 million

In each case, the value of the sure thing is 1/2000 of the value of the "winnings", so the amounts and risks are proportional.

Even a wino in the gutter wouldn't take the 5 cents. What the hell is he going to do with that?

But a poor person who is struggling to make ends meet would find the $500 hard to resist. As a previous commenter said, it's a 100% chance of living indoors for another month vs. an 85% chance of being thrown out on the street.

I'm comfortably in the middle class, so losing or finding $500 wouldn't have much of an impact on my life. But the $50,000 would look mighty tempting to me. That's more than I earn in a year, and since I'm almost 50 years old, it could make quite a big difference in my retirement.

Of course, a corporate executive making six or seven figures would consider $50,000 to be chump change. He's probably paying more than that to hangar his Gulfstream.

And age is another factor. It's well known that investment advisors will recommend a riskier portfolio for a 20 year old and a more conservative portfolio for a 50 year old. For a 20 year old at my income level, it may well be a better bet to roll the dice and go for it.

Posted by: rickl on January 27, 2006 at 10:39 PM | PERMALINK

Wow, this is an infuriating thread. Why is it that dozens and dozens of you stopped reading Kevin's original statement after the first sentence? He said:
Unless that's a misprint, I just have to wonder what kind of moron would take $500 over a 15% chance of a million bucks? That's crazy unless you're dead broke and a goon with a baseball bat is coming after you with your kneecaps in his sights.

Cue the unwashed masses: "Uhhh, Kevin, what if you're dead broke, did you ever think of that?"

Cleek:
That's a pretty big lake by the way

yeah that's what i was thinking - or reallllly small lily pads.

2^48 is 281,474,976,710,656. two hundred trillion times the original size.
That's how many lily pads would be on the lake the day AFTER it was full, assuming that the first day there was only one lily pad (2^0 = 1).

So on the 48th day, there'd only be 140 some-odd trillion there.

Posted by: Stoffel on January 27, 2006 at 10:40 PM | PERMALINK

phil:

I like the way you think. :)

Posted by: rickl on January 27, 2006 at 10:41 PM | PERMALINK

And the thing I like about that first question is that it behooves you to use algebra to get the right answer, but it's only simple arithmatic to check your work. I instantly guessed $.10 first, but then the bat would cost $1.10 and then the total would be $1.20. So if you're careful, there's no way you're gonna guess $.10. It's a great question.

Posted by: Stoffel on January 27, 2006 at 10:57 PM | PERMALINK

This discussion is fascinating, but I have to come down in general disagreement with the folks arguing that comparing the expected (or mean) pay outs $150000 and $500 is sufficient to resolve the question. Some folks have pointed out that the decision would depend upon your economic circumstances. I'd like to add some flesh to that idea. First I have to say that from my position the $500 sure thing is too low and I'd go for the 15% shot. However, it is equally clear to me that at some figure for the sure thing well below $150,000 I'd go for the sure thing. The savings on current interest payments and gains from investing that saved money would easily overcome the difference in the expected additional $50,000 for the 15% chance. Yes, I would expect to get the savings and other gains if I win, but that comes only with the 15% chance too. For those of you arguing that we should think of the expected pay out if this game is played several times over ones life. But then effect of time becomes significant. Having $500 now could be a great deal more valuable than an expected pay out in the future. If I could invest the $500 at 15% it is worth more than $1M 50 years from now. Ok so 15% is high and 50 years is long, but if the sure thing is $1000 or $2000 then the value of having that now becomes much greater than the potential win of $1M 20, 30 or 40 years in the future. My point is that if you include the rest of the real world into these risk calculations it is not absolutely clear to me that the $500 sure thing is not in some cases the better choice.

Another consideration against arguing that the solution is to simply compare the mean values is a simple thought experiment. The expected $150,000 pay out remains the same if we decrease the chance of winning by a factor of 10 and increase the pay out by a factor of 10, i.e. 1.5% chance of winning $10,000,000 or again to 0.15% chance of winning $100,000,000 and so on until the chance of actually winning anything in any one shot approaches zero. Recall that the expected pay out is staying at $150,000. I'm sorry, but although in the given example (15% chance of winning $1M) I would definitely take the chance if the odds where 1.5 in a million of winning one trillion dollars, I'd take the sure $500. I find it hard to believe that many others would choose differently.

The important issues then are not the expected pay out but the expected future value of the pay out and an assessment of whether 15% chance is too small or not.

Posted by: MSR on January 28, 2006 at 12:07 AM | PERMALINK

cmdicely: What I am simply saying is any decision rule under which a person is more likely than not to end up worse off over the course of their life than if they followed another rule is, pretty much by definition, suboptimal.

Would anyone here pay $400 each month to avoid a 1% risk of paying $30000? It's called health insurance.

Apparently if you pay it, you're a "moron", according to the analysis we've been seeing.

Posted by: eeyn524 on January 28, 2006 at 12:14 AM | PERMALINK

eeyn524,

thanks for bringing in the insurance angle. Psychologically, but not mathematically, there is a difference between avoiding losing what you have and risking getting something that you don't.

In order to risk losing a lot, many people pay premiums and salaries to other people, thereby reducing their collective net wealth. Put differently, the insured are betting that they will get sick, and the insurers are betting that they won't. The insurers are correct in total (and on the average) and there are plenty of opportunities to make and appraise the bets.

somebody said that the Cauchy distribution doesn't have a mean. I think that's incorrect but am too lazy to check; the Cauchy distribution doesn't have a variance.

Posted by: contentious on January 28, 2006 at 1:06 AM | PERMALINK

eeyn524,

Exactly, excellent point. Insurance companies only make money because people pay them to take away their worries.

Posted by: Rosey Palmer on January 28, 2006 at 1:19 AM | PERMALINK

What about the entity making the offer? They either lose small or lose big. What's in it for them?

Posted by: exasperanto on January 28, 2006 at 2:39 AM | PERMALINK

Reminds me of the time I was driving in the desert outside Vegas and saw this old bum wondering around. I was going to pick him up but I thought 'Man, it'll cost $500 bucks to get caddie cleaned up if I let him in.'

Posted by: Michael7843853 G-O in 08! on January 28, 2006 at 2:48 AM | PERMALINK

A lot of posters pointed out that whether you go for the immediate $500 or for the 15% chance depends on your wealth. One poster quantified this:

Maximizing expected log wealth is the natural decision rule for problems like this. In which case you should prefer a 15% chance at $1000000 to a sure $500 iff your initial wealth exceeds $191.45. Similarly you should prefer a 15% chance at $100000000 to a sure $1000000 iff your initial wealth exceeds $1002506.45.
Posted by: James B. Shearer on January 27, 2006 at 3:56 PM |

Does anyone know what formulas were used by JBS? (Google leads to a wild goose chase).

Posted by: JS on January 28, 2006 at 4:15 AM | PERMALINK

Google does show, however, that JBS is a mathematician working at an IBM research lab.

Posted by: JS on January 28, 2006 at 4:24 AM | PERMALINK

The post by JBS is anything but self-explanatory, and the precision of "$191.45" is most likely a joke. The point is sound: if you're starving, a meal real soon makes more difference than the remote hope of a hilltop castle.

It's a near approximation to the truth to note that a sound has to be, in some sense, ten times as loud to sound twice as loud, and a light ten times as bright to seem twice as bright. Taste and smell, in contrast, are linear. Put just one Jalapeo seed on your tongue and taste it to test.

Posted by: bad Jim on January 28, 2006 at 5:03 AM | PERMALINK

Duh, people.

If you're good at calculating mathematical risk in your head, you probably know it, and you're going to have a higher degree of confidence in your ability to do so. In other words, you can look at a situation and figure out where the higher utility is, and go with the better option.

If you -suck- at this, you learn to be conservative really fast, or you go broke. No other options. (Well, maybe "marry rich"...)

Posted by: Avatar on January 28, 2006 at 5:28 AM | PERMALINK

There is indeed no mean payout. There is a rather nice payout and a very nice payout but there is nothing mean about either of them.

Even tho right now $500 is a bit less than two weeks unemployment checks it would make no change in my life. I'd be hard pressed to come up with things to squander it on. A million, even the half million or so after taxes, would, added to my existing net worth, set me free. I'll take the odds.

Posted by: triticale on January 28, 2006 at 5:52 AM | PERMALINK

All the people talking about the answer depending on how badly someone needs the money is ignoring the fact that in the United States today, $500 just isn't very much money.

It is about 2 weeks pay at minimum wage. It is working 10 hours a week at a 2nd job for two months. Even if you are really poor and need the money right now, you could probably raise it by pawning your wedding ring, selling off the TV or the antique Dresser you inherited from Grandma, etc.

If you can't work any more hours and have already sold off all the non-essentials and still need $500 right now, you are probably in enough trouble that even with the sure thing $500 payout, you will be right back in desparate straits in a month or two at most.

On the other side, $1 million dollars is enough to change your life. Assuming that you don't go wild (lots of lottery winners do), or have bad luck, or fall afoul of scam artists (also common for those that suddenly come into money), it is enough to eliminate your money worries forever.

That is not to say that you can just multiply the odds of winning by the payout and take the higher value every time. If the choice was $5,000 right now versus a 2% chance of winning one million dollars, I would probably take the sure thing unless I had $50,000 or more in liquid assets. Even though the expected value of the million dollar chance is $20,000, the 4-1 greater mean return may not be worth 49-1 odds that you get nothing. But in the stated problem, 6-1 odds are just not that long for a 20,000 to 1 return if you win, and passing up the chance for $500 would be extremely foolish except under really unusual circumstances.

Posted by: tanj on January 28, 2006 at 6:05 AM | PERMALINK

The whole thing is also predicated upon the assumption that no people exist who have psychic powers and who can trust them. I have come to believe that most of the time I "know" when a long shot is worth taking. This knowledge doesn't mean in any way that my odds are not as long as someone else's, it simply means that when that 15% chance is spot on, sometimes I will know it with an indescribale certainty.

Posted by: Michael L. Cook on January 28, 2006 at 10:25 AM | PERMALINK

Perhaps the craziest thing about this post of Kevin's is his discussion of risk with the other math problems which have absolutely nothing to do with risk. Good one Kevin.

Posted by: MarkH on January 28, 2006 at 12:27 PM | PERMALINK

The "poor and desperate" argument doesn't justify the decision, it just illistrates how emotion can affect the decisions of the unintelligent and risk averse.

Posted by: aaron on January 28, 2006 at 2:27 PM | PERMALINK

JS: If you want to know what James B. Shearer was talking about, you can google "Kelly Criterion" (and get a lot of gambling sites), or read more here.

Posted by: Alex R on January 28, 2006 at 3:28 PM | PERMALINK

By the way, the basic idea is pretty simple: suppose you have a lot of opportunities to make winning bets. How much should you bet each time? The folks who blindly shout that you should maximize your expected value should think a bit -- maximizing your expected value for a winning bet means betting as much as you possibly can. But if you bet your entire net worth every time to place a bet, you *will* end up broke. Kelly asked: what is the proper amount of money to bet if you want to maximize your expected net worth in the *long term*, after many bets... The answer turns out to involve the log-maximization mentioned by JBS.

Posted by: Alex R on January 28, 2006 at 3:35 PM | PERMALINK

"what kind of moron would take $500 over a 15% chance of a million bucks?

With this oh-so-respectful observation, Kevin reveals himself yet again as someone who has never wanted for money."

well, I've starved in the streets during hard winters, and at those times if I could have been satisfied of the fairness of the proposition I would have taken the million every time. A 1 in 7 payoff? Those odds are very good for just about any risk in life, as for the rate of return most risks average out at much worse rates.

Posted by: bryan on January 28, 2006 at 3:38 PM | PERMALINK

I got all the questions right, but I'd take the $500 because 1) it's better than a poke in the eye with a sharp stick, and 2) I never win anything.

Posted by: Cal Gal on January 28, 2006 at 4:02 PM | PERMALINK

Steve Burnap:

My net worth is actually negative right now -- I owe more than my assets. I do have some disposable cash though.

If I had $500 in my pocket to eat for next month, and could spend it on a 1 in 7 chance for $1000000, I'd spend it and if I lost, I'd charge some groceries to my credit card, I'd go wash some dishes in a restaurant for food, or go volunteer at the food pantry. $500 of found money isn't going to fix my life, or the life of anyone reading this thread. $1000000 is another story.

Your two examples are not equivalent to the choice Kevin presented: The negative outcomes could far exceed the negative outcomes in the given example. There is no negative outcome, only $0, what you would get if you didn't get the offer.

If you read the article, you'd see that Frederick's survey showed that there were dramatic differences between risks of losses and risks of gains contrary to what you were presuming with your examples. The risks people are willing to take are very dependent on the consequenses.

The $500 or the 15% gamble options are both found money. They both would add to whatever your situation is. Both choices are better than not playing, but the $500 is nearly a consolation prize compared to the 15% lottery ticket. The negative outcome of not playing, $0, is the exact same as taking the risk and losing, which would happen 85% of the time. Most people aren't going to have their lives saved by a $500 windfall. If we they did, we'd be seeing people dropping like files. If we assumed the 20% of high scoring men who chose the $500 sure thing as making the rational decision to save themselves from starvation, we'd be seeing 20% starvation rates among survey respondents. Sure, there are some people that might die or suffer dire consequences if they don't get $500, but I don't think they are anywhere near the 20%, 62%, 60%, or 75% of the survey classes.

If the 3500 survey respondents are offered the choice of gamble by some rich, honest god, they should pool their risk and all choose the 15% chance of $1000000, and split up the 490M$-560M$ winnings. The low end result of $140000 is so much better than $500, that, if the gamble were honestly and actually offered, people would quickly figure out how to finance the risk. The first or second millionaire would start bragging about his brilliance and maybe offer to pay people to take the risk.

From one perspective, cmdiceley is right -- These opportunities don't actually come along often. If you think you see one, it is probably a scam and you should be cautious and maybe take the certain $500. But if it is an honest, fair, one-time offer, go for the $1000000 behind one of Monty Hall's doors number one through seven rather than the $500 bill. Monty isn't going to starve or kneecap you if you get the goat.

Posted by: Tony the Bat on January 28, 2006 at 4:44 PM | PERMALINK

I realize this thread is dead, but this morning I read the original research by Shane Frederick published in the Journal of Economic Perspectives, and it is quite, quite fascinating. So, just in case anyone else checks in...

Frederick's research has such an interesting angle: He finds a male-female gap! That is, 2/3's of those who got all three questions right were men, 2/3's of those who got all three questions wrong were women. This despite the fact that no significant difference between them showed up on other measures of mental ability.

The questions are interesting because each of them generates an "intuitive" wrong answer. To to get to the correct answer, you have to suppress the wrong answer long enough to figure out the right one. Men were not just more successful at doing this than women. The errors made by women were also much more likely to be the "intuitively" wrong answer.

And while the 33% of women who got all three right (A3R) were more likely to delay gratification than the A3R men (for example, on "would you prefer $3400 this month or $3800 next month," 60% A3R men v 67% A3R women chose $3800), the A3R women were comparable to the men who got all three wrong (A3W) in their willingness to take risks. For example, on the question that has been so discussed on this thread, "$500 for sure or a 15% chance of $1,000,000", 80% of A3R men and 40% of A3W men chose the gamble, versus 38% of A3R women and 25% of A3W women.

Going way, way beyond the data, what I have been pondering the long-term social implications. If women, as a group, are less willing to take even good risks ("$1000 for sure versus a 90% chance of $5000"--81% of A3R men, 59% of A3R women) what does this imply, say, for long term wealth creation?

Posted by: PTate in MN on January 28, 2006 at 6:20 PM | PERMALINK

And I just re-read Kevin's original post, and found he had already mentioned the gender gap that I describe so enthusiastically.

*Blush*

So I hope nobody reads this...please, please, please.

Posted by: PTate in MN on January 28, 2006 at 6:25 PM | PERMALINK

that's ok, PTate in MN.

You elaborated the point nicely.

Posted by: contentious on January 28, 2006 at 8:05 PM | PERMALINK

JS, the procedure for finding $191.45 is as follows. You want to maximize expected log wealth. Let x be your initial wealth. If you take the sure $500 your wealth becomes x+500 and your expected log wealth is log(x+500). If you gamble 85% of the time you receive nothing so your wealth remains x while 15% of the time you win $1000000 in which case your wealth is x+100000. Therefore if you gamble your expected log wealth is .85*log(x)+.15*log(x+1000000). A computation shows x=191.45 is the point at which the expected log wealth of gamble is the same as the expected log wealth of the sure $500 and that for x less than this the sure thing is better while for x greater than this the gamble is better (in terms of expected log wealth).

As to why you might use this decision procedure one justification is that this corresponds to assuming a log wealth utility function. A log wealth utility function asserts that for example a windfall of 10% of your wealth will feel about as good independent of your wealth which has some plausibility (ie that a gain of $100000 to a man worth $1000000 means about as much to him as a gain of $1 million to a man worth $10 million).

As alex r notes above there is also a mathematical sense in which under certain assumptions this procedure is optimal.

All of this assumes the changes in wealth capture all the utility implications of the choice. If for example your spouse is going to nag you forever if you gamble and lose this may not be true.

Posted by: James B. Shearer on January 28, 2006 at 8:35 PM | PERMALINK

Explanation much appreciated, JBS. I am still perplexed by the following:

If you gamble 85% of the time you receive nothing so your wealth remains x while 15% of the time you win $1000000 in which case your wealth is x+100000.

This seems to be based on the assumption of many repeated gambles of this type, right? At least that's what "...% of the time" seems to be implying. But then we have

Therefore if you gamble your expected log wealth is .85*log(x)+.15*log(x+1000000).

This seems to assume a single gamble. Or is it that the number of gambles, say N, would multiply both expressions so we just divide both by it?

If there are multiple gambles, I would think that x would have to be different at the beginning of each -- right? But if there is only one (which I believe the original question implied anyway) I wonder if this approach would hold.

Posted by: JS on January 29, 2006 at 1:59 AM | PERMALINK

JS, I am assuming "15% of the time", "15% chance" and "with probability .15" all mean the same thing and can refer to a single trial. Mathematically the expectation of a random variable is the sum of the value of every possible outcome weighted by the probability of that outcome.

As to whether this is a reasonable decision procedure I have found the results it gives seem plausible to me. Your opinion may be different.

Posted by: James B. Shearer on January 29, 2006 at 1:22 PM | PERMALINK

Let's say you have a choice between an electric heater (100% chance) and a 15% chance of a vacation in Aruba. Which do you pick?

Ok, now say it's winter in a northern U.S. state and your heat's been turned off (having it turned back on is impossible/possible only with ssignificant financial sacrifice). Which do you pick?

-----

I wonder how this relates to that experiement a while back about kids being offered an oreo now or two later -

Posted by: Dan S. on January 29, 2006 at 1:53 PM | PERMALINK

JBS, deciding on a decision methodology is itself a fascinating subject. Applying the plausibility test, I would say that if my net worth were $149 then accepting the certain $500 would make more sense than going at the 15% chance for much more.

I would also say that the perception of the value of a potential windfall profit, while certainly non-linear, is more likely an S-curve rather than an exponential. In other words, if you have $149 then the perceived difference between an additional $1 billion and $10 billion is not as great as between, say, $1K and $10K.

Posted by: JS on January 29, 2006 at 2:57 PM | PERMALINK

Regarding the first point above -- what I was really trying to say that my subjective crossover point in that particular dilemma would be significantly above $149 NW.

Posted by: JS on January 29, 2006 at 3:05 PM | PERMALINK

Wouldn't it be nice to have another day off of work each year? Click on www.DADAY.com and sign the petition to make the Day After the BIG GAME a national holiday!

Posted by: Angela on January 30, 2006 at 3:14 PM | PERMALINK




 

 

Read Jonathan Rowe remembrance and articles
Email Newsletter icon, E-mail Newsletter icon, Email List icon, E-mail List icon Sign up for Free News & Updates

Advertise in WM



buy from Amazon and
support the Monthly