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.

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

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.

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.

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.

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

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).

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

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

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.

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.

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).

"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.

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

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.

How's the great American novel coming Frank?

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.

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.

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.

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.

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?

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.

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

"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.

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.

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

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

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.

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?

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.

$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.

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.

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.

"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.

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.

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.

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.

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.

The article says the answers are:

5cents
5widgets
47 days

I don't get the first two answers.

Puppethead's answer doesn't help.

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.

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.

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)

"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?

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.

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

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.

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.

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...

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.

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.

Nate: Excellent post.

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.

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.

That's a pretty big lake by the way.

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,

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.

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.

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.

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.

"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.

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.)

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)

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.

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.

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.

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

We always preferred a "Bird" in the hand.

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

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.

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.

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?

"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.

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.

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.

Cleek - look up the term "opportunity cost".

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.

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.

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?"

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.

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!

"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...

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.

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

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?

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.

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.

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

(I even forgot that in my previous post)

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.

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

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.

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.

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?

:^)

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

Answer:

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

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.

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.

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.

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.

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!!!

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

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.

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/

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.

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?

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.

tinfoil -

"But regardless of math,"

I stopped reading your remark right there.

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.

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

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

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.

"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).

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.

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!

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.

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.

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.

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.)

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.

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.

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