Innumeracy is alive and well and living in the USA

Front page WSJ  story on a new game show:

"Deal or No Deal" works like this: Twenty-six models each hold a briefcase that contains a sum of money — varying from one cent to $1 million in the U.S. game. The contestant picks one briefcase as his own and then begins to open the other 25, each time, by process of elimination, revealing a little more about what his own case might hold. At the end, the contestant can also trade his briefcase for the last unopened one.

Suspense builds — and the contestant’s chance of hitting it big grows — when small sums are eliminated and the $1 million or $750,000 cases remain unopened and winnable. Periodically, as cases are eliminated, an ominously shrouded "banker" offers a deal conveyed to the contestant by Mr. Mandel. The proposal is: Stop playing now and take the money offered.

What interests Thierry Post, a professor of finance at Erasmus University in Rotterdam, the Netherlands, is how contestants respond to these offers, which are related to which dollar sums remain winnable. If the $1 million and $500,000 briefcases are left, for instance, the offer will be far higher than if they aren’t.

This can create anguishing scenarios. What to do if the last two briefcases hold $1 million and $10, and the banker offers $450,000? The contestant has a 50-50 chance at a million. Probability theory says his "expected value" is the average of the two unopened briefcases, or $500,005. Classical economic theory says that people with relatively small net worth, likely never again to see 

Its funny — we were just discussing this last month, and I put it in my TiVo queue to record. I only saw one show itself is slow and boring, requires no particular skill — other than statistical analysis. The real interesting phenomena is watching ordinairy people deal with basic mathematics. 

Why Game Shows Have Economists Glued to Their TVs
For Researchers, Players Shed Light on Decision Making;
Mr. Johnson’s ‘Gutsy’ Move

WSJ, January 12, 2006; Page A1

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  1. B commented on Jan 12

    I can’t wait until Running Man hits the game show circuit. That’s when we know that we’ve really arrived.

  2. anonimouse commented on Jan 12

    That’s “innumeracy”, not “innumercy,” you illiterte git. ;-)

    BR: That’s fat thumbed spelling, not literacy . . .

  3. PC commented on Jan 12

    You wrote:

    “This can create anguishing scenarios. What to do if the last two briefcases hold $1 million and $10, and the banker offers $450,000? The contestant has a 50-50 chance at a million. Probability theory says his “expected value” is the average of the two unopened briefcases, or $500,005.”

    Expectancy = (probability of win X gain) minus (probability of loss X loss)

    Actually another way to look at the Expectancy of this situation is this:

    50% chance of getting $1 million

    50% chance of getting $10. If you get $10 instead of $1million, you are in essence “losing out” by $999,990 compared to winning $1 million. So you can think of it as a loss of $999,990.

    So Expectancy = (50% X $1 million) minus (50% X $999,990) = $5

    If the banker offers you $450,000, grab the money and run.

  4. PC commented on Jan 12

    PS – The concept of expectancy also comes into play in trading. If you have a profit on your position now, do you take a sure profit now or wait longer in the hope of a larger profit? Agonizing, isn’t it?

  5. Preston commented on Jan 12

    I think PC’s discription of the show reveals why it is conceptually (if not actually) interesting. The ‘banker’ gives a price that is not the mathematical average of the suitcases yet many will choose to take the low settlement rather than risk losing it all.

    I don’t think it’s an issue of innumeracy at all but economic decision making.

  6. cm commented on Jan 12

    If I have a choice of $450K *for sure*, here and now, and $1M *maybe*, I for sure don’t have to think a lot!

  7. eightnine2718281828mu5 commented on Jan 12

    Preston is right; the calculation isn’t the maximization of money, it’s the maximization of happiness.

    The real calculation is 100% chance of happiness with 450k vs a 50% chance of happiness with 1m.

    No-one is gonna be miserable with 450k.

  8. john commented on Jan 12

    PC + preston are right…..classical economics is bunk when is comes to expected value and “happiness”

    The “grief” of losing out 999,900 is exponentially greater than the “grief” lost when one wins $450,000 and not $1m.

  9. RP commented on Jan 12

    I agree with Preston and the guy above me with the transcendental constant “e” in the middle of his name (wtf?)

    Remember, the expected $500,005 value happens only if the $10/$1M decision is made an infinite number of times.

    You there, squirming in the hot seat, only get to play once.

    For the average schmoe, leaving the studio with ~$50k less than “optimum” buys a lot more happiness than $10, so taking the deal is the way to go.

    A super-rich dude, say Warren Buffett, might be inclined to open the suitcase to see if he can get the extra $50k. Then again, maybe not, given that no network show runs forever :-)

  10. Trent commented on Jan 12

    However, maximizing happiness can be circumvented by the show by having the contestant then choose which case she would have picked. Then, if that case has the $1 million she can still be miserable.

  11. fiat lux commented on Jan 12

    The point here is that these “basic” mathematical concepts are not taught well, if at all, in many schools.

    I attended top-notch private schools K-12, got a BA from a very well known and respected college, yet never learned anything about game theory or statistics until I got to grad school.

  12. seamus commented on Jan 12

    A relative of mine who has the reasoning skills of a toddler adores this show. I watched it with her, and I got it.

    While “Deal or No Deal” is an undergrad-level Finance 101 exercise in option value, really it’s set up to be a game show of chance that requires no trivia knowledge, political savvy, or skills. It’s just got a bunch of pretty ladies with suitcases, and an archetype of a shadowy “banker” challenging your gutsiness. I don’t see the Every(wo)man contestants doing math. I see them judging whether they feel lucky.

  13. alea commented on Jan 12

    Innumeracy is alive and well and living in the USA

    WSJ story on economists and game show:
    Deal or No Deal works like this: Twenty-six models each hold a briefcase that contains a sum of money varying from one cent to $1 million in the U.S. game. The contestant picks one briefcas…

  14. cm commented on Jan 12

    I had to leave for work this morning, so RP beat me to it.

    To rephrase & expand on it a bit, in my system of reasoning the “expected value” argument is fallacious, because the abstraction of “expected value” is only valid as a weighted average over a large enough series of events (compare also “the law of large numbers”), and cannot be extrapolated to a single event.

    For a single try, there is no $500K “expected outcome”, there are two separate outcomes “$10” and “$1M”.

    The fallacy is not the concept of expected value, but its extrapolation to a single event.

    As much as with throwing dice once, there is no “expected outcome” of 3.5, you get either one of 1-6.

    The question is, disregarding meta-information from the size of the “sure offer”, is a 50% chance of gaining 550K over the sure bet good enough for you, versus 50% of forfeiting the $450K.

  15. Kaleberg commented on Jan 12

    A good way to look at this is to imagine you have $450,000 and someone offers to sell you a lottery ticket that could pay off either $1,000,000 or $10 with 50% odds. The expected value is $500,005, so this is about a 10% gain. This is a good return, but not a spectacular return on your investment. You can lend that money to the government and make 10% in two years or three years.

    Of course, this disregards the issue of a risk premium. It is currently not politically correct to admit that there is a preference for security and that risk incurs costs. Witness the assault on Social Security earlier this year and its rationale.

    Despite the rantings of the ideologues, in the real world, risk has its costs. The government, perceived as a low risk borrower, pays a lower interest rate than Joe Blow out of high school with no job or credit record. Banks charge higher interest rates on unsecured loans than on loans with collateral.

    If I had $450,000 and was offered an investment with a 50% chance of losing everything in the next several minutes, I would be extremely unlikely to consider it. I cannot afford to lose $450,000. If my net worth were much larger or I had a much higher tolerance for risk, I might consider this an interesting gamble.

    Consider the stock market, with its typical 7% return on investment. Most investments do not vanish completely within a year, nor do they double. Still, in the aggregate, a 7% return is considered a reasonable risk premium. If half of all companies went broke in a year, and half doubled, I’d imagine that it would require an expected return much higher than 10% to lure people into the market.

  16. trader75 commented on Jan 13

    That “running man” comment cracked me up.

    When historians study America’s “decline of empire” a hundred years from now (if not sooner), they may well point to shows like this as examples of the bread and circuses that did us in.

    The show fits the format of many others over the years — emotionally manipulating gullible people in the same way that a cruel child pokes at a bug with a stick. There is this bizarre social contract by which the contestants give up all shreds of human dignity in exchange for five minutes of fame, and the audience laughs viciously at what fools these contestants are without realizing they are laughing at themselves. This is the process by which thinking humans become technology-wielding animals.

    In college we used to have debates about whether the world would end up with Huxley’s Brave New World or Orwell’s 1984. Now it appears we galumphing towards some bizarre combination of the two.

  17. Preston commented on Jan 13

    All of the commenters have suggested that the 450,000 offer as a no-brainer. The banker is likely to respond to this by lowering the offer- would you take a sure 300,000 instead of gambling for 1 million? 200,000? Less?

    By not relying on a simple equation to determine the offer the banker can judge the capacity of risk of each contestant to make the decision as difficult as possible.

  18. cm commented on Jan 13

    Preston: I never watched the show, and probably never will. I was arguing under the assumption that the $450K was a “sure offer”, i.e. it is known that $10 and $1M are left, and the banker offers, walk with $450K now or keep/exchange your suitcase/envelope. I’m not sure that’s actually what happens.

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