## Friday, December 21, 2007

Bill Gasarch is on vacation and he had given me (Lance) a collection of posts for me to post in his absence. But then I got email from Tal Rabin who wants to get the word out about the Women in Theory workshop to be held in Princeton in June. Done. Now back to your regularly scheduled post from Bill.
I don't usually watch Deal/No Deal. I like some of the interesting math or dilemmas it brings up, but the show itself is monotonous. As Host Howie Mandel himself says "we don't ask you a bunch of trivia questions, we just ask you one question: DEAL or NO DEAL!" Here is a scenario I saw recently where I thought the contestant made the obviously wrong choice.
1. There are two numbers left on the board: \$1000 and \$200,000.
2. She is offered a \$110,000 deal.
3. She has mentioned that \$110,000 is about 5 times her salary (so this amount of money would make a huge difference in her life).
4. Usually in this show you have the audience yelling `NO DEAL! NO DEAL!' This time the audience, including her mother, her sister, and some friends, were yelling `TAKE THE DEAL! TAKE THE DEAL!'. While this is not a reason to take the deal, note that the decision to say NO DEAL is NOT a `caught up in the moment' sort of thing.
She DID NOT take the deal. We should judge if this was a good or bad decision NOT based on the final outcome (which I won't tell you). Here is why I think it was the wrong choice. Consider the following scenarios:
1. If she takes the deal, the worst case is that she gets \$110,00 instead of \$200,000.
2. If she rejects the deal, the worst case is that she gets \$1000 instead of \$110,000.
The first one is not-so-bad. The second is really really bad. Is there a rational argument for her decision? I could not come up with one, but maybe I'm just risk-averse.

1. Well somebody has take some risk.. She wanted 90K more.. yes.. the expectation of her payout is 5000 less with noDEAL than with DEAL.. perhaps not a very smart thing to do mathematically but if everybody plays a perfect game then nobody will ever win the 1,000,000 i believe. the 'banker' or whatever his name usually gives a better deal than the mean of the amount left on board..

2. It would make sense to reject the deal if she owed \$150K to a loan shark, due tomorrow. Taking the deal means a 100% chance of a broken leg, rejecting it means a 50% chance of not having a broken leg.

This, of course, is a specific example of something general. It makes sense to reject the deal if there's a threshold effect—something specific that costs significantly more than \$110K, but less than \$200K, would be life-changing, and unattainable without it.

This is Stu—too lazy to create a Blogger ID.

3. Is there a rational argument for her decision?

Yes, that she is a greedy bastard.

Seriously.

If her objective is to not be in the situation to think "gee, I could have won 200K\$", then she indeed should have rejected the deal.

4. I'm no mathmatician,however,If someone offers 110k for something I paid nothing for I'd take it.
Forget the greed.

5. Worst case scenario analysis doesn't seem enough: what if the odd is not 50:50 but 90:10? The worst case scenarios don't change, but it feels natural that the choice could be affected by the changed probability.

So 'rational' thing seems to compare banker's offer with the expectation from probability. But still people make risk-averse choices (or in some other cases risk-taking) not consistent to the 'rational' answer.

Somewhere I read an explanation that people try to maximize not the expectation of prize money but the expectation of the utility, and the utility function is not linear. But I feel this explanation lacking, because, it explained away the 'irrational' choice by introducing the utility, but how do we explain the utility function?

Could there be an evolutionary explanation to this? The theory of evolution seems a nice place to find an explanation for this kind of things.

6. My first thought was that no matter your utility curve you'd just take the
greater than expected deal and do a 50/50 bet with someone, but there's a
complication:

Taxes.

Suppose you make a whole lot of money and your combined state and federal tax
rate is 43%. You can't deduct gambling losses past gambling winnings so you
set aside \$47300 to pay taxes, leaving \$62700. If you win the bet you get an
extra \$35739 after tax, leaving total after tax winnings at \$98439 if you won
that bet. Expected after tax value there is \$49219.50

The expected after tax winnings of rejecting the deal are \$57285.

The expected winnings from standing put are \$62700, so this only makes sense if you really get more utility out of the extra money.

If you could deduct gambling losses unconditionally this anomaly would go
away leaving just the \$150K loan shark scenario.

It's also much less an issue because of her low salary, particularly since she could make the bet *next* year.

(Note: IANATL)

7. I'm not even sure we should be asking "why?" to these deal or no deal questions. Unless we already have data that tells us stuff about people's utility functions in a high range [which, since IANAE, I suppose we might], I'm of the opinion that Deal or No Deal provides raw data telling us how "average" utility functions behave. And until we know more about it, it doesn't make sense to ask why.

Also: Does Lance agreeing to make posts for Bill mean we might get to see a Lance guest post soon?

8. Does the likelihood that Bill would tell us the outcome vary depending on which result occurred?

Bill in even asking this question made one faulty assumption, that all dollars are equal. People tend to not want all things equally and there are a great many things that can't be partially purchased. Without knowing the relative weights of what she was trying to do we can not judge. But one example would be that she has a relative who needs a one of two medical procedures, the one covered by insurance has a 20% survival rate, but one that cost more then \$110000 after taxes but less then \$200000 after taxes had a 99.999% survival rate, and the survival of that relative is substantially more important then a more comfortable life style for her, then how could she not take the chance at the 200000?