## Monday, October 01, 2007

### Deal-No Deal: MORE \$ = LESS Interesting

I caught an episode of DEAL-NO DEAL recently (see this for a description). The math behind this game has been described in blog postings of Lance (see this as well as the above link)

The episode I saw showed something wrong (at least in my opinion) with the way they are promoting the show during premiere week. They have upped the amount of money to be a max of \$4,000,000 (instead of \$1,000,000). I saw the following (this might not be quite accurate but makes the point) There were 6 numbers left:
1. \$5,000
2. \$10,000
3. \$20,000
4. \$100,000
5. \$1,000,000
6. \$4,000,000
and the bankers offer was \$700,000. \$700,000 is so large and so life changing that the decision to take it (which she did) is rather obvious. Even the audience WAS NOT yelling `NO DEAL! NO DEAL!' like they usually do. To exaggerate this, imagine if the top amount was \$40,000,000 and your dilemma was whether to take \$7,000,000 or risk it to maybe do alot better but maybe do alot worse. YOU WOULD TAKE THE \$7,000,000. (or at least I would).

A question like `would you take \$70,000 or take the chance that you get \$400,000' is mildly interesting. But the \$700,000 is to large to not take. Hence the game gets less interesting mathemtatically.

Given a persons utility function (or something like it) what would be the optimal max amount (and optimal set of amounts) to maximize the games INTEREST? This question might be interesting.

1. They should just change the function that computes banker's offer to match a more realistic risk profile. I would quickly take the 700,000 but I would hate to be in that situation being offered 100,000 dollars. Well I guess not hate, but it would be a tough decision :).

Lukas

2. \$700,000 would be life-changing but not enough to, e.g., quit working for life. So there is something to be said for trying for the \$4M (depending on the odds of course).

3. This exact situation has been known to poker players for 100 years, and is part of "money management," which most authors consider to be the deepest part of gambling. The #1 factor that affects what wager is "interesting" is the size of one's starting bankroll. Imagine if all the contestants on Deal or No Deal were worth at least \$100 million each. It would be a much less dramatic program, because the players would not be facing life-changing decisions.

E.B., the pro who taught me to play poker, often told me that he always referred to chips as "units" and never considered how much they were worth until he left the table. His point was that you can't be afraid to put a whole ton of money in the pot if it's the right thing to do, nor can you play antistatistically loose because the chips are only worth a buck each so what the hell.

One book I can recommend that deals honestly with your earn rate relative to your bankroll is Matthew Hilger's Internet Texas Hold-Em. The math is clear: unless you have a lot of money to invest up front, your average hourly wage, even with world-class play, is pretty low.

Also of note here: the only time it is scientifically correct to play the lottery is when the jackpot gets phenomenally high, and an investment group can buy \$2-4 million worth of tickets. Investment groups have made a profit doing this at least twice in the last few years. I imagine, though, most readers of this blog would find \$2 million awfully interesting in itself.

4. To make the game more interesting, banker's offer must give a utility roughly equal to the expected utility of taking a random suitcase.

5. Mohammad -- just looking at utility takes away the very real aspect of risk worries. I would be hard-pressed to choose between a guaranteed \$1 million or a coin flip for \$4 million, because the latter gives me a 50% chance of nothing.

6. Hm, but thats just because your utility isn't linear. From your example, it seems your utility for n dollars might be Sqrt(n), so your choice is between Sqrt(10^6)=1000, or 1/2 * Sqrt(4*10^6) = 1000. A tough choice.

If 'interestingness' stems from making people face hard choices, it seems like an online learning problem -- the banker wants to make offers that match the player's expected utility, not knowing her utility function. Of course you can't learn much within a game -- as soon as the player takes the banker's offer, she's done. But perhaps you could hope to be competitive with the best fixed utility function in hindsight across players.

7. Since many seem to be unaware, let me note that it is not even clear that a well-defined utility function exists, at least in the way we typically think of it. (This is part of a whole research trend in economics.)

In particular, it is not clear whether there exists a function U: dollars -> reals with the property that, for *every* two distributions D, D' on dollar amounts, it holds that
Exp[U(D)] > Exp[U(D')] => person prefers distribution D to distribution D'.

(You can argue that this means people are not "rational" but that is irrelevant for the purposes of this discussion.)

For another thing, it should be obvious that even if such a utility function exists, it would be person-dependent. How would the banker possibly know an individual contestant's utility function?

8. Presumably, ABC does thorough background checks on each contestant to figure out their utility functions. Well, actually, just checking their income and net worth gives a ton of information. The poorer you are, the more nonlinear your utility function is, and the less they can offer you. If you already have \$10 million, on the other hand, then you will accept no less than \$500,000 for a coin flip between \$1 million and \$0, because your utility is nearly linear in this range.

9. Its a game show and I'm never going to be on it. I don't think anything could make it interesting to me at all.

10. To Bill Gasarch --

You need to find some really controversial topics to post to increase the popularity of this blog. Have you checked out Scott Aaronson's blog over the last two days? He's getting free printers, models etc. Don't let this blog die Bill!!! :-)