Tuesday, April 04, 2006

deal expectations

NBC has a game show called Deal Or No Deal in which the contestant picks one of 26 suitcases containing amounts of money ranging from $.01 to $1,000,000. After picking the suitcase, the contestant slowly eliminates the other suitcases, revealing their amounts. The contestant is periodically offered a "deal" to trade in their suitcase for some amount of money.
If you've eliminated a lot of the high paying suitcases, you get offered a low deal. If you've eliminated all but the most valuable suitcases, you get offered a pretty sweet deal. If you decline all the deals to the very end, you get the amount of money in the mystery suitcase.

It seems like a clear case of probabilistic expectation, so I was curious if the "deal" offers were just the expected value of the contestant's suitcase; I don't watch the show often and I don't normally carry a calculator with me, but I was watching last night and had my laptop with me, so I decided to quickly calculate some odds.

When we tuned in, the current contestant was down to 3 suitcases: $.01; $400k; and $750k. Ignoring the one cent suitcase, that's 1/3 * 400k + 1/3 * 750k or an expected value of $383k. Her deal? $375k. Pretty close. (She took the deal; her suitcase was worth $400k, so she only lost out on $25k. Chump change =).

One contestant later, they were down to 8 suitcases with one worth $750k. The offer given to the contestant was $21k. Now $750k / 8 is almost $100k, so the deal value was clearly below expectation. The contestant eliminated two more (low value) suitcases, and her deal got bumped up to $63k. One more low value suitcase down and her deal went up to $96k (her suitcase now had expected value $150k). She took that deal.
They then walked through what would have happened had she had continued the game. She kept eliminating suitcases until there were 2 suitcases (the one she'd picked and one last mystery case) and they said her offer would have been $400k. $25k above expectation.

So it looks like the game has an increasing expectation multiplier that actually hits >1. I guess if I'm ever on the show, I'll work my way down to 2 suitcases and then take the deal -- proving once and for all that mathematicians make piss poor gameshow contestants. Now if only they'd bring back Let's Make a Deal...


  1. I've watched that show. No social value whatsoever.

  2. I saw that show at the gym once. They did a viewer poll where players would text message in their choice among 6 numbered suitcases, where a random one contained a prize of some nature. What was interesting to me was that the distribution of votes was a bell curve: almost everyone picked 3 or 4, some people picked 2 or 5, and almost no one picked 1 or 6.

    Were players trying to minimize the distance between their guess and the correct answer? What psychological forces were at work?

    People are interesting creatures.

  3. Clearly a random number generator would never pick the first or last number.

    It's like in HS when you have to make a guess on a multiple choice test, you always guess "c" because a,b, and d just don't seem random enough.

  4. I don't think the game is that simple. You're analyzing each step separately, basically making a decision about whether to accept the deal based on the expected value of your suitcase. You're not considering how the expected value will change if you proceed with the game and open more suitcases.

    But I agree with the general sentiment - completely worthless show.


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