My father has been watching a new gameshow on NBC. It’s hosted by second-rate comic Howie Mandell (whose biggest claim to fame is the voice of Bobby’s Dad on the 1990s cartoon Bobby’s World), and involves about as much skill as picking lottery numbers. The shockingly creative name of the show? That’s right, Deal Or No Deal.
For those of you who don’t watch the show, it works like this: The contestant selects one of 26 numbered cases at the beginning, each case is unique and contains some cash prize ranging from $.01 to $1,000,000. After selecting a case, the player must have the buxom assistants open six of the other cases, revealing what is inside. At this point “the banker” calls Howie to make an offer of a certain cash prize for the case the player is holding. The player can accept or reject the offer, but if he/she declines he/she must open another few cases. This continues until the player takes the deal or opens every case, at which time he/she will get whatever cash prize is in the case chosen at the start. Every offer goes up or down depending on which prizes are revealed (and thusly eliminated) as the player selects cases for the models to open.
The interesting thing is that the whole show is a pretty good example of the Monty Hall problem, but it doesn’t involve any ability to change your selection. Therefore, the probability of having the $1,000,000 case is 1/26 the entire game. Each case opened tells everybody that the player certainly doesn’t have that prize, but it in no way affects the probability of the player actually having the $1,000,000. Having watched a couple of nights of this, it seems that “the banker” (a most mysterious man in a smoked glass booth) makes offers based on some percentage of what the expected value would be if the gut-feeling behind the Monty Hall problem were correct.
This is going to get complicated, so I’m putting it behind the cut, but the expected value of the entire mess from the get-go is:
(1/26)*($.01)+(1/26)*1+(1/26)*5+ +(1/26)*500,000+(1/26)*750,000+(1/26)*1,000,000
I work this out at about $131,477.54. You picked the case from a set containing all other values at the start, so if you get down to two cases your chances of having the $1,000,000 aren’t 50/50, they’re still 1/26. That may not matter all that much, depending on what you’ve learned in the process of having opened all the other cases. If you get down to $1,000,000 and $.01 and they offer you more than $38,461.54, bloody take it. You have a 25/26 chance of having the $.01 prize! If you get down to the last case, there’s always a 25/26 chance that the other case has $1,000,000 in it provided the $1,000,000 hasn’t been eliminated already, meaning that whatever the bank offers, so long as it’s better than $38,461.54 plus whatever 25/26 times the other option, you should take it.
The absolutely infuriating thing about this game, however, is that once the player has been playing a while they start showing little splash screens saying So-And-So has X/Y chance of having $1,000,000 or whatever the largest available prize is. That’s a damn lie. The player has, as outlined above, exactly a 1/26 chance at whatever the largest prize left on the board is because he/she picked the case at the start.
I guess this is why you don’t see economists on game shows, I barely qualify for the moniker and once you offered me excess of the expected value for the entire game, I’d walk.
