"Who Wants To Be a Millionaire" has had a different feature the past two weeks. They brought back the ten highest scorers over the previous two-month period for what they called the "Tournament of Champions", or "Tournament of Ten". During each show one of the ten gets to answer a question for a million dollars. If the contestant answers and gets it wrong, he/she goes down to $25,000, from the original amount won on the show.
The odds favoring answering seem overwhelming here. The first few contestants were at only $50,000, so they were risking only $25,000 for a chance to win a million. This seems to be a no-brainer at first blush, but there is a kicker. If anybody ahead of you also gets the million-dollar question, then you go back down to your original amount won.
The guy in the #8 position had the question "how many people have ever lived on the earth"? The choices were 50B, 100B, 1T and 5T. He guessed (the only contestant to do so) 100B and was right. Then he had to wait the next seven days while one-by-one the other seven took their shots at their respective questions and the chance to knock him back down to his 50K.
What is interesting mathematically is how do you assess your probabilities when guessing? Without the chance of getting knocked back down, it is easy: 3/4 chance at losing 25K, compared to 1/4 chance of winning an additional 950K. But say at position #8 you assume there is a 90% chance that you will not survive as the winner. Now the odds look like this: 3/4 chance of losing 25K, 1/40 chance of winning 950K, and a 9/40 chance of staying where you are. The odds now put you at only 5K ahead (18.75K expected loss vs. 23.75K expected gain).
As it turned out this 90% assumption would have been way too high, as all the other contestants were too risk-averse to hazard a guess (even though 7 of the 9 would have guessed the right answer had they been bold enough to guess). The bartender who sweated out all the others from the #8 position ended up winning the million!
The plans of the final contestant raised an interesting issue. She wanted to create scholarships for needy children. This altruistic motive actually presented the interesting point that the mathematical approach was valid for somebody with this motivation. That is, one can say that she will do four times the good with four times the money (she was at 250K), so the math works.
By contrast, when you are using the money for your own purposes, rather than for others, it is false to say that four times the money will make you four times happier. Indeed, I think one can assume that many lottery winners actually end up with miserable lives, because they don't know how to handle the sudden riches. If somebody offered me a sure $100K, or a 50-50 shot between nothing or a million, which would I take? Mathematically it is a no-brainer, but we are talking human reality here, not math. If I was thinking solely of myself, I would likely take the sure 100K, because that is enough for me to achieve any goals I can conjure up. This is essentially what 9 of the 10 contestants did, they took the sure money they had, rather than risking even part of it for the big score.
The final show was quite special. The last contestant was a very classy lady from the South, and though she had the right inkling she declined to guess, and it was obvious after that that she and the bartender had bonded over the period of the taping of the 10 shows, and in fact one of the things he wanted to do with his winnings was take the Southern lady skydiving. An interesting sidelight here is that the bartender said he knew the answer to the question, and expected the lady would get it.
A whole new game theory dynamic enters in here if you open up the possibility of collusion between the contestants. Total winnings for the two combined would be only $1,050,000 if #1 guessed right, while if she walked away that total would be $1,250,000. I don't for a minute think that there *was* collusion, but the game theory analysis is interesting. Given the history to that point, that the questions were hard, yes, but that most contestants had the right answer but were too risk-averse to guess, it would be rational for #8 to reason that there was a good probability that #1 would guess and get it right (she had been brilliant during her earlier time on the show). Why not offer her a good chunk of his million in exchange for her walking away?