Deal Or No Deal...The Ultimate Game?

Back in March, Mary Beth Holtzheimer (sorry if I misspelled your name) ran a nearly perfect game on Deal Or No Deal. Check out the video to see what happened.

First of all, that is the great poker player Annie Duke who is advising Mary Beth. And Annie is exactly right in her assessment of the game. With $25, $500K, and $1M cases still in play, and an offer of $404K on the table, there are three possibilities if she chooses to play on.

1. She opens up the $25 case. Then, with the $500K and $1M cases remaining, you would expect the Banker's offer to be somewhere around $750,000.
2. She opens up the $500K case. Then, with the $25 and $1M cases remaining, you would expect the Banker's offer to be somewhere around $500,000.
3. She opens up the $1M case. Then, with the $25 and $500K cases remaining, you would expect the Banker's offer to be somewhere around $250,000.

Based on this assessment, two out of the three outcomes would result in an offer higher than the current offer of $404,000. The only outcome where the offer would decrease would result in an offer of $250,000. So, this analysis suggests that the minimum possible outcomes is $250,000. At this point, my recommendation would have been to reject the offer and continue in the game.

As you can see from the video, Mary Beth did continue and she eliminated the $500K case. But the offer came in at only $341,000. What gives? The answer lies is basic financial theory.

Generally, most people tend to prefer a sure thing over taking a chance. In the case (haha) of DoND, most players prefer taking a guaranteed and immediate sum (i.e., the offer) over taking a risk to win an unknown and future amount. This concept is known as the "time value of money" and is one of the cornerstones of modern financial theory.

The amount by which one person prefers the sure thing over the gamble is known as the "discount rate". Obviously, the amount of the discount rate varies according to many factors, one of which is the individual's risk tolerance. People who are more cautious are more likely to place a higher value on an immediate and guaranteed amount, thus they have a higher discount rate.

To illustrate this point, consider Mary Beth's situation where there are only two cases remaining ($25 and $1 million). Now consider someone who had no preference between taking the guaranteed sum and taking a risk (i.e., their discount rate is zero). For such a person, their expected value of the two cases would be slightly less than $500,000. Assuming that someone's discount rate cannot be negative, then this hypothetical person would refuse any offer that was below that amount.

An analysis of past games shows that the offers that are accepted by the players tend to be 91% of the average of the remaining cases. That suggests that most people place a higher value on the guaranteed offer over the unknown risk (a discount rate of 9%).

Presumably, as the stakes get higher, people's risk tolerance tends to change. Between a choice of $25 and $100, most people probably wouldn't care much (i.e., the discount rate tends to be lower). But between a choice of $25 and $1 million, most people would probably become very cautious out of fear of losing the top prize (i.e., the discount rate tends to be higher).

The offer of $341,000 implies a discount rate of almost 32%, which suggests that the Banker believes there is a strong preference toward the guaranteed sum. Clearly, the Banker was correct as Mary Beth (wisely) took the offer.

If Mary Beth's discount rate was lower than 32%, then she would not have accepted the offer. Another way of saying this is that if Mary Beth would choose the guaranteed sum over the unknown sum less than 2 out of 3 times, then she would not take the offer. Obviously, nobody knows what their exact discount rate is but most people do have a gut feeling about their preference between a sure thing and taking a chance.

Of course, it turned out that she was one of only two players to have ever chose the Million Dollar Case. There is only a 3.8% chance of that occurring, so Mary Beth clearly beat the odds.

This was an incredibly exciting game. The "ideal" game in terms of having the most drama would probably be a situation where a contestant had the $1 million case and the $0.01 case remaining. This came pretty close!

Who is a Jeopardy cheater?

On last Thursday's Jeopardy, something very strange happened. As you can see from the clip, Paul (in the 2nd position) claimed that his electronic pen failed during Final Jeopardy and he could not write down his response. As per Alex's instructions, he used the backup pen and paper. He ended up with the correct response and wagered enough to win the game and beat Tom who was the defending champion. Roll credits and show over.

Well, the story does not end there. If you watched the show on Friday, you would have noticed that both Tom and Paul were both back.

Jeopardy takes the integrity of its show very seriously and does not want there to be any uncertainty about its clues, the responses, or the rules. Apparently, the producers conferred in between tapings and determined that Tom may have been at a disadvantage because of the extra time that Paul was given to write his response. If you watch the clip, Paul is given at least 10 extra seconds, which can make a big difference considering that the Final Jeopardy time is 30 seconds. So, it was decided that the fairest way to resolve this was to invite both players back for a rematch, which Tom won decisively.

But the story does not end there either. The big question is why didn't Paul just write down his response on the card within the 30 seconds? If he had done so, then there would be no dispute about giving one player an advantage over another.

Paul has addressed what happened on the Jeopardy message board. He concedes that there may have indeed been nothing wrong with the pen itself and that he may have caused it to malfunction by leaning against the electronic screen and they were even warned not to do so during the pre-taping instructions -- you would think Jeopardy would be able to spring for better technology!

He also concedes that they were explicitly told before the taping that the pen and paper should be used in the event of a malfunction. But Paul claims that he interpreted the instructions to require that he wait until he is told directly to use the pen and paper, hence the reason he waited until the end of the 30 second time limit to write his response.

He also claims that Tom (the defending champion who lost in the disputed game) raised a stink about the outcome and claimed that he was cheated. For the record, Tom denies making any remarks about the game until after the decision was made and indicated that he didn't even have the opportunity to do so.

But what really seems to be a sore point for Paul is that all this talk about the possibility of cheating in the 1st game was very unnerving and put him at a disadvantage in the 2nd game (the rematch). He suggests that he could have won that game without this dark cloud hanging over everything.

Contestants have had to resort to the pen and paper to write their response, so this is not a first-time occurrence. But (to my knowledge) every contestant in that situation did so within the 30-second time limit. This suggests to me that the contestants are told pretty clearly that the pen and paper are there if the electronic pen does not work. If that is the case, then I believe that Paul was given an unfair advantage and Tom should be declared the winner. Paul was given over 10 extra seconds which is a 33% increase in time that everyone else received, which is definitely an advantage.

But if Paul's was led to believe that he should wait until a clear signal to use the pen and paper, then this is the fault of the show and Paul should not be penalized.

With some possible doubt about how the instructions were given, I agree with the producers' decision to bring back both contestants and give them another shot. The alternative would have been to throw out the Final Jeopardy question entirely and do another one. But in these cases, Paul was right to think the contestants should be given some breathing time to allow them to regain their composure.

Like sports, contests, and games of all kinds, game shows rely on the integrity of the game itself. Once the rules start bending a bit, then you open up all kinds of accusations and possibilities of cheating, favoritism, and other ugly scenes. Jeopardy handled this situation as best they could and still kept the game intact.