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!