Math is Hard

Every year at work I am required to take a refresher training course in the often-bewildering ethics rules. For two out of every three years, the training consists of certifying that I have looked at a web page. However, every third year the training is in person. This is such a year for me.

In order to keep attendees paying attention, the ethics folks organize some kind of a game. One year it was “football,” in which correct answers would move the ball some random number of yards down the field. Another year it was a version of “Who Wants to be a Millionaire?” I seem to recall a version of “The Weakest Link” as well. This year it was “Deal or No Deal.” For those who haven’t seen the game show, the key feature is that the contestant selects an object of unknown value and he or she can play for the value of the object or can take the offer of some known amount of money. The offer is usually somewhere around the mean value of the remaining objects. (As an example, if the contestant knows that his object is worth either $100 or $1000, the banker might offer $525, slightly under the mean value of the two choices.)

Frankly, my aim is to get out of the room in no more than an hour, and I don’t care about the outcome of the game. My colleagues, however, are so competitive they are constitutionally unable to help themselves. This year they even argued about some of the answers in a legalistic, hair-splitting manner. As a result, one would think everyone should be strategizing about both the answer and the bet - taking the “deal” or not.

As we got to the last two questions, my team was up by about $900 and would get the last question. All of the objects near the average value were gone - all that remained were some low-valued objects ($100, $200, and $600, if I recall correctly) and some high-valued ones (three more in the $2000 range). The “banker” offered around $1000 - enough to get ahead of my team if the other team answered the question correctly. Take the deal? There was a 50% chance that the object would be worth less than $900, and thus the team would lose regardless of whether it got the question right. There was a 50% chance the object would be worth a lot, although the team would still have to get the question right to climb ahead of my team. At that point, we’d still get a chance to answer one more question for the win, and we would know whether we’d have to roll the dice on the object in order to get enough money to win.

Taking the $1000 offer by the banker is straightforward: to win, the other team needs to answer the question and hope we blow our question. If the probability of answering correctly is 50%, then the odds of the other team winning are 25%. Rejecting the offer is more complicated. Half the time the object will be worth too little, and the team loses with certainty. Half the time the object will be worth more than enough to get ahead. Depending on the draw, we might be able to win by taking the banker’s offer and answering the question - the same odds as if the other team took the $1000 banker’s offer. The rest of the time we’d be forced to reject the banker’s offer and hope for a good outcome. We don’t know the odds of this outcome, but if we’re in that branch of the decision tree there’s some chance the other team will win regardless of how we answer the question; otherwise, it still comes down to whether we can answer the question correctly. In short, the only way this choice is better is if the odds are high the first team gets a good draw and the second team gets a bad draw from the remaining objects, because all the other possible outcomes are no better (and some are substantially worse) than just taking the banker’s offer. Yet we could all see the distribution of remaining objects, so the “first team’s draw is good, second team’s draw is bad” outcome is unlikely.

Naturally, the economist-free team rejected the banker’s offer and got an object worth $600. Game over. In the end, I’d like to think I re-learned some of the ethics rules and the other team learned a little about probability theory.