Sunday, December 20, 2015

Not Exactly The Monty Hall Problem

This New Year's Eve is the opening round for this season's College Football Playoff. Oklahoma will play Clemson followed by Alabama playing Michigan State. The two winners will then face off for the National Championship 10 days later.

Since my Sooners are in the playoff this year, I've been thinking a lot about it. One random thought that crossed my mind was a bit of a logic puzzle. Read it through and answer quickly, and then think about it a while to see if you would like to revise that answer.

Suppose the following:

You are a fan of one of the teams in this year's playoff. One night while dreaming a genie with light-brown hair appears before you offering a choice among a few options*. The one you choose will be the future.

The options are:
A. Your team's total points scored in the playoff will be 75 points.
B. Your team's total points scored in the first game will be at least 50 points.
C. Your team's total points scored in the playoff will be 75 points, and at least 50 points will be scored in the first game. 
D. Your team's total points scored in the playoff will be 50 points, and you can choose how many to allot to each game.
Which do you take, and can the options be objectively ordered from best to worst?

My answers are below the jump.

*It is helpful to know a little bit about college football to fully appreciate this puzzle. The average points scored by winning teams is about 37 while the average points scored by losing teams is about 19. The lowest final score a team can have is 0 and the next lowest is 2 (a score of 1 is not possible). There are no ties in college football--overtime is played until a victor is determined. Last year out of 776 games played in the highest division of college football (the one of relevance here) a team scored 50 points or more 150 times and only 7 times did that team lose. Also last year a team scored 25 points or fewer 698 times and 564 times that team lost. Note that my data does not include the bowl and playoff games from last year.

Here is how I answer the questions:

The objectively best choice is D. Notice the qualities it possesses: You can guarantee your team advances to the second game. This is not a feature of the other choices although the likelihood is high in all three. And to maximize your team's chances of winning both games, you should elect to have your team score only 2 points in the first game (and note what a tense, odd game that will be).

After that I believe B is the next best choice since it possesses the best part of option C but with more upside (no limit on how many points your team will score in the second game). Therefore, it also contains most of the guarantee of option A along with more upside.

Now we are down to A and C. In that case I believe option A is better since it has more opportunity for a high score (more likely a winning score) in the second game. Option C leaves a lot of room for a poor performance in the second game.

My ordering (D > B > A > C) would not have been my first instinctual guess. I believe I would have recognized that D was optimal (hard to say since I was designing this as well), but C seems like a quick-response next best. I think I would have chosen A over B thinking that it makes highly likely an advancement to the second game not thinking that the higher that likelihood (a high score in game one) the lower the likelihood of a victory in game two (fewer points left to score).

I am about 90% confident in my answers/reasoning, but that leaves some room for doubt/argument.

The lesson from this is that the best option can be quite counter-intuitive at least at first glance. Even though games are won by scoring more points and more points are almost always superior to any other play-by-play outcome, points are a means to the end (winning) rather than the end in themselves. That is why at the end of a game a team might choose not to score (rather than give the opposing team possession of the ball) or why a team might choose to allow an opponent to score just to take possession back and have a last-ditch chance. If the objective is to win the games, the choice that gets you to the second game has a built-in advantage. After that, you want to maximize your chances of winning that second, championship game.

Go Sooners!