Football Scoring and the Idea of Marginal Utility

While watching "Who Wants to be a Millionaire" one night with some friends, a few friends seemed to agree that if you were reasonably certain that you knew the answer, and you only lost half your money if you got it wrong, you were an idiot if you didn't guess. I mean, if you have $64,000 and if you get this question right you get $125,000, while if you get it wrong you get $32,000. Basic math says that if you have a 50/50 shot of getting it correct, if you guess you will have an average of $78,500 ($125,000 * .5 + 32,000 * .5), while if you don't you get $64,000. And the more sure you are that you know the answer, the higher the average amount of money you would get if you guessed becomes.

So, obviously, it seemed obvious to these very intelligent kids that you should take a stab and guess. However, they were assuming that your desire for money is linear. In other words you want $100 twice as much as you want $50, and $100 million twice as much as $50 million. However, this may or may not be the case, depending on the person. Personally I really don't want $100 million dollars more than $50 million dollars at all, and I'm sure that the utility of having $100 million dollars as opposed to $50 million dollars is minimal to the majority of people. The point is that just because you are willing to spend $X of money/effort to get one of something, you are not willing to spend $2X of money/effort to get the second one, and you sure wouldn't risk the first one to get three, four, or ten of more of these 'things'.

Now, how the hell does this relate to football? I mean, having more points is always beneficial right? Well, yes; however, there is always a cost involved, and things are never what they initially seem. Assume that extra points are made with 100% frequency (which they are not, but they are close, and the math does not significantly change if I lower this to 99%, it only gets more complicated), and also assume that the probability of winning the sudden death overtime is 50/50. Now, if Team A has a 60% chance to make all two point conversions, and Team B makes 50% of all of their two point conversions, then why wouldn't you always go for two? Unless there is a good reason get the sure point (to win the game for example) this will make it so that you will win more games than your opponent if you are otherwise evenly matched right?

Well, not quite. This relates to an essay I wrote in college, and is very similar to the deeply flawed college overtime system. Going second in the college overtime system gives you a tremendous strategic advantage. You know how many points you need to win/tie, and you can play your game based on that. If you need a TD to tie, you use all four downs, if you need a FG to win, you can just sit around and do nothing and then you have a 40 yard FG at the end of it. Now this would relate to the above example in the following way:

So lets say that in a tied game team A scores a TD with 3:00 left on the clock. In this case it would be a tremendous mistake to go for two, for the following mathematical reason. If B does not march down the field and score a TD, then the point is totally moot. However, if they do and they do not leave enough time on the clock for A to come back and win then A's decision is a bad one.

If A makes the two point conversion, then if B scores a TD they will go for two. If A misses then B will kick the extra point and win.

So if A makes the two point conversion and B scores a TD to tie, then there is a 30% chance A wins outright in regulation, a 30% chance it goes to the even money overtime, and a 40% chance that B wins outright in regulation.

Now, I will freely admit that this is one mostly irrelevant example, but it shows that it is not *always* a good idea to just follow the numbers. You have to take everything into context. While I believe that if you simulated these numbers with A blindly going for two and B playing "intelligently" then A would probably win the majority of the games, but I feel that is simply due to the very large difference between A and B's proficiency in making the two point conversion. If it was closer to a 1% difference I feel then you may see B winning more games than A, despite the apparent advantage that A has by merely going for two every time.

This was an introduction to this topic, and I will expand more on this tomorrow by adding the idea of going for 7 vs. going for 3.