Wednesday, September 09, 2009

A Nerd Talks About Survivor NFL Gambling

My group of fantasy football buddies are an extraordinary collection of nerds--we've got surgeons, IT guys, professors in respected institutions, mathematicians, laboratory managers (seriously) and other sorts of nerds.

The challenge that proves that is a common sideline gamble--the Survivor NFL challenge. Pick a winner a week--you are in until the team you picked loses. You can't pick the same team twice. That's it. Whoever lasts the longest wins.

A question came up as to whether a person who waits until the last Sunday games gets any sort of advantage. Within 20 minutes, this is the answer that came back from our math guy, who I will call Prof. Jerusalem Webster Stiles:

Prepare to have your mind blown! Professor Stiles speaks:

I'm not so sure it gives an advantage in terms of probability...
Your theory is similar to the theory behind the famous Monte Hall that has confused and enraged folks with extensive mathematics training.
The basics of the problem -- i.e., the gameplay of Let's Make a Deal. You have three doors, behind one of them is a car, behind the other two are goats. The game starts with the contestant choosing a door (which stays closed). Monty then opens one of the remaining two doors. If both have goats behind them, he chooses one randomly. If one has the car and one the goat, he opens the one with the goat. The question is, do you change your door or stick with the original door?
Most people (including many mathematicians) assume that there is no advantage to changing. The goat could still be behind either closed door with equal probability. However, in reality, you double your chances of winning the car by switching doors. It's completely counterintuitive (it seems that the car is behind each door with 1/3 probability). But, once you know the goat is behind the opened door, essentially, the new information changes the probability the car is behind the other door (that you would switch to) to 2/3.
For an extensive explanation, see the ever-correct Wikipedia
Now, our situation is different. In theory, each football game is an independent event (in that the outcome of an early game shouldn't affect the outcome of an afternoon game). Or, mathematically...if team A plays B and team C plays D, then the Probability of A and C winning is P(A wins) x P(B wins). If the odds for each game are 50-50...then the probability of A and C winning is .25. Or, from combinatorics...there are four possible outcomes (4 choose 2 less the 2 impossible outcomes -- A and B win or C and D win), so any possible outcome has a .25 probability.
Moving to the real situation. Say there are 16 games each weekend, with 11 of them occurring "early", and 5 occurring "late". Thus, the early picker is presented with 2^16 = 65,536 possible outcomes whereas the late picker has only 2^5 = 32 possible outcomes -- with each outcome representing a possible set of winners). So far, it looks like being the late picker would be better. However, it would only be better if you had to correctly pick the results of ALL games. In fact, it would be far better (3.1% versus 0.002%). But, we don't have to correctly pick all the games. You only have to pick one game. So, if an early picker selects team A, then there are 2^15 possible outcomes for the other 15 irrelevant games that have team A winning. So, his chance of being correct is 32,768 / 65,536 = 0.5. For the late picker...if he picks team C, there are 2^4 possible outcomes for the other 4 irrelevant games, and his chance of being correct is 16 / 32 = 0.5.  Now, all of this assumes that each game is equal and that each person picking is equal. Handicapping can raise your advantage and can change these probabilities (making this an impossibly complex problem). In fact, if the morning games are occurring and you have chosen the Patriots to win an afternoon game, but suddenly find out that Moss ran over Tom Brady with a car and neither will play today, you could switch to one of the other games...or, you could just pick the team playing the patriots that week.
Additionally, if the afternoon games have the higher point spreads, then taking a wait and see approach might be advantageous. However, if that is the case, you're more likely to select those teams right off the bat. So, with everything else being equal, there's no advantage to choosing later. Furthermore, as the commissioner pointed out, waiting until the later games limits your choices and may back you into a corner on a pick.


Andrew Wice said...

I thought we were going to stick to 'mom' jokes on this site.

Miwacar said...

You failed to recognize me as a football coach and general bon vivant in your list of nerds. Perhaps I am not a nerd at all. Nerds!