Saturday, December 25, 2010

Game Theory

For your Christmas present this year, allow me to introduce you to a little Game Theory. Let's start with Minimax Theory. Impress your friends! Freak out your relatives! Here's a summary of that theory:
"For every two-person, zero-sum game with finite strategies, there exists a value V and a mixed strategy for each player, such that (a) Given player 2's strategy, the best payoff possible for player 1 is V, and (b) Given player 1's strategy, the best payoff possible for player 2 is −V."
Let me take a crack at it for you, without the algebra... Two opponents in a game where one will win and the other will lose will each try to minimize their bad outcomes, and maximize their good outcomes.

A gentleman named John Forbes Nash created a game solution concept for a zero-sum game (one with a winner and loser) that implies two equally matched players, each knowing the best strategies of their opponent, would adopt opposing strategies such that neither could successfully do what would be optimally best. When you get to that point, you're in what's called a Nash Equilibrium. Two tic-tac-toe players that know what they're doing can easily get to a Nash Equilibrium.

Let's think for just a moment about youth football, and the decisions of offensive and defensive coordinators. Without worrying about passing too much, since passing isn't that easy for fifth graders, then as a defensive coach you might choose to defend the outside run, or the inside run, or 'play it safe' and attempt to defend both types of running plays equally well.

As an offensive coach I can choose to run inside or run outside.  On the table below, I've shown what the defense can do on the top column labels, and what the offense can do on the row labels, and within the matrix, what the yardage gain outcome might be given the circumstances.

D Defends
D Defends
D Defends
O Runs
5 4
O Runs
50 4

There are four downs available to gain ten yards for an offense. But the risks associated with giving possession back to the opposing team on a fourth down failure make three downs the effective opportunity.

Under two out of three scenarios shown in the matrix above, as an offensive coach I can run a successful play. At first blush, that would seem to be pretty good odds. To optimize my chances for a five yard play, I need to 'keep you guessing' so let us presume that I will try to run as often inside as I do outside.

Also notice that your defense can't just sit in a defense-of-both posture. If you do that then I will gain 12 yards every three plays and march down the field.

Given my criteria, your defense would best be served to guess and defend either inside or outside on every first down. Two outcomes would be possible. If you guess correctly, I gain 0 yards. If you guess incorrectly, I gain 5 yards.

So the down and distance for the next play is either 2nd and 10, or 2nd and 5. If the distance is 10, then for both of the next two downs you should play safely against both types of runs, and force my offense into 4th and 2.

If though, you did not guess correctly on first down, you are now defending a 2nd and 5 situation. You should guess again. If you guess correctly on second down, you will force our offense to face a 3rd and 5, at which point you can safely defend against both and inside and outside run and likely force us into 4th and 1. If you guessed wrong for a second consecutive time then you have just allowed me a first down. But you should guess again, and hope you won't be wrong four times in a row.

Unfortunately for most offenses, if the defense guesses correctly even ONCE in any series of three consecutive downs, which they are statistically likely to do, then the offense will encounter a fourth down situation within four to six plays.

By guessing in this manner, your defense minimized risk over a series of three downs, and maximized the chance of creating a fourth-down during the possession. That's Minimax theory at work.

Because it gets really hard to string three first downs together, offenses reach a Nash Equilibrium in the game, and can't score.

Enter the option. (You knew it was coming, didn't you?)

The coolest thing about the triple option is that it tilts the odds in favor of the offense running the right play. The guesswork has been removed. My offense no longer cares whether your defense is defending the inside or outside, until the play has commenced.

Running the triple effectively is like being able to bet at the roulette wheel for both red and black with the same chip. Once the ball gets rolling, you're pretty sure that one or the other is going to work.

I've not just minimized, but actually eliminated the statistical advantage for the defense of defending either the inside or the outside. If you defend the outside, the triple runs inside. If you defend the inside, the option goes outside.

The test of course, especially in youth football, is that the next best strategy for your defense will then be to force my offense (or our coach) into errors. Your defense will try to get us to make the wrong read, or run the wrong version of the option, or defeat our blocks, or make a big play with some type of high-risk strategy that might leave the defense in an otherwise unsound position, but could create a big loss of yardage.

And that's another whole set of circumstances that we would have to draw out, and won't worry about today.

The triple option isn't the exclusive way to apply these game theories to football. Many modern college spread offenses have options on pass patterns as intrinsic components - so depending on how the defense is aligned, a pattern may break off or take off in different directions and timings. Zone read running plays are option running plays from shotgun, and NFL offenses throw quick slants and wide receiver zip passes off called running plays all the time, depending on what the defenses show. These are all more modern ways of using real-time decision making to maximize good outcomes.

These games within games are all Minimax efforts to tilt the Nash Equilibrium out of balance and into the favor of the offense.

To all our players, parents and friends, here's hoping your holidays are maximized for great outcomes, and  don't let the eggnog effect your equilibrium too much.

Ho, ho, ho, Merry Christmas!