Game Theory, Bayes’ Rule, and Football
http://runcodhit.blogspot.com/2010/03/game-dynamics-and-football-part-ii.html
Preparation is the key to success in American football, and studying your opponent’s tendencies plays a vital role in preparing for an opponent. As the article in the link rightfully points out is that football coaches are always trying to find ways to get a leg up on their opponent. Game theory can play a role in this. The author points out that game theory “applies to the sciences of decision making and competition games in general. Using game theory we can discover optimal choices to make, given certain circumstances.” Each play in a football game can be looked at as a mini two-player game between the offense and defense, in which each side has to choose strategies that will result in the highest payoff. Having some intuition into what strategy your opponent will use is key in determining your best response. If it’s 3rd down and long and the offense has 5 wide receivers on the field, the defense’s best response is to put extra defensive backs on the field and defend the long pass. Alternatively if it’s 3rd and goal at the 1, the offense is most likely to come out in a tight goal line formation and run the ball. The best response to this strategy is to send a couple extra big defensive linemen into the game and play for the run. The linked article goes into more depth on how game theory plays into the decision making process in specific situations in a football game.
Furthermore Bayes’ Rule can be a great tool in determining your opponent’s tendencies in specific situations in a football game and the corresponding best response. In short it is defined as the probability that an event A will happen given that some other event B has occurred (Pr(A|B)). Let’s say it’s 2nd and 2 and you’re a defensive coordinator attempting to anticipate a pass or run from the offense. You know that the offense is a bit on the run-heavy side and generally runs the ball 55% of the time (Pr(run) = 0.55 and thus, Pr(pass) = 0.45). However from watching film on this opponent, you found that in their past games this season, 10% of their run plays came on 2nd and short (2nd and 4 or less) and 20% of their pass plays came on 2nd and short (Pr(2nd&short|run) = 0.1, and Pr(2nd&short|pass) = 0.2). So probability that the offense will run the ball here, Pr(run|2nd&short), is equal to: [Pr(run)*Pr(2nd&short|run)]/Pr(2nd&short), where Pr(2nd&short) = Pr(run)*Pr(2nd&short|run) + Pr(pass)*Pr(2nd&short|pass) = 0.55*0.1+0.45*0.2 = 0.145, which represents the how much of the time the offense is in 2nd and short situations. So, Pr(run|2nd&short) = 0.55*0.1/0.145 = 0.379—so the offense will only run about 38% of the time on 2nd and short and thus you should probably expect a pass (generally teams will throw the ball deep in 2nd and short situations to try for a big play knowing that they still have 3rd and short to get the first down if they throw an incompletion).
Bayes’ Rule can be used to anticipate the opposing team’s strategies in situations like the example above or in more specific situations. (e.g. the likelihood of the defense to blitz on 3rd and medium/long yardage situations, likelihood of the offense to run the ball with a particular formation or personnel grouping, or the likelihood of the offense to do a play action pass from midfield as opposed to at its own 1-yard line, etc) Football players and coaches alike would be smart to implement this type of statistical analysis into their game planning. Overall, game theory in general and Bayes’ Rule can play a big role in any game that involves decision making, like football.