Re: Game theory football
OK, here's a general forumla for nth and X.
You need to start at 4th down.
For third down, for each situation, work out 4 probabilities: P(winning for (R,R))[A], P(winning for (R,P))[B], P(winning for (P,R))[C], P(winning for (P,P))[D] [offense's move listed first]. For (R,P) and (P,R), the winning chance will be an average of the 4th down winning chances from each die roll. For (P,P), it's the 4th down winning chance for the same yardage, and for (R,R) it's the 4th down winning chance for X-1.
In that case, equilibrium run percentage = (D-C)/(A-B-C+D). Pass percentage is one minus that.
PS, if Aaron forgot to adjust the win percentage in 4th and 2 for overthrowing, I'll be off slightly, but it's his fault. [img]/images/graemlins/smile.gif[/img]
Yardage/Run %/Pass %/Win %
3 .5807 .4193 .6832
4 .5790 .4210 .5972
5 .6190 .3810 .5344
6 .6098 .3902 .4164
7 .7057 .2943 .3194
8 .6946 .3054 .1986
9 .7403 .2597 .1555
10 .8128 .1872 .0905
I'm a bit perturbed by the fluctuations in the running frequency. Anyone have an explanation or can anyone find my error?
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