Some folks seem to be defining a trade-off: football is either won by game theory, or great execution of plays. It can't be both.
That's a goofy proposition.
Pretty similar to saying football is either won by good offense or good defense. It can't be both.
Here's a direct example of game theory in football. This is derived from Nash, who was a leading thinker in game theory until his beautiful mind imploded. We have Nash and Von Neumann to thank for the Prisoner's Dilemma, the classic game theory dilemma.
Go back in time 10 years to make this a bit more relevant.
Steroids are no good for you. They have dangerous side effects. Although they provide a significant advantage in strength and ability to recover from fatigue, often ligaments and tendons cannot take the stresses created by more powerful muscles, leading to injuries.
Steroids also lead to bigger, faster athletes in contact sports. The collisions of bigger, faster athletes lead to more serious injuries.
Further, lots of kids see those bigger, faster athletes and want to emulate them. Because athletes take steroids, kids take steroids.
So the league encourages you not to take steroids, but doesn't have an effective testing program.
You have two options:
Cooperate with the league, and resist steroids
Defect from the league, cheat, and take steroids
Let's look at your choices:
If no one else takes steroids: if you take steroids, you have a significant advantage over other players and can quickly become a starter with a huge salary; if you do not take steroids you will be a back-up player for several years and make a lower salary but your long-term health outlook is much better
If many other players take steroids (or, similarly, if the guy you are competing with for a spot takes steroids): if you take steroids, you are at parity with everyone else and can make the team but at the risk of your long-term health; if you do not take steroids you will be released for some player that did and will live a longer, poorer life
So, in your individual situation, it's always to your short term advantage to take steroids. However, from everyone's point of view, it's better if no one takes steroids. The best outcome for everyone is the first option.
Game theory, however, encourages you to defect. That's pure game theory applied to football.
Game strategy is an application of that, where you are maximizing the probability of a win. Straight probability says: if you average 6 yards per pass attempt and 4 yards per run attempt, always pass. Game theory says: if you always pass, the opponent will prepare to always defend the pass, and your average per pass will drop. So game theory will encourage you to maximize the mix of run and pass, for example, to maximize your probability of a win.
Of course proper execution is important. In the example above, if you improperly ingest the steroids, they will not have the intended effect.
Game theory has absolutely nothing to do with selecting players in the draft. The fact that Ryan Leaf failed and Tom Brady thrived has nothing to do with game theory.
Game theory is also quite different than statistical analysis. Moneyball is about statistical analysis. Everyone can see that a .300 hitter with 50 home runs a year is valuable. However, if you cannot afford those players, statistical analysis will enable you to identify undervalued assets to improve the number of wins you can achieve on a limited budget. Both are analytic tools, but otherwise not related.