I have no idea how the game is solved. Especially how do you begin with writing the norm form of the game. Appreciate your help! enter image description here

I assume the probability p of defense playing Left For the opponent's payoff if playing left: $$p * (-1) + (1-p) * (-5)$$ For the oppoent's payoff if playing right: $$p * (-5) + (1-p) *(-3.5)$$

Equating both, $p=0.27$ and $(1-p)=0.72.$ But there is no matching answer.

  • $\begingroup$ Nope. You should show your work on solving the problem, otherwise the question will be closed $\endgroup$ – user161005 Nov 20 '20 at 3:13
  • $\begingroup$ Suggestion: the line towards the end "If it makes 5 yards or more it wins, if not it loses" should help in making the game simple. Forget about yards, and write the game in two players, with two possible actions, and two possible outcomes: win/lose. You can simply define payoff as 1/0 for this. $\endgroup$ – Dayne Nov 20 '20 at 3:19
  • $\begingroup$ how then, would you deal with 0.7 probability? $\endgroup$ – linear Nov 20 '20 at 3:22
  • $\begingroup$ a payoff of 0.7 instead of 1 $\endgroup$ – Dayne Nov 20 '20 at 3:23
  • $\begingroup$ so the opponent payoff is 0.3 right~~ looks like a "one-sum" game haha $\endgroup$ – linear Nov 20 '20 at 3:25

You are making the mistake of looking at the problem, seeing some numbers, and thinking those are the payoffs. It's a common mistake for novices in game theory to treat the numerical gain as being the same as utility. The payoff for a result in this game is not the number of yards gained, but the probability of a win. That is, for each square in the payoff matrix, the entry is the probability of winning.

Also, when you post an image, you're supposed to transcribe it.

  • $\begingroup$ Thanks! I realize that. Because all problems I've seen are related to numbers as payoff. Thanks for the hint $\endgroup$ – linear Nov 20 '20 at 3:28

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