<p>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.</p>
<p>Also, when you post an image, you're supposed to transcribe it.</p>

CC BY-SA 4.0
2020-11-20T03:28:07.363Z
Thanks! I realize that. Because all problems I've seen are related to numbers as payoff. Thanks for the hint
JungleKing
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2020-11-20T03:26:26.763Z
2020-11-20T03:26:26.763Z
Acccumulation
<p>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!
<a href="https://i.sstatic.net/Xxpp5.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/Xxpp5.png" alt="enter image description here" /></a></p>
<p>I assume the probability p of defense playing Left
For the opponent's payoff if playing left:
<span class="math-container">$$p * (-1) + (1-p) * (-5)$$</span>
For the oppoent's payoff if playing right:
<span class="math-container">$$p * (-5) + (1-p) *(-3.5)$$</span></p>
<p>Equating both, <span class="math-container">$p=0.27$</span> and <span class="math-container">$(1-p)=0.72.$</span>
But there is no matching answer.</p>

CC BY-SA 4.0
2020-11-20T03:13:51.477Z
Nope. You should show your work on solving the problem, otherwise the question will be closed
KarmaPeasant
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2020-11-20T03:19:18.320Z
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.
Dayne
CC BY-SA 4.0
2020-11-20T03:22:58.430Z
how then, would you deal with 0.7 probability?
JungleKing
CC BY-SA 4.0
2020-11-20T03:23:34.703Z
a payoff of 0.7 instead of 1
Dayne
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2020-11-20T03:25:33.407Z
so the opponent payoff is 0.3 right~~ looks like a "one-sum" game haha
JungleKing
CC BY-SA 4.0
2020-11-20T03:28:38.677Z
yes. perhaps now you can see that solution is going to be $p=1/1.3$.
Dayne
<game-theory>
Question about mixed strategy game theory
JungleKing
# Question about mixed strategy game theory

(Asked by: username_0 on 2020-11-20)

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](https://i.sstatic.net/Xxpp5.png)](https://i.sstatic.net/Xxpp5.png)

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.

**Tags:** game-theory
> **Comments:**
> **username_1**: Nope. You should show your work on solving the problem, otherwise the question will be closed  
> **username_2**: 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.  
> **username_0**: how then, would you deal with 0.7 probability?  
> **username_2**: a payoff of 0.7 instead of 1  
> **username_0**: so the opponent payoff is 0.3 right~~ looks like a "one-sum" game haha  
> **username_2**: yes. perhaps now you can see that solution is going to be $p=1/1.3$.  

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## 1 answers

### Answer by username_3 on 2020-11-20. Score: 2

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.

> **Comments:**
> **username_0**: Thanks! I realize that. Because all problems I've seen are related to numbers as payoff. Thanks for the hint  

---

