<p>I'm trying to solve this problem from last year final exam in game theory:</p>
<blockquote>
<p>Consider the zero-sum game <span class="math-container">$G=(X, Y, g)$</span> where <span class="math-container">$X=Y=[0,1]$</span>, and <span class="math-container">$$\forall (x,y) \in X \times Y: g(x, y)=\max \{x(1-2 y), y(1-2 x)\}$$</span></p>
<p>Find a mixed optimal strategy for each player. (Hint: one can consider mixed
strategies of player <span class="math-container">$1$</span> which plays <span class="math-container">$x=0$</span> with some probability and <span class="math-container">$x=1$</span> with the remaining probability).</p>
</blockquote>
<p><strong>My attempt:</strong></p>
<p>Let <span class="math-container">$\sigma$</span> be a mixed optimal strategy in which player <span class="math-container">$1$</span> play <span class="math-container">$x=0$</span> with probability <span class="math-container">$p$</span> and <span class="math-container">$x=1$</span> with probability <span class="math-container">$1-p$</span>. A mixed optimal strategy <span class="math-container">$\tau$</span> of player <span class="math-container">$2$</span> is a probability measure on <span class="math-container">$[0,1]$</span>.</p>
<hr />
<p>Then I'm stuck to proceed. Could you please help me finish this exercise? Thank you so much!</p>

CC BY-SA 4.0
2019-12-06T03:33:27.103Z
to clarify, does each player choose either $0$ or $1$ or is each player choosing a value between $[0,1]$?
corran_horn
CC BY-SA 4.0
2019-12-06T03:43:15.260Z
Sorry, but it seems your attempt is just repeating the hint?
Giskard
CC BY-SA 4.0
2019-12-06T08:08:07.063Z
Hi @Giskard, Honestly, I have no idea how to solve that question :(
Akira
CC BY-SA 4.0
2019-12-06T08:09:39.333Z
Hi @corran_horn, each player choose value in the interval $[0,1]$.
Akira
CC BY-SA 4.0
2019-12-06T13:16:34.803Z
I’ll update my answer, but think through the payoff structure. For any given $x \in [0,1]$, what is the optimal response for $Y$? Is it ever optimal to play $y \in (0,1)$
corran_horn
CC BY-SA 4.0
2019-12-06T13:40:59.133Z
@corran_horn Honestly, I have no idea on how to approach this problem :(
Akira
<game-theory><mathematical-economics><nash-equilibrium>
How to find mixed optimal strategies in this zero-sum game?
Akira
# How to find mixed optimal strategies in this zero-sum game?

(Asked by: username_0 on 2019-12-05)

I'm trying to solve this problem from last year final exam in game theory:

> Consider the zero-sum game $G=(X, Y, g)$ where $X=Y=[0,1]$, and $$\forall (x,y) \in X \times Y: g(x, y)=\max \{x(1-2 y), y(1-2 x)\}$$
>
> Find a mixed optimal strategy for each player. (Hint: one can consider mixed
> strategies of player $1$ which plays $x=0$ with some probability and $x=1$ with the remaining probability).

**My attempt:**

Let $\sigma$ be a mixed optimal strategy in which player $1$ play $x=0$ with probability $p$ and $x=1$ with probability $1-p$. A mixed optimal strategy $\tau$ of player $2$ is a probability measure on $[0,1]$.

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Then I'm stuck to proceed. Could you please help me finish this exercise? Thank you so much!

**Tags:** game-theory, mathematical-economics, nash-equilibrium

> **Comments:**
> **username_1**: to clarify, does each player choose either $0$ or $1$ or is each player choosing a value between $[0,1]$?  
> **username_2**: Sorry, but it seems your attempt is just repeating the hint?  
> **username_0**: Hi @username_2, Honestly, I have no idea how to solve that question :(  
> **username_0**: Hi @username_1, each player choose value in the interval $[0,1]$.  
> **username_1**: I’ll update my answer, but think through the payoff structure. For any given $x \in [0,1]$, what is the optimal response for $Y$? Is it ever optimal to play $y \in (0,1)$  
> **username_0**: @username_1 Honestly, I have no idea on how to approach this problem :(  

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## 0 answers

