<p><strong>Hint 1:</strong> Suppose a mixed strategy NE $(\sigma_1,\sigma_2)$ exists, where $\sigma_i$ is a distribution over $[0,1]$. By the property of MSNE, 
$$u_i(x_i,\sigma_j)=u_i(x_i',\sigma_j)$$ 
for all pure strategies $x_i,x_i'$ in the support of $\sigma_i$. </p>

<p><strong>Hint 2:</strong> Note that a segment of $u_i$ is linear in strategies, and thus 
$$
x_i+\frac12(1-x_i-\mathbb E[x_j])=x_i'+\frac12(1-x_i'-\mathbb E[x_j])
$$</p>

CC BY-SA 3.0
2017-02-08T22:13:29.487Z
Sorry, I don't see the point, can you please elaborate more?
Lauren Solis
CC BY-SA 3.0
2017-02-08T23:19:00.463Z
@LaurenSolis: Hint 1 is essentially the definition of MSNE. Note that Hint 2 implies that, in a MSNE where $j$ uses a non-degenerate mixed strategy $\sigma_j$, $i$'s best response is a pure strategy, i.e. $x_i=x_i'$ and hence $\sigma_i$ is a degenerate distribution. By symmetry, the same holds for $j$. So no MSNE exists in the proper sense that players randomize over more than one pure strategy. Of course, you still need to fill in the details by discussing cases where $x_i+x_j>1$ etc.
Herr K.
CC BY-SA 3.0
2017-02-07T06:59:51.140Z
2017-02-07T06:59:51.140Z
Herr K.
<p>I'm working on a game theory problem.I'm having trouble understanding what the mixed strategy nash equilibrium is exactly in this game.</p>

<p>The game is :Two players have to choose how distribute a piece of land ($size = 1$). Each player (1, 2) send his decision $x_i \in{[0, 1]}$ to an external agent.</p>

<p>If $x_1 +x_2 \leq 1$, each player gain a land portion equal to $x_i +\frac{1-x_1-x_2}{2}$</p>

<p>If $x_1+x_2 &gt; 1$, they end up with nothing.</p>

<p>The first part of the problem is to solve the game in pure strategies. I figured out that the answer of this part is:</p>

<p>Player 1 should pick $1- x_2$, any higher and he gets 0. The game is symmetric so the same hold from player2's perspective. The optimal strategy is the intersection of these two functions.</p>

<p>The second part is to solve the game in mixed strategies (if exist). But I don't know how to aproach this part. </p>

<game-theory><nash-equilibrium>
Mixed strategies: Nash equilibrium
Lauren Solis
# Mixed strategies: Nash equilibrium

(Asked by: username_0 on 2017-02-06)

I'm working on a game theory problem.I'm having trouble understanding what the mixed strategy nash equilibrium is exactly in this game.

The game is :Two players have to choose how distribute a piece of land ($size = 1$). Each player (1, 2) send his decision $x\_i \in{[0, 1]}$ to an external agent.

If $x\_1 +x\_2 \leq 1$, each player gain a land portion equal to $x\_i +\frac{1-x\_1-x\_2}{2}$

If $x\_1+x\_2 > 1$, they end up with nothing.

The first part of the problem is to solve the game in pure strategies. I figured out that the answer of this part is:

Player 1 should pick $1- x\_2$, any higher and he gets 0. The game is symmetric so the same hold from player2's perspective. The optimal strategy is the intersection of these two functions.

The second part is to solve the game in mixed strategies (if exist). But I don't know how to aproach this part.

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

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## 1 answers

### Answer by username_1 on 2017-02-07. Score: 2

**Hint 1:** Suppose a mixed strategy NE $(\sigma\_1,\sigma\_2)$ exists, where $\sigma\_i$ is a distribution over $[0,1]$. By the property of MSNE,
$$u\_i(x\_i,\sigma\_j)=u\_i(x\_i',\sigma\_j)$$
for all pure strategies $x\_i,x\_i'$ in the support of $\sigma\_i$.

**Hint 2:** Note that a segment of $u\_i$ is linear in strategies, and thus
$$
x\_i+\frac12(1-x\_i-\mathbb E[x\_j])=x\_i'+\frac12(1-x\_i'-\mathbb E[x\_j])
$$

> **Comments:**
> **username_0**: Sorry, I don't see the point, can you please elaborate more?  
> **username_1**: @LaurenSolis: Hint 1 is essentially the definition of MSNE. Note that Hint 2 implies that, in a MSNE where $j$ uses a non-degenerate mixed strategy $\sigma_j$, $i$'s best response is a pure strategy, i.e. $x_i=x_i'$ and hence $\sigma_i$ is a degenerate distribution. By symmetry, the same holds for $j$. So no MSNE exists in the proper sense that players randomize over more than one pure strategy. Of course, you still need to fill in the details by discussing cases where $x_i+x_j>1$ etc.  

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