<p>Informally, a (possibly mixed) set of strategy choices is a Nash equilibrium if no player can expect to benefit by changing strategies while the other players keep their strategies unchanged. </p>

<p>Here I would interpret your definition as $\Delta$ being the set of all possible strategy choices across all players, $\sigma^*$ being a set of strategy choices which will be a Nash equilibrium if it satisfies the conditions, $i$ being a player, and $\sigma^* \backslash \mu_i$ being a change in player $i$'s strategy while the other players' strategies stay the same.  So I would take $\backslash$ having a type of "not included" interpretation.  </p>

CC BY-SA 3.0
2016-07-20T12:06:41.400Z
This makes sense if we define $\sigma^* \setminus \mu_i:=(\ldots,\sigma^*_{i-1}, \mu_i,\sigma^*_{i+1},\ldots)$. However, it's more conventional and intuitive if you write $U_i(\sigma^*_i, \sigma_{-i}^*) \geq U_i(\mu_i, \sigma_{-i}^*) \forall \mu_i \in \Delta_i$, where $\sigma^*_{-i}:=(\ldots,\sigma^*_{i-1}, \sigma^*_{i+1},\ldots) \in \Delta \setminus \Delta_i$.
clueless
CC BY-SA 3.0
2016-07-20T00:13:52.433Z
2016-07-20T00:13:52.433Z
Henry
<p>A definition in my book states</p>

<blockquote>
  <p>$ \sigma^* \in \Delta$ is a <strong>Nash equilibrium for $\Gamma$</strong> if for all $i \in I$ and all $\mu_i \in \Delta_i, \; U_i (\sigma^*) \geq U_i(\sigma^* \backslash \mu_i)$.  The set of Nash equilibria for $\Gamma$ is denoted $Eq(\Gamma)$</p>
</blockquote>

<p>What does the $\backslash$ here represent, divide or element not included?</p>

CC BY-SA 3.0
2016-07-18T16:33:41.203Z
What do you know about the definition of the Nash equilibrium in informal *words*, rather than the formal mathematical definition?
410 gone
CC BY-SA 3.0
2016-07-18T16:35:50.417Z
Nothing, this is the first I've heard about it.
Sunhwa
CC BY-SA 3.0
2016-07-18T16:56:30.650Z
I've skimmed the Wikipedia page, they use the notation:

$$ \forall i, x_i \in S: f_i (x_i^*, x_{-1}^*) \geq f_i (x_i, x_{-1}^*) $$ 

Where I think the $x_{-1}^*$ stands for the nash equilibrium actions (best result by not changing) of all the other players (without player $i$).  But still, I don't see how the notation in my question says this.
Sunhwa
CC BY-SA 3.0
2016-07-18T18:31:15.270Z
The definition is rather odd. What's the reference? Never seen it before. In your previous comment it should probably read $-i$.
clueless
CC BY-SA 3.0
2016-07-18T18:37:04.113Z
Concerning the wiki entry, you may define $x_i^* \in \arg\max_{x_i \in S_i}f_i(x_i,x_{-i}^*)$.
clueless
CC BY-SA 3.0
2016-07-18T19:17:42.177Z
Yes right @clueless, my haste it should be $-i$. This is from An Introduction to Mathematical Analysis for Economic Theory and Econometrics
Sunhwa
CC BY-SA 3.0
2016-07-19T04:52:33.773Z
My only encounter with that notation is as "set minus" when using any type of set theory.
123
<mathematical-economics>
"$\backslash$" notation question in definition for Nash equilibrium
Sunhwa
# "$\backslash$" notation question in definition for Nash equilibrium

(Asked by: username_0 on 2016-07-18)

A definition in my book states

> $ \sigma^\* \in \Delta$ is a **Nash equilibrium for $\Gamma$** if for all $i \in I$ and all $\mu\_i \in \Delta\_i, \; U\_i (\sigma^\*) \geq U\_i(\sigma^\* \backslash \mu\_i)$. The set of Nash equilibria for $\Gamma$ is denoted $Eq(\Gamma)$

What does the $\backslash$ here represent, divide or element not included?

**Tags:** mathematical-economics
> **Comments:**
> **username_1**: What do you know about the definition of the Nash equilibrium in informal *words*, rather than the formal mathematical definition?  
> **username_0**: Nothing, this is the first I've heard about it.  
> **username_0**: I've skimmed the Wikipedia page, they use the notation:  $$ \forall i, x_i \in S: f_i (x_i^*, x_{-1}^*) \geq f_i (x_i, x_{-1}^*) $$   Where I think the $x_{-1}^*$ stands for the nash equilibrium actions (best result by not changing) of all the other players (without player $i$).  But still, I don't see how the notation in my question says this.  
> **username_2**: The definition is rather odd. What's the reference? Never seen it before. In your previous comment it should probably read $-i$.  
> **username_2**: Concerning the wiki entry, you may define $x_i^* \in \arg\max_{x_i \in S_i}f_i(x_i,x_{-i}^*)$.  
> **username_0**: Yes right @clueless, my haste it should be $-i$. This is from An Introduction to Mathematical Analysis for Economic Theory and Econometrics  
> **username_3**: My only encounter with that notation is as "set minus" when using any type of set theory.  

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## 1 answers

### Answer by username_4 on 2016-07-20. Score: 1 | Accepted answer

Informally, a (possibly mixed) set of strategy choices is a Nash equilibrium if no player can expect to benefit by changing strategies while the other players keep their strategies unchanged.

Here I would interpret your definition as $\Delta$ being the set of all possible strategy choices across all players, $\sigma^\*$ being a set of strategy choices which will be a Nash equilibrium if it satisfies the conditions, $i$ being a player, and $\sigma^\* \backslash \mu\_i$ being a change in player $i$'s strategy while the other players' strategies stay the same. So I would take $\backslash$ having a type of "not included" interpretation.

> **Comments:**
> **username_2**: This makes sense if we define $\sigma^* \setminus \mu_i:=(\ldots,\sigma^*_{i-1}, \mu_i,\sigma^*_{i+1},\ldots)$. However, it's more conventional and intuitive if you write $U_i(\sigma^*_i, \sigma_{-i}^*) \geq U_i(\mu_i, \sigma_{-i}^*) \forall \mu_i \in \Delta_i$, where $\sigma^*_{-i}:=(\ldots,\sigma^*_{i-1}, \sigma^*_{i+1},\ldots) \in \Delta \setminus \Delta_i$.  

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