<blockquote>
<ol>
<li>then what is the set <span class="math-container">$\Sigma$</span> look like?</li>
</ol>
</blockquote>
<p><span class="math-container">$\Sigma^1$</span> is the set of all probability distributions over <span class="math-container">$\{x,y\}$</span>, as such it is given by the one dimensional unit simplex:
<span class="math-container">$$
\Sigma^1 = \{(p_1, p_2) \in \mathbb{R}^2_+| p_1 + p_2 = 1\}.
$$</span>
Here <span class="math-container">$p_1$</span> is the probability of playing <span class="math-container">$x$</span> and <span class="math-container">$p_2$</span> the probability of playing <span class="math-container">$y$</span>. If you look at <span class="math-container">$\Sigma^1$</span> as a subset of <span class="math-container">$\mathbb{R}^2$</span> it is the set of all vectors in the positive orthonant whose components sum to 1 as given in the picture below:
<a href="https://i.sstatic.net/S2QsB.png" rel="noreferrer"><img src="https://i.sstatic.net/S2QsB.png" alt="1 dimensional simplex" /></a></p>
<p>It is called the <strong>one</strong> dimensional simplex as the line is of dimension 1.</p>
<p>If there would be three strategies then we would get the two dimensional unit simplex:
<span class="math-container">$$
\Sigma^1 = \{(p_1, p_2, p_3) \in \mathbb{R}^3_+|p_1 + p_2 + p_3 = 1\}.
$$</span>
In <span class="math-container">$\mathbb{R}^3$</span>  this would look like the surface of a triangle spanned by the vectors <span class="math-container">$(1,0,0), (0,1,0), (0,0,1)$</span>. We get a 2 dimensional simplex as the surface is a 2 dimensional object.</p>
<blockquote>
<ol start="2">
<li>With the notation of <span class="math-container">$\Delta$</span> in front of a set we denote the simplex of the set? If not, how you can find it and how is it related to the pure and mixed strategies sets?</li>
</ol>
</blockquote>
<p>Yes <span class="math-container">$\Delta(S)$</span> is usually used to denote the set of all probability distributions on the set <span class="math-container">$S$</span>. If <span class="math-container">$S$</span> is finite, then <span class="math-container">$\Delta S$</span> is simply a unit simplex of dimension 1 lower than the number of elements in <span class="math-container">$S$</span>. In particular if <span class="math-container">$|S| = n$</span> then
<span class="math-container">$$
\Delta(S) = \left\{(p_1, \ldots, p_n) \in \mathbb{R}^n_+\middle| \sum_{i = 1}^n p_i = 1\right\}.
$$</span>
If <span class="math-container">$S$</span> is the set of pure strategies of a player then <span class="math-container">$\Delta(S)$</span> is the set of mixed strategies of the same player.</p>
<blockquote>
<ol start="3">
<li>Does the following hold <span class="math-container">$\Delta(S1 \times S2) = \Delta(S_1)\times \Delta(S_1)$</span> and can this result generalized for <span class="math-container">$k(&lt; +\infty)$</span> player games?</li>
</ol>
</blockquote>
<p>No, the two sets are not the same. For example, let <span class="math-container">$S_1 = \{x,y\}$</span> and <span class="math-container">$S_2 = \{u,v\}$</span> Then the elements of <span class="math-container">$S_1 \times S_2$</span> are:
<span class="math-container">$$
S_1 \times S_2 = \{(x,u), (x, v), (y,u), (y,v)\}. 
$$</span>
As such, <span class="math-container">$\Delta(S_1 \times S_2)$</span> is the 3 dimensional unit simplex with elements <span class="math-container">$(p_{(x,u)}, p_{(x,v)}, p_{(y,u)}, p_{(y,v)})$</span></p>
<p>On the other hand, <span class="math-container">$\Delta(S_1) \times \Delta(S_2)$</span> consists of all combinations <span class="math-container">$((p_x, p_y), (p_u, p_v))$</span> where <span class="math-container">$(p_x, p_y) \in \Delta(S_1)$</span> and <span class="math-container">$(p_u, p_v) \in \delta(S_2)$</span>. Although you could also see these as 4 dimensional vectors they are not in the 3 dimensional unit simplex as <span class="math-container">$p_x + p_y = 1$</span> and <span class="math-container">$p_v + p_w = 1$</span>, so their sum equals 2 (and not 1).</p>
<p>In some sense the set <span class="math-container">$\Delta(S_1 \times S_2)$</span> contains more 'distributions' than the set <span class="math-container">$\Delta(S_1) \times \Delta(S_2)$</span>. By this, I mean that there are vectors <span class="math-container">$(p_{(x,u)}, p_{(x,v)}, p_{(y,u)}, p_{(y,v)}) \in \Delta(S_1 \times S_2)$</span> for which it is <strong>impossible</strong> to find <span class="math-container">$(p_x, p_y) \in \Delta(S_1)$</span> and <span class="math-container">$(p_u, p_v) \in \Delta(S_2)$</span> such that:
<span class="math-container">$$
p_x \cdot p_u = p_{(x,u)},\\
p_x \cdot p_v = p_{(x,v)},\\
p_y \cdot p_u = p_{(y,u)},\\
p_y \cdot p_v = p_{(y,v)}.
$$</span></p>
<p>For example, <span class="math-container">$p_{(x,u)} = 0.5, p_{(x,v)} = 0, p_{(y,u)} = 0, p_{(y,v)} = 0.5$</span> is impossible to decompose in this way. In other way the set of all distributions that can be obtained by a profile of mixed strategies is a strict subset of the set of all probability distributions over the pure strategy profiles.</p>

CC BY-SA 4.0
2021-09-12T16:17:19.627Z
2021-09-12T16:17:19.627Z
tdm
<p>Suppose that we have a two players game, where <span class="math-container">$(S^i)_{i=1}^2$</span> denotes the set of pure strategies for each one. The set of mixed strategies of player <span class="math-container">$i$</span> is denoted by <span class="math-container">$\Sigma^i=\Delta(S^i)$</span> while <span class="math-container">$\Sigma=\Sigma_1\times\Sigma_2=\Delta(S_1)\times \Delta(S_2)$</span>. My questions are the following:</p>
<ol>
<li>Suppose that <span class="math-container">$S_1=\{x,y\}$</span> and <span class="math-container">$S_2=\{u,v\}$</span>, then the set <span class="math-container">$\Sigma^1=\Delta(S^1)=\{\{x\},\{y\},(p(x),(1-p)(y))\}$</span> and <span class="math-container">$\Sigma^2=\Delta(S^2)=\{\{u\},\{v\},(q(u),(1-q)(v))\}$</span>, where <span class="math-container">$p$</span> and <span class="math-container">$q$</span> denote the probability distributions upon the set of pure strategies. If I am not mistaken until here, then what is the set <span class="math-container">$\Sigma$</span> look like?</li>
<li>With the notation of <span class="math-container">$\Delta$</span> in front of a set we denote the simplex of the set? If not, how you can find it and how is it related to the pure and mixed strategies sets?</li>
<li>Does the following hold <span class="math-container">$\Delta(\underbrace{S_1\times S_2}_{\Delta(\Pi_{i=1}^2 S^i)})\underbrace{=}_{=}\underbrace{\Delta(S_1)\times\Delta(S_2)}_{\Pi_{i=1}^2 \Delta(S^i)}$</span> an can this result generalized for <span class="math-container">$k(&lt;+\infty)$</span> player games?</li>
</ol>
<p>Thank you in advance! Though the calculations or my answers may look simple, I am a bit confused with all these notation.</p>

CC BY-SA 4.0
2021-09-12T11:27:17.373Z
@Giskard you are right I am mistaken... let me fix this...in fact $\Sigma=\Sigma_1\times\Sigma_2=\Delta(S_1)\times\Delta(S_2)$
Hunger Learn
<mathematical-economics><game-theory><nash-equilibrium>
Defining the set of strategies, mixed strategies and the simplex set
Hunger Learn
# Defining the set of strategies, mixed strategies and the simplex set

(Asked by: username_0 on 2021-09-12)

Suppose that we have a two players game, where $(S^i)\_{i=1}^2$ denotes the set of pure strategies for each one. The set of mixed strategies of player $i$ is denoted by $\Sigma^i=\Delta(S^i)$ while $\Sigma=\Sigma\_1\times\Sigma\_2=\Delta(S\_1)\times \Delta(S\_2)$. My questions are the following:

1. Suppose that $S\_1=\{x,y\}$ and $S\_2=\{u,v\}$, then the set $\Sigma^1=\Delta(S^1)=\{\{x\},\{y\},(p(x),(1-p)(y))\}$ and $\Sigma^2=\Delta(S^2)=\{\{u\},\{v\},(q(u),(1-q)(v))\}$, where $p$ and $q$ denote the probability distributions upon the set of pure strategies. If I am not mistaken until here, then what is the set $\Sigma$ look like?
2. With the notation of $\Delta$ in front of a set we denote the simplex of the set? If not, how you can find it and how is it related to the pure and mixed strategies sets?
3. Does the following hold $\Delta(\underbrace{S\_1\times S\_2}\_{\Delta(\Pi\_{i=1}^2 S^i)})\underbrace{=}\_{=}\underbrace{\Delta(S\_1)\times\Delta(S\_2)}\_{\Pi\_{i=1}^2 \Delta(S^i)}$ an can this result generalized for $k(<+\infty)$ player games?

Thank you in advance! Though the calculations or my answers may look simple, I am a bit confused with all these notation.

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

> **Comments:**
> **username_0**: @Giskard you are right I am mistaken... let me fix this...in fact $\Sigma=\Sigma_1\times\Sigma_2=\Delta(S_1)\times\Delta(S_2)$  

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## 1 answers

### Answer by username_1 on 2021-09-12. Score: 4 | Accepted answer

> 1. then what is the set $\Sigma$ look like?

$\Sigma^1$ is the set of all probability distributions over $\{x,y\}$, as such it is given by the one dimensional unit simplex:
$$
\Sigma^1 = \{(p\_1, p\_2) \in \mathbb{R}^2\_+| p\_1 + p\_2 = 1\}.
$$
Here $p\_1$ is the probability of playing $x$ and $p\_2$ the probability of playing $y$. If you look at $\Sigma^1$ as a subset of $\mathbb{R}^2$ it is the set of all vectors in the positive orthonant whose components sum to 1 as given in the picture below:
[![1 dimensional simplex](https://i.sstatic.net/S2QsB.png)](https://i.sstatic.net/S2QsB.png)

It is called the **one** dimensional simplex as the line is of dimension 1.

If there would be three strategies then we would get the two dimensional unit simplex:
$$
\Sigma^1 = \{(p\_1, p\_2, p\_3) \in \mathbb{R}^3\_+|p\_1 + p\_2 + p\_3 = 1\}.
$$
In $\mathbb{R}^3$ this would look like the surface of a triangle spanned by the vectors $(1,0,0), (0,1,0), (0,0,1)$. We get a 2 dimensional simplex as the surface is a 2 dimensional object.

> 2. With the notation of $\Delta$ in front of a set we denote the simplex of the set? If not, how you can find it and how is it related to the pure and mixed strategies sets?

Yes $\Delta(S)$ is usually used to denote the set of all probability distributions on the set $S$. If $S$ is finite, then $\Delta S$ is simply a unit simplex of dimension 1 lower than the number of elements in $S$. In particular if $|S| = n$ then
$$
\Delta(S) = \left\{(p\_1, \ldots, p\_n) \in \mathbb{R}^n\_+\middle| \sum\_{i = 1}^n p\_i = 1\right\}.
$$
If $S$ is the set of pure strategies of a player then $\Delta(S)$ is the set of mixed strategies of the same player.

> 3. Does the following hold $\Delta(S1 \times S2) = \Delta(S\_1)\times \Delta(S\_1)$ and can this result generalized for $k(< +\infty)$ player games?

No, the two sets are not the same. For example, let $S\_1 = \{x,y\}$ and $S\_2 = \{u,v\}$ Then the elements of $S\_1 \times S\_2$ are:
$$
S\_1 \times S\_2 = \{(x,u), (x, v), (y,u), (y,v)\}.
$$
As such, $\Delta(S\_1 \times S\_2)$ is the 3 dimensional unit simplex with elements $(p\_{(x,u)}, p\_{(x,v)}, p\_{(y,u)}, p\_{(y,v)})$

On the other hand, $\Delta(S\_1) \times \Delta(S\_2)$ consists of all combinations $((p\_x, p\_y), (p\_u, p\_v))$ where $(p\_x, p\_y) \in \Delta(S\_1)$ and $(p\_u, p\_v) \in \delta(S\_2)$. Although you could also see these as 4 dimensional vectors they are not in the 3 dimensional unit simplex as $p\_x + p\_y = 1$ and $p\_v + p\_w = 1$, so their sum equals 2 (and not 1).

In some sense the set $\Delta(S\_1 \times S\_2)$ contains more 'distributions' than the set $\Delta(S\_1) \times \Delta(S\_2)$. By this, I mean that there are vectors $(p\_{(x,u)}, p\_{(x,v)}, p\_{(y,u)}, p\_{(y,v)}) \in \Delta(S\_1 \times S\_2)$ for which it is **impossible** to find $(p\_x, p\_y) \in \Delta(S\_1)$ and $(p\_u, p\_v) \in \Delta(S\_2)$ such that:
$$
p\_x \cdot p\_u = p\_{(x,u)},\\
p\_x \cdot p\_v = p\_{(x,v)},\\
p\_y \cdot p\_u = p\_{(y,u)},\\
p\_y \cdot p\_v = p\_{(y,v)}.
$$

For example, $p\_{(x,u)} = 0.5, p\_{(x,v)} = 0, p\_{(y,u)} = 0, p\_{(y,v)} = 0.5$ is impossible to decompose in this way. In other way the set of all distributions that can be obtained by a profile of mixed strategies is a strict subset of the set of all probability distributions over the pure strategy profiles.


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