<p>You are roughly right, but your questions also contain some slightly misleading claims. To clarify:</p>
<ul>
<li><p>The condition <span class="math-container">$p_i(t) \geq 0 \; \forall i \in \{1, 2, \dots, n\}$</span>
<em>is</em> a restriction in the sense that the state space is a <em>strict subset</em> of the affine hyperplane defined by the sum-is-equal-to-1
constraint.</p>
</li>
<li><p>The state space is a subset but <em>not</em> a sub<em>space</em> of <span class="math-container">$\mathbb{R}^n$</span>.</p>
</li>
<li><p>It is correct that every strategy corresponds to a vector <span class="math-container">$p$</span> in
<span class="math-container">$\mathbb{R}_+^n \cup \{\vec{0}\}$</span>, restricted by <span class="math-container">$\sum_{i=1}^np_i(t)
   = 1$</span>, but the <span class="math-container">$\cup \{\vec{0}\}$</span> part is redundant since <span class="math-container">$\vec{0}$</span> clearly doesn't satisfy the restriction anyway.</p>
</li>
</ul>

CC BY-SA 4.0
2024-04-05T14:14:22.400Z
Thank you. So the less-than-or-equal-to-zero condition is effectively a restriction but doesn't count as to how may dimensions the state space has, correct? (If so, why is this the case?) And yes, you're right that the state space is not a subspace because it doesn't contain $\vec{0}$ (as this of course doesn't satisfy $\sum_{i = 1}^np_i(t) = 1$). Thanks for pointing that out.
Riccardo Iorio
CC BY-SA 4.0
2024-04-06T15:19:01.460Z
@RiccardoIorio, you mean greater-than-or-equal-to zero. Yes, this doesn't change the dimension, it just cuts out some part of the hyperplane.
VARulle
CC BY-SA 4.0
2024-04-06T19:14:57.120Z
Sorry, yeah I miswrote. Thank you, got it.
Riccardo Iorio
CC BY-SA 4.0
2024-04-05T12:30:50.747Z
2024-04-05T12:30:50.747Z
VARulle
<p>Yes, the strategy space when the non-negative fractions have to add up to one is this hyperplane, frequently referred to as the unit or standard simplex.</p>
<p><a href="https://en.wikipedia.org/wiki/Simplex#Standard_simplex" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Simplex#Standard_simplex</a></p>

CC BY-SA 4.0
2024-04-05T06:38:12.433Z
2024-04-05T06:38:12.433Z
Giskard
<p>I'm reading <em>Evolutionary Game Theory</em> by Jason MacKenzie Alexander, and the author, whilst introducing the dynamical-systems treatment of evolutionary games, defines the vector <span class="math-container">$\vec{p(t)} = \langle \; p_1(t) \; p_2(t) \; \dots \; p_n(t) \; \rangle \in \mathbb{R}^n$</span>, where <span class="math-container">$p_i(t)$</span> for <span class="math-container">$i \in \{1, 2, \dots, n\}$</span> is the fraction of the population playing strategy <span class="math-container">$S_i$</span> at time <span class="math-container">$t$</span>, as the state of the system of interest at time <span class="math-container">$t$</span>. He then goes on to note that, since we impose</p>
<ul>
<li><span class="math-container">$p_i(t) \geq 0 \; \forall i \in \{1, 2, \dots, n\}$</span>, and</li>
<li><span class="math-container">$\sum_{i=1}^np_i(t) = 1$</span>,</li>
</ul>
<p>the space where <span class="math-container">$\vec{p(t)}$</span> lives, i.e. the state space of the system, is a hyperplane of <span class="math-container">$\mathbb{R}^n$</span>, i.e. an <span class="math-container">$(n - 1)$</span>-dimensional space.</p>
<p>Am I right in inferring that this is due to the fact that imposing <span class="math-container">$\sum_{i=1}^np_i(t) = 1$</span> amounts to imposing one restriction on <span class="math-container">$n$</span> variables, while the condition <span class="math-container">$p_i(t) \geq 0 \; \forall i \in \{1, 2, \dots, n\}$</span> isn't a restriction at all (so it has no relevance as to what subspace of <span class="math-container">$\mathbb{R}^n$</span> the state space is)? As a clearer way of stating the above, would it be correct to redefine the vector of strategies as an element of <span class="math-container">$\mathbb{R}_{\geq 0}^n$</span> and impose the only restriction <span class="math-container">$\sum_{i=1}^np_i(t) = 1$</span>?</p>

CC BY-SA 4.0
2024-04-05T06:33:58.567Z
"*I'm reading a book on evolutionary game theory*" I am still confused why people are reluctant to give proper references. If you phrase it like this, a person querying the book's title will not find your question, making it harder for them to benefit from the effort you and a future answerer put in.
Giskard
CC BY-SA 4.0
2024-04-05T09:18:52.010Z
Sorry, I just didn't think it was relevant for the question. The reference is Evolutionary Game Theory by Jason MacKenzie Alexander, for the CUP series Elements in Decision Theory and Philosophy.
Riccardo Iorio
<game-theory>
State space of the vector of strategies in evolutionary games
Riccardo Iorio
# State space of the vector of strategies in evolutionary games

(Asked by: username_0 on 2024-04-04)

I'm reading *Evolutionary Game Theory* by Jason MacKenzie Alexander, and the author, whilst introducing the dynamical-systems treatment of evolutionary games, defines the vector $\vec{p(t)} = \langle \; p\_1(t) \; p\_2(t) \; \dots \; p\_n(t) \; \rangle \in \mathbb{R}^n$, where $p\_i(t)$ for $i \in \{1, 2, \dots, n\}$ is the fraction of the population playing strategy $S\_i$ at time $t$, as the state of the system of interest at time $t$. He then goes on to note that, since we impose

* $p\_i(t) \geq 0 \; \forall i \in \{1, 2, \dots, n\}$, and
* $\sum\_{i=1}^np\_i(t) = 1$,

the space where $\vec{p(t)}$ lives, i.e. the state space of the system, is a hyperplane of $\mathbb{R}^n$, i.e. an $(n - 1)$-dimensional space.

Am I right in inferring that this is due to the fact that imposing $\sum\_{i=1}^np\_i(t) = 1$ amounts to imposing one restriction on $n$ variables, while the condition $p\_i(t) \geq 0 \; \forall i \in \{1, 2, \dots, n\}$ isn't a restriction at all (so it has no relevance as to what subspace of $\mathbb{R}^n$ the state space is)? As a clearer way of stating the above, would it be correct to redefine the vector of strategies as an element of $\mathbb{R}\_{\geq 0}^n$ and impose the only restriction $\sum\_{i=1}^np\_i(t) = 1$?

**Tags:** game-theory

> **Comments:**
> **username_1**: "*I'm reading a book on evolutionary game theory*" I am still confused why people are reluctant to give proper references. If you phrase it like this, a person querying the book's title will not find your question, making it harder for them to benefit from the effort you and a future answerer put in.  
> **username_0**: Sorry, I just didn't think it was relevant for the question. The reference is Evolutionary Game Theory by Jason MacKenzie Alexander, for the CUP series Elements in Decision Theory and Philosophy.  

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## 2 answers

### Answer by username_2 on 2024-04-05. Score: 1 | Accepted answer

You are roughly right, but your questions also contain some slightly misleading claims. To clarify:

* The condition $p\_i(t) \geq 0 \; \forall i \in \{1, 2, \dots, n\}$
  *is* a restriction in the sense that the state space is a *strict subset* of the affine hyperplane defined by the sum-is-equal-to-1
  constraint.
* The state space is a subset but *not* a sub*space* of $\mathbb{R}^n$.
* It is correct that every strategy corresponds to a vector $p$ in
  $\mathbb{R}\_+^n \cup \{\vec{0}\}$, restricted by $\sum\_{i=1}^np\_i(t)
  = 1$, but the $\cup \{\vec{0}\}$ part is redundant since $\vec{0}$ clearly doesn't satisfy the restriction anyway.

> **Comments:**
> **username_0**: Thank you. So the less-than-or-equal-to-zero condition is effectively a restriction but doesn't count as to how may dimensions the state space has, correct? (If so, why is this the case?) And yes, you're right that the state space is not a subspace because it doesn't contain $\vec{0}$ (as this of course doesn't satisfy $\sum_{i = 1}^np_i(t) = 1$). Thanks for pointing that out.  
> **username_2**: @username_0, you mean greater-than-or-equal-to zero. Yes, this doesn't change the dimension, it just cuts out some part of the hyperplane.  
> **username_0**: Sorry, yeah I miswrote. Thank you, got it.  

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### Answer by username_1 on 2024-04-05. Score: 0

Yes, the strategy space when the non-negative fractions have to add up to one is this hyperplane, frequently referred to as the unit or standard simplex.

<https://en.wikipedia.org/wiki/Simplex#Standard_simplex>


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