<p>I doubt that any textbook defines a strategy profile as a Cartesian product. The standard definition for a finite game is that players <span class="math-container">$i\in\{1,\ldots,n\}$</span> have <em>strategy sets</em> <span class="math-container">$S_i$</span>. Then <span class="math-container">$S=S_1\times S_2\times\cdots\times S_n$</span> is the <em>set of strategy profiles</em>, where a <em>strategy profile</em> is an <span class="math-container">$n$</span>-tuple of strategies <span class="math-container">$s\in S$</span>.</p>

CC BY-SA 4.0
2024-01-16T14:57:11.537Z
Ooops, another answer was posted while I was still typing...
VARulle
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2024-01-16T17:20:11.933Z
Room enough for both, thank you for the edit!
Giskard
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2024-01-17T03:40:40.100Z
https://youtu.be/zkO6HEdh-NU?t=546 @VARulle please comment
Narayana Srikanth Reddy
CC BY-SA 4.0
2024-01-17T16:08:46.910Z
@NarayanaSrikanthReddy I'm not sure what you want me to comment on. In this video Selcuk says almost exactly what I wrote in my answer.
VARulle
CC BY-SA 4.0
2024-01-16T14:55:19.923Z
2024-01-16T14:55:19.923Z
VARulle
<p>As @VARulle pointed out, you do give correct information, there is no contradiction.</p>
<p>A strategy profile is a list describing the strategies that each player selects, <span class="math-container">$s = (s_1, s_2, \dots,  s_n )$</span>.</p>
<p>Each player <span class="math-container">$i$</span> has a set of strategies to choose from, we will denote this with <span class="math-container">$S_i$</span>. The players can choose any of their strategies irrespective of what the others choose, so all combinations are possible. Hence the set of all possible choices is given by the Cartesian product
<span class="math-container">$$
S = S_1 \times S_2 \times \dots \times S_n.
$$</span><br />
This is called the <em>set</em> of strategy profiles.</p>
<p><strong>An example</strong>
<span class="math-container">$$S_1 = \left\{a,b\right\}, \ S_2 = \left\{\alpha,\beta\right\}$$</span>
<span class="math-container">$$S = S_1 \times S_2 =  \left\{(a,\alpha),(a,\beta),(b,\alpha),(b,\beta)\right\}$$</span></p>

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2024-01-17T03:41:35.587Z
https://youtu.be/zkO6HEdh-NU?t=546 check this out and comment please
Narayana Srikanth Reddy
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2024-01-17T07:08:51.523Z
Hi @NarayanaSrikanthReddy ! My comment is: "No way, please make questions self contained."
Giskard
CC BY-SA 4.0
2024-01-16T14:50:30.793Z
2024-01-16T14:57:25.303Z
2024-01-16T14:57:25.303Z
Giskard
<p>Which one is correct (a) or (b). If one of them is strategy profile then what is the other called. Which is the standard book to follow for such terminology</p>
<p>(a) Some textbooks describe strategy profile as:
A list describing the strategies that each player selects, s = (s1, s2, ..., sN ).For example, in a two-player game between two firms (A and B), a strategy profile could be s = (12, 8) where sA = 12 denotes firm A’s output level while sB = 8 represents firm B’s output.</p>
<p>(b) Other textbooks define it as cartesian product
Mathematically, the set of strategy profiles are defined as S = S1 × S2 × . . . × Sn ie if S1={12,8} and S2={1,10} then S= {(12,1),(12,10),(8,1),(8,10)}.</p>
<p>either s is strategy profile, if so what is S called
or S is strategy profile, if so what is s called
<a href="https://i.sstatic.net/UPXx5.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/UPXx5.png" alt="Video on Youtube" /></a></p>
<p><a href="https://youtu.be/zkO6HEdh-NU?t=549" rel="nofollow noreferrer">What is the gentleman trying to tell</a></p>

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2024-01-16T13:34:56.537Z
Did you just answer your title question yourself in the body of the text...?
VARulle
CC BY-SA 4.0
2024-01-16T14:19:17.540Z
@VARulle can you review the question again and recommend a standard textbook which lists the terminology with notation and examples
Narayana Srikanth Reddy
CC BY-SA 4.0
2024-01-17T07:11:52.040Z
@NarayanaSrikanthReddy You can find some book recommendations [here](https://economics.stackexchange.com/questions/6112/game-theory-textbooks).
Giskard
<game-theory>
What is strategy profile? Is it -a vector list of si or -the cartesian product of Si?
Narayana Srikanth Reddy
# What is strategy profile? Is it -a vector list of si or -the cartesian product of Si?

(Asked by: username_0 on 2024-01-16)

Which one is correct (a) or (b). If one of them is strategy profile then what is the other called. Which is the standard book to follow for such terminology

(a) Some textbooks describe strategy profile as:
A list describing the strategies that each player selects, s = (s1, s2, ..., sN ).For example, in a two-player game between two firms (A and B), a strategy profile could be s = (12, 8) where sA = 12 denotes firm A’s output level while sB = 8 represents firm B’s output.

(b) Other textbooks define it as cartesian product
Mathematically, the set of strategy profiles are defined as S = S1 × S2 × . . . × Sn ie if S1={12,8} and S2={1,10} then S= {(12,1),(12,10),(8,1),(8,10)}.

either s is strategy profile, if so what is S called
or S is strategy profile, if so what is s called
[![Video on Youtube](https://i.sstatic.net/UPXx5.png)](https://i.sstatic.net/UPXx5.png)

[What is the gentleman trying to tell](https://youtu.be/zkO6HEdh-NU?t=549)

**Tags:** game-theory

> **Comments:**
> **username_1**: Did you just answer your title question yourself in the body of the text...?  
> **username_0**: @username_1 can you review the question again and recommend a standard textbook which lists the terminology with notation and examples  
> **username_2**: @username_0 You can find some book recommendations [here](https://economics.stackexchange.com/questions/6112/game-theory-textbooks).  

---

## 2 answers

### Answer by username_1 on 2024-01-16. Score: 2

I doubt that any textbook defines a strategy profile as a Cartesian product. The standard definition for a finite game is that players $i\in\{1,\ldots,n\}$ have *strategy sets* $S\_i$. Then $S=S\_1\times S\_2\times\cdots\times S\_n$ is the *set of strategy profiles*, where a *strategy profile* is an $n$-tuple of strategies $s\in S$.

> **Comments:**
> **username_1**: Ooops, another answer was posted while I was still typing...  
> **username_2**: Room enough for both, thank you for the edit!  
> **username_0**: https://youtu.be/zkO6HEdh-NU?t=546 @username_1 please comment  
> **username_1**: @username_0 I'm not sure what you want me to comment on. In this video Selcuk says almost exactly what I wrote in my answer.  

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### Answer by username_2 on 2024-01-16. Score: 1

As @username_1 pointed out, you do give correct information, there is no contradiction.

A strategy profile is a list describing the strategies that each player selects, $s = (s\_1, s\_2, \dots, s\_n )$.

Each player $i$ has a set of strategies to choose from, we will denote this with $S\_i$. The players can choose any of their strategies irrespective of what the others choose, so all combinations are possible. Hence the set of all possible choices is given by the Cartesian product
$$
S = S\_1 \times S\_2 \times \dots \times S\_n.
$$  
This is called the *set* of strategy profiles.

**An example**
$$S\_1 = \left\{a,b\right\}, \ S\_2 = \left\{\alpha,\beta\right\}$$
$$S = S\_1 \times S\_2 = \left\{(a,\alpha),(a,\beta),(b,\alpha),(b,\beta)\right\}$$

> **Comments:**
> **username_0**: https://youtu.be/zkO6HEdh-NU?t=546 check this out and comment please  
> **username_2**: Hi @username_0 ! My comment is: "No way, please make questions self contained."  

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