What is strategy profile? Is it -a vector list of si or -the cartesian product of Si?

Question

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

What is the gentleman trying to tell

Answer 1

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 $s\in S$.

Answer 2

As @VARulle 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 $$S_1 = \left\{a,b\right\}, \ S_2 = \left\{\alpha,\beta\right\}$$ $$S = S_1 \times S_2 = \left\{(a,\alpha),(a,\beta),(b,\alpha),(b,\beta)\right\}$$