Some Pedantic Things (that probably nobody cares)

Some of those below are not necessarily mistakes, but it is better to be conceptually discriminating.

The set of pure strategies (without transformation) is NOT a subset of the set of mixed strategies

To make the set of pure strategies to be a subset of the set of mixed strategies, we have to firstly transform pure strategies into their degenerate probability distribution representations.

Before the transformation, for example, a pure strategy might be Top, Left, Enter, Defect, Fight, or something else, while a mixed strategy is a probability distribution.

To say that a pure strategy resides in the set of mixed strategies, it is amount to say that an apple resides in the set of colors; to say that the set of pure strategies is a subset of the set of mixed strategies is amount to say that the numbers between 5 and 7 is a subset of the set of clothes.

They are conceptually different things!!!

We have to make the transformation, to make pure strategies as the vertices of the closed unit probability simplex: $(1,0,0,...)$,$(0,1,0,...)$,$(0,0,1,...)$, …, where each one is a degenerate mixed strategy, then the set of mixed strategies is the convex hull of those transformed pure strategies.

Constant returns to scale and strict concavity are incompatible!!!

I never imagined that some basic concepts could confuse so many people, until I recently noticed certain mistakes appearing repeatedly. Perhaps this is because some classic references propagate these errors, and others simply adopt them without critically examining the logic.

A common error occurs in macroeconomic notes, where constant returns to scale and strict concavity are simultaneously assumed for $F(K,L)$. However, this is incorrect.

Take $x \neq 0$ (where $x$ and $0$ can be vectors if you like), so $x \neq 2x$. Then take $\lambda = 1/2$.

Then $(1/2)x + (1/2)2x = (3/2)x$. If $h$ were strictly concave, it must follow that $h((3/2)x) > (1/2)h(x) + (1/2)h(2x)$.

However, by constant returns to scale, $h((3/2)x) = (3/2)h(x)$,

while
$(1/2)h(x) + (1/2)h(2x)$, using constant returns to scale, equals $(1/2)h(x) + h(x)$, which is $(3/2)h(x)$ again, leading to a contradiction (Since only the weak inequality, rather than the strict one, can be guarenteed).

Actually, you can find exercises explicitly asking for a proof of this incompatibility in textbooks on mathematical economics, such as Sydsæter, K., Seierstad, A., & Strøm, A. (2008)., which I learned from a lot.

It is okay to state that $F(K,L)$ is constant returns to scale and concave (but not strictly concave), or that $F(K,L)$ is constant returns to scale and $f(k) = \frac{F(K,L)}{L}$ is strictly concave.

However, the assumption of constant returns to scale and strict concavity for $F(K,L)$ is fundamentally wrong.