We are given $n \geq 2$ positive integers on a blackboard. At each step, we pick two numbers, say $a$ and $b$, and replace them with their sum $a + b$. We aim to determine all values of $n$ for which it is possible to make all numbers on the blackboard identical after a sequence of such operations.

To solve this problem, we consider the following steps:

1. Understanding the Operation: Each operation selects two numbers and replaces them with their sum. The total sum of all numbers on the blackboard remains constant throughout all operations. Let this total sum be $S$.

2. Final Condition: For all $n$ numbers to be identical, each must equal $\frac{S}{n}$. Therefore, $S$ must be divisible by $n$.

3. Divisibility Constraint: Initially, consider numbers that are distributed arbitrarily. The sum $S$ of these numbers is fixed. To make all numbers the same, say $k$, we need $S = nk$. Hence, $n$ must divide the initial sum $S$.

4. Parity Argument: The crucial aspect is whether $n$ is even or odd:

- If $n$ is even, the sum $S$ of any set of numbers can be manipulated to be divisible by $n$ through a series of operations, stemming from the fact that each operation induces parity adjustments which explore all even divisions of the initial $S$.

- If $n$ is odd, regardless of the initial numbers, at least one number’s parity will dominate the configuration (either more odds than evens), resulting in an inability to achieve an all-even or all-odd homogeneous setup without altering $S$.

5. Conclusion:
- Operations that preserve divisibility imply that every operation change retains the evenness of $S$ concerning $n$, when $n$ is even. Hence, equalizing all numbers is feasible when $n$ is even.
- In contrast, for odd $n$, such parity balance required across the steps is generally not achievable starting from arbitrary initial conditions.

Thus, the values of $n$ for which it is possible for all numbers on the blackboard to become identical are when $n$ is even. Therefore, we conclude that the solution is:

$$\boxed{\text{n is even}}$$