So the game is each player chooses an amount between 0 and 10, if the sum is greater than 10 they win nothing. If the sum is less than or equal to 10 each player gets what they selected.
The solution states that "the Nash equilibrium of the game is any two numbers that add to 10. This is obvious because any amount greater than 10 destroys the value of the game and any amount less than 10 induces regret".
However the solution goes on to further state that "the unique Nash equilibrium is for each player to choose $5".
Where I am getting confused is that the two statements contradict each other. The first states any pair that adds to 10 is a Nash equilibrium, but then the second statement seems to suggest only 5,5 is a Nash equilibrium.
Best Answer
Here is an explanation of why $(5,5)$ is not the only nash equilibria in this problem. The important thing to remember about game theory is that the two players have no idea of what the other shall choose.
Let us consider from each player's perspective.
Player 1's perspective
From player 1's perspective player 2 has picked some choice $c_2$ such that $ 0 \leq c_2 \leq 10$. So what is player 1's most profitable decision? He/she should pick $10-c_2$. Any higher and he gets 0, any lower and he isn't maximizing profit.
Player 2's perspective
This game is symmetric so the same hold from player 2's perspective. Player 1 has picked some choice $c_1$ such that $ 0 \leq c_1 \leq 10$ and so player 2's most profitable decision is to pick $10-c_1$. Any higher and he gets 0, any lower and he isn't maximizing profit.
So what is the optimal strategy? Well if we define the optimal strategy for player 1 as a best response function of player 2's choice ($c_2$), then $f_1(c_2) = 10-c_2$ and similarly for player 2: $f_2(c_1) = 10-c_1$, then the optimal strategy is the intersection of these two functions. These are the pure strategy nash equilibria.
This is demonstrated graphically below, where the filled circles are player 1's optimal decisions at each choice x of player 2 and the gray empty circles are player 2's optimal decisions for each choice y of player 1. Clearly the best response functions intersect at every point.
Edit: I managed to find an analogous problem to this one online. You should check it out too. It's called splitting a dollar (p.23).