Let $\mathbb{N}$ be the set of all nonnegative integers. For any set $A\subset \mathbb{N}$, let $R(A,n)$ denote the number of representations of $n$ as $n=a+a^{\prime }$ with $a,a^{\prime }\in A$. There is no partition $\mathbb{N}=A\cup B$ such that $R(A,n)=R(B,n)$ for all sufficiently large integers $n$. We prove that a partition $\mathbb{N}=A\cup B$ satisfies $|R(A,n)-R(B,n)|\leq 1$ for all nonnegative integers $n$ if and only if, for each nonnegative integer $m$, exactly one of $2m+1$ and $2m$ is in $A$.