Assignment algorithm for symmetric matrices
Your problem is exactly the same as maximum weight perfect matching, solved by the blossom algorithm. Although $\delta$ is not given as symmetric, you can reduce your problem to the symmetric case by replacing $\delta$ with
$$ \delta'(\alpha,\beta) = \max(\delta(\alpha,\beta),\delta(\beta,\alpha)). $$
I'll let you figure out how to translate a solution which is optimal with respect to $\delta'$ to one which is optimal with respect to $\delta$.