The prevailaing party can normally seek payment of costs. If there is a basis to argue that neither party is a prevailiing party, or the prevailing party does not seek to recover costs, then there would not be a recovery of costs.
As recovery of costs can be modified by statute and court rule, depending upon the full facts and cause of action, you should check the state and local court rules and Code of Civil Procedure, as well as the statute for any specific cause of action, to see how they address the issue of the recovery of costs for your case.

