The give-up problem for blocked regional lists with multi-winners

► The give-up problem arises in the Italian electoral law due to the blocked lists. ► The paper presents different mathematical formulations of the give-up problem. ► Existence results and the characterization of particular solutions are given. ► The give-up problem can be solved in polynomial time using network flow techniques. ► Strong relations between the optimal solutions of the different give-up models hold.