Many democracies complement a parliamentarian system with elements of direct democracy, where the electorate decides on single issues by majority voting. A well-known paradox states that in a sequence of referenda one can get from an arbitrary original income distribution to one in which one player gets almost all the cake. In this paper we design a three-player game modelling the sequential modification mechanism. The strategic analysis reveals that the paradox survives even with rational strategic voters and though the right to propose is allocated to each player once: the last player receives almost the entire cake. The result can be extended to the three-party n-voter case and is for some cases similar when we consider a random rather than fixed sequence of proposers.