A weighted mechanism for minority voting in sequential voting

We propose a weighted minority voting mechanism within a two-round sequential voting process, in which all individuals retain their voting rights in the second round but with different weights depending on the first-round outcome. In a utilitarian framework where individuals have a given utility function that depends on the outcomes of each round, first-round winners are identified and vote with reduced weight in the second round, while losers retain full weight. By giving greater weight to first-round losers, this design ensures that first-round winners continue to contribute to the final decision without dominating it, thereby mitigating repeated disadvantages for losers. We then compare the expected aggregate utility of society across different levels of second-round weight assigned to first-round losers, including both the simple majority rule – where all voters carry equal weight in both rounds – and the limiting case of minority voting where first-round losers receive no weight in the second round. To do so, we analyze two models: one in which individual utility derives solely from material payoffs, and another in which a form of harmony is considered, whereby individuals incur a utility loss if others repeatedly belong to the losing minority. This analysis allows us to assess how strategic behavior affects the effectiveness of the proposed mechanism.