Condorcet Paradox with Side Payments

The Condorcet paradox is a case of cyclical preferences of a group of individuals and it was predicted theoretically. In recent decades, many studies have been devoted to successfully proving the occurrence of a paradox in real life. One way to resolve the Condorcet paradox is by majority rule with the casting vote of one of the players in case of equality. However, this method of resolving the paradox leads to the appearance of a new paradox, the Chairman’s paradox, when a player who is vested with the exclusive right to resolve disputed situations receives the worst result for himself. On the other hand, side payments in a collective decision environment are also quite common and are designed to manipulate voters' preferences. This work is devoted to studying from an analytical point of view the Condorcet Paradox and how to resolve this paradox when voting for the majority with the decisive vote of one of the players while adding side payments to it. In this paper, we study the solvability of the game by dominance when introducing side payments, the properties of the received game, provides an analytical solution to this game for cases where each player can make a side payment, and also assesses the change in the winnings of the players if the right to make a side payment belongs to them. A natural continuation of the study is the generalization of the game to the game of N persons, while retaining the privileges of the 1st player, which ultimately leads to a deterioration in his winnings. For the N-person game, the results obtained for the 3-person game are summarized: the properties of the game are investigated, an analytical solution and player wins are given when adding side payments.