The flow network method

We present a method which associates with any network a binary relation on its vertices which is complete and quasi-transitive, so that it admits at least a maximum. Such a method, called flow network method, is based on the concept of maximum flow. As an application, given a competition involving two or more players, we identify it with a network whose vertices are the players and use the flow network method to build a relation on the set of players. Such a relation can be interpreted as a way to establish, for every ordered pair of players, if the first one did at least as good as the second one in the competition and its maxima can be interpreted as the winners of the competition. That way to select the winners is proved to satisfy many desirable properties. Further, relying on the flow network method, we define a voting system where each individual is allowed to express as preference relation any binary relation on the set of alternatives. That voting system is proved to be decisive, anonymous, neutral, monotonic, efficient and immune to the reversal bias. Moreover, it coincides with the Borda count for preference profiles made up by linear orders.