Compliance With Voting System Criteria
Most of the mathematical criteria by which voting systems are compared were formulated for voters with ordinal preferences. Some methods, like Approval voting, requests information than can't be unambiguously inferred from a single set of ordinal preferences. The two-round system is such a method, because the voters are not forced to vote according to a single ordinal preference in both rounds.
Since the two-round system requires more information from each voter than a single ordinal ballot provides, one can't fit the criteria that are formulated expressly for voters with ordinal preferences without making a generalization as to how the voters will behave. The same problem exists in Approval voting, where one has to make assumptions as to how the voters will place their approval cutoffs.
If the voters determine their preferences before the election and always vote directly consistent to them, they will emulate the Contingent vote and get the same results as if they were to use that system. Therefore, in that model of voting behavior, the two-round system passes all criteria that the contingent vote passes, and fails all criteria the contingent vote fails.
Since the voters in the two-round system don't have to choose their second round votes while voting in the first round, they are able to adjust their votes as players in a game. More complex models consider voter behavior when the voters reach a game-theoretical equilibrium from which they have no incentive, as defined by their internal preferences, to further change their behavior. However, because these equilibria are complex, only partial results are known. With respect to the voters' internal preferences, the two-round system passes the majority criterion in this model, as a majority can always coordinate to elect their preferred candidate. Also, in the case of three candidates or less and a robust political equilibrium, the two-round system will pick the Condorcet winner whenever there is one, which is not the case in the Contingent vote model.
The equilibrium mentioned above is a perfect-information equilibrium and so only strictly holds in idealized conditions where every voter knows every other voter's preference. Thus it provides an upper bound on what can be achieved with rational (self-interested) coordination or knowledge of others' preferences. Since the voters almost surely won't have perfect information, it may not apply to real elections. In that matter, it is similar to the perfect competition model sometimes used in economics. To the extent that real elections approach this upper bound, large elections would do so less so than small ones, because it's less likely that a large electorate has information about all the other voters than that a small electorate has.
Read more about this topic: Two-round System
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