Practical Significance and Open Problems

The common reaction to social choice anomalies is that while these may be theoretically feasible, their role in practice is likely to be very limited since the profiles giving rise to the paradoxes are exceptional rather than common. We argue that while the significance and observation of the general no-show paradoxes and monotonicity failures may be limited, the same is not true of P-TOP and P-BOT paradoxes. These pertain to the voters’ best and worst alternatives. Hence, the occurrences associated with them are most likely to be observed and talked about. Vulnerability to these paradoxes creates bizarre incentives for voters. Failures on monotonicity, on the other hand, have more to do with the legitimacy of the voting outcomes. It is important to find out the structural properties of settings giving rise to monotonicity-related paradoxes. Thus far very little is known about these properties. We have related the paradoxes to the existence of majority cycles in the electorate. Our general finding is that the presence of a core or Condorcet winner does not, in general, make it harder to construct profiles that lead to paradoxes. Sometimes the opposite is true: the presence of cycles makes it harder, not easier, to construct paradoxical profiles. Finally, we speculate about reasons for continued use of non-monotonic rules.