Table 14: Start with 48 voters: Borda B, Condorcet B
| Candidates and their rank order weight | ||
Number of voters | A | B | C |
20 | 3 | 2 | 1 |
28 | 2 | 3 | 1 |
Borda weighted total | 116 | 124 | 48 |
A versus B | 20 | 28 |
|
A versus C | 48 |
| 0 |
B versus C |
| 48 | 0 |
Table 15: Add 27 ‘neutral’ others: Borda B, Condorcet A
| Candidates and their rank order weight | ||
Number of voters | A | B | C |
20 | 3 | 2 | 1 |
28 | 2 | 3 | 1 |
9 | 3 | 2 | 1 |
9 | 1 | 3 | 2 |
9 | 2 | 1 | 3 |
Borda weighted total | 170 | 178 | 102 |
A versus B | 38 | 37 |
|
A versus C | 57 |
| 18 |
B versus C |
| 66 | 9 |
Secondly, however, my problem remains that there is the phenomenon of budget changes. Note that Saari’s example uses a changing electorate rather than a changing budget. My suggestion is that a change in the electorate would require a new vote, while we would want to avoid that in case of a change in the budget. The Borda method would be best, only when the budget would be really given. When it might change, the application of cancellation to all subsets becomes doubtful, since subsets change. There is a fundamental uncertainty with respect to the future. Consider the following example. At a specific point in time, the population of a nation is given, and thus the vote for a President has a specified budget: the population. But, uncertainty sets in again, when people may withdraw from the race. Only a few actually run. Hence, we might well want a rule to deal with possible changes in the budget. Hence, it is not logically required that we cancel votes for all possible subcycles (also for candidates who are not in the race). Saari is very strong on the argument that when we accept cancellation in one case, then we should do so in all cases. I am more sensitive to the exception: when ‘if one, then all’ does not hold.
Concerning Table 14 and Table 15, my reasoning is - contrary to Saari - that the added votes cannot be neglected. The argument of rotational symmetry breaks down when we compare a winner with the alternative winner - which is a pair - while rotational symmetry requires a third candidate or more. For the pair, the addition has an effect. When we consider unrefined winner B and its alternative winner A, then the added votes are in favour of A and no longer ‘neutral’. While C is important since it shows a cycle for a subgroup of voters, another view is that C could be neglected since it is not a fixed point. Canditate C is a typical example of an irrelevant candidate that can cause a preference reversal in Borda voting. Namely, let us consider Table 15 under Borda voting, and let C decide to drop from the race: then A becomes the winner. The Borda Fixed Point method has been developed precisely to deal with that kind of preference reversal.
Thus, when you select your voting method then you must choose between the properties exemplified by this case. (1) Borda is subject to preference reversal. In the example of Table 15, when C drops out, then there would be switch from B to A. (2) The Borda Fixed Point method still depends upon the voting field. In this example, when 27 voters drop out, then there is a switch from A to B.
The choice basically is whether we attach more importance either to the voters or to the candidates. Saari suggests that the candidates are more important, since he cancels the votes of 27 voters and keeps C in the race. I would say that the voters are important and that candidate C is less relevant. The proper question would be whether the winner is a convincing winner. Of course, C can become an important candidate when we add other voters. But then the argument is that those voters count, rather than C.
Consider the impact of semantics. While it has been a long standing notion that cycles may also be taken as indifference, so that the votes cancel, Saari now rephrases this as rotational symmetry, and he suggests that acceptance of rotational symmetry implies acceptance of it for all cases and subsets. The label might be a common mathematical label, but I have a problem with that label in the realm of morality (and the implied universality). Human beings seem to have biological preference for symmetry, and by labelling something as ‘symmetry’, it becomes more attractive. When discussing the different voting schemes, we should be aware of such effects, and try to focus on what the properties really mean, and we should make a proper distinction between a property that is universal and a property that is dependent upon the situation. Perhaps it might be analysed as the ‘mathematical frame of mind’ that acceptance of a property for one set also implies acceptance for all other (sub-) sets, but my conclusion is that when we look closer, that there is room for more subtlety. Indeed, it might well be that considerations of symmetry apply to the static situation, but that we need other considerations for dynamics.
Another example for this need for subtlety is that the ‘rotational symmetry’ argument breaks down on the status quo (see below).