Party | Seats |
| Topics ordered by preference |
| Pairwise vote | |||||||||
|
|
| Low | Mid | High |
| A | B |
| B | C |
| C | A |
Red | 25 |
| A | B | C |
|
| 25 |
|
| 25 |
| 25 |
|
Green | 35 |
| C | A | B |
|
| 35 |
| 35 |
|
|
| 35 |
Blue | 40 |
| B | C | A |
| 40 |
|
|
| 40 |
|
| 40 |
Total | 100 |
|
|
|
|
| 40 | 60 |
| 35 | 65 |
| 25 | 75 |
| B | C | A |
It is, in all clarity, not that easy to aggregate votes on more than two topics. [108] For two topics one can indeed ask for pro and contra, and find a majority (and occasional ties, for which exist tie-breaking rules). For two topics one can indeed ask for pro and contra, and find a majority (and occasional ties). For more topics, votes will scatter across the topics, and there will often be no clear majority. Therefor, pairwise voting is a good strategy to get the required information on the preferences. However, pairwise voting apparently also causes problems. So, basically, the search is for a strategy without such problems. And that is, basically, also the suggested value of Arrow’s Theorem: that it states that there would be no such good strategy.
However, in this Condorcet example, we may clearly conclude that the cycle primarily means that there is a tie. The situation is in a deadlock, and the group, as a collectivity, is indifferent. That there are indifferences or ties, is nothing special. Standard economic analysis allows agents to be indifferent (we even draw indifference curves), so groups should be allowed to be indifferent too. In Condorcet’s example, indifference is even a logical choice, since when we assume something else, then we quickly run into difficulties.
There is the famous case of Buridan’s Ass (AD 1358). A donkey stands between two equal stacks of hay, at equal distances. He cannot decide which stack to take, and dies of starvation. The upshot of this parable is that rational beings can devise a decision. Constitutions generally state what happens when there are ties. Commonly the Status Quo persists. (This may happen even if it was one of the topics under ballot, and apparently was rejected at that stage.) Alternatives are that the chairman decides, or points are (re-) negotiated, and one can use dice.
It is important to see the difference between voting and deciding. In two stages, the chairperson first lists the votes, and then only secondly gives the decision with a tick of the hammer. Table 12 essentially gives a voting field, and no decision yet. There is no inconsistency as long as we record these results as voting scores, for example “B has more votes than A in a pairwise comparison”. There only arises an inconsistency when we change this into a preference, i.e. decide that “B is better than A”. There are additional rules that translate the field into a unique decision. Part of paradoxical element in voting derives from confusing voting and deciding.
We can use Condorcet’s example to give a short proof of Arrow’s Theorem, restricting our attention to majority voting.