Saari has also developed an ingenious way to depict voting schemes geometrically. For 3 candidates, this becomes a triangle, and the different procedures can be calculated from that. It appears that these triangles are a good educational tool. However, my experience is that the computer programs (Colignatus (2001) uses Mathematica) are easier to use, since they take away the need for calculations, while they are available for more dimensions and also allow for indifference and not just strict preference. A complex scheme like the Borda Fixed Point also requires more work with the triangle, while in Mathematica it is a simple procedure call. It may be noted that above discussion of the Borda Fixed Point method has been simplified by assuming single winners. In practice, there can be ties, complicating the search, and requiring tie-breaking rules.
Pareto
Another consequence of the switch of attention from statics to dynamics is the recognition of a status quo.
There appears to exist another wide-spread confusion about ‘majority voting’. This idea is that a majority result would still be democratically valid, even if the winning decision implies a real loss for the opposition. The counter-example is when the majority decides that the minority pays $1 to the majority: this is not necessarily a morally acceptable situation, even though there is a majority. From a moral point of view, each voting scheme should have two rounds: a first round to select the Pareto improving points compared to the status quo, and then a second round to select the winner from those Paretian improvements. The majority rule thus can be regarded as only a tie-breaking rule, namely for the deadlock when there are more Pareto improving points. In elections of persons, the status quo can be a vacancy, and in that respect all candidates could be taken as Paretian. But the Paretian pre-condition cannot be skipped in general.
The Paretian condition may require some subtlety. Consider the family choice for a holiday to Greece or Spain, discussed above. If little Robby considers the holiday to Spain to be a deterioration from the status quo of not having a holiday at all, then there is moral argument to say that Spain is not a valid option to take a vote on. However, if it can be established in a first round that going on a holiday is unanimously a good idea, then Robby has to accept a possible majority decision in favour of Spain and against Greece.
One argument against the selection of Pareto improving points is that people might also cheat about these points. This argument is not convincing, since Pareto improvement is in one’s own interest. Indeed, little Robby might try to veto Spain by saying that he does not want a holiday, and thus he might be trying to bargain to get everybody to accept Greece. However, this ploy can be prevented by having that first round on having a holiday, since if he really wants a holiday anyhow, then he has to show this then. Careful construction of the voting process thus remains an issue.
A note on cheating
One of the key problems in voting theory is strategic voting behaviour, better known as cheating. In a scheme like Borda, cardinal utility has already been reduced to ordinal utility, so perhaps we should be lenient and allow voters to maximize their utility from the final outcome by manipulating their vote. But our opinion on this does not matter, since the ballot generally is secret and we cannot stop people from voting strategically anyway. In fact, my Mathematica programs, Colignatus (2001), contain routines for cheating. These are simple routines that assume both full information and that others don’t cheat, since the mathematics of cheating while assuming that others cheat too is rather complex, especially when nobody has full information about the true preferences. Given all this, one surmises that election results do not reflect the true state.
Thinking about these issues gave me an idea that might be helpful to elicit the true state. Suppose that each voter is informed in advance that there is a probability p that the ranking order that is submitted will be used by the election computer for strategic voting. If the voter submits his or her true ranking, then this is rewarded with probability p to improve the election result for that voter, and much better than the voter can, since the computer knows all submitted rankings. If the voter submits a strategically adapted ranking, then this is punished with probability p namely to improve the election result for that false ranking. Likely there is a specific value of p that would generate the most truthful election result. Unfortunately, I haven’t had time to develop this idea.