Above still neglects strategic voting. This could be represented by a change in apparent position. How do we evaluate this ? It appears that the Condorcet approach is least sensitive to cheating since in a pairwise vote there is an incentive to express one’s true preferences. Pairwise voting however can be unattractive since there need not be a Condorcet winner, or, when one exists, it may conflict with the preference rankings. One way to solve the complexity of choosing between these methods is to compromise by having a run-off election. The two top outcomes of Plurality or Borda are taken and then subjected to a pairwise vote as in Condorcet. There is one final consideration. Simply taking the two ‘top outcomes’ seems unduly simple, we should consider what these actually are. In France, the election between Chirac, Jospin, Le Pen and others caused Jospin’s votes to scatter over all kinds of smaller parties so that he dropped from the race while he was the Condorcet winner of both Chirac and Le Pen. When we are compromising, we should focus on determining the two main contenders.
Borda Fixed point
Let us reconsider the dynamic process that occurs within an economy. We see that under the influence of time, the budget changes continuously. A voting scheme naturally requires that there is a list of candidates, but one cause for paradoxes is that that list is not fixed. For example, in the Borda vote above, B is selected, but if C decides to withdraw (or gets a heart attack), then we would expect B to remain the winner, but suddenly it is A (see the Condorcet vote A versus B). Remember also the Bush, Gore and Nader case. We could consider a procedure to be better when the choice is less dependent upon changes in the budget.
A way to achieve this is to use the notion of a ‘fixed point’. For a function f: D
R, for some domain D and range R, the point p is a fixed point iff f(p) = p. Let us consider this concept for voting.
Let P be the voting procedure, and let X = {x1, …, xn} be the budget with all the candidates. Let the unrefined winner be w = P(X). Let Y be the budget when w does not participate, Y = X \ {w}. Let the ‘alternative winner’ be v = P(Y) = v(w), i.e. the candidate who wins when the first winner w does not participate. This is not simply the run-off between the winner and the common runner-up, since the selection of the alternative winner requires the recalculation of the preference weights. This alternative winner can be seen as a ‘summary’ of the opposition to w. The scheme is a compromise since the Condorcet pairwise condition holds for the winner and the alternative winner. While these notions are defined with respect to the unrefined winner, we can generalise this to any winner, and in particular to our optimal winner.
An alternative condition for winning in general is the ability to win from one’s strongest opponent. This gives the fixed point condition. Define f(x) = P(x, P(X \ {x})), which is the general function ‘the vote result of x and its alternative winner’. Then w* is the solution to the fixed point condition x = f(x):
w* = P(w*, v(w*)) = P(w*, P(X \ {w*})) = f(w*)
When the unrefined winner w is not a fixed point, i.e. when the unrefined winner w = P(X) appears to lose from v, so that w