Heuristic Voting Under the Alternative Vote: The Efficiency of “Sour Grapes” Behavior

Abstract

This theoretical paper contrasts two voting heuristics: overstating and replacing. Under the Alternative Vote, overstating is useless but the replacement heuristics is consequential. The paper argues that the “replacing” manipulation corresponds to a psychologically and politically plausible voter behavior, and study its effects. The conclusion is that the Alternative Vote should not be considered as immune to manipulation.

Appendix

1.1 Appendix 1: A Successful “Overstating” Manipulation Under AV

There are 100 voters, whose preference profile is provided in Table 5. The first collum states that 26 voters rank a first, c second, b third, and d last.

Table 5 A preference profile

At such a profile, with the Hare system of elimination, d is eliminated first, with 23 transfers to a and one transfer to d. Then c is eliminated second but these votes are not transferred to d but to a and b. In the last round, a wins against b by 64 votes against 36.

Suppose now that the 10 voters with preference $c>d>b>a$ overstate their preferences between a and b and put b at their first position. They thus announce $b>c>d>a$ and the new profile is the one in Table 6.

Table 6 Modified preference profile

In that case c is eliminated first, giving votes to d. Then a is eliminated second, and b wins against d in the last round by a $61-39$ margin.

1.2 Appendix 2: Simulation Results for “Overstating” Manipulation Under AV

1.2.1 In One Dimension

I here provide simulation results for overstating, instead of replacing. The Table 7 corresponds to Table 2: same parameters, same preference profiles. But the new Table is simpler because overstating, in practice, never operates. For the simulations, “Overstating” is implemented in a manner similar to “Replacement”. Preferences are deduced from utilities defined in the specified random model. Then the precise manipulation is the following. For all individual who prefer the challenger to the winner, a bonus of $\delta = 0.2$ is added to the individual’s evaluation of the challenger. One can see that the phenomenon described in the previous paragraph—changing votes induces a change in the sequence of elimination that lead to a change of outcome—is never observed in the simulations. In this model, overstating is in practice not successful, because it does not avoid the squeezing of the center. We will see in the next section that the phenomenon is sometimes, but rarely, seen in the simulations of two-dimensional models.

Table 7 Hare transfers with sincere and tactical “overstating” votes

One could think of other implementation of “overstating” with respect to the winner and the challenger: for a disappointed voter, decreasing the evaluation of the winner by $\delta$, or increasing the evaluation of the challenger by $\delta / 2$ and decreasing the evaluation of the winner by $\delta / 2$. The obtained results are strictly identical, and they are also identical for a very large bonus $\delta =2$.

1.2.2 In Two Dimensions

The results are presented in Table 8 which is to be compared with Table 4. The parameters and profiles are the same, so that the frequencies of Condorcet winners and the numbers of potential manipulators are the same. Consider for example the case of 10 candidates, $\theta =0.75$ and $\delta =0.2$ already discussed in the text. Overstatement changes the outcome in 27 simulations (out of 1000) while Table 4 indicates that Replacement changes the outcome in 278 simulations. The Condorcet winner, when it exists is elected by the Hare process on sincere votes with frequency 467/862 = 54 %; after Overstatement this frequency decreases to 456/862 = 53 %, whereas Replacement makes it increase to 66 %. Out of the $50 \times 1000$ voters, after overstatement, only 0.9 % are better off and 0.2 % are worse off.

These simulations confirm for the two-dimensional model the idea that overstating has very limited effect on Hare elections. Usually it does not change anything, and in the rare case where it does change the winner, it is often detrimental to the manipulator.

Table 8 Hare process, the overstatement tactic. Simulation results for bi-normal profiles with 50 voters

1.3 Appendix 3: About Other Voting Rules

In Politics, for electing one candidate, most countries use either two-round majority voting or simple plurality. For instance, for Presidential Elections, 84 countries use Two-round systems,⁸ 21 countries use simple Plurality⁹ and only one country uses another system, namely Ireland using AV (Reynolds et al. 2008).

As already observed, the Two-Round system is very similar to AV, at least with respect to the ideas developed in the present paper. Here I compare the AV results with the results that would be obtained under simple Plurality and the Borda rule. Even if almost never used in politics, the Borda rule is an interesting, and well-studied, theoretical benchmark.

The replacement idea is not absurd for Plurality rule. Consider a voter who sees in the voting intentions reported by the newspapers that her preferred candidate is ranked very low. Then it is natural for this voter to give up supporting this probable loser.

The overstatement idea, in the case of Plurality voting works as follows. Suppose that A is elected, B is ranked second, and the voter prefers B to A. She thus has not voted for A. If she has voted for B she has already given maximal strength to her vote and cannot “overstate” but if she voted for some other candidate then she can upgrade her evaluation of B and finally vote for B.

Therefore the two logics—overstating and replacing—are different but eventually lead, because the expressive possibilities of simple plurality balloting are so narrow, to the same behavior: deserting low-ranked candidates in favor of one the two front-runners. This is precisely what is usually understood by “strategic voting” in the Anglo-Saxon Political Science literature, essentially devoted to simple plurality systems.

Still, it is interesting to compare, for Plurality rule, the consequences of Overstating and Replacing in the models used in this paper. For the comparison to make sense, I use the same parameters and only change the voting rule. The second ranked candidate, for plurality is naturally defined as the candidate with the second largest Plurality score. For a disappointed voter, that is a voter who prefers the second-ranked candidate (say B) to the first-ranked one, “Replacing” means lowering by $\delta$ her evaluation of B (and this may leads her to change her vote in favor of some other candidate) while “Overstating” means to increase by $\delta$ her evaluation of B (and this may leads her to change her vote in favor of B himself).

The next tables compares tactics, and voting rules. The definition of the manipulations for the Borda rule simply copies the previous ones, the Borda scores being used in place of the Plurality ones.

Table 9 Comparing rules

Table 10 Plurality rule

Table 11 Borda count

Table 9 allows comparison across voting rules. It is based on the central scenario studied in the text: 10 candidates, $\theta = 0.75$ and small perturbations ($\left| \delta \right| =0.2$). One can read that the manipulations studied in the text have always a globally positive effect on welfare: all the figures in the last line of the table are positive. This should not come as a surprise since only manipulations by voters who prefer the second-ranked candidate to the first-ranked one are studied here. But the Table shows that Overstating has only a tiny effect ($+0.009$) under the Hare process. Furthermore one can see that this is due to the fact that overstating usually doe not change the outcome of the Hare process. The lines “Frequency of outcome changes” in the table shows that specificity of the Hare process with respect to the overstatement tactic.

Table 10 compare Overstating and Replacing when ballots are counted according to Plurality rule, and Table 11 does the same for the Borda rule. These tables allow detailed comparison between Replacing (negative $\delta$) and Overstating (positive $\delta$). For instance, with 10 candidates, $\theta = 0.75$ and small perturbations ($\left| \delta \right| =0.2$) one can read that the number of potential manipulators, that is the number of voters who prefer the second-ranked candidate to the first one, are 22,490 under Plurality and 25,581 with the Borda count (out of $50 \times 1000 = 50{,}000$ voters). More often than not, the manipulation does not change the outcome, and this is true in similar proportions for Overstating and Replacing. When the outcome change, the manipulators benefit from the change in proportions which are roughly similar with Overstating and Replacing, under these voting rules.