Normed Negative Voting to Depolarize Politics

Abstract

Are the voter preferences clearly heard in a plurality voting system? Alternate voting schemes wherein the voters rank their choices instead of just voting for their first preference certainly capture the voter preferences in a richer format, but they do not explicitly capture the negative preferences. Here we propose a voting scheme that blends negative voting within a cumulative voting system. In this scheme, we can simultaneously decipher the popularity as well as the polarity of each candidate contesting in the election, giving us a two dimensional view of the candidates. An incentive structure can be built into voting system whereby we can penalize the candidates for their polarity and discourage polarizing campaign rhetorics. Theoretically, this could help depolarize the heated political environment and level the playing field for third party candidates.

Appendix

In the simulated elections presented in Table  1 of Sect. 4, individual ballots were not simulated. Instead, the aggregated positive and negative votes $(P_{\mu },N_{\mu })$ for each candidate $\mu$ were generated from a uniform distribution. Now, we shall explicitly simulate the individual ballots of the voters, namely generate $(p^{i}_{\mu }, n^{i}_{\mu })$ for each voter-i, and then construct net votes $(P_{\mu },N_{\mu })$ from them. This procedure is computationally more elaborate and time consuming, but it would let us compare NNV to ranked voting procedures.

Each simulated m-candidate election will have 100 voters. For each voter-i , the votes for each candidate $\mu$ is generated as a real number (rather than integer) from a uniform distribution between (-10,+10), and then they are normalized such that their absolute values sum to 10. If the normalized vote for a candidate-$\mu$ is positive, then it is taken as $p^{i}_{\mu }$, while if it is negative, it is taken to be $n^{i}_{\mu }$. The NNV votes are thus generated from a uniform-normed distribution. For each of these NNV ballots, an associated ranked ballot is prepared by arranging the candidates according to decreasing value of their votes in that ballot. It is ensured that no two candidates have the same rank, else that ballot is rejected. Once the 100 ballots are generated, the Instant Runoff method and the Borda method (discussed in Sect. 5) are employed to determine the winner. Further, using the aggregated $(P_{\mu },N_{\mu })$, the candidate who maximizes the voter satisfaction $S_{\mu }$ (Eq. (10)) can be determined, as well as the winning candidate according to the NNV evaluation metrics $W_{0}^{1}$ and $W_{1}^{1}$.

To determine if the winning candidate also maximizes the voter satisfaction, we simulate thousand elections and evaluate the percentage of these elections in which the winning candidate also maximizes the voter satisfaction.

Table 4 Thousand m-candidate elections were simulated with 100 voters in each election

Table 4 shows that both the Instant Runoff (IR) and the Borda methods perform worse than the NNV metrics in picking a winner who maximizes voter satisfaction. This indicates that the process of converting the cardinal NNV ballots into ranked ballots ignores some of the rich voter preference information. This is not simply a consequence of our definition of voter satisfaction by Eq. (10), since we see that the Borda and IR methods don’t align well even with $W_{0}^{1}$, nor do they align well with each other as m increases. In comparison to IR, the Borda method’s performance is much better and doesn’t deteriorate for larger m.

The above results are dependent on the probability distribution generating the ballots. Note that by aggregating the simulated individual ballots to construct the net votes, the overall variance in the aggregated $(P_{\mu },N_{\mu })$ would be reduced simply due to the law of large numbers. So, the simulations would very rarely produce situations where there are extreme differences in between the net votes of various candidates. On the other hand, directly simulating the net votes $(P_{\mu },N_{\mu })$ from a uniform distribution would produce situations with extreme differences in between the net votes of various candidates. This may potentially be the cause for the difference in the $W_{0}^{1}$, $W_{1}^{1}$ percentages observed between Tables 1 and 4.

Out of curiosity, we could tweak the distribution from which the ballots are generated to make it resemble the voter behavior. When there are many candidates, voters would tend to focus on a relatively few candidates and vote nearly zero for the other candidates. This behavior can be formalized into the distribution in the following way. (i) Generate a real number from uniform distribution between (-10,10) for each candidate-$\mu$. (ii) Cube these numbers by raising them to power 3 so that each of their sign is preserved, while the magnitude of the larger numbers are selectively exaggerated. (iii) Impose the normalization condition on the cubed numbers such that the sum of the absolute values equals 10, and then extract the $p^{i}_{\mu }$ and $n^{i}_{\mu }$. We can refer to this procedure as ballot generation from the cube-normed distribution (as opposed to uniform-normed distribution). Table 5 shows the results of the simulations using cube-normed distribution, and we can note that the performance of both Borda and IR methods have worsened in comparison to Table 4.

Table 5 Same as Table 4, but the ballots are generated from cube-normed distribution instead of uniform-normed distribution