How Many Votes is a Lie Worth? Measuring Strategyproofness through Resource Augmentation

Abstract

It is well known, by the Gibbard-Satterthwaite Theorem, that when there are more than two candidates, any non-dictatorial voting rule can be manipulated by untruthful voters. But how strong is the incentive to manipulate under different voting rules? We suggest measuring the potential advantage of a strategic voter by asking how many copies of their (truthful) vote must be added to the election in order to achieve an outcome as good as their best manipulation. Intuitively, this definition quantifies what a voter can gain by manipulating in comparison to what they would have gained by finding like-minded voters to join the election. The higher the former is, the more incentive a voter will have to manipulate, even when it is computationally costly. Using this framework, we obtain a principled method to measure and compare the manipulation potential for different voting rules. We analyze and report this potential for well-known and broad classes of social choice functions. In particular, we show that the positional scoring rule with the smallest manipulation potential will always be either Borda Count (if the number of voters outweighs the number of candidates) or Plurality (vice versa). Further, we prove that any rule satisfying a weak form of majority consistency (and therefore any Condorcet-consistent rule) cannot outperform Plurality, and that any majoritarian Condorcet rule will perform significantly worse. Consequently, out of the voting rules we analyze, Borda Count stands out as the only one with a manipulation potential that does not grow with the number of voters. By establishing a clear separation between different rules in terms of manipulation potential, our work paves the way for the search for rules that provide voters with minimal incentive to manipulate.

Figures (2)

• Figure 1: For the profile from \ref{['ex:nonmono']}, the $\textit{IRV}$ winners of $\succ_N + {k}(\succ_{1})$ and $(\succ_{1}', \succ_{-1})$ as a function of $k$. Green regions indicate the values of $k$ for which voter 1 prefers the outcome of the former to that of the latter. Even though voter 1 prefers the outcome of $\textit{IRV}(\succ_N + {k}(\succ_{1}))$ to that of $\textit{IRV}(\succ_{1}', \succ_{-1})$ for $1 < k <7$, this is not the case for $7<k<19$. As a result, the manipulation potential of $\textit{IRV}$ is at least 19 for $n=27$ and $m=5$ (see \ref{['def:manipot']}).
• Figure 2: Visualization of our results from \ref{['sec:warmup']}. For each rule, the longer the left arm of the scale, the more truthful copies a single manipulation can "lift". That is, the voter will need (in the worst case) a larger number of truthful copies to produce an outcome as desirable as that of the manipulation. The left arm of the middle scale is double that of the left scale, whereas comparing it to the rightmost scale depends on the values of ${\color{blue} n}$ and ${\color{orange} m}$.

Theorems & Definitions (50)

• Example 1.1
• Definition 1: $k$-Augmentation Strategyproofness
• Definition 2: Manipulation Potential
• Example 3.1
• Claim 4.1
• Theorem 4.2
• proof
• Theorem 4.3
• proof
• Theorem 4.4