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Approval and plurality voting with uncertainty: Info-gap analysis of robustness

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Abstract

Voting algorithms are used to choose candidates by an electorate. However, voter participation is variable and uncertain, and projections from polls or past elections are uncertain because voter preferences may change. Furthermore, electoral victory margins are often slim. Variable voter participation or preferences, and slim margins of decision, have implications for choosing a voting algorithm. We focus on approval voting (AV) and compare it to plurality voting (PV), regarding their robustness to uncertainty in voting outcomes. We ask: by how much can voting outcomes change without altering the election outcomes? We see fairly consistent empirical differences between AV and PV. In single-winner elections, PV tends to be more robust to vote uncertainty than AV in races with large victory margins, while AV tends to be more robust at low victory margins. Two conflicting concepts—approval flattening and approval magnification—explain this tendency for reversal of robust dominance between PV and AV. We also examine the robustness to vote uncertainty of PV in elections for proportional representation of parties.

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Appendices

Appendix A: Derivation of approval voting robustness function, Eq. (6)

We now derive the robustness function for approval voting, Eq. (6).

Let m(h) denote the inner minimum in Eq. (5), which occurs when:

$$\begin{aligned} a_i = ({\widetilde{a}}_i - w_i h)^+, \ \ \ a_j = \min \{T, \ {\widetilde{a}}_j + w_j h \} \end{aligned}$$
(13)

where we have defined a truncation function as \(x^+ = x\) if \(x \ge 0\) and \(x^+ = 0\) otherwise.

For \(h \le (T - {\widetilde{a}}_j)/w_j\) and \(h \le {\widetilde{a}}_i/w_i\), the inner minimum is:

$$\begin{aligned} m(h) = {\widetilde{a}}_i - w_i h - {\widetilde{a}}_j - w_j h = {\widetilde{a}}_i - {\widetilde{a}}_j - h ( w_i + w_j ) > \delta \end{aligned}$$
(14)

For \(h > (T - {\widetilde{a}}_j)/w_j\) the inner minimum is:

$$\begin{aligned} m(h) = ({\widetilde{a}}_i - w_i h)^+ - T > \delta \end{aligned}$$
(15)

The left-hand side of this inequality is negative, while \(\delta \ge 0\), so there is no non-negative solution for h. Hence the robustness is the least upper bound of solutions for h of the inequality in Eq. (14), yielding:

$$\begin{aligned} {\widehat{h}}_{ij}(\delta ) = \frac{{\widetilde{a}}_i - {\widetilde{a}}_j - \delta }{w_i + w_j} \end{aligned}$$
(16)

or zero if this is negative, where \(\delta \ge 0\). Note that the robustness is less than \((T - {\widetilde{a}}_j)/w_j\), so we needn’t consider the case where \(h > (T - {\widetilde{a}}_j)/w_j\).

Appendix B: Derivation of plurality voting robustness function, Eq. (9)

Let m(h) denote the inner minimum in the definition of the robustness, Eq. (5), which occurs when \(a_i\) is minimal and \(a_j\) is maximal, subject to the constraints of the info-gap model in Eq. (7). Recalling Eq. (8), this implies:

$$\begin{aligned} a_i = ( 1 - \gamma h)^+ {\widetilde{a}}_i \end{aligned}$$
(17)

Two conditions in the info-gap model constrain the maximum value of \(a_j\).The fractional-error condition implies that \(a_j \le (1 + \gamma h) {\widetilde{a}}_j\). The condition \(\sum _n a_n \le T\) (deriving from the plurality condition that each voter casts at most 1 vote) imposes an upper limit on \(a_j\) that is obtained if all other candidates obtain minimal votes. Thus the two conditions in the info-gap model imply that \(a_j\) is the following minimum:

$$\begin{aligned} a_j = \min \left\{ (1 + \gamma h) {\widetilde{a}}_j, \ T - \sum _{n \ne j} (1 - \gamma h)^+ {\widetilde{a}}_n \right\} \end{aligned}$$
(18)

We will now show that \((1 + \gamma h) {\widetilde{a}}_j\) is the minimum on the right-hand side of Eq. (18).

If \(h \le 1/\gamma\), then:

$$\begin{aligned} (1 + \gamma h) {\widetilde{a}}_j - \left( T - \sum _{n \ne j} (1 - \gamma h)^+ {\widetilde{a}}_n \right)= & {} - T + (1 + \gamma h) {\widetilde{a}}_j + (1 - \gamma h) \sum _{n \ne j} {\widetilde{a}}_n \end{aligned}$$
(19)
$$\begin{aligned}= & {} \underbrace{-T + \sum _{n=1} ^N {\widetilde{a}}_n}_{\le \ 0} + \gamma h \underbrace{\left( {\widetilde{a}}_j - \sum _{n \ne j} {\widetilde{a}}_n \right) }_{< \ 0} \end{aligned}$$
(20)
$$\begin{aligned}< & {} 0 \end{aligned}$$
(21)

The sum of the first two terms on the right of Eq. (20) is non-positive due to the constraint in the info-gap model on the sum of the votes. The second parenthetical term on the right of Eq. (20) is negative because \({\widetilde{a}}_i > {\widetilde{a}}_j\) and all votes are non-negative. The strict inequality in Eq. (21) results. From this we conclude that \((1 + \gamma h) {\widetilde{a}}_j\) is the minimum on the right-hand side of Eq. (18) for all \(h \le 1/\gamma\).

Thus, for \(h \le 1/\gamma\), the inner minimum in the definition of the robustness is:

$$\begin{aligned} m(h) = ( 1 - \gamma h)^+ {\widetilde{a}}_i - (1 + \gamma h) {\widetilde{a}}_j = {\widetilde{a}}_i - {\widetilde{a}}_j - ({\widetilde{a}}_i + {\widetilde{a}}_j) \gamma h \end{aligned}$$
(22)

The robustness is the least upper bound of h values for which \(m(h) > \delta\). Equating the right-hand side of Eq. (22) to \(\delta\) and solving for h yields:

$$\begin{aligned} {\widehat{h}}_{ij}(\delta ) = \frac{{\widetilde{a}}_i - {\widetilde{a}}_j - \delta }{({\widetilde{a}}_i + {\widetilde{a}}_j) \gamma } \end{aligned}$$
(23)

This is less than \(1/\gamma\), so we needn’t consider \(h > 1/\gamma\). This is Eq. (9).

Appendix C: Evaluating the robustness function for proportional representation by plurality voting

Let \(m_n(h)\) denote the inner minimum in the definition of the robustness function, \({\widehat{h}}_n({{f}_\mathrm{c}})\) in Eq. (12). A plot of h vs. \(m_n(h)\) is the same as a plot of \({\widehat{h}}_n({{f}_\mathrm{c}})\) vs. \({{f}_\mathrm{c}}\). In other words, \(m_n(h)\) is the inverse of the robustness function, and knowledge of \(m_n(h)\) is equivalent to knowledge of \({\widehat{h}}_n({{f}_\mathrm{c}})\). We will derive an expression for \(m_n(h)\).

From Eq. (10) one can readily show that:

$$\begin{aligned} \frac{{\mathrm{d}}f_n}{{\mathrm{d}}a_n} \ge 0 \ \ \ \text{ and } \ \ \ \frac{{\mathrm{d}}f_n}{{\mathrm{d}}a_j} \le 0 \ \text{ if } \ j \ne n \end{aligned}$$
(24)

From these relations we conclude that \(m_n(h)\) is obtained when \(a_n\) is chosen as small as possible, and each \(a_j\) for \(j \ne n\) is chosen as large as possible, subject to the constraints of the info-gap model, Eq. (7), and employing Eq. (8). Thus we find that \(m_n(h)\) is obtained with:

$$\begin{aligned} a_n= & {} ( 1 - \gamma h) ^+ {\widetilde{a}}_n \end{aligned}$$
(25)
$$\begin{aligned} a_j= & {} (1 + \gamma h) {\widetilde{a}}_j , \ \ \ j \ne n \end{aligned}$$
(26)

subject to the constraint on the sum of all N votes: \(\sum _{j=1} ^N a_j \le T\). Thus Eqs. (25) and (26) must satisfy:

$$\begin{aligned} \sum _{j=1} ^N a_j = \min \left( T, \ ( 1 - \gamma h) ^+ {\widetilde{a}}_n + (1 + \gamma h) \sum _{j \ne n} {\widetilde{a}}_j \right) \end{aligned}$$
(27)

where T is the total number of participating voters. \(m_n(h)\) is obtained by substituting Eqs. (25) and (27) into Eq. (10) yielding:

$$\begin{aligned} m_n(h) = \frac{ ( 1 - \gamma h) ^+ {\widetilde{a}}_n}{\min \left( T, \ ( 1 - \gamma h) ^+ {\widetilde{a}}_n + (1 + \gamma h) \sum _{j \ne n} {\widetilde{a}}_j \right) } \end{aligned}$$
(28)

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Ben-Haim, Y. Approval and plurality voting with uncertainty: Info-gap analysis of robustness. Public Choice 189, 239–256 (2021). https://doi.org/10.1007/s11127-021-00881-2

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