On the Causes and Consequences of Ballot Order Effects

Abstract

We investigate the effect of ballot order on the outcomes of California city council and school board elections. Candidates listed first win office between four and five percentage points more often than expected absent order effects. This first candidate advantage is larger in races with more candidates and for higher quality candidates. The first candidate advantage is similar across contexts: the magnitude of the effect is not statistically distinguishable in city council and in school board elections, in races with and without an open seat, and in races consolidated and not consolidated with statewide general elections. Standard satisficing models cannot fully explain ballot order effects in our dataset of multi-winner elections.

Appendices

Appendix 1: Maximum Likelihood Estimation

We wish to estimate the coefficients λ, δ, γ, and β using maximum likelihood in the model ${\text{a}}_{{{\text{p}},{\text{j}}}} = \alpha_{{{\text{p}},{\text{t}}({\text{j}})}} + {\text{Inc}}_{{{\text{p}},{\text{j}}}} \lambda_{{{\text{t}}({\text{j}})}} + \varepsilon_{{{\text{p}},{\text{j}}}}$ where $\alpha_{{{\text{p}},{\text{t}}({\text{j}})}} = \delta_{{{\text{p}},{\text{t}}({\text{j}})}} + {\text{Inc}}_{{{\text{p}},{\text{j}}}} \gamma_{{{\text{p}},{\text{t}}({\text{j}})}} + X_{j} \beta_{{{\text{p}},{\text{t}}({\text{j}})}}$.

Assume that the ε_p,j’s are drawn from a logistic distribution. Then the probability that the candidate in position p is the most attractive candidate in race j is $\Pr_{j} (p = 1) = \frac{{\exp (\alpha_{{{\text{p}},{\text{t}}({\text{j}})}} + {\text{Inc}}_{{{\text{p}},{\text{j}}}} \lambda_{{{\text{t}}({\text{j}})}} ) }}{{\sum\nolimits_{i} {\exp (\alpha_{{{\text{i}},{\text{t}}({\text{j}})}} + {\text{Inc}}_{{{\text{i}},{\text{j}}}} \lambda_{{{\text{t}}({\text{j}})}} )} }}$. Similarly, the probability that the candidate in position p is the second most attractive candidate given that the candidate in position p′ is the most attractive candidate is $\Pr_{j} (p = 2|p^{\prime} = 1) = \frac{{\exp (\alpha_{{{\text{p}},{\text{t}}({\text{j}})}} + {\text{Inc}}_{{{\text{p}},{\text{j}}}} \lambda_{{{\text{t}}({\text{j}})}} )}}{{\sum\nolimits_{{i \ne p^{\prime}}} {\exp (\alpha_{{{\text{i}},{\text{t}}({\text{j}})}} + {\text{Inc}}_{{{\text{i}},{\text{j}}}} \lambda_{{{\text{t}}({\text{j}})}} )} }}$. Finally, the probability that the candidate in position p is the third most attractive candidate given that the candidates in positions p′ and p″ are the two most attractive candidates is $\Pr_{j} (p = 3|p^{\prime}, \, p^{\prime\prime} = 1,2) = \frac{{\exp (\alpha_{{{\text{p}},{\text{t}}({\text{j}})}} + {\text{Inc}}_{{{\text{p}},{\text{j}}}} \lambda_{{{\text{t}}({\text{j}})}} )}}{{\sum\nolimits_{{i \ne p^{\prime},p^{\prime\prime}}} {\exp (\alpha_{{{\text{i}},{\text{t}}({\text{j}})}} + {\text{Inc}}_{{{\text{i}},{\text{j}}}} \lambda_{{{\text{t}}({\text{j}})}} )} }}$.

Let Y_j = [Y_1,j,…, Y_Nj,j] denote the vector of observed outcomes in race j where Y_i,j = 1 if the candidate in position i wins office and Y_i,j = 0 otherwise. We construct the likelihood function L(δ, λ, γ, β|Y_j) as follows.

In a one winner race, ${\text{L}}(\delta ,\lambda ,\gamma ,\beta |{\text{Y}}_{\text{j}} ) = \prod\nolimits_{i = 1}^{{N_{j} }} {Pr_{j} (i = 1)^{{Y_{i,j} }} }$ is the likelihood that the winning candidate is the most attractive candidate. In a two-winner race, L(δ, λ, γ, β|Y_j) is the likelihood that the two winning candidates are the two most attractive candidates. By the definition of conditional probability, the probability that the candidates in position p and p′ are the most attractive and second most attractive candidates respectively in race j, $Pr_{j} (p = 1,p^{\prime} = 2)$, is equal to $Pr_{j} (p^{\prime} = 2|p = 1)Pr_{j} (p = 1)$. Thus, ${\text{L}}(\delta ,\lambda ,\gamma ,\beta |Y_{j} ) = \prod\nolimits_{i = 1}^{{N_{j} - 1}} {\prod\nolimits_{{i^{\prime} = i + 1}}^{{N_{j} }} {(Pr_{j} (i = 1)Pr_{j} (i^{\prime} = 2|i = 1) + Pr_{j} (i^{\prime} = 2)Pr_{j} (i = 1|i = 2))}^{{Y_{i,j} Y_{{i^{\prime},j}} }} }$. Similarly, in a three-winner race, L(δ, λ, γ, β|Y_j) is the likelihood that the three winning candidates are the three most attractive candidates. Again by the definition of conditional probability $Pr_{j} (p = 1,p^{\prime} = 2,p^{\prime\prime} = 3)$ is equal to $Pr_{j} (p^{\prime\prime} = 3|p^{\prime} = 2,p = 1)Pr_{j} (p^{\prime} = 2|p = 1)Pr_{j} (p = 1)$. Thus,

$$ {\text{L}}(\delta ,\lambda ,\gamma ,|{\text{Y}}_{\text{j}} ) = \Uppi_{i = 1}^{{N_{j} - 2}} \Uppi_{{i^{\prime} = i + 1}}^{{N_{j} - 1}} \Uppi_{{i^{\prime\prime} = i^{\prime} + 1}}^{{N_{j} }} (Pr_{j} (i = 1)Pr_{j} (i^{\prime} = 2|i = 1)Pr_{j} (i^{\prime\prime} = 3|i = 1,i^{\prime} = 2) + Pr_{j} (i = 1)Pr_{j} (i^{\prime\prime} = 2|i = 1)Pr_{j} (i^{\prime} = 3|i = 1,i^{\prime\prime} = 2) + Pr_{j} (i^{\prime} = 1)Pr_{j} (i = 2|i^{\prime} = 1)Pr_{j} (i^{\prime\prime} = 3|i^{\prime} = 1,i = 2) + Pr_{j} (i^{\prime} = 1)P(i^{\prime\prime} = 2|i^{\prime} = 1)Pr_{j} (i = 3|i^{\prime} = 1,i^{\prime\prime} = 2) + Pr_{j} (i^{\prime\prime} = 1)Pr_{j} (i = 2|i^{\prime\prime} = 1)Pr_{j} (i^{\prime} = 3|i^{\prime\prime} = 1,i = 2) + Pr_{j} (i^{\prime\prime} = 1)Pr_{j} (i^{\prime} = 2|i^{\prime\prime} = 1)Pr_{j} (i = 3|i^{\prime\prime} = 1,i^{\prime} = 2))^{{Y_{i,j} Y_{{i^{\prime},j}} Y_{{i^{\prime\prime},j}} }} . $$

We solve for our parameter estimates $\hat{\delta },\hat{\lambda },\hat{\gamma },\;{\text{and}}\;\hat{\beta }$ by finding the values that maximize $\prod\nolimits_{j = 1}^{T} {L\left( {\delta ,\lambda ,\gamma ,\beta |Y_{j} } \right)}$.

Appendix 2: Constructing Counterfactuals

We illustrate how the Maximum Likelihood estimates are used to construct implied treatment effects in races with one winner and three candidates. Let C_p,p′,j be the counterfactual probability that the candidate who is actually listed in position p wins office in race j were he listed in position p’. Given the observed incumbency status of the three candidates, Inc_1,j, Inc_2,j, and Inc_3,j and race-level covariates, X _j , the counterfactual probability of winning of the candidate actually listed in position 1 in race j where he listed in position 1 is ${\text{C}}_{{ 1, 1,{\text{j}}}} = \frac{{{ \exp }(\widehat{{{\updelta}}}_{{1,{\text{t}}\left( {\text{j}} \right)}} + {\text{Inc}}_{{1,{\text{j}}}} (\widehat{{{\uplambda}}}_{{{\text{t}}\left( {\text{j}} \right)}} + \widehat{{{\upgamma}}}_{{1,{\text{t}}\left( {\text{j}} \right)}} ) + {\text{X}}_{\text{j}} \widehat{{{\upbeta}}}_{{1,{\text{t}}\left( {\text{j}} \right)}} )}}{{\exp \left( {\widehat{{{\updelta}}}_{{1,{\text{t}}\left( {\text{j}} \right)}} + {\text{Inc}}_{{1,{\text{j}}}} \left( {\widehat{{{\uplambda}}}_{{{\text{t}}\left( {\text{j}} \right)}} + \widehat{{{\upgamma}}}_{{1,{\text{t}}\left( {\text{j}} \right)}} } \right) + {\text{X}}_{\text{j}} \widehat{{{\upbeta}}}_{{1,{\text{t}}\left( {\text{j}} \right)}} } \right) + \exp \left( {\widehat{{{\updelta}}}_{{2,{\text{t}}\left( {\text{j}} \right)}} + {\text{Inc}}_{{2,{\text{j}}}} \left( {\widehat{{{\uplambda}}}_{{{\text{t}}\left( {\text{j}} \right)}} + \widehat{{{\upgamma}}}_{{2,{\text{t}}\left( {\text{j}} \right)}} } \right) + {\text{X}}_{\text{j}} \widehat{{{\upbeta}}}_{{2,{\text{t}}\left( {\text{j}} \right)}} } \right) + \exp \left( {{\text{Inc}}_{{3,{\text{j}}}} \widehat{{{\uplambda}}}_{{{\text{t}}\left( {\text{j}} \right)}} } \right)}}$.

If we rotate the candidate actually listed first to the second position, the candidate actually listed second to the third position, and the candidate actually listed third to the first position, then ${\text{C}}_{{ 1, 2,{\text{j}}}} = \frac{{{ \exp }(\widehat{{{\updelta}}}_{{2,{\text{t}}\left( {\text{j}} \right)}} + {\text{Inc}}_{{1,{\text{j}}}} (\widehat{{{\uplambda}}}_{{{\text{t}}\left( {\text{j}} \right)}} + \widehat{{{\upgamma}}}_{{2,{\text{t}}\left( {\text{j}} \right)}} ) + {\text{X}}_{\text{j}} \widehat{{{\upbeta}}}_{{2,{\text{t}}\left( {\text{j}} \right)}} )}}{{\exp \left( {\widehat{{{\updelta}}}_{{1,{\text{t}}\left( {\text{j}} \right)}} + {\text{Inc}}_{{3,{\text{j}}}} \left( {\widehat{{{\uplambda}}}_{{{\text{t}}\left( {\text{j}} \right)}} + \widehat{{{\upgamma}}}_{{1,{\text{t}}\left( {\text{j}} \right)}} } \right) + {\text{X}}_{\text{j}} \widehat{{{\upbeta}}}_{{1,{\text{t}}\left( {\text{j}} \right)}} } \right) + \exp \left( {\widehat{{{\updelta}}}_{{2,{\text{t}}\left( {\text{j}} \right)}} + {\text{Inc}}_{{1,{\text{j}}}} \left( {\widehat{{{\uplambda}}}_{{{\text{t}}\left( {\text{j}} \right)}} + \widehat{{{\upgamma}}}_{{2,{\text{t}}\left( {\text{j}} \right)}} } \right) + {\text{X}}_{\text{j}} \widehat{{{\upbeta}}}_{{2,{\text{t}}\left( {\text{j}} \right)}} } \right) + \exp \left( {{\text{Inc}}_{{2,{\text{j}}}} \widehat{{{\uplambda}}}_{{{\text{t}}\left( {\text{j}} \right)}} } \right)}}.$

Finally, if we rotate the candidates so that the candidate actually listed first is listed third, the candidate actually listed second is listed first, and the candidate actually listed third is listed second, then ${\text{C}}_{{ 1, 3,{\text{j}}}} = \frac{{{ \exp }({\text{Inc}}_{{1,{\text{j}}}} \widehat{{{\uplambda}}}_{{{\text{t}}\left( {\text{j}} \right)}} )}}{{\exp \left( {\widehat{{{\updelta}}}_{{1,{\text{t}}\left( {\text{j}} \right)}} + {\text{Inc}}_{{2,{\text{j}}}} \left( {\widehat{{{\uplambda}}}_{{{\text{t}}\left( {\text{j}} \right)}} + \widehat{{{\upgamma}}}_{{1,{\text{t}}\left( {\text{j}} \right)}} } \right) + {\text{X}}_{\text{j}} \widehat{{{\upbeta}}}_{{1,{\text{t}}\left( {\text{j}} \right)}} } \right) + \exp \left( {\widehat{{{\updelta}}}_{{2,{\text{t}}\left( {\text{j}} \right)}} + {\text{Inc}}_{{3,{\text{j}}}} \left( {\widehat{{{\uplambda}}}_{{{\text{t}}\left( {\text{j}} \right)}} + \widehat{{{\upgamma}}}_{{2,{\text{t}}\left( {\text{j}} \right)}} } \right) + {\text{X}}_{\text{j}} \widehat{{{\upbeta}}}_{{2,{\text{t}}\left( {\text{j}} \right)}} } \right) + \exp \left( {{\text{Inc}}_{{1,{\text{j}}}} \widehat{{{\uplambda}}}_{{{\text{t}}\left( {\text{j}} \right)}} } \right)}}.$

Similar counterfactual probabilities can be constructed for the candidates actually listed in position 2 and position 3 in race j. The estimated treatment effect of being listed first for the candidate actually listed in position p in race j is defined as ${\text{C}}_{{{\text{p}}, 1,{\text{j}}}} - ({\text{C}}_{{{\text{p}}, 1,{\text{j}}}} + {\text{C}}_{{{\text{p}}, 2,{\text{j}}}} + {\text{C}}_{{{\text{p}}, 3,{\text{j}}}} )/ 3$.

Appendix 3: Satisficing with a Single Vote

Consider elections with N candidates in which the qualities X₁, …, X _N of the candidates located in positions 1, …, N are random variables distributed i.i.d. with c.d.f. F(). A representative satisficing voter evaluates the candidates according to the ballot order and chooses the first candidate with quality above his aspiration threshold C. If no candidate is above C, then the voter selects the highest-quality candidate. The probability of choosing the first candidate in this model is:

$$ {\text{P(1st)}} = {\text{P}}({\text{X}}_{ 1} \ge {\text{C}}) + {\text{P}}({\text{X}}_{ 1} < {\text{C}},{\text{X}}_{ 1} > {\text{X}}_{ 2} , \ldots ,{\text{X}}_{\text{N}} ) = ( 1- {\text{F}}({\text{C}})) + ( 1/{\text{N}}){\text{F}}({\text{C}})^{\text{N}} . $$

Similarly, the probabilities of selecting the second and third candidates are:

$$ \begin{gathered} {\text{P}}(2{\text{nd}}) = {\text{P}}({\text{X}}_{ 2} \ge {\text{C}},{\text{X}}_{ 1} < {\text{C}}) + {\text{P}}({\text{X}}_{ 2} < {\text{C}},{\text{X}}_{ 2} > {\text{X}}_{ 1} , \ldots ,{\text{X}}_{\text{N}} ) = {\text{F}}({\text{C}})( 1-{\text{F}}({\text{C}})) + ( 1/{\text{N}}){\text{F}}({\text{C}})^{\text{N}} \hfill \\ {\text{P}}(3{\text{rd}}) = {\text{P}}({\text{X}}_{ 3} \ge {\text{C}},{\text{X}}_{ 1} < {\text{C}},{\text{X}}_{ 2} < {\text{C}}) + {\text{P}}({\text{X}}_{ 3} < {\text{C}},{\text{X}}_{ 3} > {\text{X}}_{ 1} , \ldots ,{\text{X}}_{\text{n}} ) = {\text{F}}({\text{C}})^{ 2} ( 1-{\text{F}}({\text{C}})) + ( 1/{\text{N}}){\text{F}}({\text{C}})^{\text{N}} . \hfill \\ \end{gathered} $$

This implies that the expected difference in the number of winners from the first and second ballot positions is P(1st) − P(2nd) = (1 − F(C))², while the expected difference in the number of winners from the second and third ballot positions is P(2nd) − P(3rd) = F(C)(1 − F(C))².

Figure 3 indicates that P(1st) − P(2nd) = 0.069 in elections with two or more winners and losers, and thus F(C) is about 0.74. This implies that the expected difference in the number of winners between the second and third ballot positions should be about three quarters of the difference between the first and second positions. As Fig. 3 indicates, however, the actual difference between the second and third is only one-fifth of the difference between the first and second candidates. Thus, the difference between the first and second candidates is larger than expected even if voters are satisificing with a single vote.