Information disclosure in elections with sequential costly participation

Abstract

Using a pivotal costly voting model of elections in which voters privately have formed preferences over two candidates and act sequentially, I study how different rules for disclosing information about the actions of early voters affect the actions of later voters, and how they ultimately affect voters’ and candidates’ welfare. Comparing three rules observed in real-life elections (no information disclosure, turnout disclosure and vote count disclosure), I find that vote count disclosure dominates the other two rules in terms of both voter welfare and the ex-ante likelihood of electing the candidate preferred by the majority. I show further that each of the rules can provide a candidate with either a greater or lesser chance of winning, depending on the levels of ex-ante support for the candidates. The findings may be useful for designing optimal voting procedures, particularly in settings with small numbers of voters.

Appendix

1.1 A. Candidate welfare

Candidate A wins when (1) both voters are A-supporters except in the case when none of them votes and the coin flip favors B; (2) both voters are B-supporters, neither of them votes and the coin flip favors A; (3) one voter supports A, one voter supports B, and only the A-supporter votes; (4) one voter supports A, one voter supports B, either both or neither of them votes, and the coin flip favors A.

No disclosure:

$$\begin{aligned} P(W_A)^N= & {} \alpha ^2\left( 1-\frac{1}{2}\left( 1-p_A^1\right) \left( 1-p_A^2\right) \right) + \frac{1}{2}\left( 1-\alpha \right) ^2\left( 1-p_B^1\right) \left( 1-p_B^2\right) + \nonumber \\&+ \alpha (1-\alpha )\left( p_A^1(1-p_B^2)+(1-p_B^1)p_A^2)\right) + \nonumber \\&+ \frac{1}{2}\alpha (1-\alpha )\left( \left( 1-p_A^1\right) \left( 1-p_B^2\right) + \left( 1-p_B^1\right) \left( 1-p_A^2\right) + p_A^1p_B^2+p_B^1p_A^2 \right) , \end{aligned}$$

(A.1)

where $p_T^t$, $T\in \{A,B\}$, $t\in \{1,2\}$, are the equilibrium participation probabilities (2). Simplifying the expression yields:

$$\begin{aligned} P(W_A)^N=\frac{1}{2}+\frac{3}{2}\frac{2\alpha -1}{(2-\alpha )^2(1+\alpha )^2}. \end{aligned}$$

(A.2)

Partial disclosure:

$$\begin{aligned} P(W_A)^P= & {} \alpha ^2\left( 1-\frac{1}{2}\left( 1-p_A^1\right) \left( 1-p_A^2(0)\right) \right) + \frac{1}{2}\left( 1-\alpha \right) ^2\left( 1-p_B^1\right) \left( 1-p_B^2(0)\right) + \nonumber \\&+ \alpha (1-\alpha )\left( p_A^1(1-p_B^2(1))+(1-p_B^1)p_A^2(0))\right) + \nonumber \\&+ \frac{1}{2}\alpha (1-\alpha )\left( \left( 1-p_A^1\right) \left( 1-p_B^2(0)\right) + \left( 1-p_B^1\right) \left( 1-p_A^2(0)\right) + p_A^1p_B^2(1)+p_B^1p_A^2(1)\right) . \end{aligned}$$

(A.3)

Plugging in the equilibrium strategies (6) yields:

$$\begin{aligned} P(W_A)^P=\alpha +\frac{1}{2}\alpha (1-\alpha )\left( (2-\gamma )p_A^1-(1+\gamma )p_B^1 \right) , \end{aligned}$$

(A.4)

where $\gamma$ is given by formula (7), and $p_A^1$ and $p_B^1$ are the equilibrium strategies (5).

Full disclosure:

$$\begin{aligned} P(W_A)^F= & {} \alpha ^2\left( 1-\frac{1}{2}\left( 1-p_A^1\right) \left( 1-p_A^2(0)\right) \right) + \frac{1}{2}\left( 1-\alpha \right) ^2\left( 1-p_B^1\right) \left( 1-p_B^2(0)\right) + \nonumber \\&+ \alpha (1-\alpha )\left( p_A^1(1-p_B^2(+))+(1-p_B^1)p_A^2(0))\right) + \nonumber \\&+ \frac{1}{2}\alpha (1-\alpha )\left( \left( 1-p_A^1\right) \left( 1-p_B^2(0)\right) + \left( 1-p_B^1\right) \left( 1-p_A^2(0)\right) + p_A^1p_B^2(+)+p_B^1p_A^2(-) \right) . \end{aligned}$$

(A.5)

Re-arranging terms:

$$\begin{aligned} P(W_A)^F=\alpha ^2 + \alpha (1-\alpha )(1-p_B^1) + \frac{1}{2}\alpha (1-\alpha )(p_A^1p_B^2(+)+p_B^1p_A^2(-)). \end{aligned}$$

(A.6)

Plugging in the equilibrium participation probabilities (8) yields:

$$\begin{aligned} P(W_A)^F = \frac{1}{2}\alpha \left( 1+\alpha +2(1-\alpha )^2\right) . \end{aligned}$$

(A.7)

1.2 B. Effect of $\alpha$ on candidate welfare

Consider candidate welfare under an N-regime as a function of $\alpha$ and participation probabilities, given by formula (A.1). Differentiating formula (A.1) with respect to $\alpha$, plugging in the equilibrium values for $p_A^1$, $p_A^2$, $p_B^1$ and $p_B^2$ given by formulas (2), and evaluating the resulting expression at $\alpha =\frac{1}{2}$, one may obtain the “support” effect:

$$\begin{aligned} \left. \frac{\partial P(W_A)^N}{\partial \alpha }\right| _{p_A=\frac{1}{1+\alpha }, p_B=\frac{1}{2-\alpha }, \alpha =\frac{1}{2}} = \frac{8}{9}. \end{aligned}$$

(A.8)

Plugging the equilibrium values for $p_A^1$, $p_A^2$, $p_B^1$ and $p_B^2$ into expression (A.1) first (the result is given by expression (A.2)), then differentiating it with respect to $\alpha$, and then evaluating the resulting expression at $\alpha =\frac{1}{2}$, one may obtain the total effect, which is the “support” effect plus the “participation” effect:

$$\begin{aligned} \left. \frac{\partial P(W_A)^N}{\partial \alpha }\right| _{\alpha =\frac{1}{2}} = \left. \frac{\partial \left( \frac{1}{2}+\frac{3}{2}\frac{2\alpha -1}{(2-\alpha )^2(1+\alpha )^2}\right) }{\partial \alpha }\right| _{\alpha =\frac{1}{2}} = \frac{16}{27}. \end{aligned}$$

(A.9)

The difference between expressions (A.9) and (A.8) is the “participation” effect: $\frac{16}{27}-\frac{8}{9}=-\frac{8}{27}.$

To evaluate the effect of $\alpha$ on candidate welfare under a P-regime at $\alpha =0.5$, one may plug $\gamma =0.5$ into expression (A.4) to obtain

$$\begin{aligned} P(W_A)^P=\alpha + \frac{3}{4}\alpha \left( 1-\alpha \right) \left( p_A^1-p_B^1\right) . \end{aligned}$$

(A.10)

Differentiating the expression above with respect to $\alpha$, entering the equilibrium values for $p_A^1$ and $p_B^1$ given by formulas (5), and evaluating the resulting expression at $\alpha =\frac{1}{2}$, one may obtain the “support” effect:

$$\begin{aligned} \left. \frac{\partial P(W_A)^P}{\partial \alpha }\right| _{p_A^1=\frac{3}{2}(1-\alpha ), p_B^1=\frac{3}{2}\alpha , \alpha =\frac{1}{2}} = 1. \end{aligned}$$

(A.11)

Plugging the equilibrium values for $p_A^1$ and $p_B^2$ into expression (A.10) first, differentiating it with respect to $\alpha$, and then evaluating the resulting expression at $\alpha =\frac{1}{2}$, one may obtain the total effect, which is the sum of the “support” effect and the “participation” effect:

$$\begin{aligned} \left. \frac{\partial P(W_A)^P}{\partial \alpha }\right| _{\alpha =\frac{1}{2}} = \left. \frac{\partial \left( \alpha +\frac{9}{8}\alpha (1-\alpha )(1-2\alpha ) \right) }{\partial \alpha }\right| _{\alpha =\frac{1}{2}} = \frac{7}{16}. \end{aligned}$$

(A.12)

The difference between expressions (A.12) and (A.11) is the “participation” effect: $\frac{7}{16}-1=-\frac{9}{16}.$

1.3 C. Voter welfare

If a voter abstains, his/her preferred candidate wins when: (1) the other voter supports the same candidate and votes; (2) the other voter abstains and the coin flip favors the preferred candidate. If a voter participates, his/her candidate always wins except when the other voter supports the opposing candidate, votes, and the coin flip favors the opposing candidate. Therefore, under no information:

$$\begin{aligned} v_A^1= & {} v_A^2=\alpha p_A+\frac{1}{2}\left( \alpha (1-p_A)+(1-\alpha )(1-p_B)\right) . \end{aligned}$$

(A.13)

$$\begin{aligned} u_A^1= & {} u_A^2=1-\frac{1}{2}\left( 1-\alpha \right) p_B. \end{aligned}$$

(A.14)

Substituting those expressions into formula (13), plugging in equilibrium values for $p_A$ and $p_B$, and re-arranging the terms, one may obtain:

$$\begin{aligned} E[W_A^1]=E[W_A^2]=\frac{1}{4}\left( 1 + \frac{\alpha ^2}{(1+\alpha )^2} + \frac{2}{2-\alpha }\right) . \end{aligned}$$

(A.15)

The expression for $E[W_B]$ can be obtained by substituting ($1-\alpha$) for $\alpha$ in the just-stated formula:

$$\begin{aligned} E[W_B^1]=E[W_B^2]=\frac{1}{4}\left( 1 + \left( \frac{1-\alpha }{2-\alpha }\right) ^2 + \frac{2}{1+\alpha }\right) . \end{aligned}$$

(A.16)

Under partial information disclosure, a second-period voter obtains utility 1 minus expected cost 0.25 if he or she observes no participation in the first period regardless of the candidate he or she supports, i.e., $EW_A^2(0)=EW_B^2(0)=EW^2(0)=0.75$. If an A-supporter observes that the first-period voter participated, his/her expected benefit is $v^2_A(1)=\gamma$ by abstaining; and it is $u^2_A(1)=\gamma +0.5(1-\gamma )$ if he or she votes. Plugging those values together with equilibrium participation $p^2_A(1)=1-\gamma$ into formula (13), expected utility is:

$$\begin{aligned} E[W_A^2](1)=\gamma + \frac{1}{2}(1-\gamma )^2-\frac{1}{4}(1-\gamma )^2=\gamma +\frac{1}{4}(1-\gamma )^2. \end{aligned}$$

(A.17)

Likewise:

$$\begin{aligned} E[W_B^2](1)=1-\gamma +\frac{1}{4}\gamma ^2. \end{aligned}$$

(A.18)

Given the first-round equilibrium participation probabilities (5), the ex-ante probability of observing positive participation after the first period is

$$\begin{aligned} P(1)={\left\{ \begin{array}{ll} 3\alpha (1-\alpha ), \alpha \in [\frac{1}{2},\frac{2}{3}); \\ 1-\alpha +\alpha (1-\alpha )(2-\gamma ), \alpha \in [\frac{2}{3},1]. \end{array}\right. } \end{aligned}$$

(A.19)

For $\alpha \in [1/2,2/3]$,

$$\begin{aligned} v_A^1= & {} \alpha p_A^2(0)+\frac{1}{2}\left( \alpha (1-p_A^2(0))+(1-\alpha )(1-p_B^2(0))\right) =\alpha . \end{aligned}$$

(A.20)

$$\begin{aligned} u_A^1= & {} 1-\frac{1}{2}\left( 1-\alpha \right) p_B^2(1)=\frac{3}{4}-\frac{1}{4}\alpha . \end{aligned}$$

(A.21)

Plugging the above expressions into formula (13) yields:

$$\begin{aligned} E[W_A^1]=\alpha +\frac{9}{16}(1-\alpha )^2. \end{aligned}$$

(A.22)

Likewise,

$$\begin{aligned} E[W_B^1]=1-\alpha +\frac{9}{16}\alpha ^2. \end{aligned}$$

(A.23)

Therefore, expected voter welfare under partial information disclosure is

$$\begin{aligned} E[W]^P= & {} \frac{1}{2}\left( \alpha E[W_A^1]+(1-\alpha )EW[_B^1]+E[W^2](0)\left( 1-3\alpha (1-\alpha )\right) +E[W^2](1)3\alpha (1-\alpha )\right) =\nonumber \\= & {} \frac{7}{8}-\alpha (1-\alpha ). \end{aligned}$$

(A.24)

Following the same logic, one may derive welfare for $\alpha >2/3$:

$$\begin{aligned} E[W]^P= & {} \frac{1-\alpha }{2}\left( \frac{3}{4}-\frac{\alpha \gamma }{2}\right) +\frac{\alpha }{2} \left( \alpha +\frac{(1-\alpha )^2(1-\gamma ^2)}{2\alpha \gamma }-\frac{(1-\alpha )^2(1-\gamma )^2}{4\alpha ^2\gamma ^2} \right) +\frac{3}{4}\left( 1-\frac{(1-\alpha )}{\gamma }\right) + \nonumber \\&+ \frac{(1-\alpha )}{\gamma }\left( (1-\alpha )\left( \gamma +\frac{(1-\gamma )^2}{4}\right) + \alpha \left( 1-\gamma +\frac{\gamma ^2}{4}\right) \right) , \end{aligned}$$

(A.25)

where $\gamma$ is the posterior probability that candidate A will lead after the first period conditional on the observed turnout being 1 given by formula (7).

Under full information, the first period’s A supporter’s expected utility is $\alpha$ if he or she abstains, and $1-0.25\alpha (1-\alpha )$ if he or she participates, since in the latter case A loses only when the second-period voter supports B, participates and the coin flip favors B. Then, from formula (13):

$$\begin{aligned} E[W_A^1]= & {} \alpha +\frac{1}{2}(1-\alpha )^2-\frac{1}{4}(1-\alpha )^2=\alpha +\frac{1}{4}(1-\alpha )^2. \end{aligned}$$

(A.26)

$$\begin{aligned} E[W_B^1]= & {} 1-\alpha +\frac{1}{4}\alpha ^2. \end{aligned}$$

(A.27)

If a second period’s A-supporter observes abstention, his/her expected utility is 1 minus the expected cost 0.25. If he or she observes A leading after the first period, his/her expected utility is 1. If he or she observes A losing, his/her expected utility is 0.25. Weighting those utilities by the ex-ante probabilities of observing each possible outcome of the first period, one may obtain:

$$\begin{aligned} E[W_A^2]=\frac{3}{4}(\alpha ^2+(1-\alpha )^2)+\alpha (1-\alpha )+\frac{1}{4}\alpha (1-\alpha )=\frac{3}{4}-\frac{1}{4}\alpha (1-\alpha ). \end{aligned}$$

(A.28)

The expression for $E[W_B^2]$ is exactly the same.

Expected voter welfare thus is

$$\begin{aligned} E[W]^F=\frac{1}{2}\left( \alpha \left( \alpha +\frac{1}{4}(1-\alpha )^2\right) +(1-\alpha )\left( 1-\alpha +\frac{1}{4}\alpha ^2\right) + \frac{3}{4}-\frac{1}{4}\alpha (1-\alpha )\right) =\frac{7}{8}-\alpha (1-\alpha ). \end{aligned}$$

(A.29)