Learning, proximity and voting: theory and empirical evidence from nuclear referenda

Abstract

This paper presents novel evidence on the pattern of voting in referenda and develops a spatial learning model that helps explain such behavior. In particular, we shed light on the determinants of voters’ choices over nuclear power using data on two Italian referenda. Exploiting the panel structure of the data, we document that voting against nuclear power increases, whenever the distance from the closest nuclear plant decreases. However, we detect a different voting behavior between municipalities close to existing reactors and those close to proposed ones. A possible explanation is that many citizens hold more precise information on nuclear safety because they have experienced the presence of a reactor in their vicinity for many years. Therefore, we propose a model of voting with endogenous information acquisition interacting both proximity and learning effects, whose results are compatible with the empirical findings. Citizens receive public and private signals and revise their beliefs on the risk of living close to a plant. Such revision process is nested into a spatial voting model establishing conditions for a similar or different voting behavior of the electorate based on the proximity from the reactor.

Appendices

Appendix A: Proof of Lemma 1

Let us start from Eq. (9) and for simplicity make a distinction between a hypothetical voter i that lives, respectively, close to an existing reactor and a proposed one, ${\upgamma } \in \{o,n\}$. It follows that,

$$\begin{aligned} \Delta U_{i}^{n}={\uppsi } \left( {\upupsilon } _{i}^{n*}-{\upvarphi } \right) +{\upkappa } ^{n}{\tilde{{\upxi }}}_{i}\text { and }\Delta U_{i}^{o}={\uppsi } \left( {\upupsilon } _{i}^{o*}-{\upvarphi } \right) +{\upkappa } ^{o}{\tilde{{\upxi }}}_{i} \end{aligned}$$

(A.1)

Starting by Eq. (7), we can refer to ${\upupsilon } _{i}^{n*}(x_{i}^{n},y)=$${\upmu } ^{n}x_{i}^{n}+(1-{\upmu } ^{n})y$ and ${\upupsilon } _{i}^{o*}(x_{i}^{o},y)={\upmu } ^{o}x_{i}^{o}+(1-{\upmu } ^{o})y$ as the posterior beliefs of voter i close to an existing reactor (o) and a proposed one (n), with ${\upmu } ^{n}=\frac{{\upkappa } ^{n}{\uptau } _{{\upgamma } }^{*}}{{\uptau } ^{\prime }+{\upkappa } ^{n}{\uptau } _{{\upgamma } }^{*}}$ and ${\upmu } ^{o}=\frac{{\upkappa } ^{o}{\uptau } _{{\upgamma } }^{*}}{{\uptau } ^{\prime }+{\upkappa } ^{o}{\uptau } _{{\upgamma } }^{*}}$. We can define $\Delta {\tilde{U}}_{i}$$=\Delta U_{i}^{n}-\Delta U_{i}^{o}$ such that,

$$\begin{aligned} \Delta {\tilde{U}}_{i}={\uppsi } \left( {\upupsilon } _{i}^{n*}(x_{i}^{n},y)-{\upupsilon } _{i}^{n*}(x_{i}^{n},y)\right) +\left( {\upkappa } ^{n}-{\upkappa } ^{o}\right) {\tilde{{\upxi }}}_{i} \end{aligned}$$

(A.2)

We may explicitly derive Eq. (A.2) as a function of the precisions of the signals ${\uptau } _{{\upgamma } }^{*}$ and ${\uptau } ^{\prime }$. By symmetry, i.e., ${\upkappa } ^{n}={\upkappa } ^{o}={\upkappa } ^{*}$, and taking the limit for ${\upkappa } ^{*}\rightarrow \infty$, it follows that,

$$\begin{aligned} \lim _{{\upkappa } ^{*}\rightarrow \infty }\text { }\Delta {\tilde{U}}_{i}=0 \end{aligned}$$

(A.3)

Appendix B: Proof of proposition 1

Starting from a hypothetical voter i that lives, respectively, close to an existing reactor and a proposed one, ${\upgamma } \in \{o,n\}$, we can define the difference in outcomes conditional on the distance from the plant as $\Delta {\tilde{U}}_{i}|_{{\upkappa } }$ and we observe the effect of varying the distance, ${\upkappa } ^{{\upgamma } }$, for ${\upkappa } ^{n}={\upkappa } ^{o}=$${\upkappa } ^{*}$,

$$\begin{aligned} \Delta {\tilde{U}}_{i}|_{{\upkappa } ^{*}}>\Delta {\tilde{U}}_{i}|_{{\upkappa } ^{*}+1} \end{aligned}$$

(B.1)

such that,

$$\begin{aligned} \left[ {\upupsilon } _{i}^{n*}(x_{i}^{n},y)-{\upupsilon } _{i}^{n*}(x_{i}^{n},y)\right] _{{\upkappa } ^{*}}>\left[ {\upupsilon } _{i}^{o*}(x_{i}^{o},y)-{\upupsilon } _{i}^{o*}(x_{i}^{o},y)\right] _{{\upkappa } ^{*}+1} \end{aligned}$$

(B.2)

Rearranging terms of Eq. (B.2), it follows that,

$$\begin{aligned} \frac{({\uptau } ^{\prime }+{\upkappa } ^{*}{\uptau } _{o}^{*})\left( y{\uptau } ^{\prime }+{\upkappa } ^{*}x_{i}^{n}{\uptau } _{n}^{*}\right) +\left( {\upkappa } ^{*}x_{i}^{o}-y\right) ({\uptau } ^{\prime }+{\upkappa } ^{*}{\uptau } _{n}^{*}) }{{\upkappa } ^{*}({\upkappa } ^{*}+1)({\uptau } ^{\prime }+{\upkappa } ^{*}{\uptau } _{n}^{*})({\uptau } ^{\prime }+{\upkappa } ^{*}{\uptau } _{o}^{*})}\lessgtr 0 \end{aligned}$$

(B.3)

The sign of Eq. (B.3) is of course uncertain although a sufficient condition to have a positive value, $\Delta {\tilde{U}} _{i}|_{{\upkappa } ^{*}}-\Delta {\tilde{U}}_{i}|_{{\upkappa } ^{*}+1}$$>0$ is that $x_{i}^{o}>y/{\upkappa } ^{*}$.

Appendix C: Proof of corollary 1

Taking the limit of the difference of Eq. (B.1) for ${\upkappa } ^{*}\rightarrow 0$, it follows that

$$\begin{aligned} \lim _{{\upkappa } ^{*}\rightarrow 0}\text { }\left[ \Delta {\tilde{U}} _{i}|_{{\upkappa } ^{*}}-\Delta {\tilde{U}}_{i}|_{{\upkappa } ^{*}+1}\right] = \frac{1}{{\uptau } ^{\prime }}\left[ (x_{i}^{n}{\uptau } _{n}^{*}-x_{i}^{o}{\uptau } _{o}^{*})+y\left( {\uptau } _{o}^{*}-{\uptau } _{n}^{*}\right) \right] \end{aligned}$$

(C.1)

thus rewriting Eq. (C.1), the difference in voting outcomes of a hypothetical voter i in the proximity of an existing and a proposed reactors appears as,

$$\begin{aligned} \Delta {\tilde{U}}_{i,{\upkappa } ^{*}\rightarrow 0}=\frac{1}{{\uptau } ^{\prime }} \left[ (x_{i}^{n}{\uptau } _{n}^{*}-x_{i}^{o}{\uptau } _{o}^{*})-y\left( {\uptau } _{n}^{*}-{\uptau } _{o}^{*}\right) \right] \end{aligned}$$

(C.2)

which suggests that the difference in outcomes at zero distance from the reactor exists when,

$$\begin{aligned} x_{i}^{o}>{\tilde{{\uptau }}}^{*}(x_{i}^{n}-y)+y \end{aligned}$$

(C.3)

where ${\tilde{{\uptau }}}^{*}={\uptau } _{o}^{*}/{\uptau } _{n}^{*}$.

Appendix D: Proof of Proposition 2

The proof is divided in two steps. In the first step, we investigate the existence and the uniqueness of a positive real root, i.e., ${\upkappa } ^{*}>0$ , that satisfies the first- and second- order conditions for a minimum. Then, in the second step, we search for an optimal solution in a closed form as the precisions of the private signals in ${\upgamma } \in \{o,n\}$ are equal, ${\uptau } _{n}^{*}={\uptau } _{o}^{*}={\uptau } ^{*}$.

Differentiating $\Delta {\tilde{U}}_{i}$ with respect to ${\upkappa } ^{*}$, it follows that,

$$\begin{aligned} \Delta {\tilde{U}}_{i}^{\prime }({\upkappa } ^{*})= & {} \frac{x_{i}^{n}{\uptau } _{n}^{*}\left( {\uptau } ^{\prime }+{\uptau } _{n}^{*}+2{\upkappa } ^{*}{\uptau } _{n}^{*}\right) \left( {\uptau } ^{\prime }+{\upkappa } ^{*}{\uptau } _{o}^{*}\right) ^{2}-x_{i}^{o}{\uptau } _{o}^{*}\left( {\uptau } ^{\prime }+{\uptau } _{o}^{*}+2{\upkappa } ^{*}{\uptau } _{o}^{*}\right) \left( {\uptau } ^{\prime }+{\upkappa } ^{*}{\uptau } _{n}^{*}\right) ^{2}}{({\upkappa } ^{*}+1)^{2}({\uptau } ^{\prime }+{\upkappa } ^{*}{\uptau } _{n}^{*})^{2}({\uptau } ^{\prime }+{\upkappa } ^{*}{\uptau } _{o}^{*})^{2}} \nonumber \\&-\frac{y{\uptau } ^{\prime }\left( {\uptau } _{n}^{*}-{\uptau } _{o}^{*}\right) \left( {\uptau } ^{\prime }+{\upkappa } ^{*}+(2+3{\upkappa } ){\uptau } _{n}^{*}{\uptau } _{o}^{*}\right) +(1-2{\upkappa } ^{*}){\uptau } ^{\prime }\left( {\uptau } _{n}^{*}+{\uptau } _{o}^{*}\right) ^{2}}{({\upkappa } ^{*}+1)^{2}({\uptau } ^{\prime }+{\upkappa } ^{*}{\uptau } _{n}^{*})^{2}({\uptau } ^{\prime }+{\upkappa } ^{*}{\uptau } _{o}^{*})^{2}} \end{aligned}$$

(D.1)

We can search for a cubic equation in ${\upkappa } ^{*}$ such that $\Delta {\tilde{U}}_{i}^{\prime }({\upkappa } ^{*})\le 0$. Applying Descartes’ rule, we notice that there is only one sign change in $\Delta {\tilde{U}} _{i}^{\prime }({\upkappa } ^{*})$ and a positive value root exists such that $\Delta {\tilde{U}}_{i}^{\prime }({\upkappa } ^{*})=0$. Indeed, $\Delta {\tilde{U}} _{i}^{\prime }(0)={\uptau } ^{\prime }(x_{i}^{n}{\uptau } _{n}^{*}({\uptau } ^{\prime }+{\uptau } _{n}^{*})-x_{i}^{o}{\uptau } _{o}^{*}({\uptau } ^{\prime }+{\uptau } _{o}^{*})-y({\uptau } ^{\prime }+{\uptau } _{n}^{*}+{\uptau } _{o}^{*})<0$ if eq. (C.3) is strictly satisfied and $\Delta {\tilde{U}} _{i}^{\prime }(\infty )={\uptau } _{n}^{2*}{\uptau } _{o}^{2*}>0$, thus the positive real root exists for ${\upkappa } ^{*}>0$.

Now let us assume that ${\uptau } _{n}^{*}={\uptau } _{o}^{*}={\uptau }$ for the sake of simplicity. Then the solution in closed form for $\Delta {\tilde{U}} _{i}^{\prime }({\upkappa } ^{*})\le 0$ is equal to:

$$\begin{aligned} {\upkappa } _{\min }=\frac{{\uptau } ^{\prime }+{\uptau } ^{*}}{2{\uptau } ^{*}} \end{aligned}$$

(D.2)

with $\Delta {\tilde{U}}_{i}^{\prime \prime }({\upkappa } _{\min })>0$. In this case the weight assigned to the public information y is zero as ${\hat{{\uptau }}} ^{*}$ collapses to zero when ${\uptau } _{n}^{*}={\uptau } _{o}^{*}={\uptau }$ and ${\tilde{{\uptau }}}^{*}=1$. Thus Eq. (D.2) shows the ${\upkappa } ^{*}-$value that satisfies the local minimum if and only if $\ x_{i}^{n}<x_{i}^{o}$ and suggests that the non monotonicity condition proposed in the main text is satisfied.

Appendix E: Tables and extra results

See Tables 3, 4, 5.

Table 3 Descriptive statistics of main variables

Table 4 Municipality level cross-sectional local analysis—old vs new

Table 5 Municipality level cross-sectional local analysis—old vs new

Appendix F: Graph and data

See Fig. 7.

Fig. 7

Turnout, the share of Yes votes over eligible voters and the share of Yes votes among the votes cast. a, b The correlations between Turnout and the share of Yes votes over eligible voters or the share of Yes votes among the votes cast. c Depicts the distribution of the share of Yes votes among the votes cast