Elections, the curse of competence and credence policies

Prope cottidie tibi hoc ad forum descendenti meditandum est: “Novus sum, consulatum peto, Roma est.”

Almost every day, when you go down to the forum, you should meditate: “I am a commoner. I want to become consul. This is Rome.”

(Q. Tullius Cicero, Commentarium Petitionis Consulatus)

Abstract

A returning political phenomenon is the impact of “office seekers” in democracies. We examine the consequences when the public faces a new type of two-dimensional uncertainty: whether a politician is competent, and whether a politician is concerned about the consequences of public utility decisions (statesman) or about public opinion (office seeker). We identify new timing distortions, as both competent and incompetent office seekers may take hasty or excessively delayed decisions in order to imitate statesmen or competent office holders. Thus, the public may benefit by disregarding the candidates’ competence and by reelecting candidates based solely on its belief as to whether a politician is a statesman. This “curse of competence” may explain why politicians are so concerned about being perceived as statesmen and why we may expect the largest distortions for credence policies.

Appendix

Proof of Proposition 1

There are three possibilities. An office seeker can prefer I over NI, prefer NI over I, or can be indifferent between I and NI. We begin with the last case. An office seeker is indifferent if and only if

$$\begin{aligned} \alpha g_{1} (I,p^{0}) + (1-\alpha )h_{1}(I,p^{0}) = \alpha g_{1} (NI,p^{0}) + (1-\alpha )h_{1}(NI,p^{0}). \end{aligned}$$

(15)

Inserting the expressions from Eqs. (5)–(8), and using the simplification

$$\begin{aligned} i:&= h_{0}g_{0}\pi _{H} + (1-h_{0})p^0,\\ 1-i&= h_{0}(1-g_{0}\pi _{H}) + (1-h_{0}) (1-p^0) \end{aligned} $$

yields

$$\begin{aligned} g_0h_0\pi _{\mathrm{H}}-ih_0=\alpha \left( i\left( g_0-h_0\right) -p^0g_0\left( 1-h_0\right) \right) . \end{aligned}$$

(16)

Rearranging terms and simplifying yields, for $\alpha <1$,

$$\begin{aligned} p^{0} = \frac{g_{0}\pi _{H}\left[ \alpha (h_{0}-g_{0}) + (1-h_{0}) \right] }{(1-\alpha )(1-h_{0}) }=g_{0}\pi _{H}\frac{ \alpha (1-g_{0}) + (1-\alpha )(1-h_{0})}{ (1-\alpha )(1-h_{0}) }=:p^*. \end{aligned}$$

(17)

In equilibrium, we must have $p^* = p^0$, where $p^*$ is the probability that the office seeker will choose I. Let us consider the boundary conditions for $p^*$.

From Eq. (17), we conclude that $p^*\ge g_0\pi _{\mathrm{H}}$ and thus $p^*$ is positive. The other boundary condition $p^*\le 1$ requires

$$\begin{aligned} \alpha \bigr [ g_{0}\pi _{H}(h_{0}-g_{0}) + (1-h_{0}) \bigl ] \le g_{0}\pi _{H}(h_{0}-1)+1-h_0, \end{aligned}$$

which implies

$$\begin{aligned} \alpha \le \frac{(1-h_0)(1-g_0\, \pi _{\mathrm{H}})}{(1-h_{0})+g_{0}\pi _{H}(h_{0}-g_{0})} =: \alpha ^{*}. \end{aligned}$$

Hence, for $\alpha \le \alpha ^{*}$ and $p^0 = p^*$ as derived in Eq. (17), an office seeker is indifferent between I and NI. Therefore, given $p^{*}$ and associated a posteriori beliefs of the public for $\alpha \le \alpha ^{*}$, a mixed strategy is optimal for an office seeker. This follows from the construction above. Moreover, one can easily verify that choosing the mixed strategy associated with $p^*$ is preferred over choosing $p^0=1$ or $p^0=0$. The case $p^0=0$ is studied below. The case $p^0=1$ is analogous. The rationality of the public’s beliefs requires that the office seeker plays I with probability $p^{*}$.

In the next step, we consider the case $\alpha >\alpha ^{*}$. According to Eq. (17), $p^{*}$ would become larger than 1. Straightforward calculations show that the office seekers are better off by selecting I over NI. Therefore, an office seeker will choose I for $\alpha > \alpha ^{*}$ with certainty.

In the last step, we observe that a pure strategy equilibrium does not exist when the office seeker selects NI. For the case of $\alpha > \alpha ^{*}$, this follows from the considerations in the last paragraph. When $\alpha \le \alpha ^{*}$, choosing NI with probability 1 would imply the following beliefs:

$$\begin{aligned} g_{1}(I,0)&= 1,\\ g_{1}(NI,0)&= \frac{h_{0}g_{0}(1-\pi _{H}) + (1-h_{0}) g_{0}}{h_{0}(1-\pi _{H}g_{0}) + (1-h_{0})},\\ h_{1}(I,0)&= 1,\\ h_{1}(NI,0)&= \frac{h_{0}(1-g_{0}\pi _{H})}{h_{0}(1-\pi _{H}g_{0}) + (1-h_{0})}. \end{aligned}$$

Hence,

$$\begin{aligned} \alpha g_{1}(I,0) + (1-\alpha ) h_{1} (I,0) = 1> \alpha g_{1} (NI,0) + (1-\alpha ) h_{1} (NI,0), \end{aligned}$$

which implies that NI is not the office seeker’s optimal choice. Given the beliefs associated with $p^{0} = 0$, it is in fact the worst choice. $\square$

Proof of Proposition 2

Suppose there is an equilibrium in which PG chooses I with probability $p_1 (0<p_1<1)$ and PB chooses I with probability $p_2 \ne p_1 (0<p_2<1)$.

Then, the posterior beliefs that a politician is competent upon choosing I and NI, respectively, are

$$\begin{aligned} g_1(I, p_1, p_2)&= \frac{h_0g_0\pi _{\mathrm{H}} + (1-h_0)g_0p_1}{h_0g_0\pi _{\mathrm{H}} + (1-h_0)(g_0p_1 + (1-g_0)p_2)},\\ g_1(NI, p_1, p_2)&= \frac{h_0g_0(1-\pi _{\mathrm{H}}) + (1-h_0)g_0(1-p_1)}{h_0(1-g_0\pi _{\mathrm{H}}) + (1-h_0)(g_0(1-p_1) + (1-g_0)(1-p_2))}. \end{aligned}$$

For $h_1(I, p_1, p_2)$ and $h_1(NI, p_1, p_2)$, only the joint probabilities $g_0p_1 + (1-g_0)p_2$ and $g_0(1-p_1) + (1-g_0)(1-p_2)$ matter, and the formulas (7) and (8) can be applied with $p^0$ and $1-p^0$ respectively, replaced by these joint probabilities.

Hence, for $\alpha >0$—and thus in the case when competence matters for voters—this can only be an equilibrium if

$$\begin{aligned}&\alpha p_1 g_1(I, p_1, p_2) + \alpha (1-p_1) g_1(NI, p_1, p_2)\\&\quad +(1-\alpha )p_1h_1(I, p_1, p_2) + (1-\alpha )(1-p_1)h_1(NI, p_1, p_2)\\&\quad = \alpha p_2 g_1(I, p_1, p_2) + \alpha (1-p_2) g_1(NI, p_1, p_2) \\&\quad +\,(1-\alpha )p_2h_1(I, p_1, p_2) + (1-\alpha )(1-p_2)h_1(NI, p_1, p_2). \end{aligned}$$

Otherwise, the competent (incompetent) office seeker could improve his reelection chances by choosing $p_2$ ($p_1)$. We write $A=h_0g_0\pi _{\mathrm{H}}+(1-h_0)(g_0p_1+(1-g_0)p_2)$ and $1-A=h_0(1-g_0\pi _{\mathrm{H}})+(1-h_0)(g_0(1-p_1)+(1-g_0)(1-p_2))$.

This yields

$$\begin{aligned}&(p_1-p_2)\Bigg (\frac{h_0g_0\pi _{H}+\alpha (1-h_0)g_0 p_1}{A}\\&\qquad -\frac{\alpha \left( h_0g_0(1-\pi _{\mathrm{H}})+(1-h_0)g_0(1-p_1)\right) +(1-\alpha )h_0(1-g_0\pi _{\mathrm{H}})}{1-A}\Bigg )=0. \end{aligned}$$

Assuming $\alpha >0$, this condition is satisfied if $p_1 = p_2$, i.e. both PG and PB choose I with the same probability. However, other equilibria are also possible. An example is the parameter constellation $h_0=g_0=\pi _{H}=\alpha =0.5$, which yields $p_2=0.5$ as a second solution. Note that it is independent of $p_1$.

For the case $\alpha =0$ the original condition can be reduced to

$$\begin{aligned} h_1(I, p_1, p_2) = h_1(NI, p_1, p_2). \end{aligned}$$

This implies

$$(p_1-p_2)\left[\frac{h_0g_0\pi_H}{A}-\frac{h_0(1-g_0\pi_H)}{1-A}\right]=0.$$

Solving for $p_2$ yields two possible solutions:

$$\begin{aligned} p_2&= p_1 \quad \text {and} \\ p_2&= \frac{g_0(\pi_H-p_1)}{1-g_0}.\end{aligned}$$

Now, we see that besides the obvious solution of $p_1=p_2$, we can express $p_2$ in terms of the parameters and $p_1$. This equation can be solved for $p_1$, $p_2 \in (0,1)$ for suitable parameter values, which means that there are other possible equilibria if $\alpha =0$. $\square$

Proof of Proposition 3

We first derive the expected welfare of the public for different choices of $\alpha$. We denote expected welfare as a function of the choice $p^*$ of office seekers by $W(p^*)$. $W(p^*)$ is the expected value of the project, given the choices of statesmen and office seekers.

If $\alpha \ge \alpha ^ *$, office seekers will play I with probability $p^*=1$ according to Proposition 1, and social welfare is given by

$$\begin{aligned} W(p^*=1) = h_0 \Bigl (\pi _{\mathrm{H}} \, V_{\mathrm{H}} \bigl (g_0 + \delta \, (1-g_0)\bigr )\Bigr ) + (1-h_0) \, \bigl (\pi _{\mathrm{H}}\,V_{\mathrm{H}} + (1-\pi _{\mathrm{H}}) \, V_{\mathrm{L}}\bigr ). \end{aligned}$$

Next, suppose office seekers play I with probability $p^*$, and $p^*\ge \pi _{\mathrm{H}}$. In this case, welfare is given by

$$\begin{aligned} W(p^*\mid p^*\ge \pi _{H})&= \Bigl (\pi _{\mathrm{H}} \, V_{\mathrm{H}} \bigl (g_0 + \delta \, (1-g_0)\bigr )\Bigr )h_0 + \Bigl [g_0 \bigl (\pi _{\mathrm{H}} \, V_{\mathrm{H}} + (1-\pi _{\mathrm{H}}) \frac{p^*- \pi _{\mathrm{H}}}{1-\pi _{\mathrm{H}}} \, V_{\mathrm{L}} \bigr )\\&\quad +\,(1-g_0) \bigl (p^*(\pi _{\mathrm{H}}\, V_{\mathrm{H}} + (1-\pi _{\mathrm{H}}) \, V_{\mathrm{L}}\bigl ) + (1-p^*)\, \delta \, \pi _{\mathrm{H}} \, V_{\mathrm{H}} \bigr )\Bigr ](1-h_0). \end{aligned} $$

For $p^*< \pi _{\mathrm{H}}$, we obtain

$$\begin{aligned} W(p^*\mid p^*< \pi _{H})&= \Bigl (\pi _{\mathrm{H}} \, V_{\mathrm{H}} \bigl (g_0 + \delta \, (1-g_0)\bigr )\Bigr )h_0+ \Bigl [g_0 \, \pi _{\mathrm{H}} \, V_{\mathrm{H}} \, \Bigl (\frac{p^*}{\pi _{\mathrm{H}}} + \frac{\pi _{\mathrm{H}} - p^*}{\pi _{\mathrm{H}}}\, \delta \Bigr )\\&\quad +\,(1-g_0)\Bigl (p^*\bigl (\pi _{\mathrm{H}} \, V_{\mathrm{H}} + (1-\pi _{\mathrm{H}}) \, V_{\mathrm{L}}\bigr ) + (1-p^*) \, \delta \, \pi _{\mathrm{H}} \, V_{\mathrm{H}} \Bigr )\Bigr ](1-h_0). \end{aligned}$$

We observe that

$$\begin{aligned} W(p^*\mid p^*\ge \pi _{H}) |_{p^{*} = \pi _{\mathrm{H}}}=\mathop{lim}_{p\!*\rightarrow\pi_H}\,W(p^*\mid p^*<\pi_H) =:W(p^*=\pi _{\mathrm{H}}). \end{aligned}$$

Note that $W(p^*\mid p^*\ge \pi _{H})$ is monotonically decreasing in $p^*$ as we assumed that $V_{\mathrm{H}}>0>V_{\mathrm{L}}$ and that (3) holds. Hence, we have

$$\begin{aligned} W(p^*\mid p^*> \pi _{H}) < W(p^*= \pi _{H}). \end{aligned}$$

Since $W(p^*\mid p^*< \pi _{H})$ is linear in $p^*$, the welfare-optimal mixing probabilities are either $p^*= \pi _{\mathrm{H}}$ or $p^*= g_0\, \pi _{\mathrm{H}}$. The probability $p^*= g_0 \pi _{\mathrm{H}}$ is the smallest possible value for $p^*$ occurring for $\alpha = 0$.

$W( p^*= \pi _{H}) < W(p^*= g_0 \pi _{\mathrm{H}})$ if and only if

$$\begin{aligned} g_0 \,\pi _{\mathrm{H}}\, V_{\mathrm{H}} \bigl [1-g_0 - (1-g_0) \, \delta \bigl ] + (1-g_0)^{2} \,\pi _{\mathrm{H}}\, \Bigl (\pi _{\mathrm{H}}\, V_{\mathrm{H}}\, (1-\delta ) + (1-\pi _{\mathrm{H}})\, V_{\mathrm{L}} \Bigr ) < 0 \end{aligned}$$

or equivalently, $K{:=}\, V_{\mathrm{H}} \, \bigl [g_0\, (1-\delta )+(1-g_0) \, \pi _{\mathrm{H}} (1-\delta )\bigr ] +(1-\pi _{\mathrm{H}})\, V_{\mathrm{L}}\, (1-g_0) <0$.

Hence, for $K \le 0$, the optimal choice of the public is $\alpha ^0 = 0$, inducing $p^*=g_0\pi _{\mathrm{H}}$. For $K >0$, the optimal choice of the public is

$$\begin{aligned} \alpha ^0 = \frac{(1-h_0)(1-g_0)}{(1-h_0)+g_0(h_0-g_0)}, \end{aligned}$$

which, using Equation (10), implies that $p^*= \pi _{\mathrm{H}}$. This completes the proof. $\square$

Proof of Proposition 4

Any equilibrium with a reelection scheme $q(g_1, h_1)$ produces a probability—or several probabilities—that an office seeker will play I. The equilibrium probabilities are denoted by ${\hat{p}}$ and are determined by

$$\begin{aligned} \mathop {\mathrm{argmax}}\limits _{p} \Bigl \{ p \Bigl ( q \bigl (g_1 (I, {\hat{p}}), h_1 (I,{\hat{p}}) \bigr )\Bigr ) + (1-p) \Bigl ( q \bigl (g_1 (NI, {\hat{p}}), h_1 (NI, {\hat{p}}) \bigr )\Bigr ) \Bigr \} = {\hat{p}}. \end{aligned}$$

We next observe from the proof of Proposition 3 that welfare is a linear function of ${\hat{p}}$ and any reelection scheme affects welfare solely through its impact on ${\hat{p}}$. Since welfare is a first-order polynomial in $p^*$, this implies that its extreme values are achieved at boundary points of the domain. Hence, the welfare-optimal mixing probabilities are either $p^* = \pi _{\mathrm{H}}$ or $p^* = \pi _{\mathrm{H}} g_0$. As the linear scheme can implement the welfare-optimal $p^*$ in both possible cases, no other scheme can perform better. $\square$