Baseline Model

The solution to the incumbent’s optimal choice of $ w\ge 0 $ in the baseline model presented in section A.3 of the supplementary appendix was based on the optimization problem

$$ \begin{array}{ccc}\underset{w}{\max }b(1\hskip0.3em -\hskip0.3em {\theta}^{*})\hskip0.3em -\hskip0.3em w\left(\frac{1}{2}+F{\displaystyle \int_0^1}\frac{(\theta +\epsilon )\hskip0.3em -\hskip0.3em {S}^{*}}{2\epsilon}\hskip0.3em d\theta \right).& & \end{array} $$

This formulation did not account for the fact that, as outlined on page 186, no agent engages in fraud in equilibrium when $ \theta <{S}^{*}\hskip0.3em -\hskip0.3em \epsilon $ , while all agents engage in fraud in equilibrium when $ \theta >{S}^{*}+\epsilon $ . In sum, the fraction of agents $ \phi $ that engage in fraud in equilibrium is

$$ \begin{array}{ccc}\phi =\left\{\begin{array}{c}0\hskip8.8em \mathrm{if}\hskip0.5em \theta <{S}^{*}\hskip0.3em -\hskip0.3em \epsilon; \qquad \\ {}\frac{(\theta +\epsilon )\hskip0.3em -\hskip0.3em {S}^{*}}{2\epsilon}\hskip4em \mathrm{if}\hskip0.5em {S}^{*}\hskip0.3em -\hskip0.3em \epsilon \le \theta \le {S}^{*}+\epsilon; \qquad \\ {}1\hskip8.8em \mathrm{if}\hskip0.5em \theta >{S}^{*}+\epsilon .\qquad \end{array}\right.& & \end{array} $$

Accounting for the above and taking an expectation with respect to the distribution of $ \theta $ , the incumbent solves

$$ {\displaystyle \begin{array}{c}\underset{w}{\max }b\left(1\hskip0.3em -\hskip0.3em {\theta}^{*}\right)\hskip0.3em -\hskip0.3em w\Big(\frac{1}{2}+F\Big[\int_0^{S^{*}\hskip0.3em -\hskip0.3em \epsilon }0\hskip0.3em d\theta \\ {}\hskip1em +\hskip2px \int_{S^{*}\hskip0.3em -\hskip0.3em \epsilon}^{S^{*}+\epsilon}\frac{\left(\theta +\epsilon \right)\hskip0.3em -\hskip0.3em {S}^{*}}{2\epsilon }d\theta +\int_{S^{*}+\epsilon}^11\hskip0.3em d\theta \Big]\hskip0.5em \Big).\end{array}} $$

Treating $ {\theta}^{*} $ and $ {S}^{*} $ as functions of w, the above simplifies to

$$ \begin{array}{ccc}\underset{w}{\max }b\left(\frac{1}{2}+F\frac{w}{c+w}\right)\hskip0.3em -\hskip0.3em w\left(\frac{1}{2}+F\left[\epsilon +\frac{1}{2}+\frac{Fw\hskip0.3em -\hskip0.3em 2\epsilon c}{c+w}\right]\right),& & \end{array} $$

where we used

$$ {\displaystyle \begin{array}{c}\int_0^{S^{*}\hskip0.3em -\hskip0.3em \epsilon }0\hskip0.3em d\theta =0,\hskip1em \int_{S^{*}\hskip0.3em -\hskip0.3em \epsilon}^{S^{*}+\epsilon}\frac{\left(\theta +\epsilon \right)\hskip0.3em -\hskip0.3em {S}^{*}}{2\epsilon }d\theta =\epsilon, \hskip1em \mathrm{and}\\ {}\int_{S^{*}+\epsilon}^11\hskip0.3em d\theta =\frac{1}{2}+\frac{Fw\hskip0.3em -\hskip0.3em 2\epsilon c}{c+w},\end{array}} $$

as well as the solutions for $ {\theta}^{*} $ and $ {S}^{*} $ stated on p. 185,

(1) $$ \begin{array}{cc}{\theta}^{*}=\frac{1}{2}\hskip0.3em -\hskip0.3em F\frac{w}{c+w}\hskip1em \mathrm{and}\hskip0.3em {S}^{*}=\frac{1}{2}\hskip0.3em -\hskip0.3em F\frac{w}{c+w}+\epsilon \frac{c\hskip0.3em -\hskip0.3em w}{c+w}\hskip0.3em .& \end{array} $$

The first-order condition for this optimization problem is

$$ {\displaystyle \begin{array}{c}b\left(\hskip0.3em -\hskip0.3em \frac{1\hskip0.3em -\hskip0.3em 2F}{2\left(c+w\right)}+\frac{c+w\hskip0.3em -\hskip0.3em 2Fw}{2{\left(c+w\right)}^2}\right)\hskip0.3em -\hskip0.3em \\ {}\Big(\frac{1}{2}+Fw\left[\hskip0.3em -\hskip0.3em \frac{1\hskip0.3em -\hskip0.3em 2F\hskip0.3em -\hskip0.3em 2\epsilon }{2\left(c+w\right)}+\frac{c+w\hskip0.3em -\hskip0.3em 2Fw\hskip0.3em -\hskip0.3em 2\epsilon c\hskip0.3em -\hskip0.3em 2\epsilon w}{2{\left(c+w\right)}^2}\right]\\ {}\hskip0.1em +\hskip2px F\left[1\hskip0.3em -\hskip0.3em \frac{c+w\hskip0.3em -\hskip0.3em 2Fw\hskip0.3em -\hskip0.3em 2\epsilon c\hskip0.3em -\hskip0.3em 2\epsilon w}{2\left(c+w\right)}\right]\Big)\hskip0.3em =0,\end{array}} $$

which simplifies to

$$ \begin{array}{ccc}{w}^2+2cw\hskip0.3em -\hskip0.3em \frac{2bcF\hskip0.3em -\hskip0.3em {c}^2\hskip0.3em -\hskip0.3em F{c}^2+2\epsilon F{c}^2}{1+F+2\epsilon F+2{F}^2}=0\hskip0.3em .& & \end{array} $$

Solving the above quadratic equation in w, we have two solutions,

$$ {\displaystyle \begin{array}{c}\frac{\hskip0.3em -\hskip0.3em 2c\hskip0.3em -\hskip0.3em \sqrt{4{c}^2+4\frac{2bcF\hskip0.3em -\hskip0.3em {c}^2\hskip0.3em -\hskip0.3em F{c}^2+2\epsilon F{c}^2}{1+F+2\epsilon F+2{F}^2}}}{2}\hskip1em \mathrm{and}\\ {}\frac{\hskip0.3em -\hskip0.3em 2c+\sqrt{4{c}^2+4\frac{2bcF\hskip0.3em -\hskip0.3em {c}^2\hskip0.3em -\hskip0.3em F{c}^2+2\epsilon F{c}^2}{1+F+2\epsilon F+2{F}^2}}}{2},\end{array}} $$

which simplify to

$$ {\displaystyle \begin{array}{c}\hskip0.3em -\hskip0.3em \hspace{2px}c\hskip0.3em -\hskip0.3em \sqrt{\frac{2cF\left(b+2\epsilon c+cF\right)}{1+F\left(1+2\epsilon +2F\right)}}\hskip1em \mathrm{and}\\ {}\hskip0.3em -\hskip0.3em \hspace{2px}c+\sqrt{\frac{2cF\left(b+2\epsilon c+cF\right)}{1+F\left(1+2\epsilon +2F\right)}}.\end{array}} $$

Of these, only the latter can be non-negative, which is the case as long as $ b\ge c\left(\frac{1+F(1\hskip0.3em -\hskip0.3em 2\epsilon )}{2F}\right) $ .

The second-order condition for this optimization problem is

$$ \begin{array}{ccc}\hskip0.3em -\hskip0.3em \frac{2cF(b+2\epsilon c+cF)}{(c+w)^3}<0\hskip0.3em ,& & \end{array} $$

which holds for any (admissible) parameter values and a non-negative w.

In sum, the equilibrium reward factor $ {w}^{*} $ is

$$ \begin{array}{ccc}{w}^{*}=\left\{\begin{array}{c}\sqrt{\frac{2cF(b+2\epsilon c+cF)}{1+F(1+2\epsilon +2F)}}\hskip0.3em -\hskip0.3em c\hskip1em \mathrm{if}\hskip0.5em b>c\left(\frac{1+F(1\hskip0.3em -\hskip0.3em 2\epsilon )}{2F}\right);\qquad \\ {}0\hskip11em \mathrm{otherwise}.\qquad \end{array}\right.& & \end{array} $$

The illustration on page 186, based on parameters $ c=1 $ , $ F=\frac{2}{10} $ , $ \epsilon =\frac{1}{10} $ , and $ b=70 $ , now yields $ {S}^{*}=0.29 $ , $ {\theta}^{*}=0.34 $ , $ {\phi}^{*}=0.78 $ , and $ {w}^{*}=3.62 $ .

The threshold $ {\theta}^{*} $ is decreasing in $ {w}^{*} $ as reported in section A.4 of the supplementary appendix. But to account for the indirect effect of the parameters b, c, F, and $ \epsilon $ on $ {\theta}^{*} $ , its total derivatives must be based on the corrected $ {w}^{*} $ . Treating $ {\theta}^{*} $ as a function of $ {w}^{*} $ , i.e. substituting the expression for $ {w}^{*} $ when it is positive into the expression for $ {\theta}^{*} $ , and differentiating with respect to each of the parameters, we get

$$ \begin{array}{l}\frac{d{\theta}^{*}}{db}=\hskip0.3em -\hskip0.3em \frac{\sqrt{cF\left(1+F+2\epsilon F+2{F}^2\right)}}{{\left[2\left(b+2\epsilon c+cF\right)\right]}^{\frac{3}{2}}}<0;\\ {}\frac{d{\theta}^{*}}{dc}=\frac{b\sqrt{cF\left(1+F+2\epsilon F+2{F}^2\right)}}{c{\left[2\left(b+2\epsilon c+cF\right)\right]}^{\frac{3}{2}}}>0;\\ {}{\displaystyle \begin{array}{rcl}\frac{d{\theta}^{*}}{d\epsilon }& =& \frac{\sqrt{cF}\left[bF\hskip0.3em -\hskip0.3em c\left(1+F+{F}^2\right)\right]}{\sqrt{2\left(1+F+2\epsilon F+2{F}^2\right){\left(b+2\epsilon c+cF\right)}^3}}\\ {}& >& 0\hskip1em \mathrm{for}\hskip1em b>c\left(1+\frac{1+{F}^2}{F}\right);\end{array}}\\ {}\frac{d{\theta}^{*}}{dF}={\displaystyle \begin{array}{l}\hskip0.3em -\hskip0.3em 1+\sqrt{\frac{\frac{1}{2}+\frac{F}{2}+\epsilon F+{F}^2}{\frac{bF}{c}+2\epsilon F+{F}^2}}\Big[1\hskip0.3em -\hskip0.3em \frac{F}{2}\Big(\frac{\frac{b}{c}+2\epsilon +2F}{\frac{bF}{c}+2\epsilon F+{F}^2}\\ {}\hskip0.3em -\hskip0.3em \hspace{2px}\frac{\frac{1}{2}+\epsilon +2F}{\frac{1}{2}+\frac{F}{2}+\epsilon F+{F}^2}\Big)\Big].\end{array}}\end{array} $$

Note that while the derivatives of $ {\theta}^{*} $ with respect to b and c have the same sign as those reported in the original appendix, the derivative of $ {\theta}^{*} $ with respect to $ \epsilon $ is positive for large enough values of b. Meanwhile, the derivative of $ {\theta}^{*} $ with respect to F is complicated.

“Fraud as Insurance” Extension

The equilibrium thresholds remain the same as in Equation 1. However, as $ \theta \sim U[\widehat{\theta}\hskip0.3em -\hskip0.3em \sigma, \widehat{\theta}+\sigma ] $ , the incumbent’s objective function becomes

$$ \begin{array}{ccc}b\frac{\widehat{\theta}+\sigma \hskip0.3em -\hskip0.3em {\theta}^{*}}{2\sigma}\hskip0.3em -\hskip0.3em w\left(E(\theta )+E[\phi (\theta )]F\right)& & \\ {}=b\frac{\widehat{\theta}+\sigma \hskip0.3em -\hskip0.3em {\theta}^{*}}{2\sigma}\hskip0.3em -\hskip0.3em w\left(\widehat{\theta}+\frac{\widehat{\theta}+\sigma \hskip0.3em -\hskip0.3em {S}^{*}}{2\sigma }F\right)& & \\ {}=b\frac{\widehat{\theta}+\sigma \hskip0.3em -\hskip0.3em \frac{1}{2}+F\frac{w}{c+w}}{2\sigma}\hskip0.3em -\hskip0.3em w\left(\widehat{\theta}+\frac{\widehat{\theta}+\sigma \hskip0.3em -\hskip0.3em \frac{1}{2}+F\frac{w}{c+w}+\epsilon \frac{w\hskip0.3em -\hskip0.3em c}{w+c}}{2\sigma }F\right).& & \end{array} $$

Multiplying by $ 2\sigma $ and dropping terms that are constant in w, the incumbent’s objective is (equivalent to)

$$ \begin{array}{ccc}bF\frac{w}{c+w}\hskip0.3em -\hskip0.3em w\left(2\sigma \widehat{\theta}+\left[\widehat{\theta}+\sigma \hskip0.3em -\hskip0.3em \frac{1}{2}+F\frac{w}{c+w}+\epsilon \frac{w\hskip0.3em -\hskip0.3em c}{w+c}\right]F\right).& & \end{array} $$

After the change of variables $ x:=c+w $ , the incumbent’s objective is

$$ {\displaystyle \begin{array}{c}bF\frac{x\hskip0.3em -\hskip0.3em c}{x}\hskip0.3em -\hskip0.3em \left(x\hskip0.3em -\hskip0.3em c\right)\left(2\sigma \widehat{\theta}+\left[\widehat{\theta}+\sigma \hskip0.3em -\hskip0.3em \frac{1}{2}+F\frac{x\hskip0.3em -\hskip0.3em c}{x}+\epsilon \frac{x\hskip0.3em -\hskip0.3em 2c}{x}\right]F\right)\\ {}\hskip1em \equiv {\displaystyle \begin{array}{l}bF\frac{x\hskip0.3em -\hskip0.3em c}{x}\hskip0.3em -\hskip0.3em \left(x\hskip0.3em -\hskip0.3em c\right)\left(2\sigma \widehat{\theta}+\left[\widehat{\theta}+\sigma \hskip0.3em -\hskip0.3em \frac{1}{2}\right]F\right)\hskip0.3em -\hskip0.3em {F}^2\frac{{\left(x\hskip0.3em -\hskip0.3em c\right)}^2}{x}\\ {}\hskip0.3em -\hskip0.3em \hspace{2px}F\epsilon \frac{\left(x\hskip0.3em -\hskip0.3em c\right)\left(x\hskip0.3em -\hskip0.3em 2c\right)}{x}\end{array}}\end{array}} $$

which is equivalent to

$$ {\displaystyle \begin{array}{c}\hskip0.3em -\hskip0.3em \frac{bFc}{x}\hskip0.3em -\hskip0.3em x\left(2\sigma \widehat{\theta}+\left[\widehat{\theta}+\sigma \hskip0.3em -\hskip0.3em \frac{1}{2}\right]F\right)\hskip0.3em -\hskip0.3em {F}^2\left(x+\frac{c^2}{x}\right)\\ {}\hskip1.30em -\hskip0.3em \hspace{2px}F\epsilon \left(x+2\frac{c^2}{x}\right).\end{array}} $$

The first-order condition is then

$$ 0={\displaystyle \begin{array}{l}\frac{bFc}{x^2}\hskip0.3em -\hskip0.3em \left(2\sigma \widehat{\theta}+\left[\widehat{\theta}+\sigma \hskip0.3em -\hskip0.3em \frac{1}{2}\right]F\right)\hskip0.3em -\hskip0.3em {F}^2\left(1\hskip0.3em -\hskip0.3em \frac{c^2}{x^2}\right)\\ {}\hskip0.3em -\hskip0.3em \hspace{2px}F\epsilon \left(1\hskip0.3em -\hskip0.3em 2\frac{c^2}{x^2}\right),\end{array}} $$

which yields

$$ \begin{array}{ccc}x=\sqrt{\frac{bFc+{F}^2{c}^2+2\epsilon F{c}^2}{2\sigma \widehat{\theta}+\left[\widehat{\theta}+\sigma \hskip0.3em -\hskip0.3em \frac{1}{2}\right]F+{F}^2+F\epsilon }},& & \end{array} $$

or

$$ \begin{array}{ccc}{w}^{*}=\sqrt{\frac{bFc+{F}^2{c}^2+2\epsilon F{c}^2}{2\sigma \widehat{\theta}+\left[\widehat{\theta}+\sigma \hskip0.3em -\hskip0.3em \frac{1}{2}\right]F+{F}^2+F\epsilon }}\hskip0.3em -\hskip0.3em c.& & \end{array} $$

This is positive when

$$ {\displaystyle \begin{array}{l}\frac{bFc+{F}^2{c}^2+2\epsilon F{c}^2}{2\sigma \widehat{\theta}+\left[\widehat{\theta}+\sigma \hskip0.3em -\hskip0.3em \frac{1}{2}\right]F+{F}^2+F\epsilon}\hskip1em >{c}^2\\ {}\hskip1em \iff bFc+{F}^2{c}^2+2\epsilon F{c}^2\\ {}\hskip1em >{c}^2\left(2\sigma \widehat{\theta}+\left[\widehat{\theta}+\sigma \hskip0.3em -\hskip0.3em \frac{1}{2}\right]F+{F}^2+F\epsilon \right)\\ {}\hskip1em \iff bF+{F}^2c+2\epsilon Fc\\ {}\hskip1em >c\left(2\sigma \widehat{\theta}+\left[\widehat{\theta}+\sigma \hskip0.3em -\hskip0.3em \frac{1}{2}\right]F+{F}^2+F\epsilon \right)\\ {}\hskip1em \iff bF\hskip1em >c\left(2\sigma \widehat{\theta}+\left[\widehat{\theta}+\sigma \hskip0.3em -\hskip0.3em \frac{1}{2}\right]F\hskip0.3em -\hskip0.3em F\epsilon \right)\\ {}\hskip1em \iff b\hskip1em >c\left(2\frac{\sigma \widehat{\theta}}{F}+\widehat{\theta}+\sigma \hskip0.3em -\hskip0.3em \frac{1}{2}\hskip0.3em -\hskip0.3em \epsilon \right).\end{array}} $$

We highlight that in the range of admissible $ \sigma $ , we have

$$ \begin{array}{ccc}\frac{\partial {w}^{*}}{\partial \sigma }<0,& & \end{array} $$

so that more prior uncertainty about the incumbent’s popularity, reduces the incentives the incumbent offers for fraud.

Furthermore, in footnote 27 (188) we should require $ \sigma >\frac{F}{2}+2\epsilon $ rather than $ \sigma >\frac{F}{2} $ , so that for appropriately chosen $ \widehat{\theta} $ , the prior covers the interval $ \left[\frac{1}{2}\hskip0.3em -\hskip0.3em F\hskip0.3em -\hskip0.3em 2\epsilon, \frac{1}{2}+2\epsilon \right] $ . This guarantees that the dominance regions cover all values of $ {S}_i $ for which the agent’s posterior about $ \theta $ is not equal to $ U[{S}_i\hskip0.3em -\hskip0.3em \epsilon, {S}_i+\epsilon ] $ .

“Differences in Competitiveness” Extension

The payoff offered to agent i should be $ w({R}_i\hskip0.3em -\hskip0.3em \alpha {P}_i) $ rather than $ w({R}_i\hskip0.3em -\hskip0.3em {P}_i) $ (p. 189), so that: (i) agents’ rewards are correctly adjusted for their district’s popularity, instead of agents in popular districts being systematically punished; (ii) as $ \alpha \to 0 $ , the model converges to the baseline. The agent’s payoff then becomes $ w\left[(1\hskip0.3em -\hskip0.3em \alpha ){S}_i+{1}_{a_i=f}F\right] $ when the incumbent wins and $ \hskip0.3em -\hskip0.3em cF{1}_{a_i=f} $ when the incumbent loses.Footnote ¹

The expressions for $ {S}^{*} $ , $ {\theta}^{*} $ and $ {\phi}^{*} $ given in page 8 of the Online Appendix are correct. The incumbent’s objective is then

$$ {\displaystyle \begin{array}{c}b\left[1\hskip0.3em -\hskip0.3em {\theta}^{*}\right]\hskip0.3em -\hskip0.3em w\left[E\left({R}_i\right)\hskip0.3em -\hskip0.3em \alpha E\left({P}_i\right)\right]\\ {}\hskip1em =b\left[1\hskip0.3em -\hskip0.3em {\theta}^{*}\right]\hskip0.3em -\hskip0.3em w\left[\left(1\hskip0.3em -\hskip0.3em \alpha \right)E\left({S}_i\right)+E\left[\phi \left(\theta \right)\right]F\right]\\ {}\hskip1em =b\left[1\hskip0.3em -\hskip0.3em {\theta}^{*}\right]\hskip0.3em -\hskip0.3em w\left[\left(1\hskip0.3em -\hskip0.3em \alpha \right)\frac{1}{2}+\left(1\hskip0.3em -\hskip0.3em {S}^{*}\right)F\right]\\ {}\hskip1em ={\displaystyle \begin{array}{l}b\left[1\hskip0.3em -\hskip0.3em \frac{\frac{1}{2}\hskip0.3em -\hskip0.3em \frac{w}{w+c}F\hskip0.3em -\hskip0.3em \alpha \pi }{1\hskip0.3em -\hskip0.3em \alpha}\right]\hskip0.3em -\hskip0.3em w\Big[\frac{1\hskip0.3em -\hskip0.3em \alpha }{2}\\ {}+\hskip2px \left(1\hskip0.3em -\hskip0.3em \frac{\frac{1}{2}\hskip0.3em -\hskip0.3em \frac{w}{w+c}F\hskip0.3em -\hskip0.3em \alpha \pi }{1\hskip0.3em -\hskip0.3em \alpha }+\epsilon \frac{w\hskip0.3em -\hskip0.3em c}{w+c}\right)F\Big]\end{array}}\end{array}} $$

Dropping terms that are constant in w, this is equivalent to

$$ {\displaystyle \begin{array}{l}\frac{bF}{1\hskip0.3em -\hskip0.3em \alpha}\frac{w}{w+c}\hskip0.3em -\hskip0.3em w\Big[\frac{1\hskip0.3em -\hskip0.3em \alpha }{2}+\Big(1\hskip0.3em -\hskip0.3em \frac{\frac{1}{2}\hskip0.3em -\hskip0.3em \frac{w}{w+c}F\hskip0.3em -\hskip0.3em \alpha \pi }{1\hskip0.3em -\hskip0.3em \alpha}\hskip1em +\hskip2px \epsilon \frac{w\hskip0.3em -\hskip0.3em c}{w+c}\Big)F\Big]\end{array}} $$

After the change of variables $ x:=w+c $ , this equals

$$ {\displaystyle \begin{array}{l}\frac{bF}{1\hskip0.3em -\hskip0.3em \alpha}\frac{x\hskip0.3em -\hskip0.3em c}{x}\hskip0.3em -\hskip0.3em \left(x\hskip0.3em -\hskip0.3em c\right)\Big[\frac{1\hskip0.3em -\hskip0.3em \alpha }{2}\hskip1em +\hskip2px \left(1\hskip0.3em -\hskip0.3em \frac{\frac{1}{2}\hskip0.3em -\hskip0.3em \frac{x\hskip0.3em -\hskip0.3em c}{x}F\hskip0.3em -\hskip0.3em \alpha \pi }{1\hskip0.3em -\hskip0.3em \alpha }+\epsilon \frac{x\hskip0.3em -\hskip0.3em 2c}{x}\right)F\Big]\end{array}} $$

which is equivalent to

$$ {\displaystyle \begin{array}{c}\hskip0.3em -\hskip0.3em \frac{bF}{1\hskip0.3em -\hskip0.3em \alpha}\frac{c}{x}\hskip0.3em -\hskip0.3em x\left[\frac{1\hskip0.3em -\hskip0.3em \alpha }{2}+\left(1\hskip0.3em -\hskip0.3em \frac{\frac{1}{2}\hskip0.3em -\hskip0.3em \alpha \pi }{1\hskip0.3em -\hskip0.3em \alpha}\right)F\right]\\ {}\hskip1.30em -\hskip0.3em \hspace{2px}\frac{{\left(x\hskip0.3em -\hskip0.3em c\right)}^2}{x}\frac{F^2}{1\hskip0.3em -\hskip0.3em \alpha}\hskip0.3em -\hskip0.3em \frac{\left(x\hskip0.3em -\hskip0.3em c\right)\left(x\hskip0.3em -\hskip0.3em 2c\right)}{x}\epsilon F\end{array}} $$

or to

$$ {\displaystyle \begin{array}{c}\hskip0.3em -\hskip0.3em \frac{bF}{1\hskip0.3em -\hskip0.3em \alpha}\frac{c}{x}\hskip0.3em -\hskip0.3em x\left[\frac{1\hskip0.3em -\hskip0.3em \alpha }{2}+\left(1\hskip0.3em -\hskip0.3em \frac{\frac{1}{2}\hskip0.3em -\hskip0.3em \alpha \pi }{1\hskip0.3em -\hskip0.3em \alpha}\right)F\right]\\ {}\hskip1.30em -\hskip0.3em \hspace{2px}\left(x+\frac{c^2}{x}\right)\frac{F^2}{1\hskip0.3em -\hskip0.3em \alpha}\hskip0.3em -\hskip0.3em \left(x+\frac{2{c}^2}{x}\right)\epsilon F\end{array}} $$

The first-order condition is then

$$ 0={\displaystyle \begin{array}{l}\frac{bF}{1\hskip0.3em -\hskip0.3em \alpha}\frac{c}{x^2}\hskip0.3em -\hskip0.3em \left[\frac{1\hskip0.3em -\hskip0.3em \alpha }{2}+\left(1\hskip0.3em -\hskip0.3em \frac{\frac{1}{2}\hskip0.3em -\hskip0.3em \alpha \pi }{1\hskip0.3em -\hskip0.3em \alpha}\right)F\right]\\ {}\hskip0.3em -\hskip0.3em \hspace{2px}\left(1\hskip0.3em -\hskip0.3em \frac{c^2}{x^2}\right)\frac{F^2}{1\hskip0.3em -\hskip0.3em \alpha}\hskip0.3em -\hskip0.3em \left(1\hskip0.3em -\hskip0.3em \frac{2{c}^2}{x^2}\right)\epsilon F\end{array}} $$

which yields

$$ \begin{array}{ccc}x=\sqrt{\frac{\frac{bFc}{1\hskip0.3em -\hskip0.3em \alpha }+\frac{F^2{c}^2}{1\hskip0.3em -\hskip0.3em \alpha }+2{c}^2\epsilon F}{\frac{1\hskip0.3em -\hskip0.3em \alpha }{2}+\left(1\hskip0.3em -\hskip0.3em \frac{\frac{1}{2}\hskip0.3em -\hskip0.3em \alpha \pi }{1\hskip0.3em -\hskip0.3em \alpha}\right)F+\frac{F^2}{1\hskip0.3em -\hskip0.3em \alpha }+\epsilon F}}& & \end{array} $$

or

$$ \begin{array}{ccc}{w}^{*}=\sqrt{\frac{\frac{bFc}{1\hskip0.3em -\hskip0.3em \alpha }+\frac{F^2{c}^2}{1\hskip0.3em -\hskip0.3em \alpha }+2{c}^2\epsilon F}{\frac{1\hskip0.3em -\hskip0.3em \alpha }{2}+\left(1\hskip0.3em -\hskip0.3em \frac{\frac{1}{2}\hskip0.3em -\hskip0.3em \alpha \pi }{1\hskip0.3em -\hskip0.3em \alpha}\right)F+\frac{F^2}{1\hskip0.3em -\hskip0.3em \alpha }+\epsilon F}}\hskip0.3em -\hskip0.3em c.& & \end{array} $$

This is positive when

$$ {\displaystyle \begin{array}{l}\frac{\frac{bFc}{1\hskip0.3em -\hskip0.3em \alpha }+\frac{F^2{c}^2}{1\hskip0.3em -\hskip0.3em \alpha }+2{c}^2\epsilon F}{\frac{1\hskip0.3em -\hskip0.3em \alpha }{2}+\left(1\hskip0.3em -\hskip0.3em \frac{\frac{1}{2}\hskip0.3em -\hskip0.3em \alpha \pi }{1\hskip0.3em -\hskip0.3em \alpha}\right)F+\frac{F^2}{1\hskip0.3em -\hskip0.3em \alpha }+\epsilon F}\hskip1em >{c}^2\\ {}\iff \frac{bFc}{1\hskip0.3em -\hskip0.3em \alpha}\hskip1em >{\displaystyle \begin{array}{l}{c}^2\left[\frac{1\hskip0.3em -\hskip0.3em \alpha }{2}+\left(1\hskip0.3em -\hskip0.3em \frac{\frac{1}{2}\hskip0.3em -\hskip0.3em \alpha \pi }{1\hskip0.3em -\hskip0.3em \alpha}\right)F+\frac{F^2}{1\hskip0.3em -\hskip0.3em \alpha }+\epsilon F\right]\hskip0.3em -\hskip0.3em \frac{F^2{c}^2}{1\hskip0.3em -\hskip0.3em \alpha}\\ {}\hskip0.3em -\hskip0.3em \hspace{2px}2{c}^2\epsilon F\end{array}}\\ {}\hskip1em \iff \frac{bFc}{1\hskip0.3em -\hskip0.3em \alpha}\hskip1em >{c}^2\left[\frac{1\hskip0.3em -\hskip0.3em \alpha }{2}+\left(1\hskip0.3em -\hskip0.3em \frac{\frac{1}{2}\hskip0.3em -\hskip0.3em \alpha \pi }{1\hskip0.3em -\hskip0.3em \alpha}\right)F\hskip0.3em -\hskip0.3em \epsilon F\right]\\ {}\hskip0.9em \iff b\hskip0.3em >c\left[\frac{{\left(1\hskip0.3em -\hskip0.3em \alpha \right)}^2}{2F}+\left(1\hskip0.3em -\hskip0.3em \alpha \hskip0.3em -\hskip0.3em \frac{1}{2}+\alpha \pi \right)\hskip0.3em -\hskip0.3em \epsilon \left(1\hskip0.3em -\hskip0.3em \alpha \right)\right].\end{array}} $$

Paralleling our last remark in the previous section, in Eq. A.2 (8, Online Appendix) we should require $ F+\alpha \pi +(1\hskip0.3em -\hskip0.3em \alpha )2\epsilon \le \frac{1}{2} $ and $ \alpha (1\hskip0.3em -\hskip0.3em \pi \hskip0.3em -\hskip0.3em 2\epsilon )\le \frac{1}{2}\hskip0.3em -\hskip0.3em 2\epsilon $ to guarantee that the dominance regions cover all values of $ {S}_i $ for which the agent’s posterior about $ \theta $ is not $ U[{S}_i\hskip0.3em -\hskip0.3em \epsilon, {S}_i+\epsilon ] $ .