The social value of information with an endogenous public signal

Abstract

This paper analyzes the equilibrium and welfare properties of an economy characterized by uncertainty and payoff externalities using a general model that nests several applications. Agents receive a private signal and an endogenous public signal, which is a noisy aggregate of individual actions, and causes an information externality. Agents in equilibrium underweight private information for a larger payoff parameter region in relation to when public information is exogenous. In addition, the socially optimal endogenous degree of coordination is lower than the socially optimal exogenous degree of coordination. The welfare effect of increasing the precision of the noise in the public signal has the same sign with endogenous or exogenous public information, but its magnitude differs. The social value of private information may be overturned in relation to when public information is exogenous: from positive to negative if agents in equilibrium coordinate more than is implied by the socially optimal exogenous degree of coordination, and the opposite if they coordinate less.

Appendix

Proof of Proposition 1

Focusing on linear arbitrary strategies of the form $k(x_{i},z)=b'+ax_{i}+c'E\left[ \theta \mid z\right]$ and using the definition of the public signal, I obtain: $w=\int k_{i}+v=b'+a\theta +c'w+v$, the informational content of which can be seen to be the same as that of $z=a\theta +v$. From the properties of the normal distribution: $E\left[ \theta |z\right] =\frac{\tau _{v}az+\tau _{\theta }\bar{\theta }}{\tau _{\theta }+a^{2}\tau _{v}}$ and $(var\left[ \theta |z\right] )^{-1}=\tau (a)=\tau _{\theta }+a{}^{2}\tau _{v}$. Hence $E\left[ \theta |x_{i},z\right] =\frac{\tau _{\xi }}{\tau _{\xi }+\tau (a)}x_{i}+\frac{\tau (a)}{\tau _{\xi }+\tau (a)}E\left[ \theta \mid z\right]$.

Using the maximization condition in the REE definition, the best response strategy $k'$ satisfies $E\left[ U_{k}(k',K,\sigma _{k},\theta )\mid x_{i},z\right] =0$ and hence $k'=E[(1-\alpha )\kappa (\theta )+\alpha K\mid x_{i},w]$. The second-order condition is satisfied since $U_{kk}<0$. Equating coefficients, I obtain: $b=\kappa _{0}$, $a=\frac{\kappa _{1}(1-\alpha )\tau _{\xi }}{(1-\alpha )\tau _{\xi }+\tau (a)}$ and $c=\frac{\kappa _{1}\tau (a)}{(1-\alpha )\tau _{\xi }+\tau (a)}$. Hence, $k(x_{i},z)=\kappa _{0}+\kappa _{1}(\gamma x_{i}+(1-\gamma )E\left[ \theta |z\right] )$, where $a=\kappa _{1}\gamma$ and $\gamma =\frac{(1-\alpha )\tau _{\xi }}{(1-\alpha )\tau _{\xi }+\tau _{\theta }+\kappa _{1}^{2}\gamma ^{2}\tau _{v}}$. The equilibrium weight to private information, $\gamma ^{m}$, is then a solution of $\Gamma (\gamma ^{m})=0$, where $\Gamma (\gamma )=\gamma ^{3}\kappa _{1}^{2}\tau _{v}+\gamma ((1-\alpha )\tau _{\xi }+\tau _{\theta })-(1-\alpha )\tau _{\xi }.$ By the Descartes’ Rule of signs, I note that there is only one sign change in $\Gamma (\gamma )$, and therefore, there is only a positive real root. Checking $\Gamma (-\gamma )$, I notice that there are no sign changes, and therefore, there are no negative real roots. Additionally, $\Gamma (0)=-(1-\alpha )\tau _{\xi }<0$ and $\Gamma (1)=\kappa _{1}^{2}\tau _{v}+\tau _{\theta }>0$, and the positive real root is between 0 and 1. $\square$

Proof of Corollary 1

Differentiating $\Gamma (\gamma ^{m})=0$ with respect to the various information and payoff parameters, and noting that $0<\gamma ^{m}<1$, I obtain:

$$\begin{aligned} \frac{\partial \gamma ^{m}}{\partial \tau _{\xi }}= & {} \frac{(1-\alpha )(1-\gamma ^{m})}{(1-\alpha )\tau _{\xi }+\tau _{\theta }+3(\gamma ^{m})^{2}\kappa _{1}^{2}\tau _{v}}>0,\\ \frac{\partial \gamma ^{m}}{\partial \tau _{\theta }}= & {} \frac{-\gamma ^{m}}{(1-\alpha )\tau _{\xi }+\tau _{\theta }+3(\gamma ^{m})^{2}\kappa _{1}^{2}\tau _{v}}<0,\\ \frac{\partial \gamma ^{m}}{\partial \tau _{v}}= & {} \frac{-(\gamma ^{m})^{3}\kappa _{1}^{2}}{(1-\alpha )\tau _{\xi }+\tau _{\theta }+3(\gamma ^{m})^{2}\kappa _{1}^{2}\tau _{v}}<0,\quad \hbox {and}\\ \frac{\partial \gamma ^{m}}{\partial \alpha }= & {} \frac{-(1-\gamma ^{m})\tau _{\xi }}{(1-\alpha )\tau _{\xi }+\tau _{\theta }+3(\gamma ^{m})^{2}\kappa _{1}^{2}\tau _{v}}<0. \end{aligned}$$

$\square$

Proof of Corollary 2

The proof follows since: $\frac{d\tau ^{m}}{d\tau _{\xi }}=\frac{(2\kappa _{1}^{2}\tau _{v}\gamma ^{m})(1-\alpha )(1-\gamma ^{m})}{(1-\alpha )\tau _{\xi }+\tau _{\theta }+3(\gamma ^{m})^{2}\kappa _{1}^{2}\tau _{v}}>0$, $0<\frac{d\tau ^{m}}{d\tau _{v}}=\kappa _{1}^{2}\gamma ^{2}(\frac{(1-\alpha )\tau _{\xi }+\tau _{\theta }+(\gamma ^{m})^{2}\kappa _{1}^{2}\tau _{v}}{(1-\alpha )\tau _{\xi }+\tau _{\theta }+3(\gamma ^{m})^{2}\kappa _{1}^{2}\tau _{v}})<\kappa _{1}^{2}$, and $\frac{d\tau ^{m}}{d\tau _{\theta }}=\frac{(1-\alpha )\tau _{\xi }+\tau ^{m}}{(1-\alpha )\tau _{\xi }+\tau _{\theta }+3(\gamma ^{m})^{2}\kappa _{1}^{2}\tau _{v}}$, which implies that $0<\frac{d\tau ^{m}}{d\tau _{\theta }}<1$. $\square$

Proof of Proposition 2

The welfare planner internalizes all externalities and maximizes ex ante utility, or equivalently minimizes welfare loss. Focusing on a linear arbitrary efficient strategy of the form $k^{*}(x_{i},z)=b^{*}+a^{*}x_{i}+c^{*}E\left[ \theta \mid z\right]$. By a similar argument as that used in Proposition 1, $b^{*}=\kappa _{0}^{*}$ and $k^{*}(x_{i},z)=\kappa _{0}^{*}+\kappa _{1}^{*}(\gamma x_{i}+(1-\gamma )E\left[ \theta |z\right] )$, where $\gamma$ is an arbitrary weight to the private signal, and $\kappa _{0}^{*},\kappa _{1}^{*}$ were found by AP and are reported in the text. The first-order condition satisfies $\frac{dL^{*}}{d\gamma }|_{\gamma =\gamma ^{*}}=0$. Hence, a candidate efficient strategy with weight to private information $\gamma$ and precision given by $\tau (\kappa _{1}^{*}\gamma )=\tau _{\theta }+(\kappa _{1}^{*}\gamma )^{2}\tau _{v}$ solves

$$\begin{aligned} \frac{\gamma }{\tau _{\xi }}-\frac{(1-\alpha ^{*})(1-\gamma )}{\tau (\kappa _{1}^{*}\gamma )}-\frac{(\kappa _{1}^{*})^{2}\tau _{v}\gamma (1-\alpha ^{*})(1-\gamma )^{2}}{(\tau (\kappa _{1}^{*}\gamma )){}^{2}}=0, \end{aligned}$$

(A1)

where the last term is due to the endogeneity of public information. The previous expression can be rewritten as

$$\begin{aligned} \gamma =\frac{(1-\alpha ^{*})\tau _{\xi }}{(1-\alpha ^{*})\tau _{\xi }+\tau _{\theta }+(\kappa _{1}^{*})^2\gamma ^{2}\tau _{v}-(1-\alpha ^{*})(1-\gamma )^{2}\frac{\tau _{v}\tau _{\xi }(\kappa _{1}^{*})^2}{(\tau _{\theta }+(\kappa _{1}^{*})^2 \gamma ^{2} \tau _{v})}}. \end{aligned}$$

(A2)

Then, $\gamma ^{*}$ is a solution of a quintic equation $\psi (\gamma ^{*})=0$, where

$$\begin{aligned} {\begin{matrix} \psi (\gamma )&{}=(\gamma )^{5}\tau _{v}^{2}(\kappa _{1}^{*})^{4}+2(\gamma )^{3}(\kappa _{1}^{*})^{2}\tau _{v}\tau _{\theta }+(\gamma )^{2}(\kappa _{1}^{*})^{2}\tau _{v}\tau _{\xi }(1-\alpha ^{*})\\ &{}\quad +\gamma (\tau _{\theta }^{2}+\tau _{\xi }(1-\alpha ^{*})(\tau _{\theta }-\tau _{v}(\kappa _{1}^{*})^{2}))-\tau _{\xi }\tau _{\theta }(1-\alpha ^{*}). \end{matrix}} \end{aligned}$$

(A3)

The SOC guarantees that the denominator of (A2) is positive and hence $\gamma ^{*}>0$. Applying Descartes’ Rule of signs to find out the number of real roots, I find that there is only one change in sign in $\psi (\gamma )$, and therefore, this polynomial has only one real positive root. Furthermore, $\psi (0)=-\tau _{\xi }\tau _{\theta }(1-\alpha ^{*})<0$, while $\psi (1)=\tau _{v}^{2}(\kappa _{1}^{*})^{4}+2(\kappa _{1}^{*})^{2}\tau _{v}\tau _{\theta }+\tau _{\theta }^{2}>0$ and the unique positive real root is between 0 and 1. $\square$

Proof of Corollary 3

Differentiating $\psi (\gamma ^{*})=0$ with respect to the various information and payoff parameters and noting that $0<\gamma ^{*}<1$, I find:

$$\begin{aligned} \frac{\partial \gamma ^{*}}{\partial \tau _{\xi }}= & {} \frac{\gamma ^{*}(1-\gamma ^{*})(1-\alpha ^{*})\tau _{v}(\kappa _{1}^{*})^{2}}{5(\gamma ^{*})^{4}\tau _{v}^{2}(\kappa _{1}^{*})^{4}+6(\gamma ^{*})^{2}(\kappa _{1}^{*})^{2}\tau _{v}\tau _{\theta }+2\gamma ^{*}(\kappa _{1}^{*})^{2}\tau _{v}\tau _{\xi }(1-\alpha ^{*})}>0, \\ \frac{\partial \gamma ^{*}}{\partial \tau _{\theta }}= & {} \frac{-2\gamma ^{*}\tau ^{*}+\tau _{\xi }(1-\alpha ^{*})(1-\gamma ^{*})}{5(\gamma ^{*})^{4}\tau _{v}^{2}(\kappa _{1}^{*})^{4}+6(\gamma ^{*})^{2}(\kappa _{1}^{*})^{2}\tau _{v}\tau _{\theta }+2\gamma ^{*}(\kappa _{1}^{*})^{2}\tau _{v}\tau _{\xi }(1-\alpha ^{*})}<0,\quad \hbox {and}\\ \frac{\partial \gamma ^{*}}{\partial \alpha ^{*}}= & {} \frac{-(\gamma ^{*}\tau _{\xi }\tau _{v}(\kappa _{1}^{*})^{2}+\tau _{\xi }\tau _{\theta })(1-\gamma ^{*})}{5(\gamma ^{*})^{4}\tau _{v}^{2}(\kappa _{1}^{*})^{4}+6(\gamma ^{*})^{2}(\kappa _{1}^{*})^{2}\tau _{v}\tau _{\theta }+2\gamma ^{*}(\kappa _{1}^{*})^{2}\tau _{v}\tau _{\xi }(1-\alpha ^{*})}<0. \end{aligned}$$

Notice that the sign of $\frac{\partial \gamma ^{*}}{\partial \tau _{v}}=\kappa _{1}^{2}\gamma ^{*}(\frac{-2(\gamma ^{*})^{2}\tau ^{*}+\tau _{\xi }(1-\alpha ^{*})(1-\gamma ^{*})}{5(\gamma ^{*})^{4}\tau _{v}^{2}(\kappa _{1}^{*})^{4}+6(\gamma ^{*})^{2}(\kappa _{1}^{*})^{2}\tau _{v}\tau _{\theta }+2\gamma ^{*}(\kappa _{1}^{*})^{2}\tau _{v}\tau _{\xi }(1-\alpha ^{*})})$ is ambiguous. $\square$

Proof of Proposition 3

The efficient strategy $\gamma ^{*}$ satisfies $\frac{dL^{*}}{d\gamma }|_{\gamma =\gamma ^{*}}=0$, where $L^{*}$ is a strictly convex function of $\gamma$. Hence ${{\mathrm{sign}}}\{\gamma ^{m}-\gamma ^{*}\}$ can be found by inspecting the sign of $\frac{dL^{*}}{d\gamma }\mid _{\gamma =\gamma ^{m}}$. Substituting for the equilibrium strategy in $\frac{dL^{*}}{d\gamma }\mid _{\gamma =\gamma ^{m}}$, I obtain that

$$\begin{aligned} {{\mathrm{sign}}}\left\{ \frac{dL^{*}}{d\gamma }\mid _{\gamma =\gamma ^{m}} \right\} ={{\mathrm{sign}}}\left\{ \gamma ^{m}-\gamma ^{*}\right\} = {{\mathrm{sign}}}\left\{ \alpha ^{*}-\alpha -\frac{(1-\alpha ^{*})(1-\alpha ) (\kappa _{1}^{2}\tau _{v}\tau _{\xi })}{((1-\alpha )\tau _{\xi }+\tau ^{m})^{2}}\right\} , \end{aligned}$$

(A4)

which implies that ${{\mathrm{sign}}}\{\gamma ^{m}-\gamma ^{*}\}={{\mathrm{sign}}}\{\hat{\alpha }-\alpha \},$ where $\hat{\alpha }=\alpha ^{*}-\frac{(1-\alpha ^{*})(1-\alpha )(\kappa _{1}^{2}\tau _{v}\tau _{\xi })}{((1-\alpha )\tau _{\xi }+\tau ^{m})^{2}}$. $\square$

Proof of Proposition 4

Follows immediately from Propositions 2 and 3. $\square$

Proof of Proposition 5

Substituting $\gamma ^{m}$ into (9), I obtain:

$$\begin{aligned} WL=\frac{\mid W_{\sigma \sigma }\mid \kappa _{1}^{2}}{2}\Big (\frac{(1-\alpha )^{2}\tau _{\xi }+(1-\alpha ^{*})\tau ^{m}}{((1-\alpha )\tau _{\xi }+\tau ^{m})^{2}}+\frac{2\phi (1-\alpha ^{*})\tau ^{m}}{(1-\alpha )\tau _{\xi }+\tau ^{m}}\Big ). \end{aligned}$$

(A5)

Comparative statics of ex ante utility with respect to $\tau _{v}$ are equivalent to the opposite comparative statics of WL. Since $\frac{dWL}{d\tau _{v}}=\Big (\frac{\partial WL}{\partial \tau ^{m}}\Big )_{\gamma \,\hbox {cons}.}\frac{d\tau ^{m}}{d\tau _{v}}$ then

$$\begin{aligned} \Big (\frac{\partial WL}{\partial \tau ^{m}}\Big )_{\gamma ^{m}\,\hbox {cons}.}=\frac{\mid W_{\sigma \sigma }\mid (\kappa _{1})^{2}}{2}\Big (\frac{\iota \tau _{\xi }+\rho \tau ^{m}}{(\tau _{\xi }(1-\alpha )+\tau ^{m})^{3}}\Big ), \end{aligned}$$

(A6)

where $\iota$ and $\rho$ are defined in Proposition 5. Recall from Corollary 2 that $0<\frac{d\tau ^{m}}{d\tau _{v}}$. Hence, I find that $\iota \gtrless 0\Leftrightarrow I=\frac{2\alpha -\alpha ^{*}-1}{2(1-\alpha ^{*})}\gtrless \phi$ and $\rho \gtrless 0\Leftrightarrow R=\frac{-1}{2}\gtrless \phi$. $\square$

Proof of Proposition 6

Noting that $\frac{dWL}{d\tau _{\xi }}=\Big (\frac{\partial WL}{\partial \tau _{\xi }}\Big )_{\tau ^{m}\,\hbox {cons}.}+\Big (\frac{\partial WL}{\partial \tau ^{m}}\Big )_{\gamma \,\hbox {cons}.}\frac{d\tau ^{m}}{d\tau _{\xi }}$, I obtain:

$$\begin{aligned} \Big (\frac{\partial WL}{\partial \tau _{\xi }}\Big )_{\tau ^{m}\,\hbox {cons}.}=\frac{\mid W_{\sigma \sigma }\mid (\kappa _{1})^{2}}{2}\Big (\frac{\varphi \tau _{\xi }+\chi \tau ^{m}}{(\tau _{\xi }(1-\alpha )+\tau ^{m})^{3}}\Big ), \end{aligned}$$

(A7)

where $\varphi$ and $\chi$ are defined in Proposition 6. Rearranging the previous equation, I obtain $\frac{d(E[u])}{d\tau _{\xi }}\gtrless 0\Leftrightarrow (\varphi +\iota \frac{d\tau ^{m}}{d\tau _{\xi }})\tau _{\xi }+(\chi +\rho \frac{d\tau ^{m}}{d\tau _{\xi }})\tau ^{m}\lessgtr 0$. Furthermore, $(\varphi +\iota \frac{d\tau ^{m}}{d\tau _{\xi }})\gtrless 0\Leftrightarrow F=\frac{-(1-\alpha )^{2}+(2\alpha -\alpha ^{*}-1)\frac{d\tau ^{m}}{d\tau _{\xi }}}{2(1-\alpha ^{*})((1-\alpha )+\frac{d\tau ^{m}}{d\tau _{\xi }})}\gtrless \phi ,$ and $(\chi +\rho \frac{d\tau ^{m}}{d\tau _{\xi }})\gtrless 0\Leftrightarrow C=\frac{(1-\alpha )(2\alpha ^{*}-\alpha -1)-(1-\alpha ^{*})\frac{d\tau ^{m}}{d\tau _{\xi }}}{2(1-\alpha ^{*})((1-\alpha )+\frac{d\tau ^{m}}{d\tau _{\xi }})}\gtrless \phi$. Notice that the denominators of F and C are always strictly positive. $\square$

Proof of Proposition 7

• (i) Cutoffs between types of games. Using the definitions of the cutoffs, I obtain the following. If $\alpha >\alpha ^{*}$ then $I>F>R>C$; if $\alpha ^{*}>\alpha$ then $C>R>F>I$; and if $\alpha ^{*}=\alpha$ then $C=R=F=I$. Manipulating the expressions previously defined, I obtain: $F>F_{\mathrm{exo}}$, $C>C_{\mathrm{exo}}$ $\Leftrightarrow (\alpha -\alpha ^{*})\frac{d\tau ^{m}}{d\tau _{\xi }}>0$. Hence, ${{\mathrm{sign}}}\{\alpha -\alpha ^{*}\}={{\mathrm{sign}}}\{F-F_{\mathrm{exo}}\}={{\mathrm{sign}}}\{C-C_{\mathrm{exo}}\},$ and the result follows.

• (ii) Cutoffs within types of games. Games of type $+{ III}$ have: $\iota>0,\varphi +\iota \frac{d\tau ^{m}}{d\tau _{\xi }}>0$ and $\rho<0,\chi +\rho \frac{d\tau ^{m}}{d\tau _{\xi }}<0,\chi <0$. Hence $\hat{X}=\frac{-\Big (\varphi +\iota \frac{d\tau ^{m}}{d\tau _{\xi }}\Big )}{\chi +\rho \frac{d\tau ^{m}}{d\tau _{\xi }}}>\frac{-\varphi }{\chi }=\hat{X}_{\mathrm{exo}}$ since $(\varphi \rho -\iota \chi )\frac{d\tau ^{m}}{d\tau _{\xi }}>0$ because $0<\varphi \rho -\iota \chi$. Similar arguments follow for games of types $+{ IV},-{ III},-{ IV}$ since: $+{ IV}$ that have $\iota>0,\varphi +\iota \frac{d\tau ^{m}}{d\tau _{\xi }}>0$ and $\rho >0,\chi +\rho \frac{d\tau ^{m}}{d\tau _{\xi }}<0,\chi <0$; games of type$-{ III}$ that have $\iota<0,\varphi +\iota \frac{d\tau ^{m}}{d\tau _{\xi }}<0$ and $\rho>0,\chi +\rho \frac{d\tau ^{m}}{d\tau _{\xi }}>0,\chi >0$; and games of type $-{ IV}$ that have $\iota<0,\varphi +\iota \frac{d\tau ^{m}}{d\tau _{\xi }}<0$ and $\rho <0,\chi +\rho \frac{d\tau ^{m}}{d\tau _{\xi }}>0,\chi >0$.

$\square$

Proof of Corollary 4

(i) Follows from Proposition 3 by noting that $\gamma ^{m}<\gamma ^{*}$ will always hold since $\alpha >\alpha ^{*}$. (ii) The parameters which determine the social value of information are as follows: $\rho =-1<0$, $\iota =-(1-\alpha )(1+2\alpha )$, $\varphi =-(1-\alpha )(1+\alpha )$, $\chi =-(1-\alpha )^{2}<0$ since $\phi =\frac{\alpha }{1-2\alpha }$ and $\alpha ^{*}=2\alpha<\alpha <0$. Note that $R=\frac{-1}{2}<\phi <0$. From Table 1, note that the game can be of types $+I,+{ II},+{ III}$. If $-\frac{1}{2}\le \alpha <0$ then both $\hat{Y},\hat{X}<0$ and the game is of type $+I$; if $\alpha <-1$ then both $\hat{Y},\hat{X}>0$ and the game is of type $+{ III}$. Due to the endogenous public signal, the range defined by $-1\le \alpha <\frac{-1}{2}$ gets divided into two: (1) if $\hat{X}_{\mathrm{exo}}<\hat{X}\le 0<\hat{Y}$ then the game is of type $+{ II}$; (2) if $\hat{X}_{\mathrm{exo}}\le 0<\hat{X}<\hat{Y}$ then the game is of type $+{ III}$. The cutoff between types $+{ II}$ and $+{ III}$ changes with endogenous public information and is given by ${{\mathrm{sign}}}\{\varphi +\iota \frac{d\tau ^{m}}{d\tau _{\xi }}\}={{\mathrm{sign}}}\{F-\phi \}$, where $F=\frac{-(1-\alpha )^{2}-\frac{d\tau ^{m}}{d\tau _{\xi }}}{2(1-2\alpha )((1-\alpha )+\frac{d\tau ^{m}}{d\tau _{\xi }})}$. Then $\phi>F\Leftrightarrow \Delta (\alpha )=(1-\alpha ^{2})+\frac{d\tau ^{m}}{d\tau _{\xi }}(1+2\alpha )>0$. Note that $\Delta (-1)<0$, $\Delta (-\frac{1}{2})>0$ and $\Delta (\alpha )$ is a strictly increasing function of $\alpha$ since some regularity conditions regarding $\frac{d\tau ^{m}}{d\tau _{\xi }}$ are satisfied. Hence, there a exists a unique $\vartheta$ such that $\Delta (\vartheta )=0$, where $-1<\vartheta <\frac{-1}{2}$. Then, if $\vartheta \le \alpha <\frac{-1}{2}$ the game is of type $+{ II}$, and if $\alpha <\vartheta$ then the game is of type $+{ III}$. From Proposition 7, the cutoff within a type $+{ III}$ game changes due to endogenous public information since in this region: $\hat{X}_{\mathrm{exo}}<\hat{X}$. $\square$

Proof of Corollary 5

• (i) Follows from Proposition 3 and by noting that $\alpha ^{*}>\alpha$. When the information externality is large, the following occurs: there exists an $\alpha ^{o}=\alpha ^{*}-\frac{(1-\alpha ^{*})(1-\alpha )(\kappa _{1}^{2}\tau _{v}\tau _{\xi })}{((1-\alpha )\tau _{\xi }+\tau ^{m})^{2}}>0$ because $\gamma (\alpha )$ and $\gamma ^{*}(\alpha ^{*})$, respectively, are both strictly decreasing functions of $\alpha$, which range from 1 to 0 as $\alpha ,\alpha ^{*}\rightarrow -\infty$ to $\alpha ,\alpha ^{*}\rightarrow 1$. Using the intermediate value theorem and Proposition 3, a unique intersection exists between the two curves, $\alpha ^{o}$, which satisfies $0<\alpha ^{o}<1$. Hence, ${{\mathrm{sign}}}\{\gamma ^{m}-\gamma ^{*}\}={{\mathrm{sign}}}\{\alpha -\alpha ^{o}\}$.

• (ii) Since $\phi >0$ and $\alpha ^{*}>\alpha >0$ then the game has:

  $$\begin{aligned} \rho= & {} \frac{-c(2c+1)b^{2}+4c(1+c)b-(c+1)(2c+1)}{(2c+1)(1+c)}, \\ \iota= & {} \frac{-((2-b)+2c(1-b))((b+1)+c(3-b^{2})+2c^{2}(1-b^{2}))}{2(1+c)^{2}(2c+1)},\\ \varphi= & {} \frac{-(b-2-2c+2bc)^{2}((b+2)+2c(3-2b)+4c^{2}(1-b))}{8(2c+1)(1+c)^{3}},\\ \chi= & {} \frac{-(2c(1-b)+(2-b))^{2}(2c(1-2b)+1)}{4(1+c)^{2}(2c+1)}, \end{aligned}$$

  where $\rho$,$\iota$, $\varphi$ are negative, and $\chi$ might be positive or negative. $\chi >0$ if and only if parameters (b, c) belong to set B defined in Corollary 5. With endogenous public information, there exists a non-empty region such that $\chi >0$ and $\chi +\rho \frac{d\tau ^{m}}{d\tau _{\xi }}<0$ if and only if the payoff parameters (b, c) and information parameters $(\tau _{\xi },\tau _{\theta },\tau _{u})$ jointly belong to set A defined in Corollary 5. Note that $A\subset B$. Hence, the game is of type $-{ IV}$ if $A^{c}\cap B$, and of type $+I$ if $A\cup B^{c}$.

$\square$