Real options signaling game models for dynamic acquisition under information asymmetry

Abstract

We construct a real options signaling game model to analyze the impact of asymmetric information on the dynamic acquisition decision made by the aggressive acquirer firm and passive target firm in the takeover terms and timing. The target firm is assumed to have partial information on the synergy factor of the acquirer firm in generating the surplus value. Our dynamic acquisition game models are based on the market valuation of the surplus value of the acquirer and target firms, where the restructuring opportunities are modeled as exchange options. We analyze the various forms of equilibrium strategies on the deal and timing of takeover in the acquisition game and provide the mathematical characterization of the pooling and separating strategies adopted by the acquirer firm. We also determine the terms of takeover in the signaling game under varying levels of information asymmetry and synergy.

Appendices

Appendix A: Proof of Lemma 2

Recall from Eq. (2.4a) that $F_A(R;\omega ,\xi )$ can be expressed as a bilinear function in R and $\xi$, where

$$\begin{aligned} F_A(R;\omega ,\xi )=\xi (A_{\omega }R-B)-K_A R, \end{aligned}$$

with $A_{\omega }=K_A+\alpha (K_A+K_T)\omega >0$ and $B=\alpha (K_A+K_T)-K_T>0$. Note that for $\xi \in (\underline{\xi }^m,\bar{\xi }^m)$, the optimal takeover threshold $R_u^*(\xi )$ lies on the indifference curve and satisfies

$$\begin{aligned} F_A(R_u^*(\xi );\omega _L,\xi )\left[ \frac{R}{R_u^*(\xi )}\right] ^{\beta }=F_A(R_L^c;\omega _L,\xi _L^c)\left( \frac{R}{R_L^c}\right) ^{\beta }. \end{aligned}$$

Suppose we write $F_A(R_u^*(\xi );\omega _L,\xi )=\xi [A_{\omega _L}R_u^*(\xi )-B]-K_AR_u^*(\xi )$, the above equation can be rewritten as

$$\begin{aligned} \xi =\frac{F_A(R_L^c;\omega _L,\xi _L^c)\left[ \frac{R_u^*(\xi )}{R_L^c}\right] ^{\beta }+K_AR_u^*(\xi )}{A_{\omega _L}R_u^*(\xi )-B}. \end{aligned}$$

By combining the above relations and observing the following properties: (i) $\displaystyle {\frac{A_{\omega _H}R-B}{A_{\omega _L}R-B}}$ is a decreasing function with respect to R, (ii) $R_u^*(\xi )>R_T^*(\xi ,\omega _H)>R_T^*(\underline{\xi }^m,\omega _H)$ for $\xi \in (\underline{\xi }^m,\bar{\xi }^m)$ , we obtain

$$\begin{aligned}&F_A(R_u^*(\xi );\omega _H,\xi )\left[ \frac{R}{R_u^*(\xi )}\right] ^{\beta }\\&\quad =\,\frac{A_{\omega _H}R_u^*(\xi )-B}{A_{\omega _L}R_u^*(\xi )-B}\left\{ F_A(R_L^c;\omega _L,\xi _L^c)\left[ \frac{R_u^*(\xi )}{R_L^c}\right] ^{\beta }+K_AR_u^*(\xi )\right\} \left[ \frac{R}{R_u^*(\xi )}\right] ^{\beta }\\&\qquad -K_AR_u^*(\xi )\left[ \frac{R}{R_u^*(\xi )}\right] ^{\beta }\\&\quad <\,\frac{A_{\omega _H}R_T^*(\underline{\xi }^m,\omega _H)-B}{A_{\omega _L}R_T^*(\underline{\xi }^m,\omega _H)-B}\left\{ F_A(R_L^c;\omega _L,\xi _L^c)\left[ \frac{R_u^*(\xi )}{R_L^c}\right] ^{\beta }+K_AR_u^*(\xi )\right\} \left[ \frac{R}{R_u^*(\xi )}\right] ^{\beta }\\&\qquad -K_AR_u^*(\xi )\left[ \frac{R}{R_u^*(\xi )}\right] ^{\beta }. \end{aligned}$$

Furthermore, by noting $\beta >1$ and $R_u^*(\xi )>R_T^*(\underline{\xi }^m,\omega _H)$, we deduce that

$$\begin{aligned}&F_A(R_u^*(\xi );\omega _H,\xi )\left[ \frac{R}{R_u^*(\xi )}\right] ^{\beta }\\&\quad<\,\frac{A_{\omega _H}R_T^*(\underline{\xi }^m,\omega _H)-B}{A_{\omega _L}R_T^*(\underline{\xi }^m,\omega _H)-B}\left[ F_A(R_L^c;\omega _L,\xi _L^c)\left( \frac{R}{R_L^c}\right) ^{\beta }\right] \\&\qquad +\left[ \frac{A_{\omega _H}R_T^*(\underline{\xi }^m,\omega _H)-B}{A_{\omega _L}R_T^*(\underline{\xi }^m,\omega _H)-B}-1\right] K_AR_u^*(\xi )\left[ \frac{R}{R_u^*(\xi )}\right] ^{\beta }\\&\quad <\,\frac{A_{\omega _H}R_T^*(\underline{\xi }^m,\omega _H)-B}{A_{\omega _L}R_T^*(\underline{\xi }^m,\omega _H)-B}\left[ F_A(R_L^c;\omega _L,\xi _L^c)\left( \frac{R}{R_L^c}\right) ^{\beta }\right] \\&\qquad +\left[ \frac{A_{\omega _H}R_T^*(\underline{\xi }^m,\omega _H)-B}{A_{\omega _L}R_T^*(\underline{\xi }^m,\omega _H)-B}-1\right] K_AR_T^*(\underline{\xi }^m,\omega _H)\left[ \frac{R}{R_T^*(\underline{\xi }^m,\omega _H)}\right] ^{\beta }\\&\quad =\,\frac{A_{\omega _H}R_T^*(\underline{\xi }^m,\omega _H)-B}{A_{\omega _L}R_T^*(\underline{\xi }^m,\omega _H)-B}\left[ F_A(R_T^*(\underline{\xi }^m,\omega _H);\omega _L,\underline{\xi }^m)\left( \frac{R}{R_T^*(\underline{\xi }^m,\omega _H)}\right) ^{\beta }\right] \\&\qquad +\left[ \frac{A_{\omega _H}R_T^*(\underline{\xi }^m,\omega _H)-B}{A_{\omega _L}R_T^*(\underline{\xi }^m,\omega _H)-B}-1\right] K_AR_T^*(\underline{\xi }^m,\omega _H)\left[ \frac{R}{R_T^*(\underline{\xi }^m,\omega _H)}\right] ^{\beta }. \end{aligned}$$

The last equality is established by observing that $(\underline{\xi },R_T^*(\underline{\xi }^m,\omega _H))$ lies in the intersection between the indifference curve and the curve of $R_T^*(\xi ,\omega _H)$.

Lastly, by expressing $F_A(R_T^*(\underline{\xi }^m,\omega _H);\omega _H:,\underline{\xi }^m)$ as a bilinear form of $\xi$ and R, we finally obtain

$$\begin{aligned}&F_A(R_u^*(\xi );\omega _H,\xi )\left[ \frac{R}{R_u^*(\xi )}\right] ^{\beta }\\&\quad <\,\left\{ \underline{\xi }^m\left[ A_{\omega _H}R_T^*(\underline{\xi }^m,\omega _H)-B\right] -K_AR_T^*(\underline{\xi }^m,\omega _H)\right\} \left[ \frac{R}{R_T^*(\underline{\xi }^m,\omega _H)}\right] ^{\beta }\\&\quad =\,F_A(R_T^*(\underline{\xi }^m,\omega _H);\omega _H,\underline{\xi }^m)\left[ \frac{R}{R_T(\underline{\xi }^m,\omega _H)}\right] ^{\beta }. \end{aligned}$$

Hence, we establish Eq. (3.10) and so the takeover package $(\xi ,R_u^*(\xi ))$, where $\xi \in (\underline{\xi }^m,\bar{\xi }^m)$, is dominated by the takeover package $(\underline{\xi }^m,R_T^*(\underline{\xi }^m,\omega _H))$.

Appendix B: Proof of Proposition 2

To establish the proof, it is necessary to use the following equivalent condition under which Eq. (3.16) holds. Suppose that $\xi _H^c\in (\underline{\xi }^m,\bar{\xi }^m)$, Eq. (3.16) holds if and only if

$$\begin{aligned}&\max _{\xi \in [\max (\underline{\xi }^m,\underline{\xi }^p),\bar{\xi }^p]}F_A(R_T^*(\xi ,\omega _p);\omega _H,\xi )\left[ \frac{R}{R_T^*(\xi ,\omega _p)}\right] ^{\beta }\nonumber \\&\quad >F_A(R_T^*(\underline{\xi }^m,\omega _H);\omega _H,\underline{\xi }^m)\left[ \frac{R}{R_T^*(\underline{\xi }^m,\omega _H)}\right] ^{\beta }. \end{aligned}$$

(B.1)

The proof of Eq. (B.1) is presented at the end of this Appendix.

We write the left-hand side of Eq. (B.1) as a function of p, where

$$\begin{aligned} f(p)=\max _{\xi \in [\max (\underline{\xi }^m,\underline{\xi }^p),\bar{\xi }^p]}F_A(R_T^*(\xi ,\omega _p);\omega _H,\xi )\left[ \frac{R}{R_T^*(\xi ,\omega _p)}\right] ^{\beta }. \end{aligned}$$

The right-hand side of Eq. (B.1) is independent of p, which is written as

$$\begin{aligned} F_A^s=F_A(R_T^*(\underline{\xi }^m,\omega _H);\omega _H,\underline{\xi }^m)\left[ \frac{R}{R_T^*(\underline{\xi }^m,\omega _H)}\right] ^{\beta }. \end{aligned}$$

In order to show that Eq. (B.1) holds when p is sufficiently large, it suffices to show that

• (a) f(p) is strictly increasing with respect to p;

• (b) $f(0)<F_A^s$;

• (c) $f(1)>F_A^s$.

To show property (a), we use the following technical results: (i) the interval $[\max (\underline{\xi }^m,\underline{\xi }^p),\bar{\xi }^p]$ becomes widened with an increasing value of p; (ii) for $0<p_2<p_1\le 1$ and $\xi >\xi ^{\ell }$, by virtue of Lemma 1, we have $R_T^*(\xi ,\omega _{p_2})>R_T^*(\xi ,\omega _{p_1})\ge R_T^*(\xi ,\omega _{H})>R_A^*(\xi ,\omega _H)$. Using these properties, we deduce that for $p_1>p_2$,

$$\begin{aligned} f(p_1)&=\,\max _{\xi \in [\max (\underline{\xi }^m,\underline{\xi }^{p_1}),\bar{\xi }^{p_1}]}F_A(R_T^*(\xi ,\omega _{p_1});\omega _H,\xi )\left[ \frac{R}{R_T^*(\xi ,\omega _{p_1})}\right] ^{\beta }\\&\ge \,\max _{\xi \in [\max (\underline{\xi }^m,\underline{\xi }^{p_2}),\bar{\xi }^{p_2}]}F_A(R_T^*(\xi ,\omega _{p_1});\omega _H,\xi )\left[ \frac{R}{R_T^*(\xi ,\omega _{p_1})}\right] ^{\beta }\\&>\,\max _{\xi \in [\max (\underline{\xi }^m,\underline{\xi }^{p_2}),\bar{\xi }^{p_2}]}F_A(R_T^*(\xi ,\omega _{p_2});\omega _H,\xi )\left[ \frac{R}{R_T^*(\xi ,\omega _{p_2})}\right] ^{\beta }>f(p_2). \end{aligned}$$

To show property (b), we observe that when $p=0$, the curve $R^*=R_T^*(\xi ,\omega _p)=R_T^*(\xi ,\omega _L)$ touches the indifference curve at $(\xi _L^c,R_L^c)$. Since $\xi _L^c\in (\underline{\xi }^m,\bar{\xi ^m})$, we deduce that

$$\begin{aligned} f(0)=\,F_A(R_L^c;\omega _H,\xi _L^c)\left( \frac{R}{R_L^c}\right) ^{\beta }&=\,F_A(R_u^*(\xi _L^c);\omega _H,\xi _L^c)\left[ \frac{R}{R_u^*(\xi _L^c)}\right] ^{\beta }\\&<\,F_A(R_T^*(\underline{\xi }^m,\omega _H);\omega _H,\underline{\xi }^m)\left[ \frac{R}{R_T^*(\underline{\xi }^m,\omega _H)}\right] ^{\beta }, \end{aligned}$$

by virtue of Eq. (3.10). To show property (c), we observe that when $p=1$, the curve $R^*=R_T^*(\xi ,\omega _p)=R_T^*(\xi ,\omega _H)$ touches the indifference curve at $(\underline{\xi }^m,R_T^*(\underline{\xi }^m,\omega _H))$ and $(\bar{\xi }^m,R_T^*(\bar{\xi }^m,\omega _H))$. Since $\xi _H^c\in (\underline{\xi }^m,\bar{\xi ^m})$, we deduce that

$$\begin{aligned} f(1)&=\,\max _{\xi \in [\underline{\xi }^m,\bar{\xi }^m]}F_A(R_T^*(\xi ,\omega _H);\omega _H,\xi )\left[ \frac{R}{R_T^*(\xi ,\omega _H)}\right] ^{\beta }\\&=\,F_A(R_T^*(\xi _H^c,\omega _H);\omega _H,\xi _H^c)\left[ \frac{R}{R_T^*(\xi _H^c,\omega _H)}\right] ^{\beta }\\&\quad >\,F_A(R_T^*(\underline{\xi }^m,\omega _H);\omega _H,\underline{\xi }^m)\left[ \frac{R}{R_T^*(\underline{\xi }^m,\omega _H)}\right] ^{\beta }. \end{aligned}$$

Once properties (a), (b) and (c) have been established, we can deduce that there exists an unique $p_0\in (0,1)$ such that $f(p_0)=F_A^s$ and $f(p)>F_A^s$ if and only if $p>p_0$. Hence, Proposition 2 is established.

Proof of Eq. (B.1)

The equivalence property between Eqs. (3.16) and (B.1) is trivial when $\underline{\xi }^m\le \underline{\xi }^p$. When $\underline{\xi }^m>\underline{\xi }^p$, by virtue of Lemma 1, we have $R_T^*(\xi ,\omega _p)\ge R_T^*(\xi ,\omega _H)>R_A^*(\xi ,\omega _H)$ for any $\xi \in [\underline{\xi }^p,\underline{\xi }^m]$. Using this fact, for any $\xi \in [\underline{\xi }^p,\underline{\xi }^m]$, we have

$$\begin{aligned} F_A(R_T^*(\xi ,\omega _p);\omega _H,\xi )\left[ \frac{R}{R_T^*(\xi ,\omega _p)}\right] ^{\beta }<\,&F_A(R_T^*(\xi ,\omega _H);\omega _H,\xi )\left[ \frac{R}{R_T^*(\xi ,\omega _H)}\right] ^{\beta }\nonumber \\ <\,&F_A(R_T^*(\underline{\xi }^m,\omega _H);\omega _H,\underline{\xi }^m)\left[ \frac{R}{R_T^*(\underline{\xi }^m,\omega _H)}\right] ^{\beta }. \end{aligned}$$

(B.2)

The last inequality is established by observing that $M_H^s=(\underline{\xi }^m,R_T^*(\underline{\xi }^m,\omega _H))$ is the least-cost separating equilibrium. Based on Eq. (B.2), we deduce that Eq. (B.1) holds if Eq. (3.16) holds. Conversely, it is quite trivial to show that if Eq. (B.1) holds, then Eq. (3.16) holds.