Learning and collusion in new markets with uncertain entry costs

Abstract

This paper analyzes an entry timing game with uncertain entry costs. Two firms receive costless signals about the cost of a new project and decide when to invest. We characterize the equilibrium of the investment timing game with private and public signals. We show that competition leads the two firms to invest too early and analyze two collusion schemes, one in which one firm pays the other to stay out of the market and one in which this buyout is mediated by a third party. We characterize conditions under which the efficient outcome can be implemented in both collusion schemes.

Appendix: Proofs

1.1 Proof of Lemma 1

Clearly, if the other firm has already entered, it is a dominant strategy to enter immediately as $\pi _d > \underline{\theta }$. Suppose that the other firm has not entered yet, and consider a firm which learns that its cost is low at period $t$. By delaying investment to period $t+1$, the firm loses the positive profit made between $t{\varDelta }$ and $(t+1){\varDelta }$, equal to $(1-{\delta })\pi _d$ or $(1-{\delta }) \pi _m$, depending on whether the other firm invests at period $t$ or not. In addition, let ${\varDelta }g(t)$ be the probability that a firm which did not learn its cost invests between $t {\varDelta }$ and $(t+1) {\varDelta }$. Recall that by the dominance argument spelled out in the text, a firm that has not learned its cost never enters after the other firm has entered. Hence, by investing at period $t$ a low-cost firm will block the entry of the competitor which has not yet learned its cost, resulting in an expected payoff of

$$\begin{aligned}&\sum _{{\tau }= t+1}^{\infty }(1-{\lambda }{\varDelta })^{{\tau }-1 - t} \frac{{\lambda }{\varDelta }}{2} (\pi _m - \underline{\theta })\nonumber \\&\quad + \sum _{{\tau }=t+1}^{\infty } (1-{\lambda }{\varDelta })^{{\tau }-1-t} \frac{{\lambda }{\varDelta }}{2} (\pi _m (1-{\delta }^{{\tau }-t}) +\pi _d {\delta }^{{\tau }-t}- \underline{\theta }). \end{aligned}$$

By contrast, if the firm waits until period $t+1$ and the competitor invests, it will get an expected payoff of $\pi _d - \underline{\theta }$. Hence, by delaying investment, the firm loses an expected payoff of

$$\begin{aligned} {\varDelta }g(T) (\pi _m - \pi _d) \left[ 1 - \sum _{{\tau }=t+1}^{\infty } (1-{\lambda }{\varDelta })^{{\tau }-1-t} \frac{{\lambda }{\varDelta }}{2}{\delta }^{{\tau }-t}\right] . \end{aligned}$$

1.2 Proof of Proposition 1

We focus attention on the optimal strategy of a firm when the other firm has not received any signal yet. For ${\varDelta }\rightarrow 0^+$, the probability that the other firm has received a signal between period $t-1$ and period $t$ goes to 0. In that case, the investment game played by the two firms is given by the following bi-matrix game, where $W(t+1)$ denotes the continuation value for both players of entering into period $t+1$ Table 5.

Table 5 Investment game played by the two firms if none of them has invested up to period $t$ and costs are not known

We first consider a symmetric equilibrium where both firms invest with positive probability $p \in (0,1)$. In that equilibrium,

$$\begin{aligned} W(t) = p(\pi _d)+(1-p)V_L - \widetilde{\theta }, \end{aligned}$$

and

$$\begin{aligned} W(t) = pV_F + (1-p) W(t+1). \end{aligned}$$

Letting the time between two successive periods go to 0, as the strategies are right continuous, the value function is right continuous and we find

$$\begin{aligned} p = \frac{V_L - \widetilde{\theta }- V_F}{V_L - \pi _d}, \end{aligned}$$

showing that an equilibrium with preemption exists if and only if $V_L - V_F = \pi _m - V_{E1} > \widetilde{\theta }$.

Next, suppose that $V_L - V_F = \pi _m - V_{E1} \le \widetilde{\theta }$. We construct a symmetric equilibrium where both firms choose to delay investment. By delaying investment by one period when the other firm does not invest, a firm obtains an expected payoff of:

$$\begin{aligned} W(t+1)&= e^{-r{\varDelta }}\left[ \left( \frac{{\lambda }{\varDelta }}{2}\right) ^2 (\pi _d - \underline{\theta }) +\frac{{\lambda }{\varDelta }}{2} (1-{\lambda }{\varDelta }) (V_L - \underline{\theta }) + \left( \frac{{\lambda }{\varDelta }}{2}\right) ^2 (\pi _m - \underline{\theta })\right. \\&\left. + \frac{{\lambda }{\varDelta }}{2} (1-{\lambda }{\varDelta }) V_F + \frac{{\lambda }{\varDelta }}{2} (1-{\lambda }{\varDelta }) V_{E1} + (1-{\lambda }{\varDelta })^2 (V_L - \widetilde{\theta })\right] . \end{aligned}$$

Letting ${\varDelta }\rightarrow 0^+$,

$$\begin{aligned} W'(T) = V_L - \widetilde{\theta }+ \frac{\lambda \varDelta }{2}(V_L + V_F - \underline{\theta }+V_{E1}) -(r+2{\lambda }){\varDelta }(V_L - \widetilde{\theta }) + \mathcal{O}(\varDelta ^2). \end{aligned}$$

Now compute

$$\begin{aligned} V_L + V_F - \underline{\theta }+V_{E1}&= \pi _m - \underline{\theta }+ \frac{{\lambda }}{{\lambda }+r} (\pi _d - \underline{\theta }) \\&> \pi _m - \underline{\theta }\\&> 2(\pi _d - \underline{\theta }) \end{aligned}$$

and, as $V_F < V_L - \widetilde{\theta },$

$$\begin{aligned} V_L - \widetilde{\theta }&< V_F \\&= \frac{{\lambda }}{2({\lambda }+r)}(\pi _d - \underline{\theta }), \end{aligned}$$

establishing that $\frac{\lambda \varDelta }{2}(V_L + V_F - \underline{\theta }+V_{E1}) -(r+2{\lambda }){\varDelta }(V_L - \widetilde{\theta }) \ge 0$ and hence, as ${\varDelta }\rightarrow 0^+$, $W'(T)>0$, so that firms always have an incentive to wait.

1.3 Proof of Theorem 1

As in the proof of Proposition 1, we first consider a symmetric equilibrium where both firms invest with positive probability $p(t) \in [0,1]$ at period $t$. Taking ${\varDelta }\rightarrow 0^+$, the probability that the rival firm has received a low-cost signal between two consecutive periods goes to 0, and we compute the expected value of a firm which invests at period $t$ as:

$$\begin{aligned} U(1) = \gamma _t(\widetilde{\theta }) p(t) (\pi _d - \widetilde{\theta }) + \gamma _t(\widetilde{\theta }) (1-p(t)) (V_L - \widetilde{\theta }) +\gamma _t(\overline{\theta })(\pi _m - \widetilde{\theta }), \end{aligned}$$

and the expected value of a firm which does not invest as

$$\begin{aligned} U(0) = \gamma _t(\widetilde{\theta }) p(t) V_F + (\gamma _t(\widetilde{\theta }) (1-p(t))+ \gamma _t(\overline{\theta })) W(t+1). \end{aligned}$$

In a symmetric equilibrium with positive exit probabilities, $W(t)=U(0)=U(1)$ and as ${\varDelta }\rightarrow 0^+$, because the value function is right continuous, $W(t+1) \rightarrow W(t)$, resulting in an exit probability at date $T$

$$\begin{aligned} p(T) = \frac{V_L(T)-V_F - \widetilde{\theta }}{\gamma _T(\widetilde{\theta }) (V_L - \pi _d)}. \end{aligned}$$

This equilibrium exists if and only if $V_L(T) \ge V_F - \widetilde{\theta }$. As $V_L(0)-V_F=\pi _m-V_{E1}$, this shows that in case (i), preemption arises at time zero and in case (ii), firms rush to invest at time $\tilde{T}$. We now consider a situation where $V_L(T) < V_F - \widetilde{\theta }$ and show that firms have an incentive to wait. Suppose that the other firm does not invest and compute the expected utility of waiting one period before investing:

$$\begin{aligned} W(t+1)&= e^{-r{\varDelta }}\left[ \left( \frac{{\lambda }{\varDelta }}{2}\right) ^2 (\pi _d - \underline{\theta }) +\frac{{\lambda }{\varDelta }}{2}\left( 1-\frac{{\lambda }{\varDelta }}{2}\right) \left[ \gamma _{t+1}(\widetilde{\theta }) (V_L - \underline{\theta }) + \gamma _{t+1}(\overline{\theta })(\pi _m - \underline{\theta })\right] \right. \\&\left. + \frac{{\lambda }{\varDelta }}{2} (1-{\lambda }{\varDelta }) V_F + \left( 1 -\frac{{\lambda }{\varDelta }}{2}\right) (1-{\lambda }{\varDelta }) (V_L(t+1) - \widetilde{\theta })\right] +\mathcal{O}({\varDelta }^2). \end{aligned}$$

As ${\varDelta }\rightarrow 0^+$,

$$\begin{aligned} W'(T)&\rightarrow \frac{{\lambda }}{2} [ V_L(T)-\underline{\theta }+ V_F - 3 (V_L(T) - \widetilde{\theta })] - r (V_L(T) - \widetilde{\theta }), \\&= \frac{{\lambda }}{2}(V_F +\widetilde{\theta }- \underline{\theta }) -(r+{\lambda })(V_L(T)- \widetilde{\theta }). \end{aligned}$$

Next,

$$\begin{aligned} \frac{2(r+{\lambda })}{{\lambda }}(V_L(T)- \widetilde{\theta })&< \frac{2(r+{\lambda })}{{\lambda }} V_F, \\&= (\pi _d - \underline{\theta }), \\&< V_F +\widetilde{\theta }- \underline{\theta }, \end{aligned}$$

where the last inequality stems from the assumption $\pi _d < \widetilde{\theta }$. This implies that $W(t+1) > W(t)$, so that the firm has an incentive to wait.

1.4 Proof of Proposition 2

In order to implement the cooperative benchmark, two conditions must be satisfied: (i) no firm must be willing to enter the market at $T < T^*$ if it has not learned its cost and (ii) a firm which learns that it has a low cost must be willing to enter the market immediately. The first condition will hold as long as:

$$\begin{aligned} U_F(T) > U_L(T) - \widetilde{\theta }. \end{aligned}$$

As $U_L(T)=\pi _m - U_F(T)$, this results in

$$\begin{aligned} 2 U_F(T) > \pi _m - \widetilde{\theta }. \end{aligned}$$

For the second condition to hold, we characterize the conditions under which an equilibrium where a firm immediately invests after it observes that its cost is low exists. The discounted expected payoff of investing at period $t$ when the other firm does not invest is:

$$\begin{aligned} W(t) = U_L(t) - \underline{\theta }, \end{aligned}$$

whereas by waiting one period the firm will obtain a discounted expected payoff of

$$\begin{aligned} W(t+1) = e^{-r\varDelta }\left[ \left( 1-\frac{e^{-{\lambda }\varDelta }}{2}\right) U_F(t+1) + \frac{e^{-{\lambda }\varDelta }}{2} U_L(t+1)\right] . \end{aligned}$$

For $\varDelta$ small enough and assuming that utilities are differentiable,

$$\begin{aligned} W'(T)= \varDelta [ (- 2r - {\lambda })U_L(T) + {\lambda }U_F(T) + U'_L(T)] \end{aligned}$$

so that the firm has an incentive to enter immediately if and only if:

$$\begin{aligned} 2 U_F(T) < \frac{2 r + {\lambda }}{r+{\lambda }} \pi _m + \frac{U'_F(T)}{r+{\lambda }}. \end{aligned}$$

1.5 Derivation of expected payoffs when ${\varDelta }\rightarrow 0^+$

Recall that, as ${\varDelta }\rightarrow 0^+$,

$$\begin{aligned} (1-{\lambda }{\varDelta })^{\frac{1}{{\varDelta }}} \rightarrow e^{- {\lambda }}, \end{aligned}$$

and that the instantaneous probability that the signal is received between two periods $t{\varDelta }$ and $(t+1) {\varDelta }$ converges to ${\lambda }$. Hence, we compute the expected payoff of a firm with high cost as

$$\begin{aligned} EV_T(\overline{\theta }) = \gamma _T(\widetilde{\theta }) \int \limits _{T}^{\infty } \frac{{\lambda }}{2} e^{-({\lambda }+ r) (t-T)} \sigma _t \mathrm{d}t. \end{aligned}$$

Similarly,

$$\begin{aligned} EV_T(\widetilde{\theta })&= \gamma _T(\overline{\theta }) \int \limits _{T}^{\infty } \frac{{\lambda }}{2} e^{-({\lambda }+r)(t-T)} \left( \pi _m - \underline{\theta }- \sigma _t\right) \mathrm{d}t \\&{+} \gamma _T(\widetilde{\theta }) \int \limits _T^{\infty } \frac{{\lambda }}{2} e^{{-}(2 {\lambda }+ r)(t-T)} \left[ \pi _m - \underline{\theta }{+} \int \limits _t^{\infty } \frac{{\lambda }}{2} e^{-({\lambda }{+}r)(\tau -t)} (\pi _m-\underline{\theta }) \mathrm{d}\tau \right] \mathrm{d}t, \\&= \gamma _T(\overline{\theta }) \int \limits _{T}^{\infty } \frac{{\lambda }}{2} e^{-({\lambda }+r)(t-T)} (\pi _m - \underline{\theta }- \sigma _t) \mathrm{d}t \nonumber \\&+ \gamma _T(\widetilde{\theta })(\pi _m - \underline{\theta })\frac{{\lambda }}{2(2{\lambda }+ r)}\left[ 1+\frac{{\lambda }}{2({\lambda }+ r)}\right] . \end{aligned}$$

Finally, we compute $IC (\underline{\theta }\rightarrow \widetilde{\theta })$ as ${\varDelta }\rightarrow 0^+$ using the fact that $\sigma _{T+1} \rightarrow \sigma _T$ and $\delta = e^{-r{\varDelta }} \equiv 1 - r {\varDelta }$,

$$\begin{aligned}&\frac{{\lambda }{\varDelta }}{2} \frac{\pi _m - \underline{\theta }}{2} + \left( 1 - \frac{{\lambda }{\varDelta }}{2}\right) (\pi _m - \underline{\theta }- \sigma _T) \ge \frac{{\lambda }{\varDelta }}{2} \sigma _T \\&\quad + \left( 1- \frac{{\lambda }{\varDelta }}{2}\right) (1-r{\varDelta }) \left[ \frac{{\lambda }{\varDelta }}{2} \frac{\pi _m - \underline{\theta }}{2} + (1- \frac{{\lambda }{\varDelta }}{2}) (\pi _m - \underline{\theta }- \sigma _T)\right] . \end{aligned}$$

Focusing on the terms in the first order in ${\varDelta }$, this inequality becomes

$$\begin{aligned} r (\pi _m - \underline{\theta }- \sigma _T) \ge \frac{{\lambda }}{2} (2 \sigma _T - \pi _m + \underline{\theta }). \end{aligned}$$

1.6 Proof of Proposition 3

When $\sigma _T=0$ for all $T$, the only binding constraint is the constraint $IC (\widetilde{\theta }\rightarrow \underline{\theta })$ which reads:

$$\begin{aligned} \gamma _T(\overline{\theta }) (\pi _m - \underline{\theta }) \frac{{\lambda }}{2({\lambda }+r)} + \gamma _T(\widetilde{\theta })(\pi _m - \underline{\theta })\frac{{\lambda }}{2(2{\lambda }+ r)}\left[ 1+\frac{{\lambda }}{2({\lambda }+ r)}\right]&\ge \pi _m - \widetilde{\theta }. \end{aligned}$$

As

$$\begin{aligned} \frac{{\lambda }}{2({\lambda }+r)} > \frac{{\lambda }}{2(2{\lambda }+ r)}\left[ 1+\frac{{\lambda }}{2({\lambda }+ r)}\right] , \end{aligned}$$

this condition is hardest to satisfy at $T=0$ when $\gamma _T(\widetilde{\theta })=1$, so that the incentive condition is satisfied for all $T$ if and only if

$$\begin{aligned} \frac{{\lambda }(\pi _m - \underline{\theta })}{2(2 {\lambda }+ r)} \left[ 1 + \frac{{\lambda }}{2({\lambda }+r)}\right] \ge \pi _m - \widetilde{\theta }. \end{aligned}$$