Joint market dominance through exclusionary compatibility

Abstract

This paper studies an infinite horizon oligopoly model in markets with network effects and segmented demands. In each period, three firms make compatibility decisions before competing in prices for a newly arrived consumer. The firm that made a sale in the last period provides a better product quality in terms of an installed base consumer, which can be shared with its rivals through compatibility. We show that compatibility can be used as an exclusionary device even though it intensifies short-run price competition when firms are sufficiently patient. Under certain conditions, this is the only stable prediction with respect to a dynamic analog of strong stability in network formation games (Dutta and Mutuswami in J Econ Theory 76:322–344, 1997).

Appendix

The following notations are used throughout the appendix. Given price distribution H, we write S(H) for the support of H, ${\bar{p}}(H)=\inf \{p>0:H([0,p])=1\}$ for the supremum of the support, $\varDelta H(p)=H(p)-\lim _{p'\nearrow p}H(p')\ge 0$ for an atom placed at price p, and $A^\circ$ for the interior of set A. In addition, ${\tilde{V}}(p';{\tilde{s}})$ denotes the expected life-time payoff of a firm with ex-post position ${\tilde{s}}\in \tilde{{\mathcal {S}}}$ when it quotes price $p'$.

Although the proofs of Lemma 1-3 rely on the standard price-undercutting argument, we introduce a complete chain of steps to clarify how this logic extends to our dynamic framework and establishes the uniqueness of equilibrium pricing strategies at each regime.

Proof of Lemma 1

Consider an empty regime where one focal firm with ex-post position ${\tilde{s}}_e^{*}$ faces two firms with ex-post position ${\tilde{s}}_e$. Let $H^{*}$ and H denote the price distributions played by the firms with ex-post positions ${\tilde{s}}_e^{*}$ and ${\tilde{s}}_e$, respectively. Note that each firm obtains a weakly higher continuation payoff when winning the current period competition since $V\left( s_e^{**}\right) \ge V\left( s_e\right)$.

It is clear that ${\bar{p}}\left( H^*\right) \le {\bar{p}}\left( H\right) +v$ since the focal firm cannot win the competition regardless of consumer type when it quotes a price higher than ${\bar{p}}\left( H\right) +v$. Now, suppose that ${\bar{p}}\left( H\right) >0$. If $\varDelta H\left( {\bar{p}}\left( H\right) \right) >0$, then each firm with ex-post position ${\tilde{s}}_e$ has an incentive to slightly undercut ${\bar{p}}\left( H\right)$, which induces a discontinuous increase in the total probability of attracting a consumer. Thus, it must be the case that $\varDelta H\left( {\bar{p}}\left( H\right) \right) =0$. This in turn implies $\varDelta H^*\left( {\bar{p}}\left( H^*\right) \right) =0$ whenever ${\bar{p}}\left( H^*\right) ={\bar{p}}\left( H\right) +v$, since the focal firm cannot win the competition regardless of a consumer type by quoting price ${\bar{p}}\left( H^*\right)$.

On the other hand, since ${\bar{p}}\left( H\right) >0$, there exists $p'\in \left( 0,{\bar{p}}\left( H\right) \right)$ such that

$$\begin{aligned} {\tilde{V}}\left( {\tilde{s}}_e\right)&\ge {\tilde{V}}\left( p';{\tilde{s}}_e\right) > \delta V\left( s_e\right) . \end{aligned}$$

This is because the probability of attracting a consumer is positive even when quoting a positive price. However, this is a contradiction since ${\bar{p}}\left( H^*\right) \le {\bar{p}}\left( H\right) +v$, $\varDelta H\left( {\bar{p}}\left( H\right) \right) =0$, and $\varDelta H^*\left( {\bar{p}}\left( H^*\right) \right) =0$ when ${\bar{p}}\left( H^*\right) ={\bar{p}}\left( H\right) +v$, which imply that the equilibrium life-time payoff for each firm with ex-post position ${\tilde{s}}_e$ is given by ${\tilde{V}}\left( {\tilde{s}}_e\right) =\delta V\left( s_e\right)$ by definition of ${\bar{p}}\left( H\right)$. Thus, it must be the case that ${\bar{p}}\left( H\right) =0$ and $\varDelta H\left( 0\right) =1$. Clearly, the unique best response of the focal firm is to quote price v with probability 1. $\square$

Proof of Lemma 2

Consider a complete regime in which every firm has the same ex-post position, ${\tilde{s}}_c^*$. Let $H^{*}$ denote the symmetric price distribution played by the firms with ex-post position ${\tilde{s}}_c^{*}$. Note that each firm obtains a weakly higher continuation payoff when winning the current period competition since $V\left( s_c^{**}\right) \ge V\left( s_c^*\right)$.

Clearly, if ${\bar{p}}\left( H^*\right) >0$, then $\varDelta H^*\left( {\bar{p}}\left( H^*\right) \right) =0$ since otherwise, each firm could profitably undercut ${\bar{p}}\left( H^*\right)$ by a small amount. In addition, ${\bar{p}}\left( H^*\right) >0$ implies that there exists $p'\in \left( 0,{\bar{p}}\left( H^*\right) \right)$ such that

$$\begin{aligned} {\tilde{V}}\left( {\tilde{s}}_c^*\right) \ge {\tilde{V}}\left( p';{\tilde{s}}_c^*\right) >\delta V\left( s_c^*\right) . \end{aligned}$$

This is because the probability of attracting a consumer is positive even when quoting a positive price. However, this is a contradiction to $\varDelta H^*\left( {\bar{p}}\left( H^*\right) \right) =0$, which implies ${\tilde{V}}\left( {\tilde{s}}_c^*\right) =\delta V\left( s_c^*\right)$ by definition of ${\bar{p}}\left( H^*\right)$. $\square$

Proof of Lemma 3

Consider a partial regime in which one firm with ex-post position ${\tilde{s}}_p$ faces two firms with ex-post position ${\tilde{s}}_p^*$. Let $H^*$ and H denote the price distributions played by the firms with ex-post positions ${\tilde{s}}_p^*$ and ${\tilde{s}}_p$, respectively. Note that each firm obtains a weakly higher continuation payoff when winning the current period competition given the condition on the continuation payoffs.

First, observe that if ${\bar{p}}\left( H^*\right) >0$, then, $\varDelta H^*\left( p\right) =0$ for all $p\in \left( 0,{\bar{p}}\left( H^*\right) \right]$ by the standard price-undercutting argument. Given this, we next show the following intermediate result.

Claim. (i) If ${\bar{p}}\left( H^*\right) >0$, then ${\bar{p}}\left( H^*\right) =v$; and (ii) $S\left( H^*\right) ^\circ$ is connected.

Proof of Claim

We first show that ${\bar{p}}\left( H^*\right) \le v$. By way of contradiction, suppose that ${\bar{p}}\left( H^*\right) >v$. In this case, the firm with ex-post position ${\tilde{s}}_{p}$ can attract a consumer with a positive probability by quoting a positive price, implying ${\tilde{V}}\left( {\tilde{s}}_p\right) >\delta V\left( s_p\right)$. Now, observe that if ${\bar{p}}\left( H^*\right) \le {\bar{p}}\left( H\right) +v$, then $\varDelta H\left( {\bar{p}}\left( H\right) \right) =0$ because the firm with ex-post position ${\tilde{s}}_p$ cannot win the competition regardless of a consumer type by quoting price ${\bar{p}}\left( H\right)$. As a result, we have $\varDelta H^*\left( {\bar{p}}\left( H^*\right) \right) =\varDelta H\left( {\bar{p}}\left( H\right) \right) =0$, which implies ${\tilde{V}}\left( {\tilde{s}}_p\right) ={\tilde{V}}\left( {\bar{p}}\left( H\right) ;{\tilde{s}}_p\right) =\delta V\left( s_p\right)$ by definition of ${\bar{p}}\left( H\right) .$ Clearly, this is a contradiction. Therefore, if ${\bar{p}}\left( H^*\right) >v$, then ${\bar{p}}\left( H^*\right) >{\bar{p}}\left( H\right) +v$, but in this case, each firm with ex-post position ${\tilde{s}}_p^*$ has no incentive to quote prices arbitrarily close or equal to ${\bar{p}}\left( H^*\right)$: for such prices, they cannot attract any consumer with probabilities arbitrarily close to 1, but quoting price v induces a positive probability of winning. Thus, we conclude that ${\bar{p}}\left( H^*\right) \le v.$ Finally, if $0<{\bar{p}}\left( H^*\right) <v$, each firm with ex-post position ${\tilde{s}}_p^*$ could profitably deviate to quoting price v since

$$\begin{aligned} {\tilde{V}}\left( {\tilde{s}}_p^*\right)&=\mu \left( {\bar{p}}\left( H^*\right) +\delta V\left( s_p^{**}\right) \right) +\left( 1-\mu \right) \delta V\left( s_p^*\right) \\&<\mu \left( v+\delta V\left( s_p^{**}\right) \right) +\left( 1-\mu \right) \delta V\left( s_p^*\right) . \end{aligned}$$

This completes the proof of the first statement.

Next, we show that $S\left( H^*\right) ^\circ$ is connected, which is trivial when ${\bar{p}}\left( H^*\right) =0$. Assuming ${\bar{p}}\left( H^*\right) >0$, note that $S\left( H^*\right) ^\circ \subset \left( 0,{\bar{p}}\left( H^*\right) \right]$ and that $H^*$ places no atom at any price in $\left( 0,{\bar{p}}\left( H^*\right) \right]$. If $S\left( H^*\right) ^\circ$ has a gap, say $\left( p,p'\right)$, then $H^*$ must be constant on $\left( p,p'\right)$, and each firm with ex-post position ${\tilde{s}}_p^*$ has a profitable deviation moving density from p to any prices in $\left( p,p'\right)$. This is a contradiction.

By these observations, $H^*$ must be continuous on $\left( 0,v\right]$, and $S\left( H^*\right) ^\circ$ is an empty or nonempty interval of the form, $\left( {\underline{p}},v\right)$. Letting $\kappa \equiv \frac{2\mu v}{\left( 1-2\mu \right) \delta }$, a tedious computation shows that ${\bar{p}}\left( H^*\right) =0$ if $\varDelta V\ge \kappa$; otherwise, we have ${\bar{p}}\left( H^*\right) =v$, and the firms with ex-post position ${\tilde{s}}_p^*$ will obtain strictly higher expected life-time payoff from quoting price 0 rather than v. By the symmetric argument, we can also check that $\varDelta V<\kappa$ implies ${\bar{p}}\left( H^*\right) > 0$.

Now, recall that if $\varDelta V<\kappa$, then ${\bar{p}}\left( H^*\right) =v$ and $\varDelta H^*\left( {\bar{p}}\left( H^*\right) \right) =0$. Thus, for all p in $S\left( H^*\right) ,$ we have

$$\begin{aligned}&\left( 1-2\mu \right) \left( \left( 1-H^*\left( p\right) \right) \left( p+\delta \varDelta V\right) +\delta V\left( s_p^*\right) \right) +\mu \left( p+\delta V\left( s_p^{**}\right) \right) +\mu \delta V\left( s_p^*\right) \\&=\mu \left( v+\delta V\left( s_p^{**}\right) \right) +\left( 1-\mu \right) \delta V\left( s_p^*\right) . \end{aligned}$$

By solving this equality, we have

$$\begin{aligned} H^*(p)= {\left\{ \begin{array}{ll} 1-\frac{\mu (v-p)}{(1-2\mu )(p+\delta \varDelta V)}, &{} \text {if } {\underline{p}}\le p< v \\ 1 &{} \text {if } p\ge v \end{array}\right. } \end{aligned}$$

where ${\underline{p}}$ is determined by $H^*\left( {\underline{p}}\right) =\varDelta H^*\left( 0\right)$. That is,

$$\begin{aligned} {\underline{p}}=\frac{\mu v-\left( 1-\varDelta H^*\left( 0\right) \right) \left( 1-2\mu \right) \delta \varDelta V}{\left( 1-\varDelta H^*\left( 0\right) \right) \left( 1-2\mu \right) +\mu } \end{aligned}$$

If $\varDelta H^*\left( 0\right) >0$, then $0\in S\left( H^*\right)$, and so

$$\begin{aligned}&\left( 1-2\mu \right) \delta \left( \frac{\varDelta H^*\left( 0\right) }{2}V\left( s_p\right) +\left( 1-\frac{\varDelta H^*\left( 0\right) }{2}\right) V\left( s_p^{**}\right) \right) +\mu \delta \left( V\left( s_p^{**}\right) +V\left( s_p^*\right) \right) \\ &\quad =\mu \left( v+\delta V\left( s_p^{**}\right) \right) +\left( 1-\mu \right) \delta V\left( s_p^*\right) \end{aligned}$$

By rearranging terms, we have $\varDelta H^*\left( 0\right) =2-\frac{2\mu v}{\left( 1-2\mu \right) \delta \varDelta V}$ and hence, $\varDelta H^*\left( 0\right) >0$ if and only if $\varDelta V>\frac{\mu v}{(1-2\mu )\delta }.$ This completes the characterization of $H^*.$$\square$

Proof of Theorem 1

To begin, we derive the expected life-time payoff at each ex-post position under $\sigma ^*$. Note that for the partial regimes, we have ${\tilde{V}}\left( {\tilde{s}}_p\right) =0$ and ${\tilde{V}}\left( {\tilde{s}}_p^*\right) =\frac{\mu v}{1-\delta }$, where the latter follows from

$$\begin{aligned}&{\tilde{V}}\left( {\tilde{s}}_p^*\right) =\mu \left( v+\delta V\left( s_p^{**}\right) \right) +\left( 1-\mu \right) \delta V\left( s_p^*\right) =\mu v+\delta {\tilde{V}}\left( {\tilde{s}}_p^*\right) \end{aligned}$$

Similarly, the expected life-time payoff at ex-post position ${\tilde{s}}_c^*$ is given by

$$\begin{aligned} {\tilde{V}}\left( {\tilde{s}}_c^*\right) =\frac{\delta }{3} V\left( s_c^{**}\right) +\frac{2\delta }{3} V\left( s_c^*\right) =\frac{\delta }{3}{\tilde{V}}\left( {\tilde{s}}_p^*\right) +\frac{2\delta }{3} \left( \frac{1}{2}{\tilde{V}}\left( {\tilde{s}}_p^*\right) +\frac{1}{2}{\tilde{V}}\left( {\tilde{s}}_p\right) \right) =\frac{2\delta }{3}{\tilde{V}}\left( {\tilde{s}}_p^*\right) . \end{aligned}$$

Finally, firms in the empty regimes obtain

$$\begin{aligned} {\tilde{V}}\left( {\tilde{s}}_e^*\right)&=\left( 1-\mu \right) \left( v+\delta V\left( s_e^{**}\right) \right) +\mu \delta V\left( s_e\right) \\&=\left( 1-\mu \right) \left( v+\delta {\tilde{V}}\left( {\tilde{s}}_p^*\right) \right) +\mu \delta \left( \frac{1}{2}{\tilde{V}}\left( {\tilde{s}}_p^*\right) +\frac{1}{2}{\tilde{V}}\left( {\tilde{s}}_p\right) \right) \\&=\frac{2v\left( 1-\mu \right) +\delta \left( 2-\mu \right) {\tilde{V}}\left( {\tilde{s}}_p^*\right) }{2}, \end{aligned}$$

and

$$\begin{aligned} {\tilde{V}}\left( {\tilde{s}}_e\right)&=\frac{\mu \delta }{2} V\left( s_e^{**}\right) +\left( 1-\frac{\mu }{2}\right) \delta V\left( s_e\right) \\&=\frac{\mu \delta }{2}{\tilde{V}}\left( {\tilde{s}}_p^*\right) +\left( 1-\frac{\mu }{2}\right) \delta \left( \frac{1}{2}{\tilde{V}}\left( {\tilde{s}}_p^*\right) +\frac{1}{2}{\tilde{V}}\left( {\tilde{s}}_p\right) \right) \\&=\frac{\delta \left( 2+\mu \right) {\tilde{V}}\left( {\tilde{s}}_p^*\right) }{4}. \end{aligned}$$

Given these, we show that there is no incentive to deviate from $\sigma ^*$, which requires the following conditions: (i) a focal firm has an incentive to share its installed base, ${\tilde{V}}\left( {\tilde{s}}_p^*\right) \ge {\tilde{V}}\left( {\tilde{s}}_e^*\right)$, (ii) a focal firm prefers a partial compatibility regime to a complete compatibility regime, ${\tilde{V}}\left( {\tilde{s}}_p^*\right) \ge {\tilde{V}}\left( {\tilde{s}}_c^*\right)$, and (iii) a non-focal firm with incompatible product is willing to accept partial compatibility, ${\tilde{V}}\left( {\tilde{s}}_p^*\right) \ge {\tilde{V}}\left( {\tilde{s}}_e\right) .$³¹

Note that condition (ii) is trivially satisfied since ${\tilde{V}}\left( {\tilde{s}}_c^*\right) =\frac{2\delta }{3}{\tilde{V}}\left( {\tilde{s}}_p^*\right) <{\tilde{V}}\left( {\tilde{s}}_p^*\right)$. Also, observe that condition (iii) is implied by condition (i) since

$$\begin{aligned} {\tilde{V}}\left( {\tilde{s}}_e^*\right) -{\tilde{V}}\left( {\tilde{s}}_e\right)&=\frac{4v\left( 1-\mu \right) +\delta \left( 2-3\mu \right) {\tilde{V}}\left( {\tilde{s}}_p^*\right) }{4}> 0. \end{aligned}$$

Therefore, it remains to show that condition (i) is satisfied when $\delta$ is high enough. To see this, observe that

$$\begin{aligned} {\tilde{V}}\left( {\tilde{s}}_p^*\right) -{\tilde{V}}\left( {\tilde{s}}_e^*\right)&=\frac{{\tilde{V}}\left( {\tilde{s}}_p^*\right) \left( 2\left( 1-\delta \right) +\delta \mu \right) -2v\left( 1-\mu \right) }{2}. \end{aligned}$$

and so,

$$\begin{aligned} M(\delta ) \equiv 2\left( 1-\delta \right) \left( {\tilde{V}}\left( {\tilde{s}}_p^*\right) -{\tilde{V}}\left( {\tilde{s}}_e^*\right) \right)&=\left( 1-\delta \right) \left( {\tilde{V}}\left( {\tilde{s}}_p^*\right) \left( 2\left( 1-\delta \right) +\delta \mu \right) -2v\left( 1-\mu \right) \right) \\&=v\left( 2 \mu \left( 1-\delta \right) +\delta \mu ^2 -2\left( 1-\delta \right) \left( 1-\mu \right) \right) . \end{aligned}$$

$M(\delta )$ is a linear function with $M(1)=\mu ^2v>0$ and $M(0)=v(4\mu -2)<0$, implying that there exists a unique threshold $\delta ^*\in (0,1)$ such that $M(\delta )\ge 0$ if and only if $\delta \ge \delta ^*.$ Note that, when $\delta \ge \delta ^*$, firms have no incentive to deviate at the pricing stage either by construction. Finally, it is immediate to check from the expression of $M(\delta )$ that $\delta ^*$ is strictly increasing in $\mu$, independent of v, and converges to 1 as $\mu$ tends to 0. $\square$

Proof of Lemma 7

Suppose that there exists an equilibrium $\sigma =\left( \sigma _{comp},\sigma _{price}\right)$ where the complete compatibility regimes are reached with positive probabilities. Note that we have either ${\tilde{V}}\left( {\tilde{s}}_c^*\right) \ge {\tilde{V}}\left( {\tilde{s}}_e^*\right)$ or ${\tilde{V}}\left( {\tilde{s}}_p^*\right) \ge {\tilde{V}}\left( {\tilde{s}}_e^*\right)$ under $\sigma$; otherwise, a focal firm has no incentive to choose $\left( c,c\right)$ at any compatibility regime. In addition, if ${\tilde{V}}\left( {\tilde{s}}_p^*\right) \le {\tilde{V}}\left( {\tilde{s}}_{c}^*\right)$, then ${\tilde{V}}\left( {\tilde{s}}_c^*\right) \ge {\tilde{V}}\left( {\tilde{s}}\right)$ for all ${\tilde{s}}\in G_2$, which implies ${\tilde{V}}\left( {\tilde{s}}_c^*\right) =0$ because ${\tilde{V}}\left( {\tilde{s}}_c^*\right) =\frac{\delta }{3}{V}\left( s_c^{**}\right) +\frac{2\delta }{3}{V}\left( s_c^{*}\right) \le \delta {\tilde{V}}\left( {\tilde{s}}_c^*\right)$. Since this is a contradiction, we have ${\tilde{V}}\left( {\tilde{s}}_p^*\right) > {\tilde{V}}\left( {\tilde{s}}_{c}^*\right)$ under $\sigma$.

Suppose that a complete compatibility regime is directly reached from an empty compatibility regime with a positive probability. Letting $\zeta \in \left( 0,1\right]$ denote the probability that a firm with ex-ante position ${\tilde{s}}_e$ chooses c, this requires

$$\begin{aligned} V(c,c;s_e^{**})&={\zeta }^2{\tilde{V}}\left( {\tilde{s}}_c^*\right) +2\zeta \left( 1-\zeta \right) {\tilde{V}} \left( {\tilde{s}}_p^*\right) +\left( 1-\zeta \right) ^2{\tilde{V}}\left( {\tilde{s}}_e^*\right) \\&\ge {\zeta }{\tilde{V}}\left( {\tilde{s}}_p^*\right) +\left( 1-\zeta \right) {\tilde{V}}\left( {\tilde{s}}_e^*\right) \\&=V\left( c,d;s_e^{**}\right) , \end{aligned}$$

where the left-hand side (resp., right-hand side) is the expected life-time payoff of a firm with ex-ante position $s_e^{**}$ when it chooses $\left( c,c\right)$ (resp., $\left( c,d\right)$). A simple computation shows that

$$\begin{aligned} V\left( c,c;s_e^{**}\right) -V\left( c,d;s_e^{**}\right)&=\zeta \left( \zeta \left( {\tilde{V}}\left( {\tilde{s}}_c^*\right) -{\tilde{V}} \left( {\tilde{s}}_{p}^*\right) \right) +\left( 1-\zeta \right) \left( {\tilde{V}} \left( {\tilde{s}}_p^*\right) -{\tilde{V}}\left( {\tilde{s}}_e^*\right) \right) \right) , \end{aligned}$$

and so

$$\begin{aligned} \zeta \left( {\tilde{V}}\left( {\tilde{s}}_c^*\right) -{\tilde{V}}\left( {\tilde{s}}_{p}^*\right) \right) +\left( 1-\zeta \right) \left( {\tilde{V}}\left( {\tilde{s}}_p^*\right) -{\tilde{V}}\left( {\tilde{s}}_e^*\right) \right) \ge 0. \end{aligned}$$

(1)

Note that this inequality cannot be satisfied if $\zeta =1$ since ${\tilde{V}}\left( {\tilde{s}}_p^*\right) >{\tilde{V}}\left( {\tilde{s}}_c^*\right)$. For the same reason, we must have ${\tilde{V}}\left( {\tilde{s}}_p^*\right) >{\tilde{V}}\left( {\tilde{s}}_e^*\right) .$

Similarly, letting $\beta , \beta ''$ and $\frac{\beta '}{2}$, respectively, denote the probabilities that a firm with ex-ante position ${\tilde{s}}_e^{**}$ chooses $\left( c,c\right)$, $\left( d,d\right)$, and $\left( c,d\right)$, the following equality should be satisfied for each non-focal firm to randomize between c and d with positive probabilities:

$$\begin{aligned} V\left( c;s_e\right)&=\beta \left( \zeta {\tilde{V}}\left( {\tilde{s}}_c^*\right) +\left( 1-\zeta \right) {\tilde{V}}\left( {\tilde{s}}_p^*\right) \right) +\frac{\beta '}{2}{\tilde{V}}\left( {\tilde{s}}_{p}^*\right) \\&\quad +\frac{\beta '}{2}\left( \zeta {\tilde{V}}\left( {\tilde{s}}_p\right) +\left( 1-\zeta \right) {\tilde{V}}\left( {\tilde{s}}_e\right) \right) +\beta ''{\tilde{V}}\left( {\tilde{s}}_e\right) \\&= \beta \left( \zeta {\tilde{V}}\left( {\tilde{s}}_{p}\right) +\left( 1-\zeta \right) {\tilde{V}}\left( {\tilde{s}}_e\right) \right) +\frac{\beta '}{2}{\tilde{V}}\left( {\tilde{s}}_e\right) \\&\quad +\frac{\beta '}{2}\left( \zeta {\tilde{V}}\left( {\tilde{s}}_p\right) +\left( 1-\zeta \right) {\tilde{V}}\left( {\tilde{s}}_e\right) \right) +\beta ''{\tilde{V}}\left( {\tilde{s}}_e\right) \\&=V\left( d;s_e\right) , \end{aligned}$$

where the left-hand side (resp., right-hand side) is the expected life-time payoff of a firm with ex-ante position $s_e$ when it chooses c (resp., d). This is equivalent to

$$\begin{aligned} V\left( c;s_e\right) -V\left( d;s_e\right)&=\beta \left( \zeta \left( {\tilde{V}}\left( {\tilde{s}}_{c}^*\right) -{\tilde{V}}\left( {\tilde{s}}_{p}\right) \right) +\left( 1-\zeta \right) \left( {\tilde{V}}\left( {\tilde{s}}_p^*\right) -{\tilde{V}}\left( {\tilde{s}}_e\right) \right) \right) \\&\quad +\frac{\beta '}{2}\left( {\tilde{V}}\left( {\tilde{s}}_{p}^*\right) -{\tilde{V}}\left( {\tilde{s}}_e\right) \right) \\&=0. \end{aligned}$$

Now, recall that ${\tilde{V}}\left( {\tilde{s}}_e^*\right) >{\tilde{V}}\left( {\tilde{s}}_e\right)$ and ${\tilde{V}}\left( {\tilde{s}}_p^*\right) \ge {\tilde{V}}\left( {\tilde{s}}_p\right)$ by Lemma 6 and that ${\tilde{V}}\left( {\tilde{s}}_p^*\right) >{\tilde{V}}\left( {\tilde{s}}_e^*\right)$ from inequality (1). Since $\zeta \in \left( 0,1\right)$ and $\beta >0$, inequality (1) implies that $V\left( c;s_e\right) -V\left( d;s_e\right)$ is strictly positive, which contradicts to $\zeta \in \left( 0,1\right) .$ Therefore, we conclude that a complete compatibility regime is not directly reached from an empty regime with a positive probability under $\sigma$.

Indeed, a similar argument shows that a complete compatibility regime is not reached from a partial compatibility regime under $\sigma$: that is, if either of two non-focal firms at a partial regime chooses c with probability 1, then a focal firm has no incentive to choose $\left( c,c\right)$, which means that both non-focal firms must randomize between c and d with positive probabilities. Given this, however, each non-focal firm should strictly prefer choosing c to choosing d whenever a focal firm is willing to play $\left( c,c\right)$ with a positive probability. Since the proof is largely similar, we omit further details and computations. $\square$

Proof of Lemma 8

Consider an equilibrium $\sigma$ reaching the partial compatibility regimes with positive probabilities. It is clear that if $G_{\sigma }=G_{\sigma ^*}$, then ${\tilde{V}}\left( {\tilde{s}}_{p}^*\right) =\frac{\mu v}{1-\delta }$ by Lemma 3. In addition, the probability that a complete compatibility regime arises is zero under $\sigma$ by Lemma 7.

To begin with, note that each non-focal firm at an empty regime should play c with probability 1. To see this, observe that

$$\begin{aligned} V\left( c;s_e\right) -V\left( d;s_e\right)&=\frac{\beta '}{2}\left( {\tilde{V}}\left( {\tilde{s}}_{p}^*\right) -{\tilde{V}}\left( {\tilde{s}}_e\right) \right) , \end{aligned}$$

where $V\left( c;s_e\right)$ (resp., $V\left( d;s_e\right)$) denotes the expected life-time payoff of a firm with ex-ante position $s_e$ when it chooses c (resp., d), and $\frac{\beta '}{2}>0$ is the symmetric probability that a firm with ex-ante position $s_e^{**}$ offers compatibility to only one of two non-focal firms. Since a focal firm is willing to achieve product compatibility with one of its rival, it must be the case where ${\tilde{V}}\left( {\tilde{s}}_{p}^*\right) \ge {\tilde{V}}\left( {\tilde{s}}_e^*\right)$. In particular, this implies that $V\left( c;s_e\right) >V\left( d;s_e\right)$ because ${\tilde{V}}\left( {\tilde{s}}_e^*\right) >{\tilde{V}}\left( {\tilde{s}}_e\right)$ by Lemma 6.

Now, we will consider two cases depending on whether the inequality ${\tilde{V}}\left( {\tilde{s}}_{p}^*\right) \ge {\tilde{V}}\left( {\tilde{s}}_e^*\right)$ is strict or not. First, consider the case where ${\tilde{V}}\left( {\tilde{s}}_{p}^*\right) >{\tilde{V}}\left( {\tilde{s}}_e^*\right)$. In this case, a focal firm with ex-ante position $s_e^{**}$ strictly prefers a partial regime to an empty regime, which implies that it assigns probability 0 to $\left( d,d\right)$, and plays $\left( c,d\right)$ and $\left( d,c\right)$ with probability $\frac{1}{2}$. As a result, firms always move to a partial regime from an empty regime with probability 1.

To see the reverse transition from a partial regime to an empty regime, let us denote by $\eta$ (resp. $\zeta$) the probability that a firm with ex-ante position $s_p^{*}$ (resp., $s_p$) chooses c. Note that if either $\eta >0$ or $\zeta >0$, then a firm with ex-ante position $s_p^{**}$ must play each of $\left( d,d\right)$ and $\left( c,c\right)$ with zero probability. Letting $\beta$ (resp., $1-\beta$) be the probability that a firm with ex-ante position $s_p^{**}$ offers compatibility only to a firm with ex-ante position $s_p^*$ (resp. $s_p$), we have

$$\begin{aligned} V\left( c;s_p^*\right) -V\left( d;s_p^*\right) =\beta \left( {\tilde{V}} \left( {\tilde{s}}_p^*\right) -{\tilde{V}}\left( {\tilde{s}}_e\right) \right) , \end{aligned}$$

where $V\left( c;s_p^*\right)$ (resp., $V\left( d;s_p\right)$) is the expected life-time payoff of a firm with ex-ante position $s_p^*$ when it chooses c (resp., d). Thus, if $\beta >0$, then $\eta =1$ since a firm with ex-ante position $s_p^*$ strictly prefers c. In this case, we have: if $\zeta <1$, then it is optimal for a focal firm to choose $\beta =1$, leading to persistent exclusionary regimes, $G_{\sigma }=G_{\sigma ^*}$; whereas if $\zeta =1$, then a focal firm is indifferent between two partial compatibility regimes, which implies that firms alternate across all the partial regimes if and only if $\beta \in \left( 0,1\right) .$ In any case, we can easily check that the conditions in Lemma 3 should be satisfied, and the price distribution played by firms with ex-post position ${\tilde{s}}_p^*$ is the highest in the first-order stochastic dominance sense when $\varDelta =V\left( s_p^{**}\right) -V\left( s_p^{*}\right) =0$. In addition, if $G_{\sigma }\ne G_{\sigma ^*}$, then each firm will occasionally be placed at ex-post position ${\tilde{s}}_p$ and makes zero profit. As a consequence, ${\tilde{V}}\left( {\tilde{s}}_p^*\right) <\frac{\mu v}{1-\delta }$. The same argument applies to the case where $1-\beta >0.$

On the other hand, if $\eta =\zeta =0$, then firms immediately terminate compatibility with probability 1, which implies that $V\left( s_p^{**}\right) ={\tilde{V}}\left( {\tilde{s}}_e^*\right)$ and $V\left( s_p^*\right) =V\left( s_p\right) ={\tilde{V}}\left( {\tilde{s}}_e\right) .$ In this case, however, we can show that ${\tilde{V}}\left( {\tilde{s}}_e^*\right) >{\tilde{V}}\left( {\tilde{s}}_p^*\right)$, which leads to a contradiction. To see this, observe that ${\tilde{V}}\left( {\tilde{s}}_{p}^*\right) \le \frac{v}{2}+\frac{\delta }{2} {\tilde{V}}\left( {\tilde{s}}_e^*\right) +\frac{\delta }{2} {\tilde{V}}\left( {\tilde{s}}_e\right)$ since v is the maximum price, and each firm with ex-post position ${\tilde{s}}_p^*$ must win with probability $\frac{1}{2}$. Therefore,

$$\begin{aligned} {\tilde{V}}\left( {\tilde{s}}_e^*\right) -{\tilde{V}}\left( {\tilde{s}}_{p}^*\right)&\ge \left( \frac{1}{2}-\mu \right) v+\left( 1-\frac{\mu }{2}\right) \delta {\tilde{V}}\left( {\tilde{s}}_{p}^*\right) +\frac{\mu \delta }{2} {\tilde{V}}\left( {\tilde{s}}_{p}\right) -\frac{\delta }{2} {\tilde{V}}\left( {\tilde{s}}_e^*\right) -\frac{\delta }{2} {\tilde{V}}\left( {\tilde{s}}_e\right) \\&>\frac{5\delta }{6} {\tilde{V}}\left( {\tilde{s}}_{p}^*\right) +\frac{\delta }{6} {\tilde{V}}\left( {\tilde{s}}_{p}\right) -\frac{\delta }{2} {\tilde{V}}\left( {\tilde{s}}_e^*\right) -\frac{\delta }{2} {\tilde{V}}\left( {\tilde{s}}_e\right) \\&=\frac{\delta }{2}\left( {\tilde{V}}\left( {\tilde{s}}_{p}^*\right) -{\tilde{V}}\left( {\tilde{s}}_e^*\right) \right) +\frac{\delta }{3} {\tilde{V}}\left( {\tilde{s}}_p^*\right) +\frac{\delta }{6} {\tilde{V}}\left( {\tilde{s}}_{p}\right) -\frac{\delta }{2} {\tilde{V}}\left( {\tilde{s}}_e\right) \\&>\frac{\delta }{6} \left( 2{\tilde{V}}\left( {\tilde{s}}_{p}^*\right) +{\tilde{V}}\left( {\tilde{s}}_p\right) -3 {\tilde{V}}\left( {\tilde{s}}_e\right) \right) , \end{aligned}$$

where the second inequality comes from $\mu \le \frac{1}{3}$. Note that

$$\begin{aligned} {\tilde{V}}\left( {\tilde{s}}_e\right) =\frac{\mu \delta }{2} {\tilde{V}}\left( {\tilde{s}}_p^*\right) +\delta \left( 1-\frac{\mu }{2}\right) \left( \frac{1}{2}{\tilde{V}}\left( {\tilde{s}}_p^*\right) +\frac{1}{2}{\tilde{V}}\left( {\tilde{s}}_{p}\right) \right) . \end{aligned}$$

Using this, one can compute to show that

$$\begin{aligned} 2{\tilde{V}}\left( {\tilde{s}}_{p}^*\right) +{\tilde{V}}\left( {\tilde{s}}_p\right) -3 {\tilde{V}}\left( {\tilde{s}}_e\right)&=\left( 2-\frac{3\delta }{2} \left( 1+\frac{\mu }{2}\right) \right) {\tilde{V}}\left( {\tilde{s}}_{p}^*\right) +\left( 1-\frac{3\delta }{2}\left( 1-\frac{\mu }{2}\right) \right) {\tilde{V}}\left( {\tilde{s}}_p\right) \\&\ge 3 \left( 1-\delta \right) {\tilde{V}}\left( {\tilde{s}}_p\right) , \end{aligned}$$

where the inequality follows from $\mu \le \frac{1}{3}.$ Thus, we have a contradiction, ${\tilde{V}}\left( {\tilde{s}}_e^*\right) >{\tilde{V}}\left( {\tilde{s}}_p^*\right)$.

Finally, let us consider the case where ${\tilde{V}}\left( {\tilde{s}}_{p}^*\right) ={\tilde{V}}\left( {\tilde{s}}_e^*\right)$, which implies that $V\left( s_p^{**}\right) ={\tilde{V}}\left( {\tilde{s}}_p^*\right)$. Note that Lemma 3 continues to apply, and if $\varDelta V=V\left( s_p^{**}\right) -V\left( s_{p}^*\right) \ge \kappa =\frac{2\mu v}{(1-2\mu )\delta }$, then firms with ex-post position ${\tilde{s}}_p^*$ must quote price 0 with probability 1. This leads to the contradiction ${\tilde{V}}\left( {\tilde{s}}_p^*\right) =0$ since

$$\begin{aligned} {\tilde{V}}\left( {\tilde{s}}_p^*\right) =\frac{\delta }{2} V\left( s_p^{**}\right) +\frac{\delta }{2} V\left( s_p^*\right) =\frac{\delta }{2} {\tilde{V}}\left( {\tilde{s}}_p^*\right) +\frac{\delta }{2} V\left( s_p^*\right) \le \delta {\tilde{V}}\left( {\tilde{s}}_p^*\right) . \end{aligned}$$

On the other hand, if $\varDelta V<\kappa$, then firms with ex-post position ${\tilde{s}}_p^*$ must randomize prices and obtain

$$\begin{aligned} {\tilde{V}}\left( {\tilde{s}}_p^*\right) =\mu \left( v+\delta V\left( s_p^{**}\right) \right) +\left( 1-\mu \right) \delta V\left( s_p^*\right)&=\mu v+\mu \delta {\tilde{V}}\left( {\tilde{s}}_p^*\right) +\left( 1-\mu \right) \delta V\left( s_p^*\right) . \end{aligned}$$

Similarly, a firm with ex-post position $s_e^*$ quotes price v and obtains

$$\begin{aligned} {\tilde{V}}\left( {\tilde{s}}_e^*\right) =\left( 1-\mu \right) \left( v+\delta V\left( s_p^{**}\right) \right) +\mu \delta V\left( s_e\right) =\left( 1-\mu \right) \left( v+\delta {\tilde{V}}\left( {\tilde{s}}_p^*\right) \right) +\mu \delta V\left( s_e\right) . \end{aligned}$$

Now, recall that if $G_{\sigma }\not =G_{\sigma ^*}$, then $V\left( s_p^*\right) <{\tilde{V}}\left( {\tilde{s}}_p^*\right)$ since ${\tilde{V}}\left( {\tilde{s}}_e\right) <{\tilde{V}}\left( {\tilde{s}}_p^*\right)$ and ${\tilde{V}}\left( {\tilde{s}}_p\right) <{\tilde{V}}\left( {\tilde{s}}_p^*\right)$. In this case, subtracting ${\tilde{V}}\left( {\tilde{s}}_p^*\right)$ from ${\tilde{V}}\left( {\tilde{s}}_e^*\right)$ yields

$$\begin{aligned} 0&=\left( 1-2\mu \right) v+\left( 1-2\mu \right) \delta {\tilde{V}}\left( {\tilde{s}}_p^*\right) +\mu \delta V\left( s_e\right) -\left( 1-\mu \right) \delta V\left( s_p^*\right) \\&>\left( 1-2\mu \right) v-\mu \delta {\tilde{V}}\left( {\tilde{s}}_p^*\right) +\mu \delta V\left( s_e\right) . \end{aligned}$$

Therefore, we have $V\left( s_e\right) <{\tilde{V}}\left( {\tilde{s}}_p^*\right) -\frac{\left( 1-2\mu \right) v}{\mu \delta }$, and so

$$\begin{aligned} {\tilde{V}}\left( {\tilde{s}}_p^*\right) ={\tilde{V}}\left( {\tilde{s}}_e^*\right)&=\left( 1-\mu \right) \left( v+\delta {\tilde{V}}\left( {\tilde{s}}_p^*\right) \right) +\mu \delta V\left( s_e\right) \\&<\left( 1-\mu \right) \left( v+\delta {\tilde{V}}\left( {\tilde{s}}_p^*\right) \right) +\mu \delta \left( {\tilde{V}}\left( {\tilde{s}}_p^*\right) -\frac{(1-2\mu )v}{\mu \delta }\right) \\&= \mu v+\delta {\tilde{V}}\left( {\tilde{s}}_p^*\right) , \end{aligned}$$

which implies that ${\tilde{V}}\left( {\tilde{s}}_p^*\right) ={\tilde{V}}\left( {\tilde{s}}_e^*\right) <\frac{\mu v}{1-\delta }$ if $G_{\sigma }\ne G_{\sigma ^*}$. This completes the proof. $\square$

Proof of Theorem 2

Let ${\tilde{V}}_{\sigma }\left( {\tilde{s}}\right)$ denotes the expected life-time payoff of a firm with ex-post position ${\tilde{s}}\in G_2$ when $\sigma$ is played. Given the previous lemmas, it sufficies to compare two strategy profiles, $\sigma ^*$ and ${\hat{\sigma }}$.

Recall that when $\sigma ^*$ is played, we have ${\tilde{V}}_{\sigma ^*}\left( {\tilde{s}}_p^*\right) =\frac{\mu v}{1-\delta }$ and ${\tilde{V}}_{\sigma ^*}\left( {\tilde{s}}_p\right) =0$. If firms play ${\hat{\sigma }},$ on the other hand, they obtain

$$\begin{aligned} {\tilde{V}}_{{\hat{\sigma }}}\left( {\tilde{s}}_e^*\right) =\frac{\left( 2-\delta \left( 2-\mu \right) \right) \left( 1-\mu \right) v}{\left( 1-\delta \right) \left( 2-\delta \left( 2-3\mu \right) \right) }, \end{aligned}$$

and

$$\begin{aligned} {\tilde{V}}_{{\hat{\sigma }}}\left( {\tilde{s}}_e\right) =\frac{\left( 1-\mu \right) \delta \mu }{\left( 1-\delta \right) \left( 2-\delta \left( 2-3\mu \right) \right) }. \end{aligned}$$

Now, suppose that ${\tilde{V}}_{\sigma ^{*}}\left( {\tilde{s}}_p^*\right) >{\tilde{V}}_{{\hat{\sigma }}}\left( {\tilde{s}}_e^*\right)$ and $\sigma ^*$ is an equilibrium. In this case, ${\hat{\sigma }}$ cannot be a stable prediction since a group consisting of a focal firm and one of its rivals would jointly deviate to a permanent exclusionary regime through $\sigma ^*.$³² On the other hand, $\sigma ^*$ remains as a stable prediction since a focal firm’s payoff cannot be further increased from any joint deviation. A tedious computation shows that ${\tilde{V}}_{\sigma ^{*}}\left( {\tilde{s}}_p^*\right) >{\tilde{V}}_{{\hat{\sigma }}}\left( {\tilde{s}}_e^*\right)$ if and only if $\delta >\frac{2-4\mu }{4\mu ^2-5\mu +2}$, where the threshold discount factor is strictly less than 1 if $\mu >\frac{1}{4}$. Since $\sigma ^*$ is an equilibrium for sufficiently high $\delta$ (Theorem 1), we conclude that any stable prediction $\sigma$ must satisfy $G_{\sigma }=G_{\sigma ^*}$ when $\mu$ and $\delta$ are high enough. $\square$

Proof of Proposition 1

Without loss of generality, we may assume $\{g_{12}^{1},g_{12}^{2}\}\subset G_{\sigma }$. Letting ${\bar{p}}_i\left( g\right)$ and ${\underline{p}}_i\left( g\right)$ denote the maximum and minimum price of firm i’s price distribution $H_i(\cdot |g)$ at regime g,  it is clear that ${\bar{p}}_1\left( g\right) ,{\bar{p}}_2\left( g\right) \le v$ at each $g\in \{g^1_{12}, g^2_{12}\}$ since $\{g_{12}^{1},g_{12}^{2}\}\subset G_{\sigma }$.

Given the condition on the continuation payoffs, we have $\varDelta H_1\left( p|g\right) \varDelta H_2\left( p|g\right) =0$ for all $p\in \left( 0,v\right]$ at each $g\in \{g^1_{12},g^2_{12}\}$ since otherwise, either firm 1 or 2 could profitably undercut p by a small amount. Now, we show that ${\bar{p}}_1\left( g\right) ={\bar{p}}_2\left( g\right)$ for each $g\in \{g^1_{12},g^2_{12}\}$. By way of contradiction, suppose that $v\ge {\bar{p}}_1\left( g\right) >{\bar{p}}_2\left( g\right)$ for some $g\in \{g^1_{12},g^2_{12}\}$. If $\varDelta H_2\left( {\bar{p}}_2\left( g\right) |g\right) =0$, then for all $p_1\in \left[ {\bar{p}}_2\left( g\right) ,{\bar{p}}_1\left( g\right) \right) ,$ firm 1 obtains

$$\begin{aligned} \mu p_1+\delta \left( \mu {\tilde{V}}_1\left( g^1_{12}\right) +\left( 1-\mu \right) {\tilde{V}}_1\left( g^2_{12}\right) \right) , \end{aligned}$$

which is strictly less than the expected life-time payoff when quoting $p_1=v$. This implies that $H_1\left( \cdot |g\right)$ is constant on $\left[ {\bar{p}}_2\left( g\right) ,{\bar{p}}_1\left( g\right) \right)$, $\varDelta H_1\left( {\bar{p}}_2\left( g\right) |g\right) =0$, and $\varDelta H_1\left( {\bar{p}}_1\left( g\right) |g\right) >0$ with ${\bar{p}}_1\left( g\right) =v.$ However, this is a contradiction since firm 2 could profitably move density in $\left( {\bar{p}}_2\left( g\right) -\epsilon ,{\bar{p}}_2\left( g\right) \right]$ to some $p_2\in \left( {\bar{p}}_2\left( g\right) ,v\right)$ for small $\epsilon >0$. Indeed, one can derive the same contradiction when $\varDelta H_2\left( {\bar{p}}_2\left( g\right) |g\right) >0$. By applying the symmetric argument to the case ${\bar{p}}_1\left( g\right) <{\bar{p}}_2\left( g\right)$, we conclude that ${\bar{p}}_1\left( g\right) ={\bar{p}}_2\left( g\right) ={\bar{p}}\left( g\right)$ for each $g\in \{g^1_{12},g^2_{12}\}$.

An immediate implication of the common upper bound is that ${\bar{p}}\left( g\right) =v$ for all $g\in \{g^1_{12},g^2_{12}\}$: if this is not the case for some $g\in \{g^1_{12},g^2_{12}\}$, then there exists a firm, say 1, such that $\varDelta H_1\left( {\bar{p}}\left( g\right) |g\right) =0$, and so

$$\begin{aligned} {\tilde{V}}_2\left( g\right)&=\mu {\bar{p}}\left( g\right) +\delta \left( \mu {\tilde{V}}_2\left( g^2_{12}\right) +\left( 1-\mu \right) {\tilde{V}}_2\left( g^1_{12}\right) \right) \\&<\mu v+\delta \left( \mu {\tilde{V}}_2\left( g^2_{12}\right) +\left( 1-\mu \right) {\tilde{V}}_2\left( g^1_{12}\right) \right) , \end{aligned}$$

where the right-hand side is the expected life-time payoff of firm 2 when it quotes price v.

We now consider lower bounds, ${\underline{p}}_1\left( g\right)$ and ${\underline{p}}_2\left( g\right)$ at $g\in \{g^1_{12},g^2_{12}\}$. First, observe that if ${\underline{p}}_1\left( g\right) <{\underline{p}}_2\left( g\right)$ at some $g\in \{g^1_{12},g^2_{12}\}$, then for all $p\in \left[ {\underline{p}}_1\left( g\right) ,{\underline{p}}_2\left( g\right) \right) ,$ firm 1 obtains

$$\begin{aligned} \left( 1-\mu \right) p+\delta \left( \left( 1-\mu \right) {\tilde{V}}_1\left( g^1_{12}\right) +\mu {\tilde{V}}_1\left( g^2_{12}\right) \right) \\ <\left( 1-\mu \right) p'+\delta \left( \left( 1-\mu \right) {\tilde{V}}_1\left( g^1_{12}\right) +\mu {\tilde{V}}_1\left( g^2_{12}\right) \right) \end{aligned}$$

for some $p'>p$, which is a contradiction. By the symmetric argument, we have ${\underline{p}}_1\left( g\right) ={\underline{p}}_2\left( g\right) ={\underline{p}}\left( g\right)$ at each $g\in \{g^1_{12},g^2_{12}\}$. In addition, it must be the case that ${\underline{p}}\left( g\right) >0$ for each $g\in \{g^1_{12},g^2_{12}\}$: if ${\underline{p}}\left( g^1_{12}\right) =0$, for instance, then ${\tilde{V}}_1\left( g^1_{12}\right) =0$ since

$$\begin{aligned} {\tilde{V}}_1\left( g^1_{12}\right) =\delta \left( \left( 1-\mu \right) {\tilde{V}}_1\left( g^1_{12}\right) +\mu {\tilde{V}}_1\left( g^2_{12}\right) \right) \le \delta {\tilde{V}}_1\left( g^1_{12}\right) , \end{aligned}$$

which is a contradiction.

Then, since ${\underline{p}}\left( g\right) >0$, it is clear that no firm places an atom at the minimum price in each $g\in \{g^1_{12},g^2_{12}\}$: if $\varDelta H_1\left( {\underline{p}}\left( g\right) |g\right) >0$ and $\varDelta H_2\left( {\underline{p}}\left( g\right) |g\right) =0$, then firm 2 could profitably move densities from $\left( {\underline{p}}\left( g\right) ,{\underline{p}}\left( g\right) +\epsilon \right)$ to a price slightly below ${\underline{p}}\left( g\right)$ for small $\epsilon >0$. Given the identical lower bounds with no atom at each $g\in \{g^1_{12},g^2_{12}\}$, we can now write

$$\begin{aligned} {\tilde{V}}_1\left( g\right) =\left( 1-\mu \right) {\underline{p}}\left( g\right) +\delta \left( \left( 1-\mu \right) {\tilde{V}}_1\left( g^{1}_{12}\right) +\mu {\tilde{V}}_1\left( g^2_{12}\right) \right) ,\\ {\tilde{V}}_2\left( g\right) =\left( 1-\mu \right) {\underline{p}}\left( g\right) +\delta \left( \left( 1-\mu \right) {\tilde{V}}_2\left( g^{2}_{12}\right) +\mu {\tilde{V}}_2\left( g^1_{12}\right) \right) , \end{aligned}$$

for each $g\in \{g^1_{12},g^2_{12}\}$. From these, we observe ${\underline{p}}\left( g^1_{12}\right) ={\underline{p}}\left( g^2_{12}\right)$ since

$$\begin{aligned} 0\le {\tilde{V}}_1\left( g^1_{12}\right) -{\tilde{V}}_1\left( g^2_{12}\right) =\left( 1-\mu \right) \left( {\underline{p}}\left( g^1_{12}\right) -{\underline{p}}\left( g^2_{12}\right) \right) ={\tilde{V}}_2\left( g^1_{12}\right) -{\tilde{V}}_2\left( g^2_{12}\right) \le 0. \end{aligned}$$

In turn, this implies that ${\tilde{V}}_1\left( g^1_{12}\right) ={\tilde{V}}_1\left( g^2_{12}\right) ={\tilde{V}}_2\left( g^1_{12}\right) ={\tilde{V}}_2\left( g^2_{12}\right)$. Now, recall that for each $g\in \{g^1_{12},g^2_{12}\}$, there exists firm $i=1,2$ such that $\varDelta H_i\left( {\bar{p}}\left( g\right) |g\right) =\varDelta H_i\left( v|g\right) =0.$ Thus, the equilibrium payoff of firm $j=1,2$ other than i should be given by $\frac{\mu v}{1-\delta }$ for both $g\in \{g^1_{12},g^2_{12}\}$. This completes the proof since ${\tilde{V}}_1\left( g\right) ={\tilde{V}}_2\left( g\right)$ for each $g\in \{g^1_{12},g^2_{12}\}$. $\square$

Proof of Proposition 2

Let us consider $g^1_{123}$ and denote by ${\bar{p}}_i$ the maximum price of firm i’s price distribution $H_i\left( \cdot \right)$ at regime $g^1_{123}.$ First, note that if ${\bar{p}}_1>{\bar{p}}_2>0$, then ${\bar{p}}_3={\bar{p}}_1\equiv {\bar{p}}$ since otherwise, firm 1 has no incentive to choose a price sufficiently close or equal to ${\bar{p}}_1$. In addition, we have $\varDelta H_1\left( {\bar{p}}\right) \varDelta H_3\left( {\bar{p}}\right) =0$ since otherwise, either firm 1 or 3 could profitably undercut ${\bar{p}}$ by a small amount.

Suppose that $\varDelta H_1\left( {\bar{p}}\right) =0.$ Then, a price arbitrarily close or equal to ${\bar{p}}$ yields

$$\begin{aligned} {\tilde{V}}_3\left( g^1_{123}\right)&=\delta \Biggl (\mu V_3\left( g^1_{123}\right) +\left( 1-\mu \right) \Bigg (\Pr \left( p_1<p_2\right) V_3\left( g^1_{123}\right) +\Pr \left( p_1>p_2\right) V_3\left( g^2_{123}\right) \\&\quad +\Pr \left( p_1=p_2\right) \left( \frac{V_3\left( g^1_{123}\right) +V_3\left( g^2_{123}\right) }{2}\right) \Bigg )\Biggl ). \end{aligned}$$

Instead, if firm 3 quotes price $p_3<{\bar{p}}_2,$ it yields strictly positive per-period expected profits, and its expected continuation payoff from the next period also weakly increases. This is a contradiction, and we have either ${\bar{p}}_i={\bar{p}}>0$ for all $i\in I$ or ${\bar{p}}_i=0$ for some $i\in I.$

Indeed, one can apply the standard price-undercutting argument to show that the former case is inconsistent with an equilibrium. To see this, suppose ${\bar{p}}_i={\bar{p}}>0$ for all $i\in I$. If no atom is placed at ${\bar{p}}$ by more than one firm, then there exists a firm that has no incentive to choose a price arbitrarily close or equal to ${\bar{p}}$. On the other hand, if more than one firm place an atom at ${\bar{p}}$, then each firm placing an atom could profitably undercut ${\bar{p}}$ by a small amount. Therefore, it must be that ${\bar{p}}_i=0$ for all $i\in I$, which completes the proof. $\square$