Capacity Constrained Firms and Expansion Subsidies: Should Governments Avoid Generous Subsidies?

Abstract

This paper examines entry deterrence and signaling when an incumbent firm experiences capacity constraints. Our results show that if the costs that constrained and unconstrained incumbents incur when expanding their facilities are substantially different, separating equilibria can be supported under large parameter values whereby information is perfectly transmitted to the entrant. If, in contrast, both types of incumbent face similar expansion costs, subsidies that reduce expansion costs can help move the industry from a pooling to a separating equilibrium with associated efficient entry. Nonetheless, our results demonstrate that if subsidies are very generous entry patterns remain unaffected, suggesting a potential disadvantage of policies that significantly reduce firms’ expansion costs.

Appendices

Technical appendix

1.1 Proof of Lemma 1

We need to show that p ^NE < p ^E. Indeed, ${{p}^{{NE}}} \equiv \frac{{F - \pi_{{ent,L}}^{{D,NE}}}}{{\pi_{{ent,H}}^{{D,NE}} - \pi_{{ent,L}}^{{D,NE}}}} < \frac{{F - \pi_{{ent,L}}^{{D,E}}}}{{\pi_{{ent,H}}^{{D,E}} - \pi_{{ent,L}}^{{D,E}}}} \equiv {{p}^E}$, and solving for the entry cost, F, we obtain

$$ F < \frac{{\pi_{{ent,H}}^{{D,E}}\pi_{{ent,L}}^{{D,NE}} - \pi_{{ent,L}}^{{D,E}}\pi_{{ent,H}}^{{D,NE}}}}{{\left( {\pi_{{ent,L}}^{{D,NE}} - \pi_{{ent,L}}^{{D,E}}} \right) - \left( {\pi_{{ent,H}}^{{D,NE}} - \pi_{{ent,H}}^{{D,E}}} \right)}} $$

and since $\pi_{{ent,H}}^{{D,NE}} = \pi_{{ent,H}}^{{D,E}}$we can reduce the above expression to $F < \pi_{{ent,H}}^{{D,E}}$, which holds by definition.

1.2 Proof of Propositions 1, 2 and 3

Pooling equilibrium with no expansion

Let us investigate if the strategy profile {NoExp _H ,NoExp _L } can be supported as a pooling PBE of this signaling game. First, the entrant’s beliefs are μ(H|NExp) = p after observing no expansion (in equilibrium) and μ(H|Exp) = μ ∊ [0,1]after observing expansion (off-the-equilibrium). Given these beliefs after observing no expansion (in equilibrium) the entrant enters if and only if

$$ p\left( {\pi_{{ent,H}}^{{D,NE}} - F} \right) + \left( {1 - p} \right)\left( {\pi_{{ent,L}}^{{D,NE}} - F} \right) \geqslant 0, $$

where the right-hand side represents the entrant’s profits from staying in the perfectly competitive market (with zero profits). Solving for p, we obtain that the entrant enters if $p \geqslant \frac{{F - \pi_{{ent,L}}^{{D,NE}}}}{{\pi_{{ent,H}}^{{D,NE}} - \pi_{{ent,L}}^{{D,NE}}}} \equiv {{p}^{{NE}}}$. Note that this cutoff is positive and smaller than one, 1 > p ^NE > 0, since entry costs, F, satisfy $\pi_{{ent,H}}^{{D,NE}} > F > \pi_{{ent,L}}^{{D,NE}}$ by definition. Hence, after observing no expansion (in equilibrium) the entrant enters the market if p ≥ p ^NEand stays out otherwise. Similarly, after observing expansion (off-the-equilibrium), the entrant enters if and only if

$$ \mu \left( {\pi_{{ent,H}}^{{D,E}} - F} \right) + \left( {1 - \mu } \right)\left( {\pi_{{ent,L}}^{{D,E}} - F} \right) \geqslant 0 $$

Solving for μ, we find that the entrant enters if $\mu \geqslant \frac{{F - \pi_{{ent,L}}^{{D,E}}}}{{\pi_{{ent,H}}^{{D,E}} - \pi_{{ent,L}}^{{D,E}}}} \equiv {{p}^E}$. Note that this cutoff is positive and smaller than one, 1 > p ^NE > 0, since $\pi_{{ent,H}}^{{D,E}} > F > \pi_{{ent,L}}^{{D,E}}$ is satisfied by definition. Indeed, $\pi_{{ent,H}}^{{D,NE}} = \pi_{{ent,H}}^{{D,E}} > F > \pi_{{ent,L}}^{{D,NE}} > \pi_{{ent,L}}^{{D,E}}$ given that the entrant’s profits are not affected by the (unconstrained) high-cost incumbent decision to expand, $\pi_{{ent,H}}^{{D,NE}} = \pi_{{ent,H}}^{{D,E}}$, and the entrant’s profits are higher when the low-cost incumbent does not expand than when she does, $\pi_{{ent,L}}^{{D,NE}} > \pi_{{ent,L}}^{{D,E}}$. Hence, after observing expansion (off-the-equilibrium) the entrant enters if μ ≥ p ^E and stays out otherwise. Given the entrant’s strategies let us now analyze the incumbent:

• If p < p ^NE and μ ≥ p ^E then the entrant does not enter after observing no expansion (in equilibrium) but enters otherwise. Hence, the low-cost incumbent prefers not to expand (as prescribed) if and only if $\matrix{ {\pi_{{inc,L}}^{{M,NE}}\left( {\overline q } \right) > \pi_{{inc,L}}^{{D,E}} - {{K}_L},}{\text{or}\;}{{{K}_L} > \pi_{{inc,L}}^{{D,E}} - \pi_{{inc,L}}^{{M,NE}}\left( {\overline q } \right)} \\ }<!end array>$, where $BC{{C}_L} - PLE_L^E \equiv \pi_{{inc,L}}^{{D,E}} - \pi_{{inc,L}}^{{M,NE}}\left( {\overline q } \right)$. Similarly, the high-cost incumbent does not expand if $\matrix{ {\pi_{{inc,H}}^{{M,NE}} > \pi_{{inc,H}}^{{D,E}} - {{K}_H},} \hfill &{\text{or}} \hfill &{{{K}_H} > \pi_{{inc,H}}^{{D,E}} - \pi_{{inc,H}}^{{M,NE}} < 0} \hfill \\ }<!end array>$is satisfied, which holds for any expansion cost K _H  > 0. Thus, the strategy profile in which both types of incumbent do not expand their facility can be supported as a pooling PBE in the signaling game if ${{K}_L} > BC{{C}_L} - PLE_L^E$; as described in Proposition 1, Part 3a.

• If p < p ^NE and μ < p ^E then the entrant does not enter after observing any action from the incumbent. Therefore, the low-cost monopolist prefers not to expand (as prescribed) if and only if $\matrix{ {\pi_{{inc,L}}^{{M,NE}}\left( {\overline q } \right) > \pi_{{inc,L}}^{{M,E}} - {{K}_L},} \hfill &{\text{or}} \hfill &{{{K}_L} > \pi_{{inc,L}}^{{M,E}} - \pi_{{inc,L}}^{{M,NE}}\left( {\overline q } \right) \equiv BC{{C}_L}} \hfill \\ }<!end array>$. Similarly, the high-cost incumbent prefers not to expand since $\matrix{ {\pi_{{inc,H}}^{{M,NE}} > \pi_{{inc,H}}^{{M,E}} - {{K}_H}} \hfill &{\text{or}} \hfill &{{{K}_H} > \pi_{{inc,H}}^{{M,E}} - \pi_{{inc,H}}^{{M,NE}} < 0} \hfill \\ }<!end array>$ which is satisfied for any K _H  > 0. Thus, this strategy profile can be sustained as a pooling PBE in the signaling game if expansion costs satisfy K _L > BCC _L ; as described in Proposition 1, Part 3b.

• If p ≥ p ^NE and μ < p ^Ethen the entrant enters after observing no expansion (in equilibrium) but does not enter otherwise. Hence, the low-cost incumbent does not expand (attracting entry) if and only if $\matrix{ {\pi_{{inc,L}}^{{D,NE}}\left( {\overline q } \right) > \pi_{{inc,L}}^{{M,E}} - {{K}_L},} \hfill &{\text{or}} \hfill &{{{K}_L} > \pi_{{inc,L}}^{{M,E}} - \pi_{{inc,L}}^{{D,NE}}\left( {\overline q } \right) \equiv BC{{C}_L} + PLE_L^{{NE}}} \hfill \\ }<!end array>$. Similarly, the high-cost incumbent does not expand if and only if $\matrix{ {\pi_{{inc,H}}^{{D,NE}} > \pi_{{inc,H}}^{{M,E}} - {{K}_H}} \hfill &{,{\text{or}}} \hfill &{{{K}_H} > \pi_{{inc,H}}^{{M,E}} - \pi_{{inc,H}}^{{D,NE}} = \pi_{{inc,H}}^{{M,E}} - \pi_{{inc,H}}^{{D,E}} = PLE_H^E} \hfill \\ }<!end array>$, since $\pi_{{inc,H}}^{{D,E}} = \pi_{{inc,H}}^{{D,NE}}$given that the high-cost incumbent is unaffected by the capacity constraint. Thus, this strategy profile can be supported as a pooling PBE in the signaling game under expansion costs $\matrix{ {{{K}_L} > BC{{C}_H} + PLE_H^{{NE}}} \hfill &{\text{and}} \hfill &{{{K}_H} > PLE_L^E} \hfill \\ }<!end array>$; as described in Proposition 2a, and Proposition 3.

• If p ≥ p ^NE and μ ≥ p ^E then the entrant enters after observing any action from the incumbent. Therefore, the low-cost incumbent does not expand (as prescribed) if and only if $\matrix{ {\pi_{{inc,L}}^{{D,NE}}\left( {\overline q } \right) > \pi_{{inc,L}}^{{D,E}} - {{K}_L}} \hfill &{,{\text{or}}} \hfill &{{{K}_L} > \pi_{{inc,L}}^{{D,E}} - \pi_{{inc,L}}^{{D,NE}}\left( {\overline q } \right) \equiv BC{{C}_L} + PLE_L^{{NE}} - PLE_L^E} \hfill \\ }<!end array>$. Similarly, the high-cost incumbent does not expand since $\matrix{ {\pi_{{inc,L}}^{{D,NE}} > \pi_{{inc,L}}^{{D,E}} - {{K}_L},} \hfill &{\text{or}} \hfill &{{{K}_L} > \pi_{{inc,L}}^{{D,E}} - \pi_{{inc,L}}^{{D,NE}} = 0} \hfill \\ }<!end array>$, which holds for any K _H >0. Thus, this strategy profile can be supported as a pooling PBE for expansion costs ${{K}_L} > BC{{C}_L} + PLE_L^{{NE}} - PLE_L^E$ and K _H   > 0; as described in Proposition 2b, and Proposition 3.

Separating equilibrium

Let us now consider the separating strategy profile where only the low-cost incumbent expands, i.e., {NotExpand _H , Expand _L }. First, entrant’s updated beliefs becomeμ(H|NExp) = 1 and μ(H|Exp) = 0. Given these beliefs, the entrant enters after observing no expansion since $\matrix{ {\pi_{{ent,H}}^{{D,E}} - F > 0,} \hfill &{\text{or}} \hfill &{\pi_{{ent,L}}^{{D,NE}} < F} \hfill \\ }<!end array>$,, which satisfies our initial assumptions. On the other hand, after observing expansion the entrant stays out since $\matrix{ {\pi_{{ent,L}}^{{D,E}} - F < 0,} \hfill &{\text{or}} \hfill &{\pi_{{ent,L}}^{{D,E}} < F} \hfill \\ }<!end array>$, which also holds by definition. Therefore, given the entrant’s responses, the high-cost incumbent does not expand (as prescribed) if and only if $\matrix{ {\pi_{{inc,H}}^{{D,NE}} > \pi_{{inc,H}}^{{M,E}} - {{K}_H},} \hfill &{\text{or}} \hfill &{{{K}_H} > \pi_{{inc,H}}^{{M,E}} - \pi_{{inc,H}}^{{D,NE}}} \hfill \\ }<!end array>$. Since $\pi_{{inc,H}}^{{M,E}} - \pi_{{inc,H}}^{{D,NE}} = \pi_{{inc,H}}^{{M,E}} - \pi_{{inc,H}}^{{D,E}} \equiv PLE_H^E$ given that $\pi_{{inc,H}}^{{D,NE}} = \pi_{{inc,H}}^{{D,E}}$, we can then conclude that the high-cost incumbent does not expand if ${{K}_H} > PLE_H^E$. By contrast, the low-cost incumbent expands (as prescribed) if and only if $\matrix{ {\pi_{{inc,L}}^{{D,NE}}\left( {\overline q } \right) < \pi_{{inc,L}}^{{M,E}} - {{K}_L}} \hfill &{\text{or}} \hfill &{{{K}_L} < \pi_{{inc,L}}^{{M,E}} - \pi_{{inc,L}}^{{D,NE}}\left( {\overline q } \right) \equiv BC{{C}_L} + PLE_L^{{NE}}} \hfill \\ }<!end array>$. Thus, this strategy profile can be sustained as a separating PBE for expansion costs $\matrix{ {{{K}_H} > PLE_H^E} \hfill &{\text{and}} \hfill &{{{K}_L} < BCC_L + PLE_L^{{NE}}} \hfill \\ }<!end array>$; as described in Proposition 1 (Part 1) and Proposition 3.

For completeness, let us check that the opposite separating strategy profile {Exp _H , NoExp _L } cannot be supported as a PBE of the signaling game. In this case, the entrant’s updated beliefs become μ(H|NExp) = 0 and μ(H|Exp) = 1. Given these beliefs, the entrant enters after observing expansion since $\pi_{{ent,H}}^{{D,E}} - F > 0\,{\text{or}}\,\pi_{{ent,H}}^{{D,E}} > F$, which holds by definition. However, the entrant does not enter after observing no expansion given that $\matrix{ {\pi_{{ent,L}}^{{D,NE}} - F < 0,} \hfill &{\text{or}} \hfill &{\pi_{{ent,L}}^{{D,NE}} < F} \hfill \\ }<!end array>$, which is satisfied by definition. Given the entrant’s responses, the low-cost incumbent does not expand (as prescribed) if and only if $\matrix{ {\pi_{{inc,L}}^{{M,NE}}\left( {\overline q } \right) > \pi_{{inc,L}}^{{D,E}} - {{K}_L},} \hfill &{\text{or}} \hfill &{{{K}_L} > \pi_{{inc,L}}^{{D,E}} - \pi_{{inc,L}}^{{M,NE}}\left( {\overline q } \right)} \hfill \\ }<!end array>$. On the other hand, the high-cost incumbent expands (as prescribed) if $\matrix{ {\pi_{{inc,H}}^{{M,NE}} < \pi_{{inc,H}}^{{D,E}} - {{K}_H},} \hfill &{\text{or}} \hfill &{{{K}_H} < \pi_{{inc,H}}^{{D,E}} - \pi_{{inc,H}}^{{M,NE}} = \pi_{{inc,H}}^{{D,E}} - \pi_{{inc,H}}^{{M,E}} < 0} \hfill \\ }<!end array>$ (since $\pi_{{inc,H}}^{{M,E}} = \pi_{{inc,H}}^{{M,NE}}$), which cannot hold for any K _H  > 0. Thus, this strategy profile cannot be supported as a separating PBE of the signaling game.

Pooling equilibrium with expansion

Let us investigate if the pooling strategy profile in which both types of incumbent expand their facility, i.e., {Exp _H , Exp _L }, can be supported as a PBE of the signaling game. First, the entrant’s beliefs are μ(H|Exp) = p after observing expansion (in equilibrium) and μ(H|NExp) = γ ∊ [0,1] after observing no expansion (off-the-equilibrium). Given these beliefs, after observing expansion, the entrant enters if

$$ p\left( {\pi_{{ent,H}}^{{D,E}} - F} \right) + \left( {1 - p} \right)\left( {\pi_{{ent,L}}^{{D,E}} - F} \right) \geqslant 0 $$

Solving for p, we obtain $p \geqslant \frac{{F - \pi_{{ent,L}}^{{D,E}}}}{{\pi_{{ent,H}}^{{D,E}} - \pi_{{ent,L}}^{{D,E}}}} \equiv {{p}^E}$, where 0 ≤ p ^E ≤ 1 from our above discussion. Hence, entry ensues after observing expansion if p > p ^E, but does not otherwise. If, instead, the entrant observes no expansion, she enters if

$$ \gamma \left( {\pi_{{ent,H}}^{{D,NE}} - F} \right) + \left( {1 - \gamma } \right)\left( {\pi_{{ent,L}}^{{D,NE}} - F} \right) \geqslant 0, $$

Solving for γ, we find $\gamma \geqslant \frac{{F - \pi_{{ent,L}}^{{D,NE}}}}{{\pi_{{ent,H}}^{{D,NE}} - \pi_{{ent,L}}^{{D,NE}}}} \equiv {{p}^{{NE}}}$, where 0 ≤ p ^NE ≤ 1 holds from our above discussion. Therefore, the entrant enters after observing no expansion if γ ≥ p ^NE, but does not otherwise. Given the entrant’s strategies let us now examine the incumbent:

• If p ≥ p ^E and γ ≥ p ^NEthen the entrant enters both after observing expansion and no expansion. Therefore, the low-cost incumbent expands (as prescribed) if and only if $\matrix{ {\pi_{{inc,L}}^{{D,NE}}\left( {\overline q } \right) < \pi_{{inc,L}}^{{D,E}} - {{K}_L},} \hfill &{\text{or}} \hfill &{{{K}_L} < \pi_{{inc,L}}^{{D,E}} - \pi_{{inc,L}}^{{D,NE}}\left( {\overline q } \right)} \hfill \\ }<!end array>$. However, the high-cost incumbent does not expand since $\matrix{ {\pi_{{inc,H}}^{{D,NE}} > \pi_{{inc,H}}^{{D,E}} - {{K}_H},} \hfill &{\text{or}} \hfill &{{{K}_H} > \pi_{{inc,H}}^{{D,E}} - \pi_{{inc,H}}^{{D,NE}} = 0} \hfill \\ }<!end array>$, which holds for any expansion costs K _H  > 0. Thus, the strategy profile in which both types of incumbent expand cannot be supported as a pooling PBE of the signaling game.

• If p ≥ p ^E but γ < p ^NEthen the entrant enters after observing expansion (in equilibrium), but does not enter otherwise. Hence, the low-cost incumbent expands (as prescribed) if and only if $\matrix{ {\pi_{{inc,L}}^{{M,NE}}\left( {\overline q } \right) < \pi_{{inc,L}}^{{D,E}} - {{K}_L},} \hfill &{\text{or}} \hfill &{{{K}_L} < \pi_{{inc,L}}^{{D,E}} - \pi_{{inc,L}}^{{M,NE}}\left( {\overline q } \right) \equiv BC{{C}_L} - PLE_L^E} \hfill \\ }<!end array>$. By contrast, the high-cost incumbent does not expand since $\matrix{ {\pi_{{inc,H}}^{{M,NE}} > \pi_{{inc,H}}^{{D,E}} - {{K}_H},} \hfill &{\text{or}} \hfill &{{{K}_H} > \pi_{{inc,H}}^{{D,E}} - \pi_{{inc,H}}^{{M,NE}} < 0} \hfill \\ }<!end array>$, which holds for all expansion costs K _H  > 0. Therefore, this strategy profile cannot be sustained as a pooling PBE of the signaling game.

• If p < p ^E but γ ≥ p ^NE then the entrant does not enter after observing expansion (in equilibrium), but enters otherwise. Hence, the low-cost incumbent expands (as prescribed) if and only if $\matrix{ {\pi_{{inc,L}}^{{D,NE}}\left( {\overline q } \right) < \pi_{{inc,L}}^{{M,E}} - {{K}_L},} \hfill &{\text{or}} \hfill &{{{K}_L} < \pi_{{inc,L}}^{{M,E}} - \pi_{{inc,L}}^{{D,NE}}\left( {\overline q } \right)} \hfill \\ }<!end array>$. Since

  $$ BC{{C}_L} + PLE_L^{{NE}} = \left( {\pi_{{inc,L}}^{{M,E}} - \pi_{{inc,L}}^{{M,NE}}\left( {\overline q } \right)} \right) + \left( {\pi_{{inc,L}}^{{M,NE}}\left( {\overline q } \right) - \pi_{{inc,L}}^{{D,NE}}\left( {\overline q } \right)} \right) = \pi_{{inc,L}}^{{M,E}} - \pi_{{inc,L}}^{{D,NE}}\left( {\overline q } \right), $$

  the low-cost incumbent expands if ${{K}_L} < BC{{C}_L} + PLE_L^{{NE}}$. Similarly, the high-cost incumbent expands (as prescribed) if and only if $\matrix{ {\pi_{{inc,H}}^{{D,NE}} < \pi_{{inc,H}}^{{M,E}} - {{K}_H},} \hfill &{\text{or}} \hfill &{{{K}_H} < \pi_{{inc,H}}^{{M,E}} - \pi_{{inc,H}}^{{D,NE}} \equiv PLE_H^{{NE}}} \hfill \\ }<!end array>$, since $\matrix{ {\pi_{{inc,H}}^{{M,NE}} = \pi_{{inc,H}}^{{M,E}}} \hfill &{\text{and}} \hfill &{\pi_{{inc,H}}^{{D,NE}} = \pi_{{inc,H}}^{{D,E}}} \hfill \\ }<!end array>$ for the unconstrained high-cost incumbent under monopoly and duopoly, respectively. Thus, this strategy profile can be supported as a pooling PBE under expansion costs $\matrix{ {{{K}_L} < BC{{C}_L} + PLE_L^{{NE}}} \hfill &{\text{and}} \hfill &{{{K}_H} < PLE_H^E} \hfill \\ }<!end array>$; as described in Proposition 1, Part 2.

• If p < p ^E and γ < p ^NE then the entrant does not enter after observing any action from the incumbent. Therefore, the low-cost incumbent expands (as prescribed) if and only if $\matrix{ {\pi_{{inc,L}}^{{M,NE}}\left( {\overline q } \right) < \pi_{{inc,L}}^{{M,E}} - {{K}_L},} \hfill &{\text{or}} \hfill &{{{K}_L} < \pi_{{inc,L}}^{{M,E}} - \pi_{{inc,L}}^{{M,NE}}\left( {\overline q } \right) \equiv BC{{C}_L}} \hfill \\ }<!end array>$. By contrast, the high-cost incumbent does not expand since $\matrix{ {\pi_{{inc,L}}^{{M,NE}}\left( {\overline q } \right) < \pi_{{inc,L}}^{{M,E}} - {{K}_L},} \hfill &{\text{or}} \hfill &{{{K}_L} < \pi_{{inc,L}}^{{M,E}} - \pi_{{inc,L}}^{{M,NE}}\left( {\overline q } \right) \equiv BC{{C}_L}} \hfill \\ }<!end array>$, which holds for any expansion cost K _H >0. Thus, this strategy profile cannot be sustained as a pooling PBE.

1.3 Proof of Proposition 4

Pooling equilibrium with no expansion

Let us investigate if the pooling strategy profile {NoExp _H ,NoExp _L } can be supported as a pooling PBE of this signaling game. First, the entrant’s beliefs are μ(H|NExp) = p after observing no expansion (in equilibrium) and μ(H|Exp) = μ ∊ [0,1] after observing expansion (off-the-equilibrium). Given these beliefs after observing no expansion (in equilibrium) the entrant enters if and only if

$$ p\left( {\pi_{{ent,H}}^{{D,NE}} - F} \right) + \left( {1 - p} \right)\left( {\pi_{{ent,L}}^{{D,NE}} - F} \right) \geqslant 0, $$

where the right-hand side represents the entrant’s profits from staying in the perfectly competitive market (with zero profits). Solving for p, we obtain that the entrant enters if $p \geqslant \frac{{F - \pi_{{ent,L}}^{{D,NE}}}}{{\pi_{{ent,H}}^{{D,NE}} - \pi_{{ent,L}}^{{D,NE}}}} \equiv {{p}^{{NE}}}$. Note that this cutoff is positive and smaller than one, 1 > p ^NE > 0, since entry costs, F, satisfy $\pi_{{ent,H}}^{{D,NE}} > F > \pi_{{ent,L}}^{{D,NE}}$ by definition. Hence, after observing no expansion (in equilibrium) the entrant enters the market if p ≥ p ^NEand stays out otherwise. Similarly, after observing expansion (off-the-equilibrium), the entrant enters if and only if

$$ \mu \left( {\pi_{{ent,H}}^{{D,E}} - F} \right) + \left( {1 - \mu } \right)\left( {\pi_{{ent,L}}^{{D,E}} - F} \right) \geqslant 0 $$

Solving for μ, we find that the entrant enters if $\mu \geqslant \frac{{F - \pi_{{ent,L}}^{{D,E}}}}{{\pi_{{ent,H}}^{{D,E}} - \pi_{{ent,L}}^{{D,E}}}} \equiv {{p}^E}$. Note that this cutoff is positive and smaller than one, 1 > p ^NE > 0, since $\pi_{{ent,H}}^{{D,E}} > F > \pi_{{ent,L}}^{{D,E}}$ is satisfied by definition. Indeed, $\pi_{{ent,H}}^{{D,NE}} > \pi_{{ent,H}}^{{D,E}} > F > \pi_{{ent,L}}^{{D,NE}} = \pi_{{ent,L}}^{{D,E}}$ given that the entrant’s profits are not affected by the (unconstrained) low-demand incumbent decision to expand, $\pi_{{ent,L}}^{{D,NE}} = \pi_{{ent,L}}^{{D,E}}$, and the entrant’s profits are higher when the high-demand incumbent does not expand than when she does, $\pi_{{ent,H}}^{{D,NE}} > \pi_{{ent,H}}^{{D,E}}$. Hence, after observing expansion (off-the-equilibrium) the entrant enters if μ ≥ p ^E and stays out otherwise. Finally, note that p ^NE < p ^E as it shown in the proof of Proposition 1.

Given the entrant’s strategies let us now analyze the incumbent:

• If p < p ^NE and μ ≥ p ^E then the entrant does not enter after observing no expansion (in equilibrium) but enters otherwise. Hence, the high-demand incumbent prefers not to expand (as prescribed) if and only if $\matrix{ {\pi_{{inc,H}}^{{M,NE}}\left( {\overline q } \right) > \pi_{{inc,H}}^{{D,E}} - {{K}_H},} \hfill &{\text{or}} \hfill &{{{K}_H} >\pi_{{inc,H}}^{{D,E}} - \pi_{{inc,H}}^{{M,NE}}\left( {\overline q } \right)} \hfill \\ }<!end array>$, where $BC{{C}_H} - PLE_H^E \equiv \pi_{{inc,H}}^{{D,E}} - \pi_{{inc,H}}^{{M,NE}}\left( {\overline q } \right)$. Similarly, the low-demand incumbent does not expand if $\matrix{ {\pi_{{inc,L}}^{{M,NE}} > \pi_{{inc,L}}^{{D,E}} - {{K}_L},}{\text{or}}\;{{{K}_L} > \pi_{{inc,L}}^{{D,E}} - \pi_{{inc,L}}^{{M,NE}} < 0} \\ }<!end array>$. is satisfied, which holds for any expansion cost K _L  > 0. Thus, the strategy profile in which both types of incumbent do not expand their facility can be supported as a pooling PBE if ${{K}_H} > BC{{C}_H} - PLE_H^E$.

• If p < p ^NE and μ < p ^E then the entrant does not enter after observing any action from the incumbent. Therefore, the high-demand monopolist prefers not to expand (as prescribed) if and only if $\matrix{ {\pi_{{inc,H}}^{{M,NE}}\left( {\overline q } \right) > \pi_{{inc,H}}^{{M,E}} - {{K}_H},} \hfill &{\text{or}} \hfill &{{{K}_H} > \pi_{{inc,H}}^{{M,E}} - \pi_{{inc,H}}^{{M,NE}}\left( {\overline q } \right) \equiv BC{{C}_H}} \hfill \\ }<!end array>$. Similarly, the low-demand incumbent prefers not to expand since $\matrix{ {\pi_{{inc,L}}^{{M,NE}} > \pi_{{inc,L}}^{{M,E}} - {{K}_L}} \hfill &{\text{or}} \hfill &{{{K}_L} > \pi_{{inc,L}}^{{M,E}} - \pi_{{inc,L}}^{{M,NE}} < 0} \hfill \\ }<!end array>$ which is satisfied for any K _L  > 0. Thus, this strategy profile can be sustained as a pooling PBE in the signaling game if expansion costs satisfy ${{K}_H} > BCC_H$.

• If p ≥ p ^NE and μ < p ^Ethen the entrant enters after observing no expansion (in equilibrium) but does not enter otherwise. Hence, the high-demand incumbent does not expand (attracting entry) if and only if $\matrix{ {\pi_{{inc,H}}^{{D,NE}}\left( {\overline q } \right) > \pi_{{inc,H}}^{{M,E}} - {{K}_H},} \hfill &{\text{or}} \hfill &{{{K}_H} > \pi_{{inc,H}}^{{M,E}} - \pi_{{inc,H}}^{{D,NE}}\left( {\overline q } \right) \equiv BC{{C}_H} + PLE_H^{{NE}}} \hfill \\ }<!end array>$. Similarly, the low-demand incumbent does not expand if and only if $\matrix{ {\pi_{{inc,L}}^{{D,NE}} > \pi_{{inc,L}}^{{M,E}} - {{K}_L},} \hfill &{\text{or}} \hfill &{{{K}_L} > \pi_{{inc,L}}^{{M,E}} - \pi_{{inc,L}}^{{D,NE}} = PLE_L^E} \hfill \\ }<!end array>$. Thus, this strategy profile can be supported as a pooling PBE in the signaling game under expansion costs $\matrix{ {{{K}_H} > BC{{C}_H} + PLE_H^{{NE}}} \hfill &{\text{and}} \hfill &{{{K}_L} > PLE_L^E} \hfill \\ }<!end array>$.

• If p ≥ p ^NE and μ ≥ p ^E then the entrant enters after observing any action from the incumbent. Therefore, the high-demand incumbent does not expand (as prescribed) if and only if $\matrix{ {\pi_{{inc,H}}^{{D,NE}}\left( {\overline q } \right) > \pi_{{inc,H}}^{{D,E}} - {{K}_H},} \hfill &{\text{or}} \hfill &{{{K}_H} > \pi_{{inc,H}}^{{D,E}} - \pi_{{inc,H}}^{{D,NE}}\left( {\overline q } \right) \equiv BC{{C}_H} + PLE_H^{{NE}} - PLE_H^E} \hfill \\ }<!end array>$. Similarly, the low-demand incumbent does not expand since $\matrix{ {\pi_{{inc,L}}^{{D,NE}} > \pi_{{inc,L}}^{{D,E}} - {{K}_L},} \hfill &{\text{or}} \hfill &{{{K}_L} > \pi_{{inc,L}}^{{D,E}} - \pi_{{inc,L}}^{{D,NE}} = 0} \hfill \\ }<!end array>$, which holds for any K _L >0. Thus, this strategy profile can be supported as a pooling PBE for expansion costs ${{K}_H} > BC{{C}_H} + PLE_H^{{NE}} - PLE_H^E$ and K _L   > 0.

Separating equilibrium

Let us now consider the separating strategy profile where only the high-demand incumbent expands, i.e., {Expand _H , NotExpand _L }. First, entrant’s updated beliefs becomeμ(H|NExp) = 0 and μ(H|Exp) = 1. Given these beliefs, the entrant enters after observing expansion since $\matrix{ {\pi_{{ent,H}}^{{D,E}} - F > 0,} \hfill &{\text{or}} \hfill &{\pi_{{ent,H}}^{{D,E}} > F} \hfill \\ }<!end array>$. On the other hand, after observing no expansion the entrant stays out since $\matrix{ {\pi_{{ent,L}}^{{D,NE}} - F < 0,} \hfill &{\text{or}} \hfill &{\pi_{{ent,L}}^{{D,NE}} < F} \hfill \\ }<!end array>$, which also holds by definition. Therefore, given the entrant’s responses, the low-demand incumbent does not expand (as prescribed) if and only if $\matrix{ {\pi_{{inc,L}}^{{D,NE}} > \pi_{{inc,L}}^{{M,E}} - {{K}_L},} \hfill &{\text{or}} \hfill &{{{K}_L} > \pi_{{inc,L}}^{{M,E}} - \pi_{{inc,L}}^{{D,NE}}} \hfill \\ }<!end array>$. Since $\pi_{{inc,L}}^{{M,E}} - \pi_{{inc,L}}^{{D,NE}} = \pi_{{inc,L}}^{{M,E}} - \pi_{{inc,L}}^{{D,E}} \equiv PLE_L^E$ given that $\pi_{{inc,L}}^{{D,NE}} = \pi_{{inc,L}}^{{D,E}}$, we can then conclude that the high-demand incumbent does not expand if ${{K}_L} > PLE_L^E$. By contrast, the high-demand incumbent expands (as prescribed) if and only if $\matrix{ {\pi_{{inc,H}}^{{D,NE}}\left( {\overline q } \right) < \pi_{{inc,H}}^{{M,E}} - {{K}_H}} \hfill &{\text{or}} \hfill &{{{K}_H} < \pi_{{inc,H}}^{{M,E}} - \pi_{{inc,H}}^{{D,NE}}\left( {\overline q } \right) \equiv BC{{C}_H} + PLE_H^{{NE}}} \hfill \\ }<!end array>$. Thus, this strategy profile can be sustained as a separating PBE for expansion costs $\matrix{ {{{K}_L} > PLE_L^E} \hfill &{\text{and}} \hfill &{{{K}_H} < BC{{C}_H} + PLE_H^{{NE}}} \hfill \\ }<!end array>$.

For completeness, let us check that the opposite separating strategy profile {NoExp _H , Exp _L } cannot be supported as a PBE of the signaling game. In this case, the entrant’s updated beliefs become μ(H|NExp) = 1 and μ(H|Exp) = 0. Given these beliefs, the entrant does not enter after observing expansion since $\matrix{ {\pi_{{ent,L}}^{{D,E}} - F < 0} \hfill &{\text{or}} \hfill &{\pi_{{ent,L}}^{{D,E}} < F} \hfill \\ }<!end array>$, which holds by definition. However, the entrant enters after observing no expansion given that $\matrix{ {\pi_{{ent,L}}^{{D,NE}} - F < 0,} \hfill &{\text{or}} \hfill &{\pi_{{ent,L}}^{{D,NE}} < F} \hfill \\ }<!end array>$, which is satisfied by definition. Given the entrant’s responses, the high-demand incumbent does not expand (as prescribed) if and only if $\matrix{ {\pi_{{inc,H}}^{{M,NE}}\left( {\overline q } \right) > \pi_{{inc,H}}^{{D,E}} - {{K}_H},} \hfill &{\text{or}} \hfill &{{{K}_H} > \pi_{{inc,H}}^{{D,E}} - \pi_{{inc,H}}^{{M,NE}}\left( {\overline q } \right)} \hfill \\ }<!end array>$. On the other hand, the low-demand incumbent expands (as prescribed) if $\matrix{ {\pi_{{inc,L}}^{{M,NE}} < \pi_{{inc,L}}^{{D,E}} - {{K}_L},} \hfill &{\text{or}} \hfill &{{{K}_L} < \pi_{{inc,L}}^{{D,E}} - \pi_{{inc,L}}^{{M,NE}} = \pi_{{inc,L}}^{{D,E}} - \pi_{{inc,L}}^{{M,E}} < 0} \hfill \\ }<!end array>$ (since $\pi_{{inc,H}}^{{M,E}} = \pi_{{inc,H}}^{{M,NE}}$), which cannot hold for any K _L  > 0. Thus, this strategy profile cannot be supported as a separating PBE of the signaling game.

Pooling equilibrium with expansion

Let us investigate if the pooling strategy profile in which both types of incumbent expand their facility, i.e., {Exp _H ,Exp _L }, can be supported as a PBE of the signaling game. First, the entrant’s beliefs are μ(H|Exp) = p after observing expansion (in equilibrium) and μ(H|NExp) = γ ∊ [0,1] after observing no expansion (off-the-equilibrium). Given these beliefs, after observing expansion, the entrant enters if

$$ p\left( {\pi_{{ent,H}}^{{D,E}} - F} \right) + \left( {1 - p} \right)\left( {\pi_{{ent,L}}^{{D,E}} - F} \right) \geqslant 0 $$

Solving for p, we obtain $p \geqslant \frac{{F - \pi_{{ent,H}}^{{D,E}}}}{{\pi_{{ent,L}}^{{D,E}} - \pi_{{ent,H}}^{{D,E}}}} \equiv {{p}^E}$, where 0 ≤ p ^E ≤ 1 from our above discussion. Hence, entry ensues after observing expansion if p ≥ p ^E, but does not otherwise. If, instead, the entrant observes no expansion, she enters if

$$ \gamma \left( {\pi_{{ent,H}}^{{D,NE}} - F} \right) + \left( {1 - \gamma } \right)\left( {\pi_{{ent,L}}^{{D,NE}} - F} \right) \geqslant 0, $$

Solving for γ, we find $\gamma \geqslant \frac{{F - \pi_{{ent,H}}^{{D,NE}}}}{{\pi_{{ent,L}}^{{D,NE}} - \pi_{{ent,H}}^{{D,NE}}}} \equiv {{p}^{{NE}}}$, where 0 ≤ p ^NE ≤ 1holds from our above discussion. Therefore, the entrant enters after observing no expansion if γ ≥ p ^NE, but does not otherwise. Given the entrant’s strategies let us now examine the incumbent:

• If p ≥ p ^E and γ ≥ p ^NEthen the entrant enters both after observing expansion and no expansion. Therefore, the high-demand incumbent expands (as prescribed) if and only if $\matrix{ {\pi_{{inc,H}}^{{D,NE}}\left( {\overline q } \right) < \pi_{{inc,H}}^{{D,E}} - {{K}_H},} \hfill &{\text{or}} \hfill &{{{K}_H} < \pi_{{inc,H}}^{{D,E}} - \pi_{{inc,H}}^{{D,NE}}\left( {\overline q } \right)} \hfill \\ }<!end array>$. However, the low-demand incumbent does not expand since $\matrix{ {\pi_{{inc,H}}^{{D,NE}} > \pi_{{inc,H}}^{{D,E}} - {{K}_H},} \hfill &{\text{or}} \hfill &{{{K}_H} > \pi_{{inc,H}}^{{D,E}} - \pi_{{inc,H}}^{{D,NE}} = 0} \hfill \\ }<!end array>$, which holds for any expansion costs K _L  > 0. Thus, the strategy profile in which both types of incumbent expand cannot be supported as a pooling PBE of the signaling game.

• If p ≥ p ^E but γ < p ^NE then the entrant enters after observing expansion (in equilibrium), but does not enter otherwise. Hence, the high-demand incumbent expands (as prescribed) if and only if $\matrix{ {\pi_{{inc,H}}^{{M,NE}}\left( {\overline q } \right) < \pi_{{inc,H}}^{{D,E}} - {{K}_H},} \hfill &{\text{or}} \hfill &{{{K}_H} < \pi_{{inc,H}}^{{D,E}} - \pi_{{inc,H}}^{{M,NE}}\left( {\overline q } \right) \equiv BC{{C}_H} - PLE_H^E} \hfill \\ }<!end array>$. By contrast, the low-demand incumbent does not expand since $\matrix{ {\pi_{{inc,L}}^{{D,NE}} > \pi_{{inc,L}}^{{D,E}} - {{K}_L},} \hfill &{\text{or}} \hfill &{{{K}_L} > \pi_{{inc,L}}^{{D,E}} - \pi_{{inc,L}}^{{D,NE}} = 0} \hfill \\ }<!end array>$, which holds for all expansion costs K _L  > 0. Therefore, this strategy profile cannot be sustained as a pooling PBE of the signaling game.

• If p < p ^E but γ ≥ p ^NE then the entrant does not enter after observing expansion (in equilibrium), but enters otherwise. Hence, the high-demand incumbent expands (as prescribed) if and only if $\matrix{ {\pi_{{inc,H}}^{{D,NE}}\left( {\overline q } \right) < \pi_{{inc,H}}^{{M,E}} - {{K}_H},} \hfill &{\text{or}} \hfill &{{{K}_H} < \pi_{{inc,H}}^{{M,E}} - \pi_{{inc,H}}^{{D,NE}}\left( {\overline q } \right)} \hfill \\ }<!end array>$. Since

  $$ BC{{C}_H} + PLE_H^{{NE}} = \left( {\pi_{{inc,H}}^{{M,E}} - \pi_{{inc,H}}^{{M,NE}}\left( {\overline q } \right)} \right) + \left( {\pi_{{inc,H}}^{{M,NE}}\left( {\overline q } \right) - \pi_{{inc,H}}^{{D,NE}}\left( {\overline q } \right)} \right) = \pi_{{inc,H}}^{{M,E}} - \pi_{{inc,H}}^{{D,NE}}\left( {\overline q } \right), $$

  the high-demand incumbent expands if ${{K}_H} < BC{{C}_H} + PLE_H^{{NE}}$. Similarly, the low-demand incumbent expands (as prescribed) if and only if $\matrix{ {\pi_{{inc,L}}^{{D,NE}} < \pi_{{inc,L}}^{{M,E}} - {{K}_L},} \hfill &{\text{or}} \hfill &{{{K}_L} < \pi_{{inc,L}}^{{M,E}} - \pi_{{inc,L}}^{{D,NE}} \equiv PLE_L^{{NE}}} \hfill \\ }<!end array>$, since $\matrix{ {\pi_{{inc,L}}^{{M,NE}} = \pi_{{inc,L}}^{{M,E}}} \hfill &{\text{and}} \hfill &{\pi_{{inc,L}}^{{D,NE}} = \pi_{{inc,L}}^{{D,E}}} \hfill \\ }<!end array>$ for the unconstrained low-demand incumbent under monopoly and duopoly, respectively. Thus, this strategy profile can be supported as a pooling PBE under expansion costs $\matrix{ {{{K}_H} < BC{{C}_H} + PLE_H^{{NE}}} \hfill &{\text{and}} \hfill &{{{K}_L} < PLE_L^E} \hfill \\ }<!end array>$.

• If p < p ^E and γ < p ^NE then the entrant does not enter after observing any action from the incumbent. Therefore, the high-demand incumbent expands (as prescribed) if and only if $\matrix{ {\pi_{{inc,H}}^{{M,NE}}\left( {\overline q } \right) < \pi_{{inc,H}}^{{M,E}} - {{K}_H},}{\text{or}}\;{{{K}_H} < \pi_{{inc,H}}^{{M,E}} - \pi_{{inc,H}}^{{M,NE}}\left( {\overline q } \right) \equiv BC{{C}_H}} \\ }<!end array>$. By contrast, the low-demand incumbent does not expand since $\matrix{ {\pi_{{inc,L}}^{{M,NE}} > \pi_{{inc,L}}^{{M,E}} - {{K}_L},} \hfill &{\text{or}} \hfill &{{{K}_L} > \pi_{{inc,L}}^{{M,E}} - \pi_{{inc,L}}^{{M,NE}} = 0} \hfill \\ }<!end array>$, which holds for any expansion cost K _L  > 0. Thus, this strategy profile cannot be sustained as a pooling PBE.

Appendix 1-Equilibrium refinement

Proposition A

All equilibria identified in Propositions 1 and 2 survive the Cho and Kreps’ (1987) Intuitive Criterion, except for:

• 1. the pooling equilibrium of no expansion followed by [NoEnter _NoExp , Enter _Exp ] as described in Proposition 1(part 3a), if expansion costs satisfy $BCC_L > {{K}_L} > BCC_L - PLE_L^E$ and K _H  > 0;

• 2. the pooling equilibrium of no expansion followed by [NoEnter _NoExp , NoEnter _Exp ]as described in Proposition 1(part 3b), if expansion costs satisfy $\matrix{ {BC{{C}_L} + PLE_L^{{NE}} > {{K}_L} > BC{{C}_L}} \hfill &{and} \hfill &{PLE_H^{{NE}} > {{K}_H} > 0} \hfill \\ }<!end array>$ ;

• 3. the pooling equilibrium of no expansion followed by [Enter _NoExp , Enter _Exp ]as described in Proposition 2b, if expansion costs satisfy $\matrix{ {BC{{C}_L} + PLE_L^{{NE}} > {{K}_L} > BC{{C}_L}} \hfill &{and} \hfill &{PLE_H^{{NE}} > {{K}_H} > 0} \hfill \\ }<!end array>$ ; and if expansion costs satisfy $\matrix{ {BC{{C}_L} + PLE_L^E > {{K}_L} > BC{{C}_L} + PLE_L^{{NE}} - PLE_L^E} \hfill &{and} \hfill &{{{K}_H} > PLE_H^{{NE}}} \hfill \\ }<!end array>$ .

2.1 Proof

Pooling equilibrium (Proposition 2b)

If the high-cost incumbent deviates towards expansion, the highest payoff she can obtain is $\pi_{{inc,H}}^{{M,E}} - {{K}_H}$, which strictly exceeds her equilibrium payoff of $\pi_{{inc,H}}^{{D,NE}}$ if and only if $\pi_{{inc,H}}^{{M,E}} - \pi_{{inc,H}}^{{D,NE}} > {{K}_H}$, and since $\pi_{{inc,H}}^{{M,E}} = \pi_{{inc,H}}^{{M,NE}}$ this condition implies $PLE_H^{{NE}} \equiv \pi_{{inc,H}}^{{M,NE}} - \pi_{{inc,H}}^{{D,NE}} > {{K}_H}$. Hence, considering the equilibrium condition for the high-cost incumbent (K _H  > 0), she deviates towards expansion if and only if $PLE_H^{{NE}} > {{K}_H} > 0$. Similarly, if the low-cost incumbent deviates towards expansion, the highest payoff she can obtain is $\pi_{{inc,L}}^{{M,E}} - {{K}_L}$, which strictly exceeds her equilibrium payoff of $\pi_{{inc,L}}^{{D,NE}}$ if and only if ${{K}_L} < \pi_{{inc,L}}^{{M,E}} - \pi_{{inc,L}}^{{D,NE}}\left( {\overline q } \right) \equiv BC{{C}_L} + PLE_L^E$. Considering the equilibrium condition for the low-cost incumbent (K _L > BCC _L ), she deviates towards expansion if and only if $BCC_L + PLE_L^{{NE}} - PLE_L^E < {{K}_L} < BCC_L + PLE_L^E$. Hence, the following cases can arise:

• If $\matrix{ {BC{{C}_L} + PLE_L^{{NE}} - PLE_L^E < {{K}_L} < BC{{C}_L} + PLE_L^E} \hfill &{\text{and}} \hfill &{0 < {{K}_H} < PLE_H^{{NE}}} \hfill \\ }<!end array>$ for the low-cost and high-cost incumbent, respectively, then both types of incumbent deviate towards expansion. Then the entrant’s off-the-equilibrium beliefs are updated to μ(H|Exp) = p, where p ^NE ≤ p < p ^E, leading the entrant to stay out after observing this expansion. But then both types of incumbent have incentives to deviate towards expansion, and the pooling equilibrium without expansion violates the Intuitive Criterion for all $\matrix{ {BC{{C}_L} + PLE_L^{{NE}} - PLE_L^E < {{K}_L} < BC{{C}_L} + PLE_L^E} \hfill &{\text{and}} \hfill &{0 < {{K}_H} < PLE_H^{{NE}}} \hfill \\ }<!end array>$and intermediate priors p ^NE ≤ p < p ^E. If, instead, priors are relatively high, p ≥ p ^E, the entrant enters after observing the deviation towards expansion. Hence, the high-cost incumbent does not deviate since its profit from deviating towards expansion, $\pi_{{inc,H}}^{{D,E}} - {{K}_H}$, is lower than her equilibrium profit from not expanding, $\pi_{{inc,H}}^{{D,NE}}$, since $\pi_{{inc,H}}^{{D,E}} = \pi_{{inc,H}}^{{D,NE}}$. Similarly, the low-cost incumbent does not deviate towards expansion either if her profits from doing so, $\pi_{{inc,L}}^{{D,E}} - {{K}_L}$, is lower than her equilibrium profit, $\pi_{{inc,L}}^{{D,NE}}\left( {\overline q } \right)$ or ${{K}_L} > \pi_{{inc,L}}^{{D,E}} - \pi_{{inc,L}}^{{D,NE}}\left( {\overline q } \right) \equiv BC{{C}_L} + PLE_L^{{NE}} - PLE_L^E$, which holds in this pooling equilibrium. Therefore, no type of incumbent deviates and the pooling equilibrium without expansion of Proposition 2b survives the Intuitive Criterion for all $\matrix{ {BC{{C}_L} + PLE_L^{{NE}} - PLE_L^E < {{K}_L} < BC{{C}_L} + PLE_L^E} \hfill &{\text{and}} \hfill &{0 < {{K}_H} < PLE_H^{{NE}}} \hfill \\ }<!end array>$ and priors are relatively high, i.e., p ≥ p ^E.

• If ${{K}_H} > PLE_H^{{NE}}$ but $BCC_L + PLE_L^{{NE}} - PLE_L^E < {{K}_L} < BCC_L + PLE_L^E$, then the high-cost incumbent does not deviate towards expansion, but the low-costs incumbent does deviate. The entrant’s off-the-equilibrium beliefs (after observing a expansion) become μ(H|Exp) = 0, whereas his beliefs after observing no expansion are μ(H|NoExp) = p ∊ [0,1]. Since p > p ^NE holds in this equilibrium, the entrant does not enter after observing expansion, but enters otherwise, i.e., [NoEnter _exp, Enter _noexp]. Given this response by the entrant after updating his off-the-equilibrium beliefs, the high-cost incumbent does not deviate towards expansion if $\matrix{ {\pi_{{inc,H}}^{{D,NE}}\left( {\overline q } \right) > \pi_{{inc,H}}^{{M,E}} - {{K}_H}} \hfill &{\text{or}} \hfill &{{{K}_H} > PLE_H^{{NE}}} \hfill \\ }<!end array>$, which is satisfied in the case we consider. However, the low-cost incumbent deviates towards expansion since $\pi_{{inc,L}}^{{D,NE}}\left( {\overline q } \right) < \pi_{{inc,L}}^{{M,E}} - {{K}_L}$, or

  $$ 0 < {{K}_L} < \pi_{{inc,L}}^{{M,E}} - \pi_{{inc,L}}^{{D,NE}}\left( {\overline q } \right) \equiv BC{{C}_L} + PLE_L^E. $$

• Considering, in addition, this incumbent’s equilibrium conditions $\left( {BC{{C}_L} + PLE_L^{{NE}} - PLE_L^E < {{K}_L}} \right)$, the above condition becomes $BCC_L < {{K}_L} < BCC_L + PLE_L^E$, which indeed holds in the case we consider. As a consequence, the low-cost incumbent deviates towards expansion. Hence, the pooling equilibrium (NoExp_H, NoExp_L) with (Enter_NoExp, Enter_Exp), as described in Proposition 2b, violates the Intuitive Criterion when expansion costs satisfy$\matrix{ {{{K}_H} > PLE_H^{{NE}}} \hfill &{\text{and}} \hfill &{BC{{C}_L} + PLE_L^{{NE}} - PLE_L^E < {{K}_L} < BC{{C}_L} + PLE_L^E} \hfill \\ }<!end array>$.

• If $\matrix{ {{{K}_L} > BC{{C}_L} + PLE_L^{{NE}}} \hfill &{\text{and}} \hfill &{{{K}_H} > PLE_H^{{NE}}} \hfill \\ }<!end array>$, then no incumbent deviates towards expansion. This implies that the entrant does not update his beliefs and therefore he responds by using the prescribed strategy (Enter_EXP, Enter_NEXP). Hence, the pooling (NoExp_H;NoExp_L) with (Enter_EXP; Enter_NEXP) survives the Intuitive Criterion.

• If $\matrix{ {{{K}_L} > BC{{C}_L} + PLE_L^{{NE}}} \hfill &{\text{but}} \hfill &{0 < {{K}_H} < PLE_H^{{NE}}} \hfill \\ }<!end array>$, then the low-cost incumbent does not deviate towards expansion, but the high-cost incumbent does deviate. The entrant’s off-the-equilibrium beliefs become μ(H|Exp) = 1, whereas his beliefs after observing no expansion (in equilibrium) are μ(H|NoExp) = p ∊ [0,1]. Since p > p ^NE holds in this equilibrium, the entrant enters after observing expansion, and also enters after observing no expansion, i.e., [Enter _Exp , Enter _NoExp ]. Hence, the entrant’s strategy coincides with that in equilibrium, and therefore both types of incumbent’s equilibrium strategy are unaffected. Thus, this pooling equilibrium survives the Intuitive Criterion under expansion costs $\matrix{ {{{K}_L} > BC{{C}_L} + PLE_L^{{NE}}} \hfill &{\text{and}} \hfill &{0 < {{K}_H} < PLE_H^{{NE}}} \hfill \\ }<!end array>$.

Pooling equilibrium (Proposition 1, Part 3a)

If the low-cost incumbent deviates towards expansion the highest payoff she can obtain is $\pi_{{inc,L}}^{{M,E}} - {{K}_L}$, which strictly exceeds her equilibrium payoff of $\pi_{{inc,L}}^{{M,NE}}$ if and only if ${{K}_L} < \pi_{{inc,L}}^{{M,E}} - \pi_{{inc,L}}^{{M,NE}} \equiv BCC_L$. Combining this condition with the parameter values under which this equilibrium is supported $\left( {{{K}_L} > BC{{C}_L} - PLE_L^E} \right)$, we obtain that the low-cost incumbent deviates towards expansion if $BC{{C}_L} > {{K}_L} > BC{{C}_L} - PLE_L^E$. Similarly, if the high-cost incumbent deviates towards expansion, the highest payoff she can obtain is $\pi_{{inc,H}}^{{M,E}} - {{K}_H}$, which does not exceed her equilibrium payoff of $\pi_{{inc,H}}^{{M,NE}}$ since $\pi_{{inc,H}}^{{M,E}} - {{K}_H} > \pi_{{inc,H}}^{{M,NE}}$ implies $0 = \pi_{{inc,H}}^{{M,E}} - \pi_{{inc,H}}^{{M,NE}} > {{K}_H}$. As consequence, the high-cost incumbent does not deviate under any parameter values. The following two cases can hence arise:

• If $BC{{C}_L} > {{K}_L} > BC{{C}_L} - PLE_L^E$only the low-cost incumbent has incentives to deviate towards expansion, which helps the entrant restrict his off-the-equilibrium beliefs to μ(H|Exp) = 0. These beliefs induce no entry after observing expansion (and no entry after expansion either since μ(H|NoExp) = p < p ^NEin this equilibrium), i.e., [NoEnter _NoExp , NoEnter _Exp ]. Given this strategy for the entrant, the low-cost incumbent deviates towards expansion since $\pi_{{inc,L}}^{{M,E}} - {{K}_L} > \pi_{{inc,L}}^{{M,NE}}\left( {\overline q } \right)$given that K _L < BCC _L holds in this case. By contrast, the high-cost incumbent does not deviate towards expansion since $\pi_{{inc,H}}^{{M,E}} - {{K}_H} < \pi_{{inc,H}}^{{M,NE}}$for all K _H  > 0. Therefore, only the low-cost incumbent deviates towards expansion, and the pooling PBE where (NoExp_H, NoExp_L) with (NEnter_NoExp, Enter_Exp), as described in Proposition 1, Part 3a, violates the Intuitive Criterion if expansion costs satisfy$BC{{C}_L} > {{K}_L} > BC{{C}_L} - PLE_L^E$and K _H  > 0.

• If, instead, ${{K}_L} > BC{{C}_L} > BC{{C}_L} - PLE_L^E$, then no type of incumbent has incentives to deviate towards no expansion. Hence, the entrant’s beliefs are unaffected, his strategy still coincides with that in the pooling equilibrium, i.e., [NEnter _NoExp , Enter _Exp ], and this pooling equilibrium survives the Intuitive Criterion.

Pooling equilibrium (Proposition 2a)

If the low-cost incumbent deviates towards expansion the highest payoff she can obtain is $\pi_{{inc,L}}^{{M,E}} - {{K}_L}$, which strictly exceeds her equilibrium payoff of $\pi_{{inc,L}}^{{D,NE}}\left( {\overline q } \right)$ if and only if ${{K}_L} < \pi_{{inc,L}}^{{M,E}} - \pi_{{inc,L}}^{{D,NE}}\left( {\overline q } \right) \equiv BC{{C}_L} + PLE_L^{{NE}}$. This inequality, however, contradicts the parameter condition for the low-cost incumbent supporting this pooling PBE. As a consequence, she does not deviate towards expansion. Similarly, if the high-cost incumbent deviates towards expansion, the highest payoff she can obtain is $\pi_{{inc,H}}^{{M,E}} - {{K}_H}$, which strictly exceeds her equilibrium payoff of $\pi_{{inc,H}}^{{D,NE}}$ if and only if ${{K}_H} < \pi_{{inc,H}}^{{M,E}} - \pi_{{inc,H}}^{{D,NE}} \equiv PLE_H^E$. This inequality also contradicts the parameter condition for the high-cost incumbent supporting this pooling PBE. Therefore, she does not deviate towards expansion either. Hence, no type of incumbent has incentives to deviate towards expansion, and the pooling equilibrium (NoExp_H, NoExp_L) with (E_NEXP; NE_EXP) survives the Intuitive Criterion.

Pooling equilibrium (Proposition 1b)

If the low-cost incumbent deviates towards expansion the highest payoff she can obtain is $\pi_{{inc,L}}^{{M,E}} - {{K}_L}$, which strictly exceeds her equilibrium payoff of $\pi_{{inc,L}}^{{M,NE}}$ if and only if ${{K}_L} < \pi_{{inc,L}}^{{M,E}} - \pi_{{inc,L}}^{{M,NE}}\left( {\overline q } \right) \equiv BC{{C}_L}$. This inequality, however, contradicts the parameter condition for the high-cost incumbent supporting this pooling PBE (K _L > BCC _L ). As a consequence, she does not deviate towards expansion. Similarly, if the high-cost incumbent deviates towards expansion, the highest payoff she can obtain is $\pi_{{inc,H}}^{{M,E}} - {{K}_H}$, which does not exceed her equilibrium payoff of $\pi_{{inc,H}}^{{M,NE}}\left( {\overline q } \right)$for any K _H  > 0. Therefore, the high-cost incumbent does not deviate towards expansion either, and the pooling equilibrium (NoExp_H, NoExp_L) with (NoEntry_NEXP; NoEntry_EXP) survives the Intuitive Criterion.

Pooling equilibrium (Proposition 1, Part 3b)

If the high-cost incumbent deviates towards no expansion, the highest payoff she can obtain is $\pi_{{inc,H}}^{{M,NE}}$, which strictly exceeds her equilibrium payoff of $\pi_{{inc,L}}^{{M,E}} - {{K}_L}$ for any K _H  > 0, i.e., $\pi_{{inc,H}}^{{M,NE}} > \pi_{{inc,H}}^{{M,E}} - {{K}_H},{\text{or}}\,{{K}_H} > \pi_{{inc,H}}^{{M,E}} - \pi_{{inc,H}}^{{M,NE}} = 0$. Considering the equilibrium condition for the high-cost incumbent $\left( {{{K}_H} < PLE_H^{{NE}}} \right)$, this implies that she deviates towards no expansion if and only if $0 < {{K}_H} < PLE_H^{{NE}}$. Regarding the low-cost incumbent, if she deviates towards no expansion the highest payoff she can obtain is $\pi_{{inc,L}}^{{M,NE}}\left( {\overline q } \right)$, which strictly exceeds her equilibrium payoff of $\pi_{{inc,L}}^{{M,E}} - {{K}_L}$ if and only if ${{K}_L} > \pi_{{inc,L}}^{{M,E}} - \pi_{{inc,L}}^{{M,NE}}\left( {\overline q } \right) \equiv BC{{C}_L}$. Considering the equilibrium condition for the low-cost incumbent $\left( {{{K}_L} < BC{{C}_L} + PLE_L^{{NE}}} \right)$, she deviates towards no expansion if and only if $BCC_L < {{K}_L} < BCC_L + PLE_L^{{NE}}$. Hence, the following cases can arise:

• If $\matrix{ {BC{{C}_L} < {{K}_L} < BC{{C}_L} + PLE_L^{{NE}}} \hfill &{\text{and}} \hfill &{0 < {{K}_H} < PLE_H^{{NE}}} \hfill \\ }<!end array>$, then both incumbents deviate towards no expansion, and the entrant’s off-the-equilibrium beliefs become μ(H|NoExp) = p, where p < p ^NE (the pooling equilibrium of Proposition 1, part 3b holds only for relatively low priors). The entrant hence stays out after observing a deviation towards no expansion. Therefore, both types of incumbents have incentives to deviate towards no expansion if $\matrix{ {\pi_{{inc,H}}^{{M,NE}} > \pi_{{inc,H}}^{{M,E}} - {{K}_H},} \hfill &{\text{or}} \hfill &{{{K}_H} > \pi_{{inc,H}}^{{M,E}} - \pi_{{inc,H}}^{{M,NE}} = 0} \hfill \\ }<!end array>$ for the high-cost incumbent and $\matrix{ {\pi_{{inc,L}}^{{M,NE}}\left( {\overline q } \right) > \pi_{{inc,L}}^{{M,E}} - {{K}_L},} \hfill &{\text{or}} \hfill &{{{K}_L} > \pi_{{inc,L}}^{{M,E}} - \pi_{{inc,L}}^{{M,NE}}\left( {\overline q } \right) \equiv BC{{C}_L}} \hfill \\ }<!end array>$ for the low-cost incumbent. Since both conditions on the expansion costs of the low and high-cost incumbent are satisfied, both types of incumbent deviate towards no expansion, and the pooling equilibrium with expansion of Proposition 1 (part 3b) violates the Intuitive Criterion for all: $\matrix{ {BC{{C}_L} < {{K}_L} < BC{{C}_L} + PLE_L^{{NE}}} \hfill &{\text{and}} \hfill &{0 < {{K}_H} < PLE_H^{{NE}}} \hfill \\ }<!end array>$.

• If, instead, $\matrix{ {{{K}_L} < BC{{C}_L} < BC{{C}_L} + PLE_L^{{NE}}} \hfill &{\text{but}} \hfill &{0 < {{K}_H} < PLE_H^{{NE}}} \hfill \\ }<!end array>$, then the low-cost incumbent does not deviate towards no expansion, while the high-cost incumbent does deviate. The entrant’s off-the-equilibrium beliefs become μ(H|NExp) = 1, whereas his beliefs after observing expansion are μ(H|Exp) = p ∊ [0,1]. Since in this equilibrium priors satisfyp <p ^E, the entrant stays out after observing expansion but enters after observing no expansion, i.e., (NoEnter _Exp ,Enter _NExp ). The optimal response by the entrant after updating his beliefs, however, coincides with his response in this pooling equilibrium. As a consequence, the incumbent’s expansion decision is unaffected, and we can conclude that this pooling equilibrium survives the Intuitive Criterion if expansion costs satisfy $\matrix{ {{{K}_L} < BC{{C}_L} < BC{{C}_L} + PLE_L^{{NE}}} \hfill &{\text{and}} \hfill &{0 < {{K}_H} < PLE_H^{{NE}}} \hfill \\ }<!end array>$.

Appendix 2-Semiseparating equilibria

Proposition B

The following strategy profiles can be supported as semi-separating PBEs of the game:

• 1. A strategy profile where the incumbent expands her facility when her costs are low, p _L   = 1, but expands with probability p _H ∊ (0,1)when her costs are high, where ${{p}_H} = \frac{{{{p}^E}}}{{1 - {{p}^E}}}\frac{{1 - p}}{p}.$ In this equilibrium, after observing no expansion the entrant enters, s = 1, and after observing expansion, the entrant enters with probability $r = 1 - \frac{{{{K}_H}}}{{PLE_H^E}}$ , given beliefs μ(H|NoExp) = 1 and μ(H|Exp) = p ^E . This equilibrium can be only supported if priors are relatively high, p > p ^E , and expansion costs satisfy $0 < {{K}_H} < \min \left\{ {PLE_H^E,{{K}_H}^1} \right\}$ , where $\matrix{ {K_H^1 \equiv - \frac{B}{{PLE_L^E}} + \frac{{PLE_H^E}}{{PLE_L^E}}{{K}_L}} \hfill &{and} \hfill &{B \equiv PLE_H^E\left[ {BC{{C}_L} + \left( {PLE_L^{{NE}} - PLE_L^E} \right)} \right]} \hfill \\ }<!end array>$ .

• 2. A strategy profile where the incumbent expands her facility when her costs are high, p _H   = 1, but expands with probability p _L ∊ (0,1)when her costs are low, where ${{p}_L} = \frac{p}{{1 - p}}\frac{{1 - {{p}^E}}}{{{{p}^E}}}.$ In this equilibrium, after observing no expansion the entrant stays out, s = 0, and after observing expansion, the entrant enters with probability $r = 1 - \frac{{{{K}_H}}}{{PLE_H^E}}$ , given beliefs μ(H|NoExp) = 0 and μ(H|Exp) = p ^E . This equilibrium can be only supported if priors are relatively high, p > p ^E , and expansion costs satisfy ${{K}_L} < PLE_L^E$ .

• 3. A strategy profile where the high-cost incumbent expands with probability ${{p}_H} = \frac{{{{p}^E}\left( {p - {{p}^{{NE}}}} \right)}}{{\left( {{{p}^E} - {{p}^{{NE}}}} \right)p}}$ ,and the low-cost incumbent expands with probability ${{p}_L} = \frac{{\left( {1 - {{p}^E}} \right)\left( {p - {{p}^{{NE}}}} \right)}}{{\left( {{{p}^E} - {{p}^{{NE}}}} \right)\left( {1 - p} \right)}}$ , where ${{p}_H},{{p}_L} \in \left( {0,1} \right)$ After observing expansion (no expansion) the entrant enters with probability r (s, respectively), where

  $$ r = \left[ {{{K}_L} - BC{{C}_L} - \frac{{{{K}_H} \times PLE_L^{{NE}}}}{{PLE_H^E}}} \right] \times \frac{1}{{PLE_L^{{NE}} - PLE_L^E}}\,and\, s = \left[ {{{K}_L} - BC{{C}_L} - \frac{{{{K}_H} \times PLE_L^E}}{{PLE_H^E}}} \right] \times \frac{1}{{PLE_L^{{NE}} - PLE_L^E}} $$

  given beliefs μ(H|NoExp) = p ^NE and μ(H|Exp) = p ^E . This equilibrium can be only supported if priors are intermediate, p ^NE ≥ p > p ^E , and: (1) expansion costs satisfy $\matrix{ {0 < {{K}_H} < \frac{C}{{PLE_L^{{NE}}}}} \hfill &{and} \hfill &{0 < {{K}_L} < \frac{{PLE_L^E}}{{PLE_H^E}}{{K}_H} + \frac{B}{{PLE_H^E}}} \hfill \\ }<!end array>$ if $PLE_L^{{NE}} > PLE_L^E$ ; (2) expansion costs satisfy K _H   > 0 for all K _L ≤ BCC _L and ${{K}_H} > \frac{C}{{PLE_L^E}}$ otherwise if $PLE_L^{{NE}} \leqslant PLE_L^E$ , where $\matrix{ {B \equiv PLE_H^E\left[ {BC{{C}_L} + \left( {PLE_L^{{NE}} - PLE_L^E} \right)} \right]} \hfill &{and} \hfill &{C \equiv PLE_H^E\left( {{{K}_L} - BC{{C}_L}} \right)} \hfill \\ }<!end array>$ .

• 4. A strategy profile where the incumbent does not expand her facility when her costs are high, p _H   = 0, but expands with probability p _L ∊ (0,1)when her costs are low, where ${{p}_L} = \frac{{{{p}^{{NE}}} - p}}{{{{p}^{{NE}}}\left( {1 - p} \right)}}$ In this equilibrium, after observing expansion the entrant stays out, r = 0, and after observing no expansion the entrant enters with probability $s = \frac{{BC{{C}_L}}}{{PLE_L^E}} + \frac{{{{K}_L}}}{{PLE_L^E}}$ , given beliefs μ(H|Exp) = 0 and μ(H|NoExp) = p ^NE . This equilibrium can be only supported if priors are relatively low, p < p ^NE , and expansion costs satisfy $PLE_L^E - BC{{C}_L} > {{K}_L} > \frac{{PLE_L^E}}{{PLE_H^E}}{{K}_H} - BC{{C}_L}$ .

• 5. A strategy profile where the incumbent does not expand her facility when her costs are low, p _L   = 0, but expands with probability p _H ∊ (0,1)when her costs are low, where ${{p}_H} = \frac{{p - {{p}^{{NE}}}}}{{p\left( {1 - {{p}^{{NE}}}} \right)}}$ In this equilibrium, after observing expansion the entrant enters, r = 0, and after observing no expansion the entrant enters with probability $s = \frac{{{{K}_H}}}{{PLE_H^E}}$ , given beliefs μ(H|Exp) = 1 and μ(H|NoExp) = p ^NE . This equilibrium can be only supported for priors, p > p ^NE , and expansion costs satisfying

  $$ PLE_H^E > {{K}_H} > \frac{{PLE_H^E}}{{PLE_L^E}}{{K}_L} + \frac{{BC{{C}_L} \times PLE_H^E}}{{PLE_L^E}} $$

3.1 Proof

p _L  = 1 and p _H ∊ (0,1)

In this equilibrium, the entrant’s beliefs after observing no expansion become μ(H|NoExp) = 1, which leads him to enter since $\pi_{{ent,H}}^{{D,NE}} > F$. In the case that the entrant observes expansion, he mixes if his beliefs μ(H|Exp) satisfy

$$ \mu (H|Exp) \times \left( {\pi_{{ent,H}}^{{D,E}} - F} \right) + \left( {1 - \mu \left( {H|Exp} \right)} \right)\left( {\pi_{{ent,L}}^{{D,E}} - F} \right) = 0 $$

and solving for μ(H|Exp), we obtain $\mu \left( {H|Exp} \right) = \frac{{F - \pi_{{ent,H}}^{{D,E}}}}{{\pi_{{ent,L}}^{{D,E}} - \pi_{{ent,H}}^{{D,E}}}} \equiv {{p}^E}$. We can now use the entrant’s posterior beliefs μ(H|Exp) = p ^Ein order to find the probability, p _H , with which the high-cost incumbent randomizes, by using Bayes’ rule as follows

$$ \mu \left( {H|Exp} \right) = {{p}^E} = \frac{{p \times {{p}_H}}}{{\left( {p \times {{p}_H}} \right) + \left( {1 - p} \right)}} $$

Solving for p _H we obtain ${{p}_H} = \frac{{{{p}^E}}}{{\left( {1 - {{p}^E}} \right)}} \times \frac{{\left( {1 - p} \right)}}{p}$, where p _H ∊ (0,1) for all p ^E < p. In addition, note that p _H is decreasing in p since $\frac{{\partial {{p}_H}}}{{\partial p}} = \frac{{{{p}^E}}}{{\left( {{{p}^E} - 1} \right) \times {{p}^2}}} < 0$, starting from $\mathop{{\lim }}\limits_{{p \to {{p}^E}}} {{p}_H} = 1$ and converging to $\mathop{{\lim }}\limits_{{p \to 1}} {{p}_H} = 0$.

Regarding the incumbent, when her costs are high, she mixes as prescribed, p _H ∊ (0,1), if and only if

$$ r \times \left( {\pi_{{inc,H}}^{{D,E}} - {{K}_H}} \right) + \left( {1 - r} \right) \times \left( {\pi_{{inc,H}}^{{M,E}} - {{K}_H}} \right) = \pi_{{inc,H}}^{{D,NE}}, $$

where r is the probability with which the entrant enters after observing expansion. Solving for r, we obtain $r = \frac{{\pi_{{inc,H}}^{{M,E}} - \pi_{{inc,H}}^{{D,NE}} - {{K}_H}}}{{\pi_{{inc,H}}^{{M,E}} - \pi_{{inc,H}}^{{D,E}}}}$, and since $\matrix{ {PLE_H^E = \pi_{{inc,H}}^{{M,E}} - \pi_{{inc,H}}^{{D,E}}} \hfill &{\text{and}} \hfill &{\pi_{{inc,H}}^{{D,E}} = \pi_{{inc,H}}^{{D,NE}}} \hfill \\ }<!end array>$, we can rewrite this probability as $r = 1 - \frac{{{{K}_H}}}{{PLE_H^E}}$, where r ∊ (0,1) only if expansion costs satisfy ${{K}_H} < PLE_H^E$. On the other hand, the low-cost incumbent expands as prescribed (p _L  = 1) if and only if

$$ r \times \left( {\pi_{{inc,L}}^{{D,E}} - {{K}_L}} \right) + \left( {1 - r} \right) \times \left( {\pi_{{inc,L}}^{{M,E}} - {{K}_L}} \right) > \pi_{{inc,L}}^{{D,NE}}. $$

which implies

$$ r > \frac{{\pi_{{inc,L}}^{{M,E}} - \pi_{{inc,L}}^{{D,NE}} - {{K}_L}}}{{\pi_{{inc,L}}^{{M,E}} - \pi_{{inc,L}}^{{D,E}}}} = \frac{{BC{{C}_L} + PLE_L^{{NE}} - {{K}_L}}}{{PLE_L^E}} \equiv \widehat{r}, $$

Hence, probability r must satisfy $r > \widehat{r}$which implies ${{K}_H} < - \frac{B}{{PLE_L^E}} + \frac{{PLE_H^E}}{{PLE_L^E}}{{K}_L} \equiv {{K}^1}_H$ where$B \equiv PLE_H^E\left[ {BC{{C}_L} + \left( {PLE_L^{{NE}} - PLE_L^E} \right)} \right]$. Hence, this semiseparating strategy profile can be supported as an equilibrium if expansion costs satisfy $0 < {{K}_H} < \min \left\{ {PLE_H^E,{{K}_H}^1} \right\}$ and priors are relatively high, i.e. p > p ^E.

p _H  = 1 and p _L ∊ (0,1)

In this equilibrium, the entrant’s posterior beliefs after observing no expansion are μ(H|NoExp) = 0, which leads him to stay out since $\pi_{{ent,L}}^{{D,NE}} < F$. In the case that the entrant observes expansion, he mixes if his beliefs μ(H|Exp) satisfy

$$ \mu \left( {H|Exp} \right) \times \left( {\pi_{{ent,H}}^{{D,E}} - F} \right) + \left( {1 - \mu \left( {H|Exp} \right)} \right)\left( {\pi_{{ent,L}}^{{D,E}} - F} \right) = 0 $$

and solving for μ(H|Exp), we obtain μ(H|Exp) = p ^E. Using Bayes’ rule, $\mu \left( {H|Exp} \right) = {{p}^E} = \frac{p}{{p + \left( {1 - p} \right){{p}_L}}}$ and solving for p _L we obtain ${{p}_L} = \frac{p}{{1 - p}} \times \frac{{1 - {{p}^E}}}{{{{p}^E}}}$, where p _L ∊ (0,1) for all p < p ^E. In addition, p _L is increasing in p since $\frac{{\partial {{p}_L}}}{{\partial p}} = \frac{{1 - {{p}^E}}}{{{{p}^E}{{{\left( {1 - p} \right)}}^2}}} > 0$, starts at $\mathop{{\lim }}\limits_{{p \to 0}} {{p}_L} = 0$and converges to $\mathop{{\lim }}\limits_{{p \to {{p}^E}}} {{p}_L} = 1$.

Regarding the incumbent, when her costs are high, she expands as prescribed (p _H  = 1) if and only if

$$ r \times \left( {\pi_{{inc,H}}^{{D,E}} - {{K}_H}} \right) + \left( {1 - r} \right) \times \left( {\pi_{{inc,H}}^{{M,E}} - {{K}_H}} \right) > \pi_{{inc,H}}^{{M,NE}}, $$

solving for r, and using the property that $\pi_{{inc,H}}^{{M,NE}} = \pi_{{inc,H}}^{{M,E}}$, we find

$$ r > \frac{{ - {{K}_H}}}{{\pi_{{inc,H}}^{{M,E}} - \pi_{{inc,H}}^{{D,E}}}} = - \frac{{{{K}_H}}}{{PLE_H^E}} \equiv \widetilde{r} $$

where cutoff $\widetilde{r} < 0$ for all parameter values. On the other hand, the low-cost incumbent randomizes as prescribed, p _L ∊ (0,1), if and only if

$$ r \times \left( {\pi_{{inc,L}}^{{D,E}} - {{K}_L}} \right) + \left( {1 - r} \right) \times \left( {\pi_{{inc,L}}^{{M,E}} - {{K}_L}} \right) = \pi_{{inc,L}}^{{D,E}}. $$

which implies

$$ r = \frac{{\pi_{{inc,L}}^{{M,E}} - \pi_{{inc,L}}^{{D,E}} - {{K}_L}}}{{\pi_{{inc,L}}^{{M,E}} - \pi_{{inc,L}}^{{D,E}}}} = 1 - \frac{{{{K}_L}}}{{PLE_L^E}}, $$

where r ∊ (0,1) if ${{K}_L} < PLE_L^E$. Finally, since cutoff $\widetilde{r} < 0,r > \widetilde{r}$under all parameter values. Hence, this semiseparating strategy profile can be supported as an equilibrium if expansion costs satisfy ${{K}_L} < PLE_L^E$for low and intermediate priors, i.e., p < p ^E.

p _L , p _H ∊ (0,1)

In this equilibrium, after observing an expansion, the entrant is indifferent between entering and not entering the incumbent’s market if and only if his posterior beliefs μ(H|Exp) satisfy

$$ \mu (H|Exp) \times \left( {\pi_{{ent,H}}^{{D,E}} - F} \right) + \left( {1 - \mu \left( {H|Exp} \right)} \right)\left( {\pi_{{ent,L}}^{{D,E}} - F} \right) = 0 $$

and solving for μ(H|Exp), we obtain $\mu \left( {H|Exp} \right) = \frac{{F - \pi_{{ent,H}}^{{D,E}}}}{{\pi_{{ent,L}}^{{D,E}} - \pi_{{ent,H}}^{{D,E}}}} \equiv {{p}^E}$. We can then use the entrant’s posterior beliefs μ(H|Exp) = p ^Ein order to find probability, p _H , with which the incumbent randomizes when her costs are high, by using Bayes’ rule, as follows

$$ \mu \left( {H|Exp} \right) = {{p}^E} = \frac{{p \times {{p}_H}}}{{\left( {p \times {{p}_H}} \right) + \left( {\left( {1 - p} \right) \times {{p}_L}} \right)}} $$

Solving for p _H we obtain ${{p}_H}\left( {{{p}_L}} \right) = \frac{{{{p}^E} \times {{p}_L} \times \left( {1 - p} \right)}}{{p \times \left( {1 - {{p}^E}} \right)}}$. Similarly, after observing that the incumbent does not expand, the entrant is indifferent between entering and not entering the incumbent’s market if and only if his posterior beliefs μ(H|NoExp) satisfy

$$ \mu \left( {H|NoExp} \right) \times \left( {\pi_{{ent,H}}^{{D,NE}} - F} \right) + \left( {1 - \mu \left( {H|NoExp} \right)} \right)\left( {\pi_{{ent,L}}^{{D,NE}} - F} \right) = 0 $$

and solving for μ(H|NoExp) we obtain $\mu \left( {H|NoExp} \right) = \frac{{F - \pi_{{ent,L}}^{{D,NE}}}}{{\pi_{{ent,H}}^{{D,NE}} - \pi_{{ent,L}}^{{D,NE}}}} \equiv {{p}^{{NE}}}$. We can hence use the entrant’s posterior beliefs μ(H|NoExp) = p ^NEin order to find probability, p _L , with which the incumbent randomizes when her costs are low, by using Bayes’ rule, as follows

$$ \mu \left( {H|NoExp} \right) = {{p}^{{NE}}} = \frac{{p \times \left( {1 - {{p}_H}} \right)}}{{\left( {p \times \left( {1 - {{p}_H}} \right)} \right) + \left( {\left( {1 - p} \right) \times \left( {1 - {{p}_L}} \right)} \right)}} $$

Solving for p _L we obtain ${{p}_L}\left( {{{p}_H}} \right) = \frac{{{{p}^{{NE}}} + \left( {{{p}_H} - 1} \right)p - {{p}_H} \times {{p}^{{NE}}} \times p}}{{{{p}^{{NE}}} \times \left( {1 - p} \right)}}$. Solving for p _H and p _L simultaneously, we obtain $\matrix{ {{{p}_L} = \frac{{\left( {1 - {{p}^E}} \right) \times \left( {p - {{p}^{{NE}}}} \right)}}{{\left( {{{p}^E} - {{p}^{{NE}}}} \right) \times \left( {1 - p} \right)}}} \hfill &{\text{and}} \hfill &{{{p}_H} = \frac{{{{p}^E} \times \left( {p - {{p}^{{NE}}}} \right)}}{{\left( {{{p}^E} - {{p}^{{NE}}}} \right) \times p}}} \hfill \\ }<!end array>$. First, note that p _L  ≥ 0 if and only if p ≥ p ^NE, given that p ^E > p ^NE and p, p ^E, p ^NE ∊ (0,1) under all parameter values. In addition, p _L  < 1 for all p < p ^E. Therefore, p _L ∊ (0,1) if and only if priors are intermediate, i.e., p ^NE ∊ p < p ^E. Second, note that p _H  ≥ 0if and only if p ≥ p ^NE. Furthermore, p _H  < 1 for all p < p ^E. Therefore, p _H ∊ (0,1) only if priors are intermediate, i.e., p ^NE ≤ p < p ^E. We can therefore conclude that under intermediate priors both types of incumbent randomize their expansion decisions, p _H , p _L ∊ (0,1), where note that p _H ≥ p _L for all p < p ^E, which holds in this regime of intermediate priors. We next show that these probabilities are both increasing in p, since

$$ \frac{{\partial {{p}_{L}}}}{{\partial p}} = \frac{{\left( {1 - {{p}^{E}}} \right)\left( {1 - {{p}^{{NE}}}} \right)}}{{{{{\left( {1 - p} \right)}}^{2}}\left( {{{p}^{E}} - {{p}^{{NE}}}} \right)}} > 0\;{\text{and}}\;\frac{{\partial {{p}_{H}}}}{{\partial p}} = \frac{{{{p}^{{NE}}}{{p}^{E}}}}{{{{p}^{2}}\left( {{{p}^{E}} - {{p}^{{NE}}}} \right)}} > 0. $$

Finally, note that at the lower bound of p ∊ [p ^NE, p ^E), the incumbent’s randomization becomes $\mathop{{\lim }}\limits_{{p \to {{p}^{{NE}}}}} {{p}_L} = \mathop{{\lim }}\limits_{{p \to {{p}^{{NE}}}}} {{p}_H} = 0$ whereas at the upper bound, we obtain $\mathop{{\lim }}\limits_{{p \to {{p}^E}}} {{p}_L} = \mathop{{\lim }}\limits_{{p \to {{p}^E}}} {{p}_H} = 1$.

Let us now examine the incumbent’s strategy in this equilibrium. If the high-cost incumbent expands with probability p _H ∊ (0,1), as prescribed, it must be that the entrant makes her indifferent between expanding and not expanding her facility,

$$ r \times \left( {\pi_{{inc,H}}^{{D,E}} - {{K}_H}} \right) + \left( {1 - r} \right) \times \left( {\pi_{{inc,H}}^{{M,E}} - {{K}_H}} \right) = s \times \pi_{{inc,H}}^{{D,NE}} + \left( {1 - s} \right) \times \pi_{{inc,H}}^{{M,NE}}, $$

where r and s are the probability with which the entrant enters after observing expansion and no expansion, respectively. Solving for r, we obtain

$$ r(s) = \frac{{\pi_{{inc,H}}^{{M,E}} - \pi_{{inc,H}}^{{M,NE}} - {{K}_H}}}{{\pi_{{inc,H}}^{{M,E}} - \pi_{{inc,H}}^{{D,E}}}} + s\left( {\frac{{\pi_{{inc,H}}^{{M,NE}} - \pi_{{inc,H}}^{{D,NE}}}}{{\pi_{{inc,H}}^{{M,E}} - \pi_{{inc,H}}^{{D,E}}}}} \right) $$

and since $\matrix{ {\pi_{{inc,H}}^{{M,E}} = \pi_{{inc,H}}^{{M,NE}}} \hfill &{\text{and}} \hfill &{\pi_{{inc,H}}^{{D,E}} = \pi_{{inc,H}}^{{D,NE}}} \hfill \\ }<!end array>$, the above expression simplifies to $r(s) = s - \frac{{{{K}_H}}}{{PLE_H^E}}$.

Similarly, when the incumbent’s costs are low, the entrant makes the incumbent indifferent between expanding and not expanding her facility,

$$ r \times \left( {\pi_{{inc,L}}^{{D,E}} - {{K}_L}} \right) + \left( {1 - r} \right) \times \left( {\pi_{{inc,L}}^{{M,E}} - {{K}_L}} \right) = s \times \pi_{{inc,L}}^{{D,NE}} + \left( {1 - s} \right) \times \pi_{{inc,L}}^{{M,NE}}. $$

Solving for s, we obtain

$$ s(r) = \frac{{\pi_{{inc,L}}^{{M,NE}} + {{K}_L} - \pi_{{inc,L}}^{{M,E}}}}{{\pi_{{inc,L}}^{{M,NE}} - \pi_{{inc,L}}^{{D,NE}}}} - r\left( {\frac{{\pi_{{inc,L}}^{{D,E}} - \pi_{{inc,L}}^{{M,E}}}}{{\pi_{{inc,L}}^{{M,NE}} - \pi_{{inc,L}}^{{D,NE}}}}} \right), $$

which simplifies to $s(r) = \frac{{{{K}_L} - BC{{C}_L}}}{{PLE_L^{{NE}}}} + r\left( {\frac{{PLE_L^E}}{{PLE_L^{{NE}}}}} \right)$. Solving for probabilities s and r simultaneously, we obtain

$$ \matrix{ {s = \left[ {{{K}_L} - BC{{C}_L} - \frac{{{{K}_H} \times PLE_L^E}}{{PLE_H^E}}} \right] \times \frac{1}{{PLE_L^{{NE}} - PLE_L^E}}} \\ {r = \left[ {{{K}_L} - BC{{C}_L} - \frac{{{{K}_H} \times PLE_L^{{NE}}}}{{PLE_H^E}}} \right] \times \frac{1}{{PLE_L^{{NE}} - PLE_L^E}}} \\ }<!end array> $$

First, note, that probability s is positive if and only if $\matrix{ {PLE_L^{{NE}} > PLE_L^E} \hfill &{\text{and}} \hfill &{\frac{{PLE_H^E \times \left( {{{K}_L} - BC{{C}_L}} \right)}}{{PLE_L^E}} > {{K}_H}} \hfill \\ }<!end array>$ hold. Secondly, s < 1 if and only if

$$ {{K}_H} > \frac{{PLE_H^E}}{{PLE_L^E}}{{K}_L} - \frac{{PLE_H^E \times \left( {BC{{C}_L} + \left( {PLE_L^{{NE}} - PLE_L^E} \right)} \right)}}{{PLE_L^E}} $$

(1)

Similarly, note that probability r is positive if and only if $\matrix{ {PLE_L^{{NE}} > PLE_L^E} \hfill &{\text{and}} \hfill &{\frac{{PLE_H^E \times \left( {{{K}_L} - BC{{C}_L}} \right)}}{{PLE_L^{{NE}}}} > {{K}_H}} \hfill \\ }<!end array>$hold. Finally, note that r < 1 if and only if

$$ {{K}_H} > \frac{{PLE_H^E}}{{PLE_L^{{NE}}}}{{K}_L} - \frac{{PLE_H^E\left( {BC{{C}_L} + \left( {PLE_L^{{NE}} - PLE_L^E} \right)} \right)}}{{PLE_L^{{NE}}}} $$

(2)

Let us first analyze the case in which condition$PLE_L^{{NE}} > PLE_L^E$holds. In this case, probabilities r,s ∊ (0,1) if expansion costs satisfy

$$ \matrix{ {\frac{C}{{PLE_L^E}} > {{K}_H} > \frac{{PLE_H^E}}{{PLE_L^E}}{{K}_L} - \frac{B}{{PLE_L^E}}}{\text{\;and\;}}{\frac{C}{{PLE_L^{{NE}}}} > {{K}_H} > \frac{{PLE_H^E}}{{PLE_L^{{NE}}}}{{K}_L} - \frac{B}{{PLE_L^{{NE}}}}} \\ }<!end array> $$

where

$$ \matrix{ {B \equiv PLE_H^E\left[ {BC{{C}_L} + \left( {PLE_L^{{NE}} - PLE_L^E} \right)} \right]}{\text{and}}{C \equiv PLE_H^E\left( {{{K}_L} - BC{{C}_L}} \right).} \\ }<!end array> { } $$

First, note that $\frac{C}{{PLE_L^E}} > \frac{C}{{PLE_L^{{NE}}}}$since $PLE_L^{{NE}} > PLE_L^E$ in the case we consider. Hence $\frac{C}{{PLE_L^{{NE}}}} > {{K}_H}$ is more restrictive than $\frac{C}{{PLE_L^E}} > {{K}_H}$. In order to rank expressions$\matrix{ {\frac{{PLE_H^E}}{{PLE_L^E}} \times {{K}_L} - \frac{B}{{PLE_L^E}}} \hfill &{\text{and}} \hfill &{\frac{{PLE_H^E}}{{PLE_L^{{NE}}}} \times {{K}_L} - \frac{B}{{PLE_L^{{NE}}}}} \hfill \\ }<!end array>$, note that vertical intercepts satisfy $0 > \frac{B}{{PLE_L^{{NE}}}} > \frac{B}{{PLE_L^E}}$ and the equation in condition (2) is flatter than that in (1) since $\frac{{PLE_H^E}}{{PLE_L^E}} > \frac{{PLE_H^E}}{{PLE_L^{{NE}}}}$. Hence, condition (1) is more restrictive than (2). Thus, we only need to use ${{K}_H} > \frac{{PLE_H^E}}{{PLE_L^E}}{{K}_L} - \frac{B}{{PLE_L^E}}$. Regarding condition $\frac{C}{{PLE_L^{{NE}}}} > {{K}_H}$, note that it can be expressed as

$$ \frac{{PLE_H^E}}{{PLE_L^{{NE}}}}{{K}_L} - \frac{{BC{{C}_L} \times PLE_H^E}}{{PLE_L^{{NE}}}} > {{K}_H} $$

which is flatter than condition (1) since $\frac{{PLE_H^E}}{{PLE_L^{{NE}}}} < \frac{{PLE_H^E}}{{PLE_L^E}}$, and the vertical intercept is also smaller (absolute value) than that of condition (1) since $\matrix{ {BC{{C}_L} \times PLE_H^E < B} \hfill &{\text{and}} \hfill &{PLE_L^{{NE}} < PLE_L^E} \hfill \\ }<!end array>$, as depicted in Fig. 4.

Fig. 4

Costs supporting the semiseparating PBE

Hence, when condition$PLE_L^{{NE}} > PLE_L^E$holds, this semiseparating equilibrium with p _H , p _L ∊ (0,1) can be supported for intermediate priors p ^NE ≥ p > p ^E and expansion costs satisfying

$$ \matrix{ {0 < {{K}_H} < \frac{C}{{PLE_L^{{NE}}}}}{\text{and}}{0 < {{K}_L} < \frac{{PLE_L^E}}{{PLE_H^E}}{{K}_H} + \frac{B}{{PLE_H^E}}.} \\ }<!end array> { } $$

If, by contrast, $PLE_L^{{NE}} < PLE_L^E$holds, probabilities r,s∊ (0,1) if expansion costs satisfy

$$ \matrix{ {{{K}_H} > \frac{C}{{PLE_L^E}}}\;{\text{and}}\;{{{K}_H} > \frac{{PLE_H^E}}{{PLE_L^E}}{{K}_L} - \frac{B}{{PLE_L^E}}} \\ }<!end array> $$

and

$$ \matrix{ {{{K}_H} > \frac{C}{{PLE_L^{{NE}}}}}\;{\text{and}}\;{{{K}_H} > \frac{{PLE_H^E}}{{PLE_L^{{NE}}}}{{K}_L} - \frac{B}{{PLE_L^{{NE}}}}} \\ }<!end array> $$

From our previous discussion, $\frac{C}{{PLE_L^E}} > \frac{C}{{PLE_L^{{NE}}}}$. In addition, ${{K}_H} > \frac{C}{{PLE_L^E}}$ is more restrictive than ${{K}_H} > \frac{{PLE_H^E}}{{PLE_L^E}}{{K}_L} - \frac{B}{{PLE_L^E}}$ since both expressions have the same slope but the former originates at a higher vertical intercept than the latter since $BC{{C}_L} \times PLE_H^E < B$. Therefore, when $PLE_L^{{NE}} < PLE_L^E$holds this semiseparating equilibrium with p _H , p _L ∊ (0,1) can be supported for intermediate priors p ^NE ∊ p > p ^E and expansion costs satisfying K _H  > 0 for all K _L ≤ BCC _L , and ${{K}_H} > \frac{C}{{PLE_L^E}}$ otherwise.

p _H =0 and p _L ∊ (0,1)

We now check other semiseparating strategy profiles where either type of incumbent does not expand her facility. Let us first analyze the case where p _H =0 and p _L ∊ (0,1). In this case, the entrant’s posterior beliefs become μ(H|Exp) = 0 after observing an expansion, which leads him to stay out since $\pi_{{ent,L}}^{{D,E}} < F$. In the case that the entrant observes no expansion from the incumbent, the entrant mixes if and only if his beliefs μ(H|Exp) satisfy

$$ \mu \left( {H|NoExp} \right) \times \left( {\pi_{{ent,H}}^{{D,NE}} - F} \right) + \left( {1 - \mu \left( {H|NoExp} \right)} \right)\left( {\pi_{{ent,L}}^{{D,NE}} - F} \right) = 0 $$

and solving for μ(H|NoExp), we obtain μ(H|NoExp) = p ^NE. We can hence use Bayes’ rule to find

$$ \mu \left( {H|NoExp} \right) = {{p}^{{NE}}} = \frac{p}{{p + \left( {1 - p} \right)\left( {1 - {{p}_L}} \right)}} $$

Solving for p _L we obtain ${{p}_L} = \frac{{{{p}^{{NE}}} - p}}{{{{p}^{{NE}}}\left( {1 - p} \right)}}$, where p _L ∊ (0,1) for low priors, i.e., p < p ^NE. In addition, p _L is decreasing in p since $\frac{{\partial {{p}_L}}}{{\partial p}} = \frac{{{{p}^{{NE}}} - 1}}{{{{p}^{{NE}}}{{{\left( {1 - p} \right)}}^2}}} < 0$ starting at $\mathop{{\lim }}\limits_{{p \to {{p}^{{NE}}}}} {{p}_L} = 0$and converging to $\mathop{{\lim }}\limits_{{p \to 0}} {{p}_L} = 1$.

Regarding the high-cost incumbent, she does not expand as prescribed, p _H  = 0, if

$$ \pi_{{inc,H}}^{{M,E}} - {{K}_H} < s \times \pi_{{inc,H}}^{{D,NE}} + \left( {1 - s} \right) \times \pi_{{inc,H}}^{{M,NE}}. $$

Solving for s, we obtain

$$ s > \frac{{\pi_{{inc,H}}^{{M,NE}} - \pi_{{inc,H}}^{{M,E}}}}{{\pi_{{inc,H}}^{{M,NE}} - \pi_{{inc,H}}^{{D,NE}}}} + \frac{{{{K}_H}}}{{\pi_{{inc,H}}^{{M,NE}} - \pi_{{inc,H}}^{{D,NE}}}} $$

and since $\matrix{ {PLE_H^{{NE}} = \pi_{{inc,H}}^{{M,NE}} - \pi_{{inc,H}}^{{D,NE}},} \hfill &{\pi_{{inc,H}}^{{M,NE}} = \pi_{{inc,H}}^{{M,E}}} \hfill &{\text{and}} \hfill &{PLE_H^{{NE}} = PLE_H^E} \hfill \\ }<!end array>$, then this expression reduced to $s > \frac{{{{K}_H}}}{{PLE_H^E}} \equiv \hat{s}$. On the other hand, the low-cost incumbent mixes as prescribed, p _L ∊ (0,1), if and only if

$$ \pi_{{inc,L}}^{{M,E}} - {{K}_L} = s \times \pi_{{inc,L}}^{{D,NE}} + \left( {1 - s} \right) \times \pi_{{inc,L}}^{{M,NE}}. $$

Solving for s, we obtain

$$ s = \frac{{\pi_{{inc,L}}^{{M,NE}} - \pi_{{inc,L}}^{{D,E}}}}{{\pi_{{inc,L}}^{{M,NE}} - \pi_{{inc,L}}^{{D,NE}}}} + \frac{{{{K}_L}}}{{\pi_{{inc,L}}^{{M,NE}} - \pi_{{inc,L}}^{{D,NE}}}}, $$

and since $\matrix{ {BC{{C}_L} = \pi_{{inc,L}}^{{M,NE}} - \pi_{{inc,L}}^{{M,E}}} \hfill &{\text{and}} \hfill &{PLE_L^{{NE}} = \pi_{{inc,L}}^{{M,NE}} - \pi_{{inc,L}}^{{D,NE}}} \hfill \\ }<!end array>$, this expression becomes $s = \frac{{BC{{C}_L}}}{{PLE_L^E}} + \frac{{{{K}_L}}}{{PLE_L^E}} \equiv \widetilde{s}$, where probability cutoff $\widetilde{s}$satisfies $\widetilde{s} \in \left( {0,1} \right)$ if ${{K}_L} < PLE_L^E - BC{{C}_L}$. Hence, we need $\widetilde{s} > \widehat{s}$, or ${{K}_L} > \frac{{PLE_L^E}}{{PLE_H^E}}{{K}_H} - BC{{C}_L}$. Hence, this semiseparating equilibrium can be sustained for relatively low priors, p < p ^NE, and expansion costs satisfying $PLE_L^E - BC{{C}_L} > {{K}_L} > \frac{{PLE_L^E}}{{PLE_H^E}}{{K}_H} - BC{{C}_L}$.

p _L =0 and p _H ∊ (0,1)

Let us finally check the strategy profile where only the high-cost incumbent randomizes and the low-cost incumbent does not expand. In this case, the entrant’s posterior beliefs after observing expansion are μ(H|Exp) = 1, which leads him to enter since $\pi_{{ent,H}}^{{D,E}} > F$. In the case that the entrant observes no expansion from the incumbent, the entrant mixes if his beliefs μ(H|NoExp) are such that

$$ \mu \left( {H|NoExp} \right) \times \left( {\pi_{{ent,H}}^{{D,NE}} - F} \right) + \left( {1 - \mu \left( {H|NoExp} \right)} \right)\left( {\pi_{{ent,L}}^{{D,NE}} - F} \right) = 0 $$

and solving for μ(H|NoExp), we obtain μ(H|NoExp) = p ^NE. Using Bayes’ rule we have $\mu \left( {H|NoExp} \right) = {{p}^{{NE}}} = \frac{{p\left( {1 - {{p}_H}} \right)}}{{p\left( {1 - {{p}_H}} \right) + \left( {1 - p} \right)}}$, and solving for p _H we obtain ${{p}_H} = \frac{{p - {{p}^{{NE}}}}}{{p\left( {1 - {{p}^{{NE}}}} \right)}}$, where p _H ∊ (0,1) for all p > p ^NE. In addition, p _H is increasing in p since $\frac{{\partial {{p}_H}}}{{\partial p}} = \frac{{{{p}^{{NE}}}}}{{\left( {1 - {{p}^{{NE}}}} \right) \times {{p}^2}}} > 0$, starting at $\mathop{{\lim }}\limits_{{p \to {{p}^{{NE}}}}} {{p}_H} = 0$and converging to $\mathop{{\lim }}\limits_{{p \to 1}} {{p}_H} = 1$.

Regarding the low-cost incumbent, she does not expand as prescribed (p _L  = 0) if and only if

$$ \pi_{{inc,L}}^{{M,E}} - {{K}_L} < s \times \pi_{{inc,L}}^{{D,NE}} + \left( {1 - s} \right) \times \pi_{{inc,L}}^{{M,NE}}, $$

and solving for s, we find $s > \frac{{BC{{C}_L}}}{{PLE_L^E}} + \frac{{{{K}_L}}}{{PLE_L^E}} \equiv \widetilde{s}$. On the other hand, the high-cost incumbent mixes as prescribed, p _H ∊ (0,1), if and only if

$$ \pi_{{inc,H}}^{{M,E}} - {{K}_H} = s \times \pi_{{inc,H}}^{{D,NE}} + \left( {1 - s} \right)\pi_{{inc,H}}^{{M,NE}}. $$

Solving for s, we obtain $s = \frac{{{{K}_H}}}{{PLE_H^E}} \equiv \widehat{s}$, where $\widehat{s} \in \left( {0,1} \right)$for all ${{K}_H} < PLE_H^E$. Hence, we need that $\matrix{ {\widehat{s} > \widetilde{s},} \hfill &{\text{or}} \hfill &{{{K}_H} > \frac{{PLE_H^E}}{{PLE_L^E}}{{K}_L} + \frac{{BC{{C}_L} \times PLE_H^E}}{{PLE_L^E}}} \hfill \\ }<!end array>$. Hence, this semiseparating strategy profile can be sustained for priors p > p ^NE and expansion costs satisfying

$$ PLE_H^E > {{K}_H} > \frac{{PLE_H^E}}{{PLE_L^E}}{{K}_L} + \frac{{BC{{C}_L} \times PLE_H^E}}{{PLE_L^E}}. $$