Figure 18: Firm best response correspondence for ex-ante low demand 2’s Figure 18 illustrates firm best response correspondence. The capacity game 2’s equilibria is obtained by intersecting each best response correspondence. firm’s Appendix 3.2: Proof of Theorem 3 The case where > reduces the number of options firm 2 has in declaring capacity, D and eliminates the regime. All best responses from section 4.1 are applicable here, only there are now less cases to analyse. If firm 2 declares > , firm best response is to declare and if D K2 K1 D, 1’s firm 2 declares > 0, firm best response is to declare > (D The best K2 K1 K2). 1’s – response correspondence for firm 1 is Firm best responses are as follows, keeping in mind is no longer an K2 D 2’s option. If firm 1 declares firm best response is to declare . If firm 1 K1 D, K2 2’s declares > , firm best response is to declare . Finally, if firm 1 declares D K1 2’s K2 > 0 firm best response is to declare (D Firm best response K1 K2 K1). 2’s – 2’s correspondence is The capacity game equilibria is obtained by intersecting each best response firm’s correspondence. Appendix 3.3: Proof of Theorem 4 When > > 0, the result is a further reduction of firm capacity options. D 2’s Firm best response is to declare > (D Firm best response if > (D K1 K2). K1 1’s – 2’s is to declare its actual capacity, because its profits in the area, , are strictly K2), – increasing in K2. Appendix 3.4: Proof of Theorem 5 The case where > reduces the number of options firm 1 has in declaring D capacity, and eliminates the possibility of the regime occurring. For firm 1 K2 D, prefers to declare > , and be in profit region 2, than declare > 0, and be in K1 K1 profit region 4 if >
Figure 18: Firm 2’s best response correspondence for ex-ante low demand Figure 18 illustrates firm 2’s best response correspondence. The capacity game equilibria is obtained by intersecting each firm’s best response correspondence. Appendix 3.2: Proof of Theorem 3 The case where D > reduces the number of options firm 2 has in declaring capacity, and eliminates the regime. All best responses from section 4.1 are applicable here, only there are now less cases to analyse. If firm 2 declares D > K2 , firm 1’s best response is to declare K1 D, and if firm 2 declares > K2 0, firm 1’s best response is to declare > K1 (D – K2). The best response correspondence for firm 1 is Firm 2’s best responses are as follows, keeping in mind K2 D is no longer an option. If firm 1 declares K1 D, firm 2’s best response is to declare K2 . If firm 1 declares D > K1 , firm 2’s best response is to declare K2 . Finally, if firm 1 declares > K1 0 firm 2’s best response is to declare K2 (D – K1). Firm 2’s best response correspondence is The capacity game equilibria is obtained by intersecting each firm’s best response correspondence. Appendix 3.3: Proof of Theorem 4 When D > > 0, the result is a further reduction of firm 2’s capacity options. Firm 1’s best response is to declare > K1 (D – K2). Firm 2’s best response if > K1 (D – K2), is to declare its actual capacity, because its profits in the area, , are strictly increasing in K2. Appendix 3.4: Proof of Theorem 5 The case where D > reduces the number of options firm 1 has in declaring capacity, and eliminates the possibility of the regime occurring. For K2 D, firm 1 prefers to declare > K1 , and be in profit region 2, than declare > K1 0, and be in profit region 4 if >
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