Appendix B. Simultaneous Entry
This proof is valid for the main result in the second and third versions of the multi-market game.
Proof. Step 1. Begin in stage two. The objective of player m in the second stage is to solve the
following program:
max υ (x™, xf ) + V (x™, xf) + ∙∙ + v (ʃ^, x”)
o f ™n ™7l I I ™n < ‰
s.t. X1 + Xf + ∙∙ + X^ ≤ rv
If x™ + x™ + ∙∙ + xr^ < k, then ∂v (x^,ln, xf^) ∕∂x™ = 0 for t = 1, ..., n. If x™ + x™ = k, then
∂v (x™, x↑) ∕∂x™ = ∂v (x™, xf) ∕dx™ = ∙∙ = ∂V (x™, x”) ∕∂x™.
Step 2. In the last stage there are 2” subgames. First, if entry does not occur in any market and
k > nx™, then ∂v (x^,∣n, 0) ∕∂x^ra = 0 for all t = 1, ..., n ⇒ x™ = x™ for all t. If k ≤ nx™, then
∂v (x™, x1) ∕∂x™ = ∂v (x™1,x2) ∕∂xιy = ∙∙ = ∂V (x™, xS) ∕∂x^ ⇒ x2l = - for all t.
∖l^l∕/ I ∖ c_ 1 C_ / I ” \”^”// ” l ”
Step 3. Second, if one player enters (w.l.o.g. player 1) and k > x™+(n — 1) x™, then ∂v (x™, xf) ∕∂x1y =
0 and ∂V (x™, 0) ∕∂x^ = 0 ⇒ x™ = x™ and x™ = x™ for t = 2, ∙∙, n. If k ≤ x™ + (n — 1) x™,
then from (S) ∂v (x™, xf) ∕∂xl-∖ = ∂v (x™, 0) ∕∂x^ for t = 2, ∙∙∙,n ⇒ x™ < k∕n and x™ > k∕n. To
deter the entry of a single entrant while n — 1 players stays out, the incumbent must install
∂V (x™,xe) = dv (⅛≡d, 0)
dx™ dx™ l )
and from (S) k > nx.
Step X Next, if capacity k deters the entry of a single entrant, k deters the entry of more than
one player, which is shown with induction. Assume k deters the entry of t players. Then
where entry occurs in i and no entry occurs in market j. If t +1 players enter, deterrence is credible
if
∂v (x™, xe) /dx™ — ∂v
/(k — tx™)
∖ (n — t)
0) ∕∂x™
(19)
∂v (X™, xe) /dx™ — ∂v ((k(~ (t +^1ζap, 0) ∕∂x™ > 0 (20)
where entry occurs in i and no entry occurs in market j. The last inequality holds as long as
k > nx. Hence, we have shown that if capacity k deters the entry of a single entrant, then k deters
the entry of more than one entrant.
28
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