Appendix A. Sequential Entry
Proof. Step 1. Start in period n. Let the remaining capacity be kn = kn-1 — x^-↑ ■ There are
two subgames; either player n enters or stays out of market n. In the subgame with entry, the
unique Nash equilibrium is {xt^ (kn), -r^}, where xt^ (kn) = min {kn,x™} and "xrn = βe (xtβ (kn)).
If player n decides to stay out of market n, we have the following limit Nash equilibrium [x™', θ},
where ^ym (x™, θ) = 0. The unique Nash equilibrium in the subgame with no entry is {x)(λ (k), 0},
where x™ (kn) = min {kn,x™}. From S, it follows that x™ > x™.
Step 2. Player n would enter if kn < x and stay out as long as kn ≥ x. To deter entry, player m
would need kn ≥ x. Now, assume that enough unused capacity remains to deter entry. Rewrite the
equilibrium output of player m in period n as a function of kn-ι and x%-ι, i.e. x™ (kn-ι, x%-ι ) =
min {kn-i — xrβ-1,χm}.
Step 3. Working backwards to period n — 1, we have two subgames; either player n — 1 enters or
stays out of market n — 1. First, capacity kn-ι would ensure a successful commitment by player
m in market n — 1 to an output x™—i, if and only if:
°v °x™ °V
-----(T™ 1,β1 (T™ 1)) + —ɪ ∙ -— (x™ (k,T™ 1), 0) ≥ 0 (13)
я„,™ ∖ n-1, ∖ n- 1)) а^т Ягг-т ∖ n V , n-1 ) , ) — V >
°xn-1 °xn-1 °xn
Now, ∂χm = —1 if k ≤ x™ + τm and g∂⅛ (k, 2:^-1 ) = 0 if k > x™ + xrβ^. To deter entry, player
m has to commit to T in the subgame with entry. The following inequality must be satished:
°V °V
-°^(x, в™(T)) = ° °c, 0) (14)
If xn-1 > 0, it follows from (S) that xrβ > x ⇒ kn-1 > 2T. If (D) holds with equality, i.e.
T = x™, then the LHS of equality [14] is equal to zero and the equality is satished if and only if
, ~ =rn , , ~ , =rn
kn-1 — x = x ⇒ kn-1 = x + x .
Step ^. Working backwards to period n — 2, we have two subgames; either player n — 2 enters or
stays out of market n — 2. First, capacity kn-2 would deter entry if:
°v ° V
∂x^- (t, в™ (T)) = ∂x^
°xn-2 °xn-1
° ° °V
(x™.,, 0 = (x™, 0)
∖ n- 1 ’ / rnrm × n ’ '
^x >x ^n
(15)
26
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