To prove the existence of a two party equilibrium, not only must (A10) be
verified, but P =3and 4 must prefer to be merged rather than to stay split given
that P = 1, 2 are merged. When merged, the payoff to P = 3 or 4 will be (1γψγ)
and when split, it will be 2(γψγ) — 3φ. The condition to merge given that 1 and 2
have merged is thus
1+ Y φ< 2
(A11)
γ ψ 3
Note that inequality (A10) is more stringent than (A11).
For a three party equilibrium with P =3, 4 merging to exist, one must verify
whether P =3and 4 prefer to merge when 1 and 2 prefer to remain split. Reversed
inequality (A10) gives the condition for 1 and 2 to remain split when P =3, 4
have merged.
Underathreepartyequilibrium, 2IIIWG4 = 4 [1+E(msS4)γ] γ4ψ+4E(IIIsO4) γ4ψγ.
We derive:
1 W34 =(1+γ) l 1 , 2 - β
2 III G γψ 6φ 2φ(6 + β )
With four parties, the payoff to party P = 3,4 is IV WP = 2 [1+E(IVsP)γ2] γψ +
2 E (IV sO) γψ γ. We get
IVWGP
2(1 + γ) 2β
Yψ φ(4 + β )
Therefore, IVWp < 2IIIWG4 if and only if
1 + Y φ < 2β + 2 — β + 1
γ ψ (4 + β) + 2(6 + β )+6
Note that the right hand side increases with β.
For a three party equilibrium, given (A10), we must thus have
2 — β + 1 < 2β + 2 — β + 1
6 + β + 3 (4 + β ) + 2(6 + β )+6
i.e. (4+β) — 2(26+β) > 1. This inequality will not hold for β = 0 but will hold
for β =1. One also sees that the left-hand side monotonically increases with β.
There is thus a threshold β* ∈ (0,1) above which a three party equilibrium will
exist. Heterogeneity can thus lead to a three party equilibrium.
51
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