The name is absent

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 (¹_γψ^γ)
and when split, it will be ²(γψ^γ) — 3φ. The condition to merge given that 1 and 2
have merged is thus

1+ Y φ< 2

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. W^e get

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.

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