Proof by Backward Induction: Effects of Great-Power Inter vention on Challenger and Defender Behavior

Great Power UG(I) =−ci UG(∼I) =− |p0−xg| ⇒I iff p0 ∈/ xg ±ci (20) Defender UD,t=w−1(A|I) = ∞ X t=0 δt(1−xg) = 1−xg 1−δ UD,t=w−1(∼A|I) = ∞ X t=0 δt(1−x∗0) = 1−x ∗0 1−δ

Therefore, given fixed δ, D choosesA|I iff xg < x∗0.

Remembering Equation 3, (x∗0= (p0−c0_c)1/ρc_),

A|I iff p0 > xρc

g +c 0

c (21)

and ∼ A|I otherwise. As greater risk-aversion (ρc → 0) causes this threshold to rise, the

lowest threshold occurs at xg+c0c for ρc= 1.

Under the conditions ∼I, Dbehaves as it would without the presence ofGin the game. Therefore, for p0 ∈xg±ci, Dchooses A| ∼I iff p0 > p0+ ∆pc (cf. appendix A.1 above).

Challenger UC,t=w−1(N|A, I) =xg UC,t=w−1(∼N|A, I) =qρc ⇒N|A, I iff xg > qρc (22) UC,t=w−1(N| ∼A, I) =x∗0 UC,t=w−1(∼N| ∼A, I) =qρc ⇒N| ∼A, I iff p0 > qρ2c _+c0 c (23)

This condition (as does the condition, described below, for N| ∼A,∼I) holds at the lower bound of p0 =q+c0_c for ρc = 1.

Under the conditions∼I, C behaves as it would without the presence ofGin the game. Therefore, for p0 ∈xg±ci, C behaves as follows:

UC,t=w−1(N|A,∼I) =p0−cc UC,t=w−1(∼N|A,∼I) =qρc ⇒N|A,∼I iff p0 > qρc+cc (24) UC,t=w−1(N| ∼A,∼I) =x∗0 UC,t=w−1(∼N| ∼A,∼I) =qρc ⇒N| ∼A,∼I iff p0 > qρ2c _+c0 c (25)

These conditions lead to the deviations from Equation 6 under triadic conditions, as described in Figures 5 and 6.

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