To complete the proof of Proposition 2.1, note that the cutoff points in expressions (2.11) and (2.12) are readily derived by substituting *F = θ* in the decision rules for the U.S. in expression (2.5) and for the challenger in expression (2.8), respectively. To derive the equilibrium conditions, note that substituting *F = θ* in expression (2.5) yields *c*_{ U }\( \le \frac{p\theta +q\left(1-\theta \right)}{1-\theta \left(1-{\delta}_U\right)}\equiv {\widehat{c}}_U. \) For \( {\widehat{c}}_U \) to be feasible, it must be that 1 − *θ*(1 − *δ*_{ U }) > 0, which implies *θ* < \( \frac{1}{1-{\delta}_U}. \) Substituting for *θ* ≡ *sψ* − *v*_{ J } + *a*_{ J } and simplifying yields *s*\( <\psi \left(\ \frac{1}{1-{\delta}_U}-{a}_J\right). \)*Q.E.D.*

### Proof of Proposition 2.2

The preliminary analysis in Sect.

2.3 has already characterized the best responses Japan’s intervention decision, the resistance decision for the U.S., and the challenger’s initiation decision. Thus, it suffices here to derive the cut-points with the posterior beliefs induced by the proposed signaling strategy, given

*θ*. The proposition postulates the following signaling strategy for Japan: the resolute type with

*c*_{ J } ≤

*θ* sets

*v*_{ J } = 0 and intervenes; some irresolute type

*c*_{ J } ∈ (

*θ, β*) sets

*v*_{ J } = 0 and stays out; and other irresolute type

*c*_{ J } >

*β* sets

*v*_{ J } =

\( {\overline{v}}_J \) and stays out. Observing the signal

*v*_{ J } = 0, the posterior beliefs are

*F*(·) =

\( \frac{\theta }{\beta } \) and 1 −

*F* =

\( \frac{\beta -\theta }{\beta } \) (with the uniform distribution). For the U.S. decision, substituting (

2.5) for the posterior gives

$$ {c}_U\le \frac{\theta \psi +\beta q}{\beta -\theta \left(1-{\delta}_U\right)}\equiv {\widehat{c}}_U. $$

Solving for

*β* gives its critical value:

*β* =

\( \frac{\theta \left(\psi +{c}_U\left(1-{\delta}_U\right)\right)}{c_U-q}\equiv {\widehat{\beta}}_U \). Since the postulated signaling strategy requires that

*β > θ*, substituting for

*β* =

\( {\widehat{\beta}}_U \) gives

\( \frac{\theta \left(\psi +{c}_U\left(1-{\delta}_U\right)\right)}{c_U-q}>\uptheta . \) Solving for

*c*_{ U } yields

*c*_{ U } <

\( \frac{p}{\delta_U} \) if

*c*_{ U } –

*q >* 0. If

*c*_{ U } –

*q* < 0 instead, the condition

*β* >

*θ* would hold for

*c*_{ U } >

\( \frac{p}{\delta_U} \). But there is no

*c*_{ U } that both satisfies

*c*_{ U } <

*q* and

*c*_{ U }*>*\( \frac{p}{\delta_U} \). Thus, for the proposed signaling strategy to be part of a PBE, it must be that

$$ {c}_U\in \left(q,\frac{p}{\delta_U}\right). $$

For the challenger’s best response, substituting (

2.8) for the posterior belief yields

$$ {c}_C\le \frac{\beta \left(1-q\right)-\theta \psi}{\beta}\equiv {\widehat{c}}_C. $$

Solving for

*β* yields

*β* =

\( \frac{\theta \psi}{1-q-{c}_C} \)≡

\( {\widehat{\beta}}_C \). As before, this proposed equilibrium requires

*β* > θ. Substituting for

\( {\widehat{\beta}}_C \) gives

\( \frac{\psi }{1-q-{c}_C} \) > 1. This inequality holds either if 1 −

*q* –

*c*_{ C } > 0 and

*c*_{ C } > 1−

*p*, or if 1 −

*q* –

*c*_{ C } < 0 and

*c*_{ C } < 1 −

*p*. The latter case implies that

*c*_{ C } ∈ (1 −

*q*, 1 −

*p*), which is not true since

*p* >

*q* by assumption. Thus, the proposed signaling strategy constitutes a PBE if

$$ {c}_C\in \left(1-p,1-q\right). $$

Observing

\( {v}_J^{*} \), the posterior beliefs are

*F*(·) = 0. The best responses by the U.S. and the challenger with this belief are already given by the preliminary analysis in Sect.

2.3.

Lemma 2.4 has already derived the condition under which the proposed signaling strategy is incentive compatible.

To assess if Japan’s intervention decision is sequential rational given its signaling strategy, let \( {c}_J^1 \) define as any *c*_{ J } ≤ *θ*, \( {c}_J^2 \) as *c*_{ J } ∈ (*θ, β*], and \( {c}_J^3 \) as *c*_{ J } > *β*, where \( {c}_{{\it J}}^1 \) < \( {c}_J^2 \) < \( {c}_J^3 \)_{.} For the resolute type *c*_{ J } ≤ *θ* to signal *v*_{ J } = 0 and intervene in equilibrium, it must be that *sp-*\( {c}_J^1 \) – 0 ≥ *sq* – *a*_{ J }, or \( {c}_J^1 \) ≤ *sψ* + *a*_{ J } − 0. This holds for all *c*_{ J } ≤ *θ* ≡ *sψ* + *a*_{ J }. Similarly, for some irresolute type *c*_{ J } ∈ (*θ, β*] to set *v*_{ J } = 0 and stay out, it must be that *sp-*\( {c}_J^2 \) – 0 < *sq* – *a*_{ J }, or \( {c}_J^2 \) > *sψ* + *a*_{ J }. This holds for all *c*_{ J } > *θ* ≡ *sψ* + *a*_{ J }. Lastly, for some irresolute type *c*_{ J } > *β* to stay out after signaling \( {v}_J^{*} \) in equilibrium, it must be that *sp* − \( {c}_{\mathrm{J}}^3 \) − \( {v}_J^{*} \) < *sq* – *a*_{ J }, or \( {c}_J^3 \) > *sψ* + *a*_{ J } - \( {v}_J^{*} \). This holds for all *c*_{ J } > *β* because *β* > *θ* ≡ *sψ* + *a*_{ J } - \( {v}_J^{*} \)_{.}*Q.E.D.*