Signaling Game of Collective Self-Defense in the U.S.-Japan Alliance

Shuhei Kurizaki

doi:10.1007/978-4-431-56466-9_2

Games of Conflict and Cooperation in Asia pp 31-55 | Cite as

Signaling Game of Collective Self-Defense in the U.S.-Japan Alliance

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Shuhei Kurizaki

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First Online: 08 February 2017

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Part of the The Political Economy of the Asia Pacific book series (PEAP)

Abstract

What are the implications of collective self-defense for Japan’s security environment? This chapter considers invoking the right of collective self-defense as intervention into an armed conflict on behalf of an ally. Since the Japanese government’s reinterpretation of the constitutional constraint on collective self- defense is perceived by neighboring countries as a revelation of underlying militarism and expansionism, I analyze Japan’s decision on its constitutional constraints as a costly signal that Japan sends to deter the challenges against the U.S.-Japan alliance. The equilibrium analysis suggests that the manipulation of collective self-defense does not help Prime Minister Abe and his government achieve the security policy objectives that they claim invoking collective self- defense would deliver.

Keywords

Alliance Collective self-defense Extended deterrence Signaling game U.S.-Japan Security Treaty

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Notes

Acknowledgements

I am grateful for the comments and discussion by Motoshi Suzuki, Atsushi Tago, Yukari Iwanami, Atsushi Ishida, Keisuke Iida, Akira, Okada, Daisuke Oyama, Yasushi Asako, and the seminar participants at the Workshop on Politics & Economics at Keio University in July 2014 and the Game Theory Workshop at Kyoto University in March 2015. Research assistance by Arisa Fukuoka is gratefully acknowledged.

Appendix

Proof of Lemma 2.1

Suppose that Japan sends some separating signal v_J. Since the cut-point configuration θ ≤ 0 induces the posterior belief to be fixed at F(v_J, θ) = 0, any separating strategy cannot be consistent with this fixed belief. Now consider a pooling signal v_J = 0. Substituting for expression (2.3), the condition θ ≤ 0 yields θ ≡ sψ − v_J + a_J < 0, or sψ < − a_J. Since s > 0, ψ > 0, and a_J > 0, this condition never holds. Also, consider a pooling signal of the other extreme v_J = $ {\overline{v}}_J $. Since θ ≡ sψ − v_J + a_J from Eq. (2.3), substituting for $ {\overline{v}}_J $ ≡ sψ – c_J + a_J from Eq. (2.4) gives

$$ \begin{array}{r}\theta \equiv s\psi -{\overline{v}}_J+{a}_J\le 0\\ {} s\psi -\left( s\psi -{c}_J+{a}_J\right)+{a}_J\le 0\\ {}{c}_J\le 0.\end{array} $$

This condition contradicts the assumption c_J > 0. Finally, consider a pooling signal on $ {v}_J^{\prime } $ > 0. Since the condition θ ≤ 0 implies that intervention is strictly dominated, the pooling signal is equivalent to $ {\overline{v}}_J $ that satisfies the Self-Defense Constraint. Q.E.D.

Proof of Lemma 2.2

Suppose to the contrary that the challenger strictly prefers keeping the status quo to making a challenge. Then, it must follow that $ {\widehat{c}}_C $ ≤ 0. With expression (2.8) for $ {\widehat{c}}_C $, this condition holds iff 1− q − F(v_J, θ)ψ ≤ 0, or equivalently iff F(v_J, θ) ≥ $ \frac{1-q}{\psi } $ = $ \frac{1-q}{p-q} $. Since p < 1 and F(·) < 1 by construction, it follows that F(v_J, θ) ≥ $ {\frac{1-q}{\psi }} $ > 1. Thus, this last condition is never attainable. Q.E.D.

Proof of Lemma 2.3

Consider an out-of-equilibrium signal $ {v}_J^{\dagger } $ < $ {v}_J^{\prime } $. The PBE with a partition signal v_J = {$ {v}_J^{\prime } $, $ {\overline{v}}_J $}, where $ {v}_J^{\prime } $ = 0, requires that the challenger and the U.S. observe places the zero weight on such highly-resolved type. This is not “reasonable” since a highly-resolved type could benefit from setting v_J =$ {v}_J^{\dagger } $. To see this, note that the payoffs for a highly-resolved type $ {v}_J^{\dagger } $ < θ from signaling $ {v}_J^{\prime } $ and $ {v}_J^{\dagger } $ are given by

$$ {u}_J\left({v}_J^{\prime}\right)=\left\{\begin{array}{c}\hfill s\hfill \\ {}\hfill 0\hfill \\ {}\hfill sp-{c}_{J}-{v}_{J}^{\prime}\hfill \end{array}\right.\kern1em \begin{array}{c}\hfill \mathrm{if}\ {c}_U\le p\ \mathrm{and}\ {c}_C > 1-p\hfill \\ {}\hfill \mathrm{if}\ {c}_U>p\hfill \\ {}\hfill \mathrm{if}\ {c}_U\le p\ \mathrm{and}\ {c}_C\le 1-p\hfill \end{array} $$

$$ {u}_J\left({v}_J^{\dagger}\right)=\left\{\begin{array}{c}\hfill s\hfill \\ {}\hfill 0\hfill \\ {}\hfill sp-{c}_J-{v}_J^{\dagger}\hfill \end{array}\right.\kern1em \begin{array}{c}\hfill \mathrm{if}\ {c}_U\le q\ \mathrm{and}\ {c}_C > 1-q\hfill \\ {}\hfill \mathrm{if}\ {c}_U>q\hfill \\ {}\hfill \mathrm{if}\ {c}_U\le q\ \mathrm{and}\ {c}_C\le 1-q\hfill \end{array} $$

Clearly, if c_U ≤ q and c_C ≤ 1 − q, type $ {c}_J^{\dagger } $ can profitably deviate to $ {v}_J^{\dagger } $ until $ {v}_J^{\prime } $ = 0. Q.E.D.

Proof of Lemma 2.4

To identify the conditions under which the divided alliance commitment can occur in the equilibria specified in Proposition 2.2, it suffices to identify the condition under which the signal v_J = 0 is incentive compatible for types c_J ∈(θ, β]. That is, for these types, it must be that u_J (0) ≥ u_J ($ {v}_J^{*} $), where

$$ {u}_J(0)=\left\{\begin{array}{c}\hfill s\hfill \\ {}\hfill 0\hfill \\ {}\hfill sq-{a}_J\hfill \end{array}\right.\kern1em \begin{array}{c}\hfill \mathrm{if}\ {c}_U\le {\widehat{c}}_U\ \mathrm{and}\ {c}_C > {\widehat{c}}_C\hfill \\ {}\hfill \mathrm{if}\ {c}_U>{\widehat{c}}_U\hfill \\ {}\hfill \mathrm{if}\ {c}_U\le {\widehat{c}}_U\ \mathrm{and}\ {c}_C\le {\widehat{c}}_U\hfill \end{array} $$

$$ {u}_J\left({v}_J^{*}\right)=\left\{\begin{array}{c}\hfill s\hfill \\ {}\hfill 0\hfill \\ {}\hfill sp-{a}_J\hfill \end{array}\right.\kern1em \begin{array}{c}\hfill \mathrm{if}\ {c}_U\le q\ \mathrm{and}\ {c}_C > 1-q\hfill \\ {}\hfill \mathrm{if}\ {c}_U>q\hfill \\ {}\hfill \mathrm{if}\ {c}_U\le q\ \mathrm{and}\ {c}_C\le 1-q\hfill \end{array} $$

If c_U ∈ (q, $ {\widehat{c}}_U $] and c_C ≤ $ {\widehat{c}}_C $ , types c_J ∈(θ, α] may have an incentive to deviate since, under these conditions, the equilibrium payoff with a signal v_J = 0 is sq − a_J and the payoff from the deviation to $ {v}_J^{*} $ is 0. Hence, these types c_J ∈ (θ, β] may have an incentive to deviate unless sq − a_J ≥ 0, or s ≥ $ \frac{a_J}{q} $. Q.E.D.

Proof of Proposition 2.1

To show the results, I establish. Lemmas 2.5 through 2.7 regarding the equilibrium behavior.

Lemma 2.5

There is no PBE in which all government types of Japan pool on a signal v _J = 0 if all types stay out of a bilateral conflict.

Proof of Lemma 2.5

Note that for all types of Japan to stay out, it must be that θ < 0. Substituting θ ≡ sψ – v_J + a_J , where v_J = 0, we rewrite this condition as

$$ \begin{array}{r}\uptheta \equiv s\psi \hbox{--} {v}_J+{a}_J<0\\ {} s\psi -0+{a}_J<0\\ {} s\psi <-{a}_J.\end{array} $$

Since s > 0 and ψ > 0, and a_J > 0 by definition, this condition never holds. Q.E.D.

Lemma 2.6

All government types of Japan pool on a signal v _J = 0 and subsequently intervene in a multilateral conflict in equilibrium if the salience of the disputed good is sufficiently high or if the U.S.-Japan alliance improves their fighting capability.

Proof of Lemma 2.6

This case is equivalent to a trivial case where there is no irresolute type so that c_J ≤ θ. For all government types of Japan to intervene, it must be true that EU_J (IN) ≥ EU_J (OUT) conditional on the signal v_J = 0 for all c_J types. This inequality holds if sp – c_J ≥ sq − a_J or c_J ≤ sψ + a_J. This condition holds for all c_J types in this case because they are below θ ≡ sψ + a_J. Finally, since it must be that F(·) ≥ 1 in this case, it implies that θ ≥ 1, which in turn implies that s ≥ $ \frac{1-{a}_J}{\psi } $. Q.E.D.

Lemma 2.7

A pooling signal with $ {v}_J^{\prime } $ > 0 cannot constitute a PBE, regardless of the underlying resolve level for Japan (i.e., regardless of the relative magnitude of θ).

Proof of Lemma 2.7

Suppose θ > 0 so that there exists the resolute type c_J ≤ θ that prefers intervening to staying out. Then, this type has an incentive to deviate to any v_J < $ {v}_J^{\prime } $ if the history of the signaling game calls upon Japan to intervene. Similarly, suppose that θ ≤ 0 so that all government types of Japan are irresolute so that they prefer staying out to intervening. By Lemma 2.1, there is no equilibrium with this condition. Thus, a pooling strategy with $ {v}_J^{\prime } $ cannot constitute an equilibrium. Q.E.D.

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.

Lemma 2.8

A separating signal with partition v_J = {0, $ {v}_J^{{\prime\prime} } $, $ {\overline{v}}_J $}, where 0 < $ {v}_J^{{\prime\prime} } $< $ {\overline{v}}_J $, cannot constitute a PBE. All c_J ≤ α set v_J = 0, all c_J ∈ (α, γ] choose $ {v}_J^{{\prime\prime} } $, and all c_J > γ choose the self-defense constraint $ {\overline{v}}_J $.

Proof of Lemma 2.8

(Sketch) Suppose that θ < 0 so that all government types of Japan stay out of a bilateral conflict. Then, all types c_J ≤ α can profitably deviate to some $ {v}_J^{{\prime\prime} } $ > 0 if the history of the signaling game reaches Japan’s decision to intervene. Suppose that 0 < θ < α so that some types below α would not intervene. Then, types c_J ∈ (θ, α] would deviate to some $ {v}_J^{{\prime\prime} } $ > 0; otherwise, this cut-point strategy contradicts the Japan’s intervention decision rule in expression (2.3). Suppose that α < θ < γ. Then, types above γ will have an incentive to deviate from $ {\overline{v}}_J $ to $ {v}_J^{{\prime\prime} } $ because a signal $ {v}_J^{{\prime\prime} } $ would induce the challenger to make a challenge at the lower rate than would a signal $ {\overline{v}}_J $. Finally, suppose that θ > γ. Then types below γ would profitably deviate from $ {v}_J^{{\prime\prime} } $ to v_J = 0. 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.

Lemma 2.9

The PBE with a separating signal v_J= {$ {v}_J^{\prime } $ , $ \overline{v} $_J} cannot be sustained if the resolute type sets v_J = $ \overline{v} $_Jwith a positive probability.

Proof of Lemma 2.9

The signal v_J = $ \overline{v} $_J is not incentive compatible for some resolute types c_J ∈ (α, θ], where α is defined as the highest-cost type of Japanese government that prefers the signal $ {v}_J^{\prime } $. To prove this, it suffices to show that these types can profitably deviate to the signal $ {v}_J^{\prime } $. The signals $ \overline{v} $_J and $ {v}_J^{\prime } $, respectively, yield the following payoffs:

$$ {u}_J\left({\overline{v}}_J\right)=\kern0.5em \left\{\begin{array}{c}\hfill s\hfill \\ {}\hfill 0\hfill \\ {}\hfill sp-{c}_J-{\overline{v}}_J\hfill \end{array}\ \right.\kern0.75em \begin{array}{c}\hfill \mathrm{if}\ {c}_U\le p\ \mathrm{and}\ {c}_C>1-p\hfill \\ {}\hfill \mathrm{if}\ {c}_U>p\hfill \\ {}\hfill \mathrm{if}\ {c}_U\le p\ \mathrm{and}\ {c}_C\le 1-p\hfill \end{array} $$

$$ {u}_J\left({v}_J^{\prime}\right)=\left\{\begin{array}{c}\hfill s\hfill \\ {}\hfill 0\hfill \\ {}\hfill sp-{c}_J-{v}_J^{\prime}\hfill \end{array}\ \right.\kern0.75em \begin{array}{c}\hfill \mathrm{if}\ {c}_U\le {\widehat{c}}_U\ \mathrm{and}\ {c}_C>{\widehat{c}}_C\hfill \\ {}\hfill \mathrm{if}\ {c}_U>{\widehat{c}}_U\hfill \\ {}\hfill \mathrm{if}\ {c}_U\le {\widehat{c}}_U\ \mathrm{and}\ {c}_C\le {\widehat{c}}_C\hfill \end{array} $$

Since $ \overline{v} $_J > $ {v}_J^{\prime } $ by definition, types c_J ∈ (α, θ] can profitably deviate to the signal $ {v}_J^{*} $. Q.E.D.

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Shuhei Kurizaki
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1.School of Political Science and EconomicsWaseda UniversityTokyoJapan

Signaling Game of Collective Self-Defense in the U.S.-Japan Alliance

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