Korean Democracy in Transition: A Rational Blueprint for Developing Societies

Appendix B: Solution for the Equilibrium of the U.S.–North Korea Bayesian Game

Appendix B Solution for the Equilibrium of the U.S.–North Korea Bayesian Game The equilibrium of the U.S.–North Korea Bayesian game is solved here. Again, I use the notion of Perfect Bayesian Equilibrium (PBE). First of all, it is easy to see that North Korea has a dominant action on its last move. The sincere type will always choose A’ while the deceitful type will choose ~A’. Now I narrow our attention down to North Korea’s first move. [1] AA (pooling on A) When NK is pooling on A, the U.S. equilibrium strategy is ~AA~AA. Since at least the deceitful NK always has an incentive to switch, there is no PBE under this scenario. [2] A~A (separating with the sincere type choosing A) When the U.S. observes NK’s choice of A, then p’ = 1 and the U.S.’s best response is ~AA. When the U.S. observes NK’s choice of ~A, then p’ = 0 and the US’s best response is ~A~A. The deceitful NK does not have incentive to switch. For sincere NK, UNK (A) = qUNK (REJUS ) + (1-q) UNK (RECONNK ) and UNK (~A) = qUNK (SQ) + (1-q)UNK (RECONUS ). For sincere NK not to have incentive to switch from its equilibrium strategy , UNK (A) ≥ UNK (~A) must hold. It is true when q ≥ [UNK (RECONUS ) - UNK (RECONNK )] / [UNK (RECONUS ) + UNK (REJUS ) - UNK (RECONNK ) - UNK (SQ)]. Therefore a PBE is: {(A/A’~A/~A’), (~AA~A~A), q ≥ [UNK (RECONUS ) 110 Appendix B - UNK (RECONNK )] / [UNK (RECONUS ) + UNK (REJUS ) - UNK (RECONNK ) - UNK (SQ)]}. [3] ~AA (separating with the sincere type choosing ~A) When the U.S. observes NK’s choice of A, then p’ = 0 and the U.S.’s best response is ~AA. When the U.S. observes NK’s choice of ~A, then p’ = 1 and the U.S.’s best response is ~A~A. Since at least the deceitful NK always has an incentive to switch its action, there is no PBE under this scenario. [4] ~A~A (pooling on ~A) The hawkish U.S.’s best response is ~A. For the dovish U.S., its best response is A if p’ ≥ [UUS (SQ) - UUS (REJNK )] / [UUS (RECONUS ) - UUS (REJNK )]. Otherwise, it is ~A. The deceitful NK does not have an incentive to switch. If the dovish U.S.’s equilibrium action is A, then the sincere NK does not have an incentive to switch, either. Therefore a PBE is {(~A/A’~A/~A’), (~AA~AA), p’ ≥ [UUS (SQ) UUS (REJNK )] / [UUS (RECONUS ) - UUS (REJNK )], q}. If, on the other hand, the dovish U.S. responds with ~A (p’ < [UUS (SQ) - UUS (REJNK )] / [UUS (RECONUS ) - UUS (REJNK )]), the sincere NK has incentive to switch to A if UNK (A) ≥ UNK (~A). To prevent this, UNK (~A) ≥ UNK (A) must hold for the sincere NK. This is true when q ≥ [UNK (RECONNK ) UNK (SQ)] / [UNK (RECONNK ) - UNK (REJUS )]. Therefore, another PBE is {(~A/A’~A/~A’), (~A~A~A~A), p’ < [UUS (SQ) - UUS (REJNK )] / [UUS (RECONUS ) - UUS (REJNK )], q ≥ [UNK (RECONNK ) - UNK (SQ)] / [UNK (RECONNK ) - UNK (REJUS )]}. The equilibrium solution above leads to the following observations: Observation 4: The hawkish US’s equilibrium strategy does not depend on NK’s type. Simple backward induction shows that the hawkish U.S.’s optimal strategy is always ~A regardless of NK’s type. Equilibrium of the U.S.–North Korea Bayesian Game 111 Observation 5: The dovish U.S. will always reciprocate the accommodation initiated by NK. Whether the dovish U.S. will initiate accommodation in the absence of NK’s initial accommodation will depend on the value of p’. Observation 6: The deceitful NK never initiates A. This comes from the three PBEs under [2] and [4] above. Observation 7: Long-term reconciliation is possible if NK is sincere and the U.S. is dovish. Among the three PBEs above, only the combination of a sincere NK and a dovish U.S. leads to RECON outcomes. Observation 8: The combination of a deceitful NK and a hawkish U.S. always leads to SQ. From the three PBEs above. ...