Rational Appeasement, Daniel Treisman, International Organization. Proofs of Propositions 4 and 5

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1 Raional Appeaseen, Daniel Treisan, Inernaional Organizaion. Proofs of Proposiions 4 and 5 Proposiion 4: In he gae wih endogenous sakes, a deerrene equilibriu an exis only if he ener s fixed os of fighing, k, is in he range: πk πk k in [, ]. π π Proof: Denoe he equilibriu value of * and he equilibriu values of on he op, b iddle and boo branhes, and respeively. In an on-equilibriu pah DE, L hallenges, boh srong and weak C s figh and se srily posiive s, and L aquieses. Working bakwards, L only aquieses a h if k and µ k µ ( / ) µ k. In equilibriu, I will show ha boh C s play idenially up o ie 4, so he only onsisen belief a h is µ = π ; he previous ondiion π k beoes π. If boh C s se = *, and given ha boh play F a ie, he weak C will se = if he srong C does (raher han reveal hiself o be weak proping a hallenge). The srong C an never benefi by seing <. Migh he se > (in whih ase, he will opiize by seing = )? For a sequenial equilibriu we jus need o show ha here is soe belief suh ha he prefers no o deviae. We are ineresed hroughou in defining he broades se of paraeer values suh ha an equilibriu an exis. I an be heked ha he

2 belief ha suppors a DE under he broades range of is ha any C who ses C is weak. Given his belief, a DE exiss if + k + k k. In equilibriu, a weak C will always figh a ie if he srong C does (raher han reveal herself weak and ge a axiu payoff of zero). For he srong C o figh a ie, here us be soe belief assoiaed wih A suh ha he prefers F. The belief ha yields he broades range of in DE is ha any C who plays A is weak. This would suppor a DE so long as k + k +. (If C deviaes he will opiize by seing =.) Given ha boh C s se = * in equilibriu, L s belief a ie us be he prior, p. k For L o play, π( * k) + ( π)( k) * *. If he srong C ses π * =, he weak one will do he sae sine his payoff fro his is k > 0 (reall ha k ), while if he deviaes and reveals hiself o be weak, his highes payoff is 0. For he srong C o prefer o se = *, here us be soe belief ha akes deviaion unaraive. The belief ha yields he broades ondiion for equilibriu is ha any C who deviaes is weak. The srong C will prefer no o deviae only if he equilibriu payoff * + k + k * + + k. (If she deviaes, she will opiize by seing = =, and boh loal aors will hallenge.) So, for an on-equilibriu-pah DE, we have he following neessary ondiions: k k ; π ; + ; π k * k ; * π + + and, whih ogeher wih * and iply: πk πk π π ax [, ] k in [, ] and k ( π ).

3 In an off-equilibriu-pah DE, L aquieses beause she orrely believes ha if she hallenged, boh srong and weak C s would figh. Noe ha for boh C s o play F a ie off he equilibriu pah, boh C s equilibriu sraegies us se he sae * and. Oherwise, he weak C reveals hiself o be weak, L would hallenge, and he weak C would raher have appeased a ie. Given his, a h he only onsisen belief for L is he prior, µ = π, and he only onsisen belief for L a ie is also p. L s expeed payoff fro playing is π( k ) + ( π)( k ) and her payoff fro playing a is. Thus, for her o play a, * k. π Given ha boh C s would figh off he equilibriu pah a ie, he weak C will se a he sae level as he srong C o avoid revealing his weakness. The weak C will only figh a ie as required if L will aquiese. (If L would hallenge, he weak C would be beer off playing A, even if his props a hallenge.) L will only aquiese if µ µ k π k ( k) = µ π, and k. Would he srong C se a πk = π? If no, an off-equilibriu-pah DE anno exis beause he weak C would never figh. For a srong C o prefer a =, here us be soe belief assoiaed wih seing > ha akes his unaraive. The belief ha yields he broades range of is ha any C who ses π k > is weak. Then C S will prefer o se = π so long as * + k * + k k. For he srong C o play F a ie off he equilibriu pah, here us be a belief assoiaed wih A ha akes hi prefer F. The belief ha s suppors he broades range of is ha any C who plays A is weak. Then he srong C will

4 prefer F only if * + k k * +. If he srong C plays F a ie, he weak C will prefer o do he sae raher han reveal his weakness. For he srong C o se = * here us be soe belief ha akes deviaing unaraive. The belief ha yields he looses possible onsrain is ha any C who deviaes is weak. Given his, C S will only se = * if k * + ax[ π, k] + k. The π π k lef-hand side is C S s axiu payoff given ha L plays a ( π is he axiu if C W pools wih C S, k is he axiu if he does no); he righ hand-side is her axiu payoff given ha she ses > *, proping he belief she is weak. So, for an off-equilibriu pah DE, he following ondiions are neessary: k k * ; π ; k; k; π π k * + ; and * + ax[ π, k] + k. Togeher wih π * and, hese iply: π k k π, k π in [ k k, ] π π π π and k k k eiher k > in [ π, π ] or k < π π ( π) π and eiher πk [ k πk ] and > + k k π π π or k π k k < π π. So, for boh on- and off-equilibriu pah deerrene equilibria, neessary (hough no suffiien). Q.E.D. πk πk k in [, ] π π is

5 Proposiion 5: A leas one appeaseen equilibriu exiss if πk πk k π π and eiher π k + k < ( π) ax[, ( ) ], or k. Proof: No off-equilibriu-pah appeaseen equilibria exis for he reason given in he ex. In an on-equilibriu-pah appeaseen equilibriu, L us hoose a a h, boh ypes of C us hoose A a ie, > 0, and L us hoose. Using he sae noaion as before, for L o hoose a a h, ( ) µ k µ k, and µ k. In equilibriu, I will show ha boh ypes of C play idenially up o ie 4, so he belief a h us be µ = π. Thus, µ k µ beoes πk. For a srong C o se π =, here us be soe belief assoiaed wih ha akes deviaion unaraive. I an be heked ha he belief ha will yield he broades range of possible equilibiru s is ha any C who ses is weak. Then, C S will prefer no o deviae only if k (assuing if he deviaes he opiizes and ses = ). If C S ses =, C W will prefer o do he sae raher han prop he belief he is weak and ge a payoff of zero. For he srong C o prefer o play A a ie, here us be a belief assoiaed wih playing F ha akes hi prefer A. The belief ha will yield he broades range of possible equilibriu s is ha any C who plays F is weak. Given his, he srong C prefers A only if ax[ + k, k]. Bu noe ha k is iplied by k and =, required already, so his ondiion beoes:

6 + k if > k. ( > k is required for + k > k.) C W will also prefer o play A here if C S does, raher han prop a hallenge. Given ha boh C s play A a ie, L will only play if * > 0. For he srong C o prefer no o se *, here us be a belief assoiaed wih his ha akes deviaion unaraive. The belief ha yields he broades ondiions on * is ha any C who deviaes fro * = is weak. Suppose C S deviaes o soe > 0 (in whih ase, I assue he opiizes by seing = ). The greaes payoff C S an ge is ax[ + k, k]. So for hi no o deviae in his way, ax[ + k, k], whih given k and, redues o + k if > k. Given ha he srong C ses = *, he weak C will do he sae raher han be believed weak. So, neessary ondiions for an AE are: πk k; ; k; * > 0; π and + k if > k. Togeher wih and *, hese iply: πk πk k π π and eiher k < and k π k ( + ) or k ( π). Q.E.D. We have assued > 0, bu in fa his ould be derived as follows. Suppose C S deviaes o < 0. L would aquiese, C would se =, L would hallenge, yielding a payoff o C of ax [ k +, ], whih is never greaer han he srong C s equilibriu payoff of > k. (Reall ha in equilibriu k.) So he would never se < 0.