(a) The first best decision rule is to choose d=0 in and d=1 in s. (b) If we allocate the decision right to A, the per period expected total payoff

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Tyler York
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1 Solution to PS 5 Poblem : daptation (a The fit bet deciion ule i to chooe d0 in and d in. (b If we allocate the deciion ight to, the pe peiod expected total payoff ( EU [ + U] i ( If we allocate the deciion ight to, the pe peiod expected total payoff i ( Theefoe, the econd bet allocation of the deciion ight goe to. (c (i Note that own the deciion ight at time 0 of peiod. We aume that whoeve ha the deciion ight in a peiod inheit the deciion ight at the beginning of the next peiod. Conide the following tigge tategy: On the equilibium path: In each peiod, ( obtain the deciion ight though bagaining. ( pay k>0 (3 chooe d0 in and d in. (4 pay m( Off the equilibium path: ( No bagaining occu. The owne of the deciion ight keep the deciion ight. ( No tanfe of money take place between patie. (3 The owne of the deciion ight chooe a deciion favoable to him. (4 No tanfe take place. F F F F F (ii Denote U, U a the equilibium payoff fo and. U + U U Define U, U a the off-equilibium payoff fo and. U + U U tategy above to be a SPE, we need to check the following IC containt. Fit, i willing to give k. Thi i the ame a F U U Second, i willing to give m( to fo all, i.e. N E Fo the m ( F ( U U The left hand ide i cuent peiod gain if he fail to pay fo m(. The ight hand i the lo of continuation payoff. Thid, will want to chooe d0 in, i.e. F m ( ( U U

2 The left hand ide i gain in the cuent peiod if he deviate. eceive by chooing d and loe the payment fom m (. The ight hand ide i the lo of continuation value. y adding the above inequalitie, a neceay condition i F ( U U (5 5 Note that in thi equilibium ha the deciion ight on the off-equilibium path, o 5, U U 0. Note that the neceay condition i alo ufficient. In paticula, conide the cae that alway pay m to afte the bagaining. nd no othe tanfe i involved. Fo to be willing to pay fo m, we need, m+ (6 + ( m+ (6 0 whee the left hand ide i and the ight hand ide i. In othe wod, we need F U m 3 On the othe hand, incentive containt i + ( 5 ( m 4 + U y chooing d0, eceive 0 fom the action and a continuation payoff of ( m + * 4. If chooe d, he eceive fom the action, and he continuation payoff will be 5. The inequality above can be ewitten a ( m. Let m3, we ee that the maximum value of i 5. (iii If the initial aignment of deciion ight goe to, then U 0, U 4. imila neceay condition a in (ii ay that F I ( U U (5 4 whee i gain fom deviation in. Theefoe,. a eult, it i bette to allocate the initial deciion ight to.

3 Poblem : Contacting fo Contol (a Let * be the cutoff tate in which and have identical payoff: σ * + ρ σ * + ρ then the fit bet ule i to pick d d if <* d d if >* The expected total payoff i then ( * L[ ρ + σ ( L + *] + ( H *[ ρ + σ( H + *] ρ ρ whee *. σ σ (b In the pot veion, if contol the deciion ight, he expected uplu i ρ + σ ( L + H If contol the deciion ight, he expected payoff i ρ + σ( L + H Theefoe, contol the deciion ight if and only if ρ ρ ( σ σ ( L + H (c Conide the following tigge tategy. On the equilibium path: In each peiod, ( contol the deciion ight. ( pay t. (3 chooe the fit bet deciion. (4 pay T(d, Off the equilibium path: ( The owne of the deciion ight keep the deciion ight. ( No tanfe of money take place between patie. (3 The owne of the deciion ight chooe a deciion favoable to he. (4 No tanfe take place. Fo the tigge tategy to be a SPE, thee ae eveal poibilitie that will deviate. (i Fail to pay t. (ii Fail to cay out the coect deciion. (iii Fail to pay T(d, F F F F F Denote U, U a the equilibium payoff fo and. U + U U Define N E. U, U a the off-equilibium payoff fo and. U + U U Then the IC fo (i i like an IR containt, i.e. F U U

4 Denote ( a the exta gain fo fom deviation fom the fit bet deciion. Then the IC fo (ii i F ( + T( d, ( U F Note that IC in (ii implie the IC in (iii.( T( d, ( U U Now fo, thee ae two place he might want to deviate. Fit, he might fail to pay t. Thi i like an IR containt that equie F U U Second, he may deviate i failue to pay T(d,. The IC fo theefoe i F T( d, ( U U dding the two inequalitie above how that a neceay condition fo the tigge tategy to be a SPE i F up ( ( U U in poblem, it i eay to ee that thi i a ufficient condition a well. To ee thi, let F T(d,0 fo all d,, and let t ( H *[ ρ + σ( H + * ], o U U 0. It i taightfowad to check that will not deviate in thi ituation povided σ ρ σ ρ F up ( ( * ( U σ σ (d Fom the analyi in (c, we ee that the maximum dicount ate fo fit bet when deciion ight goe to i F U up ( When ha the deciion ight, the maximum dicount ate fo fit bet i F U up ( hould be given the deciion ight if and only if. In thi cae, ince up ( up (, the maximum temptation to deviate happen at *, hould have the deciion ight if and only if U U, i.e. ρ ρ ( σ σ ( L + H (e In thi cae, the ame analyi goe though. We have F U up ( and F U up ( nd hould be given the deciion ight if and only if. The only diffeence i that up ( now occu in L intead of *.

5 Poblem 3: lliance (a To chaacteize the SPE of thi game, we tat backwad. In tage (3, the only action pofile that fom a Nah equilibium of the ubgame i m m 0 Theefoe in tage (, we mut have * d ( agmax U ( d *, d (, and (b (i: The fit bet deciion ule d d ( agmax U ( d, d (, * * d F d ( atifie that ( d (, d ( ag max( U ( d (, d (, + U ( d (, d (, F F F F F F d, d (ii: Conide the following tigge tategy: On the equilibium path: F ( Playe chooe d ( ( Playe pay m( epectively. Off the equilibium path: ( Playe chooe d ( ( Playe pay m m 0. Note that in decibing the tategy, we can alway aume m 0. Now define the maximum gain fom deviation fo playe i in tate a F F F ( max( U ( d (, d (, + U ( d (, d (, i di i i j i F F F U + U U F F nd let U, U a the equilibium payoff fo and. Define U, U a the off-equilibium payoff fo and. U + U U Then a in poblem and, the neceay condition become F ( + m( ( U F ( m( ( U Then a neceay condition i F up( ( + ( ( U U In geneal, thi i a ufficient condition a well. To ee thi, denote F U up( ( + ( Now we can chooe m(, fo,,,n uch that F ( ( + m( ( U U Note that F U i a function of m( a well.

6 Thi can be done in geneal, becaue we have n unknown fo n equation. With thee m( choen, we ee that F F ( + m( ( U U ( U U fo all. Theefoe, the IC fo ae all atified. We now claim that with thee m( choen, we mut alo have F ( ( + m( ( U U Thi would implie that the IC fo i atified a well and thu pove the ufficiency. We pove the claim above by contadiction. If the claim i fale, then thee exit one tate uch that F ( ( ' + m( ' > ( U U Since F ( ( ' + m( ' ( U U Summing the two equation above give that F ( ( ' + ( ' > ( U U In othe wod, F F ( U U > ( ' + ( ' up( ( + ( nd thi contadict the definition of. So we pove the claim and thu pove the ufficiency. Note that thi way of poving the ufficiency i not vey contuctive becaue it elie on the exitence of olution fo n linea equation. (c: Tigge Stategy: On the equilibium path: F Playe chooe d ( Off the equilibium path: Playe chooe d ( To detemine the lage, we ee that fo playe, the IC ae fo all, F ( ( U Fo playe, the IC ae fo all, F ( ( U Theefoe, the maximum dicount F Ui i Min( i { a, b} up( ( i

Player B ensure a. is the biggest payoff to player A. L R Assume there is no dominant strategy That means a

Player B ensure a. is the biggest payoff to player A. L R Assume there is no dominant strategy That means a Endogenou Timing irt half baed on Hamilton & Slutky. "Endogenizing the Order of Move in Matrix Game." Theory and Deciion. 99 Second half baed on Hamilton & Slutky. "Endogenou Timing in Duopoly Game: Stackelberg