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

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1 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 or Cournot Equilibria." Game and Economic ehavior. Vol., No., March 990, pp. 9-6 Thee were originally written a one paper, but they were publihed eparately. Endogenou Timing - give player choice of when they move e.g., period or period ; player don't chooe role leader/follower, but when they move; if both chooe ame time, it' a imultaneou game; if player chooe firt period and player chooe econd period then i the leader and i the follower Endogenou Deciion Type - give player choice on whether to et price or quantity or for international: chooe whether to ue tariff or quota Player, Strategy Game Strictly Ordered - player have no tie o a a a a and b b b ; only have type to worry about: b i unique pure trategy equilibrium e.g., prioner' dilemma ii unique mixed trategy equilibrium e.g., matching pennie iii multiple equilibria; will be pure and mixed e.g., coordination problem U a, b a, b D a, b a, b Pure Strategy emma - there exit a unique pure trategy equilibrium if and only if at leat one player ha a dominant trategy Proof: a Dominant trategy unique PSE One player having a dominant trategy mean we can eliminate one row column from conideration in finding an equilibrium Since the payoff are trictly ordered, the column row player will have a definite choice when facing two payoff in the ingle row column a ingle pure trategy equilibrium exit b Unique PSE dominant trategy Proof with contra poitive no dominant trat no unique PSE We can alway wap row and column a neceary to enure a i the bigget payoff to player ume there i no dominant trategy U a, b a, b That mean a > a becaue we jut aid a i bigget payoff D a, b a, b and a > a otherwie U would be a dominant trategy There are poibilitie for player : i b > b b < b aumed no dominant trategie U a, b a, b multiple pure trategy equilibria D a ii b < b b > b aumed no dominant trategie, b a, b no pure trategy equilibrium of 8

2 Mixed Strategy emma - in mixed trategy equilibrium, each player' expected payoff i greater than two of hi payoff and le than the other two payoff Proof: or and to ue a mixed trategy it mut be the cae that pa + p a = pa + p a and qb + q b = qb + q b Jut conider player imilar argument for player a = q pa + p a + q pa + p a Expected payoff i [ ] [ ] pa + p a = pa + p a, we can ay + p a = pa + p a Uing fact that a = pa That i, ' expected payoff i the average of two of player ' payoff; ince we aumed trictly ordered payoff, either a or a mut be greater than a and the other i le than a there are two equation o it turn into two payoff greater and two le Game - for endogenou timing we have two option: announce timing or don t announce nnounce Timing - player have imultaneou choice to pick whether the move in firt round or econd round; the player ee the reult of their choice and then play the two trategy game from previou page uing the timing they picked p p q U a, b a, b q D a, b a, b Simultaneou choice for timing Play original game with elected timing U D S U D U D U D U D a b a b a b Subgame Perfect - ue refinement we haven't tudied formally yet; look at lower tree twotage game and get equilibria for each one Simultaneou Move - for leftmot and rightmot tree, player have imultaneou game; thi ha the three option we covered earlier unique PSNE, unique MSNE, multiple equilibria... we'll aume a unique equilibrium: a,b Sequential Move - the middle game are Stackelberg game unique equilibria; in firt one i leader and i follower o label equilibrium a l,b f ; in econd one i leader and i follower o label equilibrium a f,b l educed Extenive orm - uing ubgame perfect refinement educed Normal orm - uing ubgame perfect refinement Pure Strategy Equilibrium - aume the unique equilibrium in the imultaneou move game i a pure trategy; further, ince the pure trategy lemma ay one of the player ha a dominant trategy, aume it' the row player and row i dominant a > a and a > a a b a b a b a b of 8 a b a b a b a b a b a b a b a b a b a b S a l b f S a f b l S a b

3 Column Player - we can alway label the column o that b > b o the imultaneou equilibrium a, b = a, b our Cae - baed on relationhip between b & b, a & a, and b & b b > b - both player have dominant trategy; timing doen't matter b > b & a > a - timing doen't matter irt, Second - if he play U, player play ; reult in a, b ; if he play D, player play ; reult in a, b ; ince a > a, will play U... get a, b irt, Second - ince will play U for either of ' trategie and b > b, play... get a, b U a, b a, b D a, b a, b U a, b a, b D a, b a, b b > b, a > a, & b > b - player chooe imultaneou irt, Second - if he play U, player play ; reult in a, b ; if he play D, player play ; reult in a, b ; ince a > a, will play D... get a, b irt, Second - ame a cae U a, b a, b a, b a, b D a, b a, b Timing Game - S i weakly dominated for player and for... technically we'd have to how that with the full extenive form tree, but it' weakly dominated in the reduced form Slutky got grief for that in the paper... reult i both player chooing the firt period imultaneou b > b, a > a, & b > b - player chooe firt round, chooe econd Sequential - ame reult a cae pure trategy Nah U a, b a, b a, b a, b equilibrium; S, i D a, b a, b S a, b a, b weakly dominated Timing Game - ame weakly dominated argument from cae of 8 a, b a, b S a, b a, b a, b a, b S a, b a, b S a, b a, b pure trategy Nah equilibrium; S, i weakly dominated

4 eult - if there i a payoff that Pareto dominate like a, b in cae, player will prefer to chooe timing to get away from imultaneou equilibrium Mixed Strategy Equilibrium - aume the unique equilibrium in the imultaneou move game i a mixed trategy; there are two cae with no pure trategy equilibrium, we'll look at arrow going clockwie ymmetric argument for arrow going counter-clockwie; U a, b a, b D a, b a, b aumption: - a > a and a > a Guarantee no pure trategy equilibrium - b > b and b > b - a, b Pareto dominate a, b the mixed trategy expected payoff; require thing: i b > b > b > b... Mixed Strategy emma p. ay b ha to be one of top two payoff ii a > a... Mixed Strategy emma again; don't know how a & a are related irt, Second - if chooe U, prefer... reult in a, b ; Depend on if chooe D, prefer... reult in U a, b a, b a, b a > or < a a, b ; ' choice in U or D depend D a on relationhip between a & a, but it, b a, b won't matter when we look at the timing game below ' prefer to a, b a, b move to equential irt, Second - if chooe, prefer U... reult in a, of 8 b prefer D... reult in a, b ; ince b > b > b > b, will chooe Timing Game - It' clear that, and S,S will not be equilibria becaue from,, player prefer to move down and from S,S, player prefer to move left; to find equilibrium we have to look cloer at,s: ; if chooe, If it' a, b, we know a > a > a a > a, o player prefer to move down...,s i not an equilibrium If it' a, b, we know b > b > b > b b > b, o player prefer to move left...,s i not an equilibrium S a, b a, b We end up with one of the player having a dominant trategy and S, i the unique pure trategy equilibrium eult - if the expected payoff from the mixed trategy equilibrium i Pareto dominated and player don't both have a dominant trategy, the player will prefer to chooe timing to get away form the mixed trategy Plauibility of Mixed Strategie - Slutky ay thi lend credibility to uing mixed trategie if the expected payoff i not Pareto dominated x Only - thi rule work for x becaue the Pareto dominant payoff will be one player' bet reply; for x thi in't alway the cae: a, b l f f = a, b = a, b l = 0,0 a, b won't matter in timing game a, b a, b or a, b Nah Eq. Pareto dominate the Nah eq. M U 0, 0, 8, 7 0, 0 M 9, 5, -0,0-0,0 D 9,, - 0, 0 9, 0, 0, -, 7

5 Don't nnounce Timing - alo called "action commitment"; there' ambiguity in firt period a to who will do what, but any firm that move in the econd period will know that it i either the follower or will be competing imultaneouly a b a b W a b a b U a b D One Dominant Strategy - aume only player ha a dominant trategy o there' a unique pure trategy equilibrium; aume it' the row player o a > a and a > a ; we can arrange column o that b > b ; ince column player doen't have dominant trategy we alo know b > b Simultaneou Equilibrium - by contruction a, b = a, b uing ubgame perfection U a, b a, b D a, b a, b a, b a, b a, b a, b Weak Dominance - now apply another refinement: Column Player - W weakly dominate and ow Player - W i the ame a weakly dominate U eult - if a > a timing ha no effect; if a > a, row player move firt and play the dominated trategy Summary - With nnouncement - player only ue timing choice if they have a Pareto dominant payoff that' a bet reply Without nnouncement - don't need a Pareto dominant payoff to ue the timing choice, jut need a > a Gator' Example - why announce firing Coach Zook in mid-eaon and let him finih the eaon? U committed to hire a new coach. a b W a b W a b a b a b W U D a b Subgame perfect refinement thi become imultaneou equilibrium: a, b a b a b a b W U a, b a, b a, b D a, b a, b a, b W a, b a, b a, b 5 of 8

6 Continuou Strategy Space - baically doing the ame thing uing continuou trategy pace intead of x game; firm chooe to move in firt or econd period imultaneouly; then they chooe their output or price in their deignated period; thi tart the econd paper Subgame Perfect - apply refinement: Simultaneou - if firm move in ame period, reult i Cournot equilibrium Sequential - if firm move in equential period, reult i Stackelberg equilibrium No Tie - unlike x game, we don t have to worry about tie in thi cae make it eaier: l f f l a, b a, b a, b Never S, S - becaue leader can alway chooe imultaneou l l level, a > a and b > b a, b S a l, b f Payoff Cae - f f If a > a and b > b, then, i unique equilibrium f If a f > a and b > b, then S, and, S are multiple equilibria alo have mixed trategy equilibrium; imilar to coordination game f If a f > a and b > b, then S, i unique equilibrium Pareto dominate a, b If f a > a and f b > b, then, S i unique equilibrium Pareto dominate, b a player ue timing choice if there i a Pareto improvement... thi will be more clear hopefully uing ome pecific cae Specific Cae - the choice on timing will depend on thee two thing: whether the bet replie lope in the ame direction and which direction the indifference curve face et eplie - can lope up or down o there are four cae: up, up, up, down, down, up, down, down Indifference Curve - can face left or right for player on vertical axi; up or down for player on horizontal axi; again there are four cae: a b a l b f a f b l a b Not an Eq. oth Upward Sloping - a Indiff Curve up and right - both bet replie enter the Pareto improvement et; multiple equilibria: S, and, S... both are better than imultaneou equilibrium b Indiff curve down and left - ame reult a a i follower Ioprofit = f = g i leader Pareto improvement = f Simultaneou Ioprofit = g 6 of 8

7 c Indiff curve down and right - neither bet reply enter the Pareto improvement et; leader i better off, but follower i wore off o, i unique equilibrium d Indiff curve up and left - ame reult a c = f = f = g = g oth Downward Sloping - ame reult a both upward loping: cae with multiple equential equilibria and cae with unique imultaneou equilibrium a Indiff Curve up and right - neither bet reply enter the Pareto improvement et; leader i better off, but follower i wore off o, i unique equilibrium b Indiff curve down and left - ame reult a a c Indiff curve down and right - both bet replie enter the Pareto improvement et; multiple equilibria: S, and, S... both are better than imultaneou equilibrium d Indiff curve up and left - ame reult a c = f =f = f = f = g =g = g = g et eplie Have Oppoite Slope - in all cae only bet reply goe into the Pareto improvement et o there i alway a unique equential move equilibrium a Indiff Curve up and right - only one bet reply enter the Pareto improvement et; player i better off a the leader and i better off a the follower o, S i unique equilibrium b Indiff curve down and left - ame reult a a = f = g = f = g 7 of 8

8 c Indiff curve down and right - only one bet reply enter the Pareto improvement et; player i bet off a the follower and i bet off a the leader o S, i unique equilibrium d Indiff curve up and left - ame reult a c = f = g = f = g eult - Qualitatively Symmetric - bet replie lope in ame direction and ui / j and u j i / have ame ign i.e., Pareto improvement et i up-right or down-left, then Slope Up - S, and, S are multiple equilibria trategic complement Slope Down -, i unique equilibrium trategic ubtitute ee ECO 798 Product Differentiation, "Strategic Move" note for more detail on trategic ubtitute and complement Endogenou Stackelberg - only way to get unique leader-follower equilibrium i to have bet replie lope in oppoite direction Point - make ome factor endogenou by adding a prior tage; can make almot anything a deciion variable in an earlier tage... ometime get intereting reult 8 of 8