1Recap:{ Last class (January 21, 2010)z Examples of games with continuous action sets{ Tragedy of the commonsz Duopoly models: Cournot and Bertrand { Today (January 26, 2010)z Duopoly models{ Comparison of duopoly models with MonopolyStackelberg 1{Stackelberg z & comparison with Cournot, Bertrand, and Monopolyz Multistage games with observed actionsStackelberg Model{ Two competing firms, selling a homogeneous good {The marginalcostof producing each unit of the good: {The marginalcostof producing each unit of the good: c1and c2{ Firm 1 moves first and decides on the quantity to sell: q1{ Firm 2 moves next and after seeing q1, decides on the quantity to sell: q2 { Q= q1+q2 total market demand{ The market price, P is determined by (inverse) market 2p, y( )demand: z P=a-bQ if a>bQ, P=0 otherwise.{ Both firms seek to maximize profits2Stackelberg Model{ Qj: the space of feasible qj’s, j=1,2St t i f fi 2{Strategies of firm 2:s2: Q1→ Q2(That is, the strategy of firm 2 can depend on firm 1’s decision){ Strategies of firm 1: q1∈Q1{Outcomes and payoffs in pure strategies3{Outcomes and payoffs in pure strategies(q1, q2) = (q1, s2(q1))πj(q1, q2) = [a-b(q1+ q2)- cj] qjStackelberg Model: Strategy of Firm 2{ Suppose firm 1 produces q1{Firm 2’s profits if it produces qare:{Firm 2s profits, if it produces q2are:π2= (P-c)q2= [a-b(q1+q2)]q2 – c2q2 = (Residual) revenue – Cost{ First order conditions:d π2/dq2= a - 2bq2 – bq1– c2= = RMRMC= 0 →4= RMR–MC= 0 →q2=(a-c2)/2b – q1/2= R2(q1)s2= R2(q1) Strategy of firm 23Stackelberg Model: Firm 1’s decision{ Firm 1’s profits, if it produces q1are:π1= (P-c)q1= [a-b(q1+q2)]q1 –c1q1 π1 (Pc)q1 [ab(q1+q2)]q1 c1q1 { (Now, what is different from the Cournot game?){ We know that from the best response of Firm 2: q2=(a-c2)/2b – q1/2{ Substitute q2into π1:5q21π1= [a-b(q1+(a-c2)/2b- q1/2)]q1 – c1q1 = [(a+ c2)/2-(b/2) q1-c1]q1{ From FOC: dπ1/dq1= (a+ c2)/2-b q1-c1= 0 →q1= (a-2c1+c2)/2b Stackelberg Equilibrium{ We have Firm 1’s profits, if it produces q1:q= (a2c+c)/2b q1= (a-2c1+c2)/2b And firm 2’s best responseq2=(a-c2)/2b – q1/2{ Therefore: q2=(a+2c1-3c2)/4b {If c= c= c6{If c1= c2= cq1= (a-c)/2b q2= (a-c)/4b Q = 3(a-c)/4bRecall: qci= (a-c)/3b4Cournot vs. Stackelberg vs. BertrandBertrand Stackelberg Cournot MonopolyPrice c (a+3c)/4 (a+2c)/3 (a+c)/2Quantity(a-c)/b3(a-c)/4b ((a-c)/2b+(a-c)/4b)2(a-c)/3b (a-c)/2bTotal Firm f03(a-c) 2/16b2(a-c) 2/9b(a-c) 2/4b7Profits03(ac) /16b2(ac) /9b(ac) /4bExample: Stackelberg Competition{ P = 130-(q1+q2), so a=130, b=1{c= c= c = 10{c1= c2= c = 10{ Firm 2: q2=(a-c2)/2b – q1/2 = 60 - q1/2{ Firm 1:z Π1= [a-b(q1+q2)]q1– c1q1z Π1= [(a+c2)/2-(b/2)q1]q1–c1q1z Π1= [70-q1/2]q1– c1q18→ q_1 = 60{ Market price and demandQ=90 P=405Monopoly vs. Cournot vs. Bertrand vs. StackelbergBertrandStackelbergCournotMonopolyBertrandStackelbergCournotMonopolyPrice10 40 50 70Quantity12090 (60+30)80 60Total Firm Profits02700 (1800+900)3200 36009(1800+900){ Firm profits and prices:Bertrand ≤ Stackelberg ≤ Cournot ≤ MonopolyStackelberg competitionP130P=130-QConsumer surplus=4050Firm profits=270010qMC=101304090Deadweight loss=4506Monopoly vs. Cournot vs. Bertrand vs. StackelbergBertrand Stackelberg Cournot MonopolyConsumer surplus7200 4050 3200 1800Deadweight loss0 450 800 180011Total Firm Profits02700 (1800+900)3200 3600Stackelberg and Information{ Does player 2 do better or worse in this St k lb d t th C t Stackelberg game compared to the Cournot game?{ Does player 2 have more or less information in the Stackelberg game compared to the Cournot game?127Multi-Stage Games with Observed ActionsThese games have “stages” such thatSt k 1 i l d ti ll ft t k{Stage k+1 is played sequentially after stage k{ In each stage k, every player knows all the actions (including those by Nature) that were taken at any previous stagez Players can move simultaneously in each stage k{ Some players may be limited to action set “do nothing” in some stages{ Each player moves at most once within a given 13py gstage{ Players’ payoffs are common knowledgeStackelberg game{ Stage 1z Firm 1 chooses its quantity q1; Firm 2 does nothing{ Stage 2z Firm 2, knowing q1, chooses its own quantity q2; Firm 1 does nothing148Multi-Stage Games with Observed ActionskkKkhkhk1)( stage ofstart at theHistory :110−kkikishkihAKkaaahik specifies that player for strategy Pure : history given stagein player toavailable actions ofSet :)(,...,1),,...,,(110==15kkihkhAahistory each and each for )( action an ∈Q: why would the strategy need to specific an action for each stage and history? Finite games of perfect information{ A multistage game has perfect information if zeach player knows all previous moves when making eac p aye o s a p e ous o es e a ga decisionz for every stage k and history hk, exactly one player has a nontrivial action set, and all other players have one-element action set “do nothing”{ In a finite game of perfect information, the number of stages and the number of actions at any stage are finite.16at any stage are finite.{ Theorem (Zermelo 1913; Kuhn 1953): A finite game of perfect information has a pure-strategy Nash equilibriumz (Note, this is a little stronger than the Nash result!)9Backward inductionDetermine the optimal action(s) in the final stage K for each history hKgyFor each stage j=K-1,…,1{ Determine the optimal action(s) in stage j for each possible hjgiven the optimal actions determined for stages j+1,…,K.17The strategy profile constructed by backward induction is a Nash Equilibrium.Each player’s actions are optimal at every possible history.Example: Stackelberg competition{ P = 130-(q1+q2), c1= c2= c = 10Backward inductionz Firm 2 strategy: s2(q1)= q2= 60 - q1/2 z Firm 1 strategy: q1= 60z The outcome (60,30) is a Nash equilibrium (Stackelberg outcome)18Is (60,30) the unique equilibrium in this game?Let’s consider the Cournot equilibrium (40,40) for the Stackelberg game s2(q1)= 40 q1= 4010Stackelberg: other equilibria{ Suppose that player 2 commits to choosing the C t tit tt h tCournot quantity no matter whatz s_2(q_1)=q_2^C, for all q_1{ Then player 1’s strategy should also be the Cournot quantityz q_1=q_1^C{Cournot equilibrium (40 40) is also an equilibrium 19{Cournot equilibrium (40,40) is also an equilibrium for the Stackelberg game! s2(q1)= 40 q1= 40{ Will this happen?{ Implications?Classroom exercise: Strategic