Mastering Dynamic Games: Nash and Subgame Equilibria Explained
School
University of Wisconsin, Madison**We aren't endorsed by this school
Course
ECON 701
Subject
Economics
Date
Dec 12, 2024
Pages
3
Uploaded by PresidentTurkeyPerson1308
Econ 701Handout 11Rodrigo, OH: M-W:10-11, Office 7439 SSCOverview•Dynamic games•Subgame•Subgame perfect Nash equilibriumExercisesNE vs. SPNEConsider the game:12(7,6)left(5,4)rightLeft2(4,3)left(2,4)rightRighta) What is the set of actions of agents?b) What is the set of strategies of agents?c) What are all the subgames?d) Find the normal form of the game.e) Find all the Nash equilibria.f) Find all the SPNE, compare with the set of NE.a)Agent 1 has two actions: Left and Right, while agent 2 has four possible actions: ”leftif player 1 plays left”, ”right if player 1 plays left”, ”left if player 1 plays right” and”right if player 1 plays right”b) What is the set of strategies of agents?c)Agent 1 has two pure strategies:Left and Right, while agent 2 has four possiblestrategies: (”left if player 1 plays left” and ”left if player 1 plays right”), (”left if player1 plays left” and ”right if player 1 plays right”), (”right if player 1 plays left” and”left if player 1 plays right”), (”right if player 1 plays left” and ”right if player 1 playsright”)1
Econ 701Handout 11Rodrigo, OH: M-W:10-11, Office 7439 SSCd) What are all the subgames?e)One that starts after player one plays ”Left, One that starts after player 1 plays ”Right”and the whole game.f) Find the normal form of the game.One that starts after player one plays ”Left, One that starts after player 1 plays ”Right”and the whole game.1/2LLLRRLRRL7,67,65,45,4R4,32,44,32,4All the best responses are:1/2LLLRRLRRL7*,6*7*,6*5*,45*,4R4,32,4*4,32,4*g) Find all the Nash equilibria.So all the Nash equilibria are (L, LL) and (L, LR).h) Find all the SPNE, compare with the set of NE.Using backward induction we find that:12(7,6)left(5,4)rightLeft2(4,3)left(2,4)rightRightIt is optimal for player 2 to play left if player 1 plays left and right if player 1 playsright.Given that, it is optimal for player 1 to play Left.So, unique SPNE is (L,(L,R)).Cournot and Stackelberg.Consider an asymmetric duopoly. The market demand isp= 12-Q. Firms have constantmarginal costc= 3.undefined.ΩΩSee the enumitem package documentation for explanation.ΩType H ¡return¿ for immedia2
Econ 701Handout 11Rodrigo, OH: M-W:10-11, Office 7439 SSC1. Find Cournot equilibrium. Find prices and profits earned by all firms.Profit functions of two firms are:π1= (12-q1-q2)q1-3q1→maxπ2= (12-q1-q2)q2-3q2→maxThe first order conditions are:π1∂q1= (12-2q1-q2)-3 = 0→maxπ2∂q2= (12-q1-2q2)-3 = 0→maxThus,q1= 3,q2= 3.p= 6,π1= 9,π2= 9.2. Assume firm 1 is a Stackelberg leader, while firm 2 is a follower.Find equilibriumquantities, prices and profits of two firms.π1= (12-q1-q2)q1-3q1→maxπ2= (12-q1-q2)q2-3q2→maxUsing backward induction we find that firm 2 choosesq2=9-q12.Plugging back to the first-order condition of firm one we have:(12-q1-9-q12)q1-3q1→maxSo,q1= 4.5,q2= 2.25,p= 5.25,π1= 14.625,π2= 7.31253