Advanced Microeconomics II Game Theory Fall

Transcription

1 Advanced Microeconomics II Game Theory 2016 Fall LIJUN PAN GRADUATE SCHOOL OF ECONOMICS NAGOYA UNIVERSITY 1

2 Introduction

3 What is ame theory? A Motivatin Example Friends - S02, Ep05 To celebrate Monica's promotion, everyone decides to o out and eat at a restaurant. This, however, causes financial issues with, Phoebe, Joey, and Rachel who are not as well off as the rest of their friends (Ross, Monica, and Chandler). Poor Rich 3

4 What is ame theory? We focus on ames where: There are at least two rational players Each player has more than one choices The outcome depends on the strateies chosen by all players; there is strateic interaction Example: Five people o to a restaurant. Each person pays his/her own meal a simple decision problem Before the meal, every person arees to split the bill evenly amon them a ame 4

5 What is ame theory? Game theory is a formal way to analyze strateic interaction amon a roup of rational players (or aents) who behave strateically Game theory has applications Economics Politics Law International Relations Sports 5

6 What is Game Theory? No man is an island John Donn Study of rational behavior in interactive or interdependent situations Bad news: Knowin ame theory does not uarantee winnin Good news: Framework for thinkin about strateic interaction 6

7 Strateies for Studyin Games of Stratey Two eneral approaches Case-based Pro: Relevance, connection of theory to application Con: Generality Theory Pro: General principle is clear Con: Application to reality may not be feasible 7

8 Terminoloy Strateies Choices available to each of the players Payoffs Some numerical representation of the objectives of each player Could take account of fairness/reputation, etc. Does not mean players are narrowly selfish 8

9 Standard Assumptions Rationality Players are perfect calculators and implementers of their desired stratey Common knowlede of rules All players know the ame bein played Equilibrium Players play strateies that are mutual best responses 9

10 The Use of Game Theory Explanatory A lens throuh which to view and learn from past neotiations/conflicts Predictive With many caveats Prescriptive The main thin you ll take out of the course is an ability to think strateically 10

11 Classification of Games In this course, we classify the ames from the perspective of move orders and information Simultaneous move Sequential moves Complete information Static ame of complete information Dynamic ame of complete information Incomplete information Static ame of incomplete information Dynamic ame of incomplete information

12 Lecture 1 Static (or Simultaneous-Move) Games of Complete Information 12

13 Outline of Static Games of Complete Information Normal-form (or strateic-form) representation Iterated elimination of strictly dominated strateies Nash equilibrium Applications of Nash equilibrium Mixed stratey Nash equilibrium 13

14 Aenda Examples Prisoner s dilemma The battle of the sexes Matchin pennies Static (or simultaneous-move) ames of complete information Normal-form or strateic-form representation Dominated strateies Iterated elimination of strictly dominated strateies Nash equilibrium 14

15 Classic Example: Prisoners Dilemma Two suspects held in separate cells are chared with a major crime. However, there is not enouh evidence. Both suspects are told the followin policy: If neither confesses then both will be convicted of a minor offense and sentenced to one month in jail. If both confess then both will be sentenced to jail for six months. If one confesses but the other does not, then the confessor will be released but the other will be sentenced to jail for nine months. Prisoner 1 Mum Confess Prisoner 2 Mum Confess -1, -1-9, 0 0, -9-6, -6 15

16 Example: The battle of the sexes At the separate workplaces, Chris and Pat must choose to attend either an opera or a prize fiht in the evenin. Both Chris and Pat know the followin: Both would like to spend the evenin toether. But Chris prefers the opera. Pat prefers the prize fiht. Chris Opera Prize Fiht Pat Opera Prize Fiht 2, 1 0, 0 0, 0 1, 2 16

17 Example: Matchin pennies Each of the two players has a penny. Two players must simultaneously choose whether to show the Head or the Tail. Both players know the followin rules: If two pennies match (both heads or both tails) then player 2 wins player 1 s penny. Otherwise, player 1 wins player 2 s penny. Player 1 Head Tail Player 2 Head Tail -1, 1 1, -1 1, -1-1, 1 17

18 Static (or simultaneous-move) ames of complete information A static (or simultaneous-move) ame consists of: A set of players (at least two players) For each player, a set of strateies/actions Payoffs received by each player for the combinations of the strateies, or for each player, preferences over the combinations of the strateies {Player 1, Player 2,... Player n} S 1 S 2... S n u i (s 1, s 2,...s n ), for all s 1 S 1, s 2 S 2,... s n S n. 18

19 Static (or simultaneous-move) ames of complete information Simultaneous-move Each player chooses his/her stratey without knowlede of others choices. Complete information Each player s strateies and payoff functions are common knowlede amon all the players. Assumptions on the players Rationality Players aim to maximize their payoffs Players are perfect calculators Each player knows that other players are rational 19

20 Static (or simultaneous-move) ames of complete information The players cooperate? No. Only noncooperative ames The timin Each player i chooses his/her stratey s i without knowlede of others choices. Then each player i receives his/her payoff u i (s 1, s 2,..., s n ). The ame ends. 20

21 Definition: normal-form or strateicform representation The normal-form (or strateic-form) representation of a ame G specifies: A finite set of players {1, 2,..., n}, players stratey profiles S 1 S 2... S n and their payoff functions u 1 u 2... u n where u i : S 1 S 2... S n R. 21

22 Normal-form representation: 2-player ame Bi-matrix representation 2 players: Player 1 and Player 2 Each player has a finite number of strateies Example: S 1 ={s 11, s 12, s 13 } S 2 ={s 21, s 22 } Player 2 s 21 s 22 s 11 u 1 (s 11,s 21 ), u 2 (s 11,s 21 ) u 1 (s 11,s 22 ), u 2 (s 11,s 22 ) Player 1 s 12 u 1 (s 12,s 21 ), u 2 (s 12,s 21 ) u 1 (s 12,s 22 ), u 2 (s 12,s 22 ) s 13 u 1 (s 13,s 21 ), u 2 (s 13,s 21 ) u 1 (s 13,s 22 ), u 2 (s 13,s 22 ) 22

23 Classic example: Prisoners Dilemma: normal-form representation Set of players: {Prisoner 1, Prisoner 2} Sets of strateies: S 1 = S 2 = {Mum, Confess} Payoff functions: u 1 (M, M)=-1, u 1 (M, C)=-9, u 1 (C, M)=0, u 1 (C, C)=-6; u 2 (M, M)=-1, u 2 (M, C)=0, u 2 (C, M)=-9, u 2 (C, C)=-6 Players Strateies Mum Prisoner 1 Confess Prisoner 2 Mum Confess -1, -1-9, 0 0, -9-6, -6 Payoffs 23

24 Example: The battle of the sexes Chris Opera Prize Fiht Pat Opera Prize Fiht 2, 1 0, 0 0, 0 1, 2 Normal (or strateic) form representation: Set of players: { Chris, Pat } (={Player 1, Player 2}) Sets of strateies: S 1 = S 2 = { Opera, Prize Fiht} Payoff functions: u 1 (O, O)=2, u 1 (O, F)=0, u 1 (F, O)=0, u 1 (F, O)=1; u 2 (O, O)=1, u 2 (O, F)=0, u 2 (F, O)=0, u 2 (F, F)=2 24

25 Example: Matchin pennies Player 1 Head Tail Player 2 Head Tail -1, 1 1, -1 1, -1-1, 1 Normal (or strateic) form representation: Set of players: {Player 1, Player 2} Sets of strateies: S 1 = S 2 = { Head, Tail } Payoff functions: u 1 (H, H)=-1, u 1 (H, T)=1, u 1 (T, H)=1, u 1 (H, T)=-1; u 2 (H, H)=1, u 2 (H, T)=-1, u 2 (T, H)=-1, u 2 (T, T)=1 25

26 Example: Cournot model of duopoly A product is produced by only two firms: firm 1 and firm 2. The quantities are denoted by q 1 and q 2, respectively. Each firm chooses the quantity without knowin the other firm has chosen. The market price is P(Q)=a-Q, where Q=q 1 +q 2. The cost to firm i of producin quantity q i is C i (q i )=cq i. The normal-form representation: Set of players: { Firm 1, Firm 2} Sets of strateies: S 1 =[0, + ), S 2 =[0, + ) Payoff functions: u 1 (q 1, q 2 )=q 1 (a-(q 1 +q 2 )-c), u 2 (q 1, q 2 )=q 2 (a-(q 1 +q 2 )-c) 26

27 Static (or simultaneous-move) ames of complete information - Recap A static (or simultaneous-move) ame consists of: A set of players (at least two players) For each player, a set of strateies/actions Payoffs received by each player for the combinations of the strateies, or for each player, preferences over the combinations of the strateies {Player 1, Player 2,... Player n} S 1 S 2... S n u i (s 1, s 2,...s n ), for all s 1 S 1, s 2 S 2,... s n S n. 27

28 Solvin Prisoners Dilemma Confess always does better whatever the other player chooses Dominated stratey There exists another stratey which always does better reardless of other players choices Players Strateies Mum Prisoner 1 Confess Prisoner 2 Mum Confess -1, -1-9, 0 0, -9-6, -6 Payoffs 28

29 Definition: strictly dominated stratey In the normal-form ame {S 1, S 2,..., S n, u 1, u 2,..., u n }, let s i ', s i " S i be feasible strateies for player i. Stratey s i ' is strictly dominated by stratey s i " if u i (s 1, s 2,... s i-1, s i ', s i+1,..., s n ) < u i (s 1, s 2,... s i-1, s i ", s i+1,..., s n ) for all s 1 S 1, s 2 S 2,..., s i-1 S i-1, s i+1 S i+1,..., s n S n. s i is strictly better than s i reardless of other players choices Prisoner 1 Mum Confess Prisoner 2 Mum Confess -1, -1-9, 0 0, -9-6, -6 29

30 Example Two firms, Reynolds and Philip, share some market Each firm earns $60 million from its customers if neither do advertisin Advertisin costs a firm $20 million Advertisin captures $30 million from competitor Philip Reynolds No Ad Ad No Ad 60, 60 30, 70 Ad 70, 30 40, 40 30

31 2-player ame with finite strateies S 1 ={s 11, s 12, s 13 } S 2 ={s 21, s 22 } s 11 is strictly dominated by s 12 if u 1 (s 11,s 21 )<u 1 (s 12,s 21 ) and u 1 (s 11,s 22 )<u 1 (s 12,s 22 ). s 21 is strictly dominated by s 22 if u 2 (s 1i,s 21 ) < u 2 (s 1i,s 22 ), for i = 1, 2, 3 Player 2 s 21 s 22 s 11 u 1 (s 11,s 21 ), u 2 (s 11,s 21 ) u 1 (s 11,s 22 ), u 2 (s 11,s 22 ) Player 1 s 12 u 1 (s 12,s 21 ), u 2 (s 12,s 21 ) u 1 (s 12,s 22 ), u 2 (s 12,s 22 ) s 13 u 1 (s 13,s 21 ), u 2 (s 13,s 21 ) u 1 (s 13,s 22 ), u 2 (s 13,s 22 ) 31

32 Definition: weakly dominated stratey In the normal-form ame {S 1, S 2,..., S n, u 1, u 2,..., u n }, let s i ', s i " S i be feasible strateies for player i. Stratey s i ' is weakly dominated by stratey s i " if u i (s 1, s 2,... s i-1, s i ', s i+1,..., s n ) (but not always =) u i (s 1, s 2,... s i-1, s i ", s i+1,..., s n ) for all s 1 S 1, s 2 S 2,..., s i-1 S i-1, s i+1 S i+1,..., s n S n. s i is at least as ood as s i reardless of other players choices Player 1 U B Player 2 L R 1, 1 2, 0 0, 2 2, 2 32

33 Strictly and weakly dominated stratey A rational player never chooses a strictly dominated stratey. Hence, any strictly dominated stratey can be eliminated. A rational player may choose a weakly dominated stratey. 33

34 Iterated elimination of strictly dominated strateies Approach If a stratey is strictly dominated, eliminate it The size and complexity of the ame is reduced Eliminate any strictly dominated strateies from the reduced ame Continue doin so successively 34

35 Iterated elimination of strictly dominated strateies Implicit assumptions A rational player will never play a strictly dominated stratey Common knowlede of rationality: The structure of the ame and the rationality of the players are common knowlede amon the players All the players know that each player will never play a strictly dominated stratey => they can effectively inore those strictly dominated strateies that opponents will never play 35

36 Iterated elimination of strictly dominated strateies: an example Player 1 Up Down Player 2 Left Middle Riht 1, 0 1, 2 0, 1 0, 3 0, 1 2, 0 Player 1 Up Down Player 2 Left Middle 1, 0 1, 2 0, 3 0, 1 36

37 Example: Tourists & Natives Only two bars (bar 1, bar 2) in a city Can chare price of $2, $4, or $ tourists pick a bar randomly 4000 natives select the lowest price bar Example 1: Both chare $2 each ets 5,000 customers and $10,000 Example 2: Bar 1 chares $4, Bar 2 chares $5 Bar 1 ets =7,000 customers and $28,000 Bar 2 ets 3000 customers and $15,000 37

38 Example: Tourists & Natives Bar 2 Bar 1 Bar 1 $2 $4 $5 $2 10, 10 14, 12 14, 15 $4 12, 14 20, 20 28, 15 $5 15, 14 15, 28 25, 25 Payoffs are in thousands of dollars Bar 2 $4 $5 $4 20, 20 28, 15 $5 15, 28 25, 25 38

39 New solution concept: Nash equilibrium Player 1 Player 2 L C R T 0, 4 4, 0 5, 3 M 4, 0 0, 4 5, 3 B 3, 5 3, 5 6, 6 The combination of strateies (B, R) has the followin property: Player 1 CANNOT do better by choosin a stratey different from B, iven that player 2 chooses R. Player 2 CANNOT do better by choosin a stratey different from R, iven that player 1 chooses B. 39

40 Nash Equilibrium: idea Nash equilibrium A set of strateies, one for each player, such that each player s stratey is best for her, iven that all other players are playin their equilibrium strateies 40

41 John Nash John Nash (June 13, 1928 May 23, 2015) A mathematician and an economist He developed several theories that were relevant in understandin economic interactions His important contribution was the famous Nash equilibrium He won the Nobel Memorial Prize in Economic Sciences in 1994 He suffered from mental illness from 1959, and his recovery became the basis for his bioraphy, A Beautiful Mind He died of a car accident with his wife on May 23, 2015

42 John Nash A Beautiful Mind: a 2001 American bioraphical drama film based on the life of John Nash

43 John Nash John Nash and his wife In the movie

44 Definition: Nash Equilibrium In the normal-form ame {S 1, S 2,..., S n, u 1, u 2,..., u n }, a combination * * of strateies ( s,..., s ) is a Nash equilibrium if, for every player i, for all s S i i 1 n * ui ( s1 u i ( s. That is,,..., s * 1 * i 1,..., s * i Prisoner 1, s * i 1 s solves * i, s, s i * i 1, s,..., s * i 1 * n ),..., s * * * * 1 i 1 i i 1 n Maximize u ( s,..., s, s, s,..., s ) Subject to i s S i i * n ) Mum Given others choices, player i cannot be betteroff if she deviates from s i * Prisoner 2 Confess Mum -1, -1-9, 0 Confess 0, -9-6, -6 44

45 2-player ame with finite strateies S 1 ={s 11, s 12, s 13 } S 2 ={s 21, s 22 } (s 11, s 21 )is a Nash equilibrium if u 1 (s 11,s 21 ) u 1 (s 12,s 21 ), u 1 (s 11,s 21 ) u 1 (s 13,s 21 ) and u 2 (s 11,s 21 ) u 2 (s 11,s 22 ). Player 2 s 21 s 22 s 11 u 1 (s 11,s 21 ), u 2 (s 11,s 21 ) u 1 (s 11,s 22 ), u 2 (s 11,s 22 ) Player 1 s 12 u 1 (s 12,s 21 ), u 2 (s 12,s 21 ) u 1 (s 12,s 22 ), u 2 (s 12,s 22 ) s 13 u 1 (s 13,s 21 ), u 2 (s 13,s 21 ) u 1 (s 13,s 22 ), u 2 (s 13,s 22 ) 45

46 Findin a Nash equilibrium: cell-by-cell inspection Player 1 Up Down Player 2 Left Middle Riht 1, 0 1, 2 0, 1 0, 3 0, 1 2, 0 Player 1 Up Down Player 2 Left Middle 1, 0 1, 2 0, 3 0, 1 46

47 Usin best response function to find Nash equilibrium: example Player 1 Player 2 L C R T 0, 4 4, 0 3, 3 M 4, 0 0, 4 3, 3 B 3, 3 3, 3 3.5, 3.6 M is Player 1 s best response to Player 2 s stratey L T is Player 1 s best response to Player 2 s stratey C B is Player 1 s best response to Player 2 s stratey R L is Player 2 s best response to Player 1 s stratey T C is Player 2 s best response to Player 1 s stratey M R is Player 2 s best response to Player 1 s stratey B 47

48 Excercise: Tourists & Natives Bar 2 Bar 1 $2 $4 $5 $2 10, 10 14, 12 14, 15 $4 12, 14 20, 20 28, 15 $5 15, 14 15, 28 25, 25 Payoffs are in thousands of dollars Use best response function to find the Nash equilibrium. 48

49 Example: The battle of the sexes Chris Opera Prize Fiht Opera Pat Prize Fiht 2, 1 0, 0 0, 0 1, 2 Opera is Player 1 s best response to Player 2 s stratey Opera Opera is Player 2 s best response to Player 1 s stratey Opera Hence, (Opera, Opera) is a Nash equilibrium Fiht is Player 1 s best response to Player 2 s stratey Fiht Fiht is Player 2 s best response to Player 1 s stratey Fiht Hence, (Fiht, Fiht) is a Nash equilibrium 49

50 Example: Matchin pennies Player 1 Head Tail Player 2 Head Tail -1, 1 1, -1 1, -1-1, 1 Head is Player 1 s best response to Player 2 s stratey Tail Tail is Player 2 s best response to Player 1 s stratey Tail Tail is Player 1 s best response to Player 2 s stratey Head Head is Player 2 s best response to Player 1 s stratey Head Hence, NO Nash equilibrium 50

51 Definition: best response function In the normal-form ame {S 1, S 2,..., S n, u 1, u 2,..., u n }, if player 1, 2,..., i-1, i+1,..., n choose strateies s 1,..., si 1, si1,..., sn, respectively, then player i's best response function is defined by B ( s,..., s, s,..., s ) i { s i 1 S i i1 : u i ( s u i i1 1 ( s,..., s 1 i1,..., s n, s i1 i, s i1, s, s i,..., s i1 n ),..., s n ), for all s S i i Given the strateies chosen by other players } Player i s best response 51

52 Definition: best response function An alternative definition: Player i's stratey si Bi ( s1,..., si1, si1,... sn) if and only if it solves (or it is an optimal solution to) Maximize ui ( s1,..., si1, si, si1,..., sn) Subject to si Si where s 1,..., si 1, si1,..., sn are iven. Player i s best response to other players strateies is an optimal solution to 52

53 Usin best response function to define Nash equilibrium In the normal-form ame {S 1,..., S n, u 1,..., u n }, * * a combination of strateies ( s 1,..., s n) is a Nash equilibrium if for every player i, s * i B i ( s * 1,..., s * i 1, s * i 1,..., s A set of strateies, one for each player, such that each player s stratey is best for her, iven that all other players are playin their strateies, or A stable situation that no player would like to deviate if others stick to it * n ) 53

54 Cournot model of duopoly A product is produced by only two firms: firm 1 and firm 2. The quantities are denoted by q 1 and q 2, respectively. Each firm chooses the quantity without knowin the other firm has chosen. The market priced is P(Q)=a-Q, where a is a constant number and Q=q 1 +q 2. The cost to firm i of producin quantity q i is C i (q i )=cq i. 54

55 Cournot model of duopoly The normal-form representation: Set of players: { Firm 1, Firm 2} Sets of strateies: S 1 =[0, + ), S 2 =[0, + ) Payoff functions: u 1 (q 1, q 2 )=q 1 (a-(q 1 +q 2 )-c) u 2 (q 1, q 2 )=q 2 (a-(q 1 +q 2 )-c) 55

56 Cournot model of duopoly How to find a Nash equilibrium Find the quantity pair (q 1 *, q 2 *) such that q 1 * is firm 1 s best response to Firm 2 s quantity q 2 * and q 2 * is firm 2 s best response to Firm 1 s quantity q 1 * That is, q 1 * solves Max u 1 (q 1, q 2 *)=q 1 (a-(q 1 +q 2 *)-c) subject to 0 q 1 + and q 2 * solves Max u 2 (q 1 *, q 2 )=q 2 (a-(q 1 *+q 2 )-c) subject to 0 q

57 Usin best response function to find Nash equilibrium In a 2-player ame, ( s 1, s 2 ) is a Nash equilibrium if and only if player 1 s stratey s 1 is her best response to player 2 s stratey s 2, and player 2 s stratey s 2 is her best response to player 1 s stratey s 1. Prisoner 1 Mum Confess Prisoner 2 Mum Confess -1, -1-9, 0 0, -9-6, -6 57

58 Cournot model of duopoly How to find a Nash equilibrium Find the quantity pair (q 1 *, q 2 *) such that q 1 * is firm 1 s best response to Firm 2 s quantity q 2 * and q 2 * is firm 2 s best response to Firm 1 s quantity q 1 * That is, q 1 * solves Max u 1 (q 1, q 2 *)=q 1 (a-(q 1 +q 2 *)-c) subject to 0 q 1 + and q 2 * solves Max u 2 (q 1 *, q 2 )=q 2 (a-(q 1 *+q 2 )-c) subject to 0 q

59 Cournot model of duopoly How to find a Nash equilibrium Solve Max u 1 (q 1, q 2 *)=q 1 (a-(q 1 +q 2 *)-c) subject to 0 q 1 + FOC: a - 2q 1 - q 2 *- c = 0 q 1 = (a - q 2 *- c)/2 59

60 Cournot model of duopoly How to find a Nash equilibrium Solve Max u 2 (q 1 *, q 2 )=q 2 (a-(q 1 *+q 2 )-c) subject to 0 q 2 + FOC: a - 2q 2 q 1 * c = 0 q 2 = (a q 1 * c)/2 60

61 Cournot model of duopoly How to find a Nash equilibrium The quantity pair (q 1 *, q 2 *) is a Nash equilibrium if q 1 * = (a q 2 * c)/2 q 2 * = (a q 1 * c)/2 Solvin these two equations ives us q 1 * = q 2 * = (a c)/3 61

62 Cournot model of duopoly Best response function Firm 1 s best function to firm 2 s quantity q 2 : R 1 (q 2 ) = (a q 2 c)/2 if q 2 < a c; 0, otherwise Firm 2 s best function to firm 1 s quantity q 1 : R 2 (q 1 ) = (a q 1 c)/2 if q 1 < a c; 0, otherwise a c q 2 Nash equilibrium (a c)/2 (a c)/2 a c q 1 62

63 Cournot model of oliopoly A product is produced by only n firms: firm 1 to firm n. Firm i s quantity is denoted by q i. Each firm chooses the quantity without knowin the other firms choices. The market priced is P(Q)=a-Q, where a is a constant number and Q=q 1 +q q n. The cost to firm i of producin quantity q i is C i (q i )=cq i. 63

64 Cournot model of oliopoly The normal-form representation: Set of players: { Firm 1,... Firm n} Sets of strateies: S i =[0, + ), for i=1, 2,..., n Payoff functions: u i (q 1,..., q n )=q i (a-(q 1 +q q n )-c) for i=1, 2,..., n 64

65 Cournot model of oliopoly How to find a Nash equilibrium Find the quantities (q 1 *,... q n *) such that q i * is firm i s best response to other firms quantities That is, q 1 * solves Max u 1 (q 1, q 2 *,..., q n *)=q 1 (a-(q 1 +q 2 * +...+q n *)-c) subject to 0 q 1 + and q 2 * solves Max u 2 (q 1 *, q 2, q 3 *,..., q n *)=q 2 (a-(q 1 *+q 2 +q 3 *+...+ q n *)-c) subject to 0 q

66 Bertrand model of duopoly (differentiated products) Two firms: firm 1 and firm 2. Each firm chooses the price for its product without knowin the other firm has chosen. The prices are denoted by p 1 and p 2, respectively. The quantity that consumers demand from firm 1: q 1 (p 1, p 2 ) = a p 1 + bp 2. The quantity that consumers demand from firm 2: q 2 (p 1, p 2 ) = a p 2 + bp 1. The cost to firm i of producin quantity q i is C i (q i )=cq i. 66

67 Bertrand model of duopoly (differentiated products) The normal-form representation: Set of players: { Firm 1, Firm 2} Sets of strateies: S 1 =[0, + ), S 2 =[0, + ) Payoff functions: u 1 (p 1, p 2 )=(a p 1 + bp 2 )(p 1 c) u 2 (p 1, p 2 )=(a p 2 + bp 1 )(p 2 c) 67

68 Bertrand model of duopoly (differentiated products) How to find a Nash equilibrium Find the price pair (p 1 *, p 2 *) such that p 1 * is firm 1 s best response to Firm 2 s price p 2 * and p 2 * is firm 2 s best response to Firm 1 s price p 1 * That is, p 1 * solves Max u 1 (p 1, p 2 *) = (a p 1 + bp 2 * )(p 1 c) subject to 0 p 1 + and p 2 * solves Max u 2 (p 1 *, p 2 ) = (a p 2 + bp 1 * )(p 2 c) subject to 0 p

69 Bertrand model of duopoly (differentiated products) How to find a Nash equilibrium Solve firm 1 s maximization problem Max u 1 (p 1, p 2 *) = (a p 1 + bp 2 * )(p 1 c) subject to 0 p 1 + FOC: a + c 2p 1 + bp 2 * = 0 p 1 = (a + c + bp 2 *)/2 69

70 Bertrand model of duopoly (differentiated products) How to find a Nash equilibrium Solve firm 2 s maximization problem Max u 2 (p 1 *, p 2 )=(a p 2 + bp 1 * )(p 2 c) subject to 0 p 2 + FOC: a + c 2p 2 + bp 1 * = 0 p 2 = (a + c + bp 1 *)/2 70

71 Bertrand model of duopoly (differentiated products) How to find a Nash equilibrium The price pair (p 1 *, p 2 *) is a Nash equilibrium if p 1 * = (a + c + bp 2 *)/2 p 2 * = (a + c + bp 1 *)/2 Solvin these two equations ives us p 1 * = p 2 * = (a + c)/(2 b) 71

72 The problems of commons n farmers in a villae. Each summer, all the farmers raze their oats on the villae reen. Let i denote the number of oats owned by farmer i. The cost of buyin and carin for a oat is c, independent of how many oats a farmer owns. The value of a oat is v(g) per oat, where G = n There is a maximum number of oats that can be razed on the reen. That is, v(g)>0 if G < G max, and v(g)=0 if G G max. Assumptions on v(g): v (G) < 0 and v (G) < 0. Each sprin, all the farmers simultaneously choose how many oats to own. 72

73 The problems of commons The normal-form representation: Set of players: { Farmer 1,... Farmer n} Sets of strateies: S i =[0, G max ), for i=1, 2,..., n Payoff functions: u i ( 1,..., n )= i v( n ) c i for i = 1, 2,..., n. 73

74 The problems of commons How to find a Nash equilibrium Find ( 1 *, 2 *,..., n *) such that i * is farmer i s best response to other farmers choices. That is, 1 * solves Max u 1 ( 1, 2 *,..., n *)= 1 v( *...+ n *) c 1 subject to 0 1 < G max and 2 * solves Max u 2 ( 1 *, 2, 3 *,..., n *)= 2 v( 1 * *+...+ n *) c 2 subject to 0 2 < G max... 74

75 The problems of commons How to find a Nash equilibrium and n * solves Max u n ( 1 *,..., n-1 *, n )= n v( 1 *+...+ n-1 *+ n ) c n subject to 0 n < G max... 75

76 The problems of commons FOCs: 0 ) *... * ( ) *... * (... 0 *)... * * ( *)... * * ( 0 *)... * ( *)... * ( c v v c v v c v v n n n n n n n n n 76

77 The problems of commons How to find a Nash equilibrium ( 1 *, 2 *,..., n *) is a Nash equilibrium if 0 *) *... * ( *) *... * (... 0 *)... * * * ( *)... * * * ( 0 *)... * * ( *)... * * ( c v v c v v c v v n n n n n n n n n 77

78 The problems of commons Summin over all n farmers FOCs and then dividin by n yields 1 v( G*) G * v( G*) c 0 n where G* * * n * 78

79 The problems of commons The social problem Max s.t. Gv( G) Gc 0 G G max FOC : v( G) Gv( G) c 0 Hence, the optimal solution G ** satisfies v( G **) G ** v( G **) c 0 79

80 The problems of commons v( G*) v( G **) 1 n G G * v( G*) c 0 ** v( G **) c 0 G* G **? 80

81 Summary Nash Equilibrium Best Response Function Cournot models of duopoly and oliopoly Bertrand model of duopoly The problems of commons Next time Mixed stratey Nash equilibrium 81