Static (or Simultaneous- Move) Games of Complete Information

Static (or Simultaneous- Move) Games of Complete Information Introduction to Games Normal (or Strategic) Form Representation Teoria dos Jogos - Filomena Garcia 1

Outline of Static Games of Complete Information Introduction to games Normal-form (or strategic-form) representation Iterated elimination of strictly dominated strategies Nash equilibrium Review of concave functions, optimization Applications of Nash equilibrium Mixed strategy Nash equilibrium Teoria dos Jogos - Filomena Garcia 2

Agenda What is game theory Examples Prisoner s dilemma The battle of the sexes Matching pennies Static (or simultaneous-move) games of complete information Normal-form or strategic-form representation Teoria dos Jogos - Filomena Garcia 3

What is game theory? We focus on games where: There are at least two rational players Each player has more than one choices The outcome depends on the strategies chosen by all players; there is strategic interaction Example: Six people go to a restaurant. Each person pays his/her own meal a simple decision problem Before the meal, every person agrees to split the bill evenly among them a game Teoria dos Jogos - Filomena Garcia 4

What is game theory? Game theory is a formal way to analyze strategic interaction among a group of rational players (or agents) who behave strategically Game theory has applications Economics Politics etc. Teoria dos Jogos - Filomena Garcia 5

Classic Example: Prisoners Dilemma Two suspects held in separate cells are charged with a major crime. However, there is not enough evidence. Both suspects are told the following 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 Teoria dos Jogos - Filomena Garcia 6

Example: The battle of the sexes At the separate workplaces, Chris and Pat must choose to attend either an opera or a prize fight in the evening. Both Chris and Pat know the following: Both would like to spend the evening together. But Chris prefers the opera. Pat prefers the prize fight. Chris Opera Prize Fight Pat Opera Prize Fight 2, 1 0, 0 0, 0 1, 2 Teoria dos Jogos - Filomena Garcia 7

Example: Matching 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 following 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 Teoria dos Jogos - Filomena Garcia 8

Static (or simultaneous-move) games of complete information A static (or simultaneous-move) game consists of: A set of players (at least two players) For each player, a set of strategies/actions Payoffs received by each player for the combinations of the strategies, or for each player, preferences over the combinations of the strategies {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. Teoria dos Jogos - Filomena Garcia 9

Static (or simultaneous-move) games of complete information Simultaneous-move Each player chooses his/her strategy without knowledge of others choices. Complete information Each player s strategies and payoff function are common knowledge among 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 Teoria dos Jogos - Filomena Garcia 10

Static (or simultaneous-move) games of complete information The players cooperate? No. Only noncooperative games The timing Each player i chooses his/her strategy s i without knowledge of others choices. Then each player i receives his/her payoff u i (s 1, s 2,..., s n ). The game ends. Teoria dos Jogos - Filomena Garcia 11

Definition: normal-form or strategicform representation The normal-form (or strategic-form) representation of a game G specifies: A finite set of players {1, 2,..., n}, players strategy spaces S 1 S 2 their payoff functions u 1 u 2... u n where u i : S 1 S 2... S n R.... S n and Teoria dos Jogos - Filomena Garcia 12

Normal-form representation: 2-player game Bi-matrix representation 2 players: Player 1 and Player 2 Each player has a finite number of strategies Example: S 1 ={s 11, s 12, s 13 } S 2 ={s 21, s 22 } Player 2 s 21 s 11 u 1 (s 11,s 21 ), u 2 (s 11,s 21 ) s 22 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 ) Teoria dos Jogos - Filomena Garcia 13

Classic example: Prisoners Dilemma: normal-form representation Set of players: {Prisoner 1, Prisoner 2} Sets of strategies: 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 Strategies Mum Prisoner 1 Confess Prisoner 2 Mum Confess -1, -1-9, 0 0, -9-6, -6 Payoffs Teoria dos Jogos - Filomena Garcia 14

Example: The battle of the sexes Chris Opera Prize Fight Pat Opera Prize Fight 2, 1 0, 0 0, 0 1, 2 Normal (or strategic) form representation: Set of players: { Chris, Pat } (={Player 1, Player 2}) Sets of strategies: S 1 = S 2 = { Opera, Prize Fight} 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 Teoria dos Jogos - Filomena Garcia 15

Example: Matching pennies Player 1 Head Tail Player 2 Head Tail -1, 1 1, -1 1, -1-1, 1 Normal (or strategic) form representation: Set of players: {Player 1, Player 2} Sets of strategies: 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 Teoria dos Jogos - Filomena Garcia 16

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 knowing 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 producing quantity q i is C i (q i )=cq i. The normal-form representation: Set of players: { Firm 1, Firm 2} Sets of strategies: 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) Teoria dos Jogos - Filomena Garcia 17

One More Example Each of n players selects a number between 0 and 100 simultaneously. Let x i denote the number selected by player i. Let y denote the average of these numbers Player i s payoff = x i 3y/5 The normal-form representation: Teoria dos Jogos - Filomena Garcia 18

Solving Prisoners Dilemma Confess always does better whatever the other player chooses Dominated strategy There exists another strategy which always does better regardless of other players choices Players Strategies Mum Prisoner 1 Confess Prisoner 2 Mum Confess -1, -1-9, 0 0, -9-6, -6 Payoffs Teoria dos Jogos - Filomena Garcia 19

Definition: strictly dominated strategy In the normal-form game {S 1, S 2,..., S n, u 1, u 2,..., u n }, let s i ', s i " S i be feasible strategies for player i. Strategy s i ' is strictly dominated by strategy 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 regardless of other players choices Prisoner 1 Mum Confess Prisoner 2 Mum Confess -1, -1-9, 0 0, -9-6, -6 Teoria dos Jogos - Filomena Garcia 20

Summary Static (or simultaneous-move) games of complete information Normal-form or strategic-form representation Next time Dominated strategies Iterated elimination of strictly dominated strategies Nash equilibrium Reading lists Sec 1.1.A and1.1.b of Gibbons and Sec 2.1-2.5, 2.9.1 and 2.9.2 of Osborne Teoria dos Jogos - Filomena Garcia 21

Static (or Simultaneous- Move) Games of Complete Information Dominated Strategies Nash Equilibrium Teoria dos Jogos - Filomena Garcia 22

Today s Agenda Dominated strategies Iterated elimination of strictly dominated strategies Nash equilibrium Teoria dos Jogos - Filomena Garcia 24

Solving Prisoners Dilemma Confess always does better whatever the other player chooses Dominated strategy There exists another strategy which always does better regardless of other players choices Prisoner 2 Mum Confess Prisoner 1 Mum Confess -1, -1 0, -9-9, 0-6, -6 Teoria dos Jogos - Filomena Garcia 25

Definition: strictly dominated strategy In the normal-form game {S 1, S 2,..., S n, u 1, u 2,..., u n }, let s i ', s i " S i be feasible strategies for player i. Strategy s i ' is strictly dominated by strategy 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 regardless of other players choices Prisoner 1 Mum Confess Mum -1, -1 0, -9 Prisoner 2 Confess -9, 0-6, -6 Teoria dos Jogos - Filomena Garcia 26

Example Two firms, Reynolds and Philip, share some market Each firm earns $60 million from its customers if neither do advertising Advertising costs a firm $20 million Advertising captures $30 million from competitor Philip No Ad Ad Reynolds No Ad Ad 60, 60 70, 30 30, 70 40, 40 Teoria dos Jogos - Filomena Garcia 27

2-player game with finite strategies 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 11 u 1 (s 11,s 21 ), u 2 (s 11,s 21 ) s 22 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 ) Teoria dos Jogos - Filomena Garcia 28

Definition: weakly dominated strategy In the normal-form game {S 1, S 2,..., S n, u 1, u 2,..., u n }, let s i ', s i " S i be feasible strategies for player i. Strategy s i ' is weakly dominated by strategy 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 good as s i regardless of other players choices Player 1 U B Player 2 L R 1, 1 2, 0 0, 2 2, 2 Teoria dos Jogos - Filomena Garcia 29

Strictly and weakly dominated strategy A rational player never chooses a strictly dominated strategy. Hence, any strictly dominated strategy can be eliminated. A rational player may choose a weakly dominated strategy. Teoria dos Jogos - Filomena Garcia 30

Iterated elimination of strictly dominated strategies If a strategy is strictly dominated, eliminate it The size and complexity of the game is reduced Eliminate any strictly dominated strategies from the reduced game Continue doing so successively Teoria dos Jogos - Filomena Garcia 31

Iterated elimination of strictly dominated strategies: an example Player 2 Left Middle Right Player 1 Up Down 1, 0 0, 3 1, 2 0, 1 0, 1 2, 0 Player 1 Up Down Player 2 Left Middle 1, 0 1, 2 0, 3 0, 1 Teoria dos Jogos - Filomena Garcia 32

One More Example The normal-form representation: Players: {player 1, player 2,..., player n} Strategies: S i =[0, 100], for i = 1, 2,..., n. Payoff functions: u i (x 1, x 2,..., x n ) = x i 3y/5 Is there any dominated strategy? What numbers should be selected? Teoria dos Jogos - Filomena Garcia 34

New solution concept: Nash equilibrium Player 2 L C R T 0, 4 4, 0 5, 3 Player 1 M 4, 0 0, 4 5, 3 B 3, 5 3, 5 6, 6 The combination of strategies (B, R) has the following property: Player 1 CANNOT do better by choosing a strategy different from B, given that player 2 chooses R. Player 2 CANNOT do better by choosing a strategy different from R, given that player 1 chooses B. Teoria dos Jogos - Filomena Garcia 35

New solution concept: Nash equilibrium Player 2 L C R T 0, 4 4, 0 3, 3 Player 1 M 4, 0 0, 4 3, 3 B 3, 3 3, 3 3.5, 3.6 The combination of strategies (B, R ) has the following property: Player 1 CANNOT do better by choosing a strategy different from B, given that player 2 chooses R. Player 2 CANNOT do better by choosing a strategy different from R, given that player 1 chooses B. Teoria dos Jogos - Filomena Garcia 36

Nash Equilibrium: idea Nash equilibrium A set of strategies, one for each player, such that each player s strategy is best for her, given that all other players are playing their equilibrium strategies Teoria dos Jogos - Filomena Garcia 37

Definition: Nash Equilibrium In the normal-form game {S 1, S 2,..., S n, u 1, u 2,..., u n }, a combination of strategies ( s 1,..., sn) is a Nash equilibrium if, for every player i, u i ( s 1,..., s 1 i 1, s i 1 i, s i+ 1,..., s i+ 1 ui ( s,..., s, si, s,..., s ) for all si Si. That is, s i solves Maximize u i( s1,..., si 1, si, si+ 1,..., sn) Subject to s S i i Prisoner 1 Mum Confess n ) n Mum -1, -1 0, -9 Given others choices, player i cannot be betteroff if she deviates from s i Prisoner 2 Confess -9, 0-6, -6 Teoria dos Jogos - Filomena Garcia 38

2-player game with finite strategies 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 11 u 1 (s 11,s 21 ), u 2 (s 11,s 21 ) s 22 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 ) Teoria dos Jogos - Filomena Garcia 39

Finding a Nash equilibrium: cell-bycell inspection Player 2 Left Middle Right Player 1 Up Down 1, 0 0, 3 1, 2 0, 1 0, 1 2, 0 Player 1 Up Down Player 2 Left Middle 1, 0 1, 2 0, 3 0, 1 Teoria dos Jogos - Filomena Garcia 40

Summary Dominated strategies Iterated elimination Nash equilibrium Next time Nash equilibrium Best response function Reading lists Sec 1.1.C and 1.2.A of Gibbons and Sec 2.6-2.8 of Osborne Teoria dos Jogos - Filomena Garcia 41

Static (or Simultaneous- Move) Games of Complete Information Nash Equilibrium Best Response Function Teoria dos Jogos - Filomena Garcia 42

Today s Agenda Nash equilibrium Best response function Use best response function to find Nash equilibria Examples Teoria dos Jogos - Filomena Garcia 44

Definition: strictly dominated strategy In the normal-form game {S 1, S 2,..., S n, u 1, u 2,..., u n }, let s i ', s i " S i be feasible strategies for player i. Strategy s i ' is strictly dominated by strategy 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 regardless of other players choices Prisoner 1 Mum Confess Mum -1, -1 0, -9 Prisoner 2 Confess -9, 0-6, -6 Teoria dos Jogos - Filomena Garcia 45

One More Example The normal-form representation: Players: {player 1, player 2,..., player n} Strategies: S i =[0, 100], for i = 1, 2,..., n. Payoff functions: u i (x 1, x 2,..., x n ) = x i 3y/5 What is the Nash equilibrium? Teoria dos Jogos - Filomena Garcia 46

Best response function: example Player 2 L C R T 0, 4 4, 0 3, 3 Player 1 M 4, 0 0, 4 3, 3 B 3, 3 3, 3 3.5, 3.6 If Player 2 chooses L then Player 1 s best strategy is M If Player 2 chooses C then Player 1 s best strategy is T If Player 2 chooses R then Player 1 s best strategy is B If Player 1 chooses B then Player 2 s best strategy is R Best response: the best strategy one player can play, given the strategies chosen by all other players Teoria dos Jogos - Filomena Garcia 47

2-player game with finite strategies S 1 ={s 11, s 12, s 13 } S 2 ={s 21, s 22 } Player 1 s strategy s 11 is her best response to Player 2 s strategy s 21 if u 1 (s 11,s 21 ) u 1 (s 12,s 21 ) and u 1 (s 11,s 21 ) u 1 (s 13,s 21 ). Player 2 s 21 s 11 u 1 (s 11,s 21 ), u 2 (s 11,s 21 ) s 22 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 ) Teoria dos Jogos - Filomena Garcia 48

Using best response function to find Nash equilibrium In a 2-player game, ( s 1, s 2 ) is a Nash equilibrium if and only if player 1 s strategy s 1 is her best response to player 2 s strategy s 2, and player 2 s strategy s 2 is her best response to player 1 s strategy s 1. Prisoner 1 Mum Confess Prisoner 2 Mum Confess -1, -1-9, 0 0, -9-6, -6 Teoria dos Jogos - Filomena Garcia 49

Using best response function to find Nash equilibrium: example Player 2 L C R T 0, 4 4, 0 3, 3 Player 1 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 strategy L T is Player 1 s best response to Player 2 s strategy C B is Player 1 s best response to Player 2 s strategy R L is Player 2 s best response to Player 1 s strategy T C is Player 2 s best response to Player 1 s strategy M R is Player 2 s best response to Player 1 s strategy B Teoria dos Jogos - Filomena Garcia 50

Example: The battle of the sexes Chris Opera Prize Fight Pat Opera Prize Fight 2, 1 0, 0 0, 0 1, 2 Opera is Player 1 s best response to Player 2 s strategy Opera Opera is Player 2 s best response to Player 1 s strategy Opera Hence, (Opera, Opera) is a Nash equilibrium Fight is Player 1 s best response to Player 2 s strategy Fight Fight is Player 2 s best response to Player 1 s strategy Fight Hence, (Fight, Fight) is a Nash equilibrium Teoria dos Jogos - Filomena Garcia 51

Example: Matching 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 strategy Tail Tail is Player 2 s best response to Player 1 s strategy Tail Tail is Player 1 s best response to Player 2 s strategy Head Head is Player 2 s best response to Player 1 s strategy Head Hence, NO Nash equilibrium Teoria dos Jogos - Filomena Garcia 52

Definition: best response function In the normal-form game {S 1, S 2,..., S n, u 1, u 2,..., u n }, if player 1, 2,..., i-1, i+1,..., n choose strategies s 1,..., si 1, si+ 1,..., sn, respectively, then player i's best response function is defined by B ( s,..., s, s,..., s ) = i { s i 1 S i i 1 : u i ( s u i i+ 1 1 ( s,..., s 1 i 1,..., s n, s i 1 i, s i+ 1, s, s i,..., s i+ 1 n ),..., s n ), for all s S i i Given the strategies chosen by other players } Player i s best response Teoria dos Jogos - Filomena Garcia 53

Definition: best response function An alternative definition: Player i's strategy s i Bi ( s1,..., si 1, si+ 1,... sn) if and only if it solves (or it is an optimal solution to) Maximize u i( s1,..., si 1, si, si+ 1,..., sn) Subject to si Si where s 1,..., si 1, si+ 1,..., sn are given. Player i s best response to other players strategies is an optimal solution to Teoria dos Jogos - Filomena Garcia 54

Using best response function to define Nash equilibrium In the normal-form game {S 1,..., S n, u 1,..., u n }, a combination of strategies ( s 1,..., s n) is a Nash equilibrium if for every player i, 1 s i Bi ( s1,..., si, si+ 1,..., s n ) A set of strategies, one for each player, such that each player s strategy is best for her, given that all other players are playing their strategies, or A stable situation that no player would like to deviate if others stick to it Teoria dos Jogos - Filomena Garcia 55

Summary Nash equilibrium Best response function Using best response function to define Nash equilibrium Using best response function to find Nash equilibrium Next time Applications Reading lists Sec 1.2.A and 1.2.B of Gibbons Teoria dos Jogos - Filomena Garcia 56

Static (or Simultaneous- Move) Games of Complete Information Concave Function and Maximization Cournot models of duopoly and oligopoly Teoria dos Jogos - Filomena Garcia 57

Agenda Solving maximization problem Cournot models of duopoly and oligopoly Bertrand model of duopoly (differentiated products) (sec 1.2.B of Gibbons) Bertrand model of duopoly (homogeneous products) (sec 3.2 of Osborne) Contributing to a public good (sec 2.8.4 of Osborne) Teoria dos Jogos - Filomena Garcia 59

Nash equilibrium survive iterated elimination of strictly dominated strategies Player 2 Left Middle Right Player 1 Up Down 1, 0 0, 3 1, 2 0, 1 0, 1 2, 0 Player 1 Up Down Player 2 Left Middle 1, 0 1, 2 0, 3 0, 1 Teoria dos Jogos - Filomena Garcia 61

The strategies that survive iterated elimination of strictly dominated strategies are not necessarily are Nash equilibrium strategies Player 2 L C R T 0, 4 4, 0 3, 3 Player 1 M 4, 0 0, 4 3, 3 B 3, 3 3, 3 3.5, 3.6 Teoria dos Jogos - Filomena Garcia 62

Summary In an n-player normal-form game, if iterated elimination of strictly strategies eliminates all but the strategies ( s 1, s 2,..., s n ), then ( s 1, s 2,..., s n ) is the unique Nash equilibrium. In an n-player normal-form game, if the strategies ( s 1, s 2,..., s n ) is a Nash equilibrium then they survive iterated elimination of strictly strategies. But the strategies that survive iterated elimination of strictly dominated strategies are not necessarily are Nash equilibrium strategies. Teoria dos Jogos - Filomena Garcia 63

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 knowing 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 producing quantity q i is C i (q i )=cq i. Teoria dos Jogos - Filomena Garcia 64

Cournot model of duopoly The normal-form representation: Set of players: { Firm 1, Firm 2} Sets of strategies: 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) Teoria dos Jogos - Filomena Garcia 65

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 2 + Teoria dos Jogos - Filomena Garcia 66

Concave function A function f (x) is concave if, for any x and y, f ( αx + (1 α) y) αf ( x) + (1 α) f ( y) for all 0 α 1. f(x) 0 x Teoria dos Jogos - Filomena Garcia 67

Convex function A function f (x) is convex if, for any x and y, f ( αx + (1 α ) y) αf ( x) + (1 α ) f ( y) for all 0 α 1. f(x) 0 x Teoria dos Jogos - Filomena Garcia 68

Concavity and convexity Some useful results: A function f (x) is concave if and only if f ( x) A function f (x) is convex if and only if (x) f ( x) f is concave if and only if f (x) is convex. 0. 0. Teoria dos Jogos - Filomena Garcia 69

Concavity and convexity 2 Example 1: f ( x) = 3x + 6x 4, f ( x) = 6x + 6, f ( x) = 6 < 0. Hence, f (x) is concave. x Example 2: f ( x) = e, f ( x) = f ( x) = e > 0. Hence, f (x) is convex. x Example 3: f ( x, y) = 2xy + 4y x + 3y 2 2 For any fixed y, f ( x, y) is concave in x because fx ( x, y) = 2y 2x and fx ( x, y) = 2 < 0 For any fixed x, f ( x, y) is convex in y because f ( x, y) = 2x + 4 + 6y and f ( x, y) = 6 > 0 y y Teoria dos Jogos - Filomena Garcia 70

Maximum and minimum A maximum of f (x) is a point x' such that f ( x ) f ( x) for any x. A minimum of f (x) is a point x such that f ( x) f ( x) for any x. f(x) 0 x x x Teoria dos Jogos - Filomena Garcia 71

Maximum and minimum First Order Condition: if a point x is a maximum or a minimum then x satisfies the first order condition (FOC): f ( x) = 0. If f (x) is concave and x satisfies the FOC, then x is a maximum. If f (x) is convex and x satisfies the FOC, then x is a minimum. f(x) f(x) 0 x x 0 x x Teoria dos Jogos - Filomena Garcia 72

Finding a maximum of a concave function Find a maximum of a concave function f (x) Compute f (x) Compute f (x ) and check whether f ( x) 0 If f ( x) 0 then solve f ( x) = 0. The solution is a maximum 2 Example 1: f ( x) = 3x + 6x 4, f ( x) = 6x + 6, f ( x) = 6 < 0. Hence, f (x) is concave. Solving f ( x) = 0 gives x=1, a maximum. Example 3: f ( x, y) = 2xy + 4y x + 3y For any fixed y, f ( x, y) is concave in x. For any fixed y, find a maximum of f ( x, y) 2 2 Teoria dos Jogos - Filomena Garcia 73

Maximum and minimum A maximum of f (x) in the domain [x 1, x 2 ] is a point x' in [x 1, x 2 ] such that f ( x ) f ( x) for all x [ x 1, x2]. A minimum of f (x) in the domain [x 1, x 2 ] is a point x in [x 1, x 2 ] such that f ( x) f ( x) for all x x 1, x ]. [ 2 f(x) x 1 0 x x x 2 x Teoria dos Jogos - Filomena Garcia 74

Finding a maximum of a concave function with constraints Find a maximum of a concave f (x) in the domain [ x 1, x2] Find a maximum x of f (x) without constraints If x is in [ x 1, x2] then x is also a maximum for the constrained problem Otherwise, a. if f ( x1 ) > f ( x2) or x < x1 then x 1 is a maximum; b. if f ( x1 ) < f ( x2) or x > x2 then x 2 is a maximum; c. if f ( x1 ) = f ( x2) then any point in [ x 1, x2] is a maximum. Teoria dos Jogos - Filomena Garcia 75

Finding a maximum of a concave function with constraints Example 4: 2 Max f ( x) = 3x + 6x 4 subject to 2 x 2 Solving the unconstrained problem gives us x=1. Since 1 is in the domain, so 1 is also the maximum for the constrained problem. Example 5: Max f ( x) = 3x x 2 2 + 6x 4 Teoria dos Jogos - Filomena Garcia 76

Finding a maximum of a concave function with constraints f(x)=-3x 2 +6x-4 x Teoria dos Jogos - Filomena Garcia 77

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 Teoria dos Jogos - Filomena Garcia 80

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 Teoria dos Jogos - Filomena Garcia 81

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 Solving these two equations gives us q 1 = q 2 = (a c)/3 Teoria dos Jogos - Filomena Garcia 82

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, othwise 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, othwise a c q 2 Nash equilibrium (a c)/2 (a c)/2 a c Teoria dos Jogos - Filomena Garcia 83 q 1

Cournot model of oligopoly 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 knowing the other firms choices. The market priced is P(Q)=a-Q, where a is a constant number and Q=q 1 +q 2 +...+q n. The cost to firm i of producing quantity q i is C i (q i )=cq i. Teoria dos Jogos - Filomena Garcia 84

Cournot model of oligopoly The normal-form representation: Set of players: { Firm 1,... Firm n} Sets of strategies: S i =[0, + ), for i=1, 2,..., n Payoff functions: u i (q 1,..., q n )=q i (a-(q 1 +q 2 +...+q n )-c) for i=1, 2,..., n Teoria dos Jogos - Filomena Garcia 85

Cournot model of oligopoly 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 2 +... Teoria dos Jogos - Filomena Garcia 86

Summary Nash equilibrium Concave function and maximization Cournot model of duopoly and oligopoly Next time Bertrand model of Duopoly Reading lists Sec 1.2.B of Gibbons and Sec 3.1-3.2 of Osborne Teoria dos Jogos - Filomena Garcia 87

Static (or Simultaneous- Move) Games of Complete Information The Problems of Commons Mixed Strategy Equilibrium Teoria dos Jogos - Filomena Garcia 88

Today s Agenda The problems of commons (sec 1.2.D of Gibbons) Mixed strategies Solving matching pennies Teoria dos Jogos - Filomena Garcia 90

The problems of commons n farmers in a village. Each summer, all the farmers graze their goats on the village green. Let g i denote the number of goats owned by farmer i. The cost of buying and caring for a goat is c, independent of how many goats a farmer owns. The value of a goat is v(g) per goat, where G = g 1 + g 2 +... + g n There is a maximum number of goats that can be grazed on the green. 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 spring, all the farmers simultaneously choose how many goats to own. Teoria dos Jogos - Filomena Garcia 91

The problems of commons The normal-form representation: Set of players: { Farmer 1,... Farmer n} Sets of strategies: S i =[0, G max ), for i=1, 2,..., n Payoff functions: u i (g 1,..., g n )=g i v(g 1 +...+ g n ) c g i for i = 1, 2,..., n. Teoria dos Jogos - Filomena Garcia 92

The problems of commons How to find a Nash equilibrium Find (g 1, g 2,..., g n ) such that g i is farmer i s best response to other farmers choices. That is, g 1 solves Max u 1 (g 1, g 2,..., g n )= g 1 v(g 1 + g 2...+ g n ) c g 1 subject to 0 g 1 < G max and g 2 solves Max u 2 (g 1, g 2, g 3,..., g n )= g 2 v(g 1 +g 2 +g 3 +...+ g n ) cg 2 subject to 0 g 2 < G max... Teoria dos Jogos - Filomena Garcia 93

The problems of commons How to find a Nash equilibrium and g n solves Max u n (g 1,..., g n-1, g n )= g n v(g 1 +...+ g n-1 + g n ) cg n subject to 0 g n < G max... Teoria dos Jogos - Filomena Garcia 94

Teoria dos Jogos - Filomena Garcia 95 The problems of commons FOCs: 0 )... ( )... (... 0 )... ( )... ( 0 )... ( )... ( 1 1 1 1 3 2 1 2 3 2 1 2 1 1 2 1 = + + + + + + + = + + + + + + + + + = + + + + + + + c g g g v g g g g v c g g g g v g g g g g v c g g g v g g g g v n n n n n n n n n

Teoria dos Jogos - Filomena Garcia 96 The problems of commons How to find a Nash equilibrium (g 1, g 2,..., g n ) is a Nash equilibrium if 0 )... ( )... (... 0 )... ( )... ( 0 )... ( )... ( 1 1 1 1 3 2 1 2 3 2 1 2 1 1 2 1 = + + + + + + + = + + + + + + + + + = + + + + + + + c g g g v g g g g v c g g g g v g g g g g v c g g g v g g g g v n n n n n n n n n

The problems of commons Summing over all n farmers FOCs and then dividing by n yields 1 v( G) + G v ( G) c = 0 n where G = g + g +... + g 1 2 n Teoria dos Jogos - Filomena Garcia 97

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 Teoria dos Jogos - Filomena Garcia 98

Teoria dos Jogos - Filomena Garcia 99 The problems of commons? 0 ) ( ) ( 0 ) ( 1 ) ( G G c G v G G v c G v G n G v > = + = +

More on weakly dominated strategy In the normal-form game {S 1, S 2,..., S n, u 1, u 2,..., u n }, let s i ', s i " S i be feasible strategies for player i. Strategy s i ' is weakly dominated by strategy 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 good as s i, but not always equal. regardless of other players choices Player 1 U B Player 2 L R 1, 1 2, 0 0, 2 2, 2 Teoria dos Jogos - Filomena Garcia 100

Matching 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 strategy Tail Tail is Player 2 s best response to Player 1 s strategy Tail Tail is Player 1 s best response to Player 2 s strategy Head Head is Player 2 s best response to Player 1 s strategy Head Hence, NO Nash equilibrium Teoria dos Jogos - Filomena Garcia 101

Solving matching pennies Player 1 Head Tail Head -1, 1 1, -1 Player 2 Tail 1, -1-1, 1 r 1-r q 1-q Randomize your strategies to surprise the rival Player 1 chooses Head and Tail with probabilities r and 1-r, respectively. Player 2 chooses Head and Tail with probabilities q and 1-q, respectively. Mixed Strategy: Specifies that an actual move be chosen randomly from the set of pure strategies with some specific probabilities. Teoria dos Jogos - Filomena Garcia 102

Mixed strategy A mixed strategy of a player is a probability distribution over player s (pure) strategies. A mixed strategy for Chris is a probability distribution (p, 1-p), where p is the probability of playing Opera, and 1-p is that probability of playing Prize Fight. If p=1 then Chris actually plays Opera. If p=0 then Chris actually plays Prize Fight. Battle of sexes Pat Opera Prize Fight Chris Opera (p) Prize Fight (1-p) 2, 1 0, 0 0, 0 1, 2 Teoria dos Jogos - Filomena Garcia 103

Solving matching pennies Player 1 Head Tail Player 2 Head Tail -1, 1 1, -1 1, -1-1, 1 q 1-q r 1-r Expected payoffs 1-2q 2q-1 Player 1 s expected payoffs If Player 1 chooses Head, -q+(1-q)=1-2q If Player 1 chooses Tail, q-(1-q)=2q-1 Teoria dos Jogos - Filomena Garcia 104

Solving matching pennies Player 1 Head Tail Player 2 Head Tail -1, 1 1, -1 1, -1-1, 1 q 1-q r 1-r Expected payoffs 1-2q 2q-1 Player 1 s best response B 1 (q): 1 r For q<0.5, Head (r=1) For q>0.5, Tail (r=0) 1/2 For q=0.5, indifferent (0 r 1) 1/2 1 q Teoria dos Jogos - Filomena Garcia 105

Solving matching pennies Player 1 Expected payoffs Head Tail Player 2 Head Tail -1, 1 1, -1 1, -1-1, 1 q 1-q 2r-1 1-2r r 1-r Expected payoffs 1-2q 2q-1 Player 2 s expected payoffs If Player 2 chooses Head, r-(1-r)=2r-1 If Player 2 chooses Tail, -r+(1-r)=1-2r Teoria dos Jogos - Filomena Garcia 106

Solving matching pennies Player 1 Expected payoffs Head Tail Head -1, 1 1, -1 Player 2 s best response B 2 (r): For r<0.5, Tail (q=0) For r>0.5, Head (q=1) Player 2 For r=0.5, indifferent (0 q 1) Tail 1, -1-1, 1 q 1-q 2r-1 1-2r 1 1/2 r Expected payoffs 1-2q 2q-1 Teoria dos Jogos - Filomena Garcia 107 r 1-r 1/2 1 q

Solving matching pennies Player 1 s best response B 1 (q): For q<0.5, Head (r=1) For q>0.5, Tail (r=0) For q=0.5, indifferent (0 r 1) Player 2 s best response B 2 (r): For r<0.5, Tail (q=0) For r>0.5, Head (q=1) For r=0.5, indifferent (0 q 1) Check r = 0.5 B 1 (0.5) q = 0.5 B 2 (0.5) Player 1 Head Tail 1/2 1 r Head -1, 1 1, -1 Player 2 1/2 Tail 1, -1-1, 1 q 1-q 1 q Teoria dos Jogos - Filomena Garcia 108 r 1-r Mixed strategy Nash equilibrium

Summary The problems of commons Mixed strategies Solving matching pennies Next time Mixed strategy Nash equilibrium Reading lists Chapter 1.3 of Gibbons Teoria dos Jogos - Filomena Garcia 109