Classic example: Prisoners'Dilemma normal-form representation ■ Set of players: Prisoner 1, Prisoner 2) ■ Sets of strategies: S,=S2=Mum, Confess) Payoff functions 1(M,M=1,1(M,C)=9,u1(C,M=0,1(C,C)=6; l2(,M)=1,u2(M,C)=0,2(C,M)=9,2(C,C)=-6 Players Prisoner 2 Strategies →Mum Confess Mum 1 19 96 0 Prisoner 1 Confess 0 6 Payoffs
18 Classic example: Prisoners’ Dilemma: normal-form representation ◼ Set of players: {Prisoner 1, Prisoner 2} ◼ Sets of strategies: S1 = S2 = {Mum, Confess} ◼ Payoff functions: u1 (M, M)=-1, u1 (M, C)=-9, u1 (C, M)=0, u1 (C, C)=-6; u2 (M, M)=-1, u2 (M, C)=0, u2 (C, M)=-9, u2 (C, C)=-6 -1 , -1 -9 , 0 0 , -9 -6 , -6 Prisoner 1 Prisoner 2 Confess Mum Confess Mum Players Strategies Payoffs
Example: The battle of the sexes Pat Opera Prize Fight Opera 2 10 0 Chris Prize Fight[0,01,2 Normal (or strategic) form representation > Set of players: Chris, Pat(Player 1, Player 2)) Sets of strategies: S= S2=, Opera, Prize eight) Payoff functions l1(O,O)=2,u1(O,F)=0,u1(F,O)=0,1(F,O=1; 12(,O)=1,u2(O,F)=0,u2(F,O=0,u2(F,F)=2 19
19 Example: The battle of the sexes ◼ Normal (or strategic) form representation: ➢ Set of players: { Chris, Pat } (={Player 1, Player 2}) ➢ Sets of strategies: S1 = S2 = { Opera, Prize Fight} ➢ Payoff functions: u1 (O, O)=2, u1 (O, F)=0, u1 (F, O)=0, u1 (F, O)=1; u2 (O, O)=1, u2 (O, F)=0, u2 (F, O)=0, u2 (F, F)=2 2 , 1 0 , 0 0 , 0 1 , 2 Chris Pat Prize Fight Opera Prize Fight Opera
Example: Matching pennies Player 2 Head Tail Head 1 11 1 Player 1 1 1|-1 1 Normal (or strategic) form representation > Set of players: ( Player 1, Player 2) > Sets of strategies: S=S2=(Head, Tail e Payoff functions l1(H,H)=1,u1(H,T=1,1(T,H)=1,u1(H,T)=1; l2(H,H)=1,u2(H,T)=1,u2(,H)=-1,a2(1,T)=1
20 Example: Matching pennies ◼ Normal (or strategic) form representation: ➢ Set of players: {Player 1, Player 2} ➢ Sets of strategies: S1 = S2 = { Head, Tail } ➢ Payoff functions: u1 (H, H)=-1, u1 (H, T)=1, u1 (T, H)=1, u1 (H, T)=-1; u2 (H, H)=1, u2 (H, T)=-1, u2 (T, H)=-1, u2 (T, T)=1 -1 , 1 1 , -1 1 , -1 -1 , 1 Player 1 Player 2 Tail Head Tail Head
Example: Tourists natives Only two bars(bar 1, bar 2 ) in a city Can charge price of $2, $4, or $5 a 6000 tourists pick a bar randomly 4000 natives select the lowest price bar ■ Examp|e1 Both charge $2 each gets 5,000 customers and $10,000 ■ EXample2 Bar 1 charges $4, Bar 2 charges $5 Bar 1 gets 3000+4000=7, 000 customers and $28,000 Bar 2 gets 3000 customers and $15,000
21 Example: Tourists & Natives ◼ Only two bars (bar 1, bar 2) in a city ◼ Can charge price of $2, $4, or $5 ◼ 6000 tourists pick a bar randomly ◼ 4000 natives select the lowest price bar ◼ Example 1: Both charge $2 ➢ each gets 5,000 customers and $10,000 ◼ Example 2: Bar 1 charges $4, Bar 2 charges $5 ➢ Bar 1 gets 3000+4000=7,000 customers and $28,000 ➢ Bar 2 gets 3000 customers and $15,000
Example: Cournot model of duopoly A product is produced by only two firms: firm 1 and firm 2. The quantities are denoted by q, and g2 respectively. Each firm chooses the quantity without knowing the other firm has chosen a The market price is P(2=a-g, Where 2=q,+q2 a The cost to firm i of producing quantity i is Ci=cqi The normal-form representation Set of players: f Firm 1, Firm 2) Sets of strategies:Sr=0,+∞),S2=1[0,+∞) Payoff functions 1(b,q2)=q1(a-(qr+q2)-C),u2(b,q2)=q2(-(q1+q2)-C)
22 Example: Cournot model of duopoly ◼ A product is produced by only two firms: firm 1 and firm 2. The quantities are denoted by q1 and q2 , respectively. Each firm chooses the quantity without knowing the other firm has chosen. ◼ The market price is P(Q)=a-Q, where Q=q1+q2 . ◼ The cost to firm i of producing quantity qi is Ci (qi )=cqi . The normal-form representation: ➢ Set of players: { Firm 1, Firm 2} ➢ Sets of strategies: S1=[0, +∞), S2=[0, +∞) ➢ Payoff functions: u1 (q1 , q2 )=q1 (a-(q1+q2 )-c), u2 (q1 , q2 )=q2 (a-(q1+q2 )-c)