Static (or simultaneous-move)games of complete information ■Simultaneous-move -Each player chooses his/her strategy without knowledge of others'choices. Complete information (on game's structure) >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 14
14 Static (or simultaneous-move) games of complete information ◼ Simultaneous-move ➢ Each player chooses his/her strategy without knowledge of others’ choices. ◼ Complete information (on game’s structure) ➢ 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
Static (or simultaneous-move)games of complete information The players cooperate? -No.Only non-cooperative games Methodological individualism ■The timing >Each player i chooses his/her strategy s;without knowledge of others'choices. -Then each player i receives his/her payoff uS1,S2,,Sn）. >The game ends. 6
15 Static (or simultaneous-move) games of complete information ◼ The players cooperate? ➢ No. Only non-cooperative games ➢ Methodological individualism ◼ The timing ➢ Each player i chooses his/her strategy si without knowledge of others’ choices. ➢ Then each player i receives his/her payoff ui (s1 , s2 , ..., sn ). ➢ The game ends
Definition:normal-form or strategic- form 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 S2...S,and -their payoff functions uu...u where u;S XS2 X...XSR max ui SiESi S.t., Sj∈Sj,j∈{1,2，，n/i, 16
16 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 S1 S2 ... Sn and ➢ their payoff functions u1 u2 ... un where ui : S1 × S2 × ...× Sn→R
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: S1={S11,S12,S13JS2=(s21,S22] Player 2 521 522 S11 u1（s11,S21),u2（s11,S21) u1（S11,S22,u2（s11,S22) Player 1 512 u1（S12,S21),u2（S12,s21) u1（s12,S22),2（S12,S22） S13 u1(s13,S21),u2（S13/S21) u1（S131s22),2（S131S22) Bi-matrix:a matrix with two elements per cell
17 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: S1={s11, s12, s13} S2={s21, s22} Player 2 s21 s22 Player 1 s11 u1(s11,s21), u2(s11,s21) u1(s11,s22), u2(s11,s22) s12 u1(s12,s21), u2(s12,s21) u1(s12,s22), u2(s12,s22) s13 u1(s13,s21), u2(s13,s21) u1(s13,s22), u2(s13,s22) Bi-matrix: a matrix with two elements per cell
Classic example:Prisoners'Dilemma: normal-form representation ■Set of players: {Prisoner 1,Prisoner 2) Sets of strategies: S=S2=(Mum,Confess ■Payoff functions: uM,M0=-1,u1M,C)=-9,4(C,M0=0,u1(C,C)=-6; u2M,M0)=-1,u2M,C)=0,u2(C,M0=-9,u2(C,C)=-6 Players Prisoner 2 Strategies Mum Confess Mum -1 -1 -9 0 Prisoner 1 Confess 0 -9 -6,-6 Payoffs 18
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