Static (or simultaneous-move) games of complete information A static (or simultaneous-move) game consists of A set of players(at least>Player 1, Player 2 two players) ayer For each player, a set of 12 s strategies/actions Payoffs received by (S,S2…Sn, for all each player for the n∈SpS2∈S2y…Sn∈S combinations of the strategies, or for each player, preferences over the combinations of the strategies
13 Static (or simultaneous-move) games of complete information ◼ 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} ➢ S1 S2 ... Sn ➢ ui (s1 , s2 , ...sn ), for all s1S1 , s2S2 , ... snSn . A static (or simultaneous-move) game consists of:
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 players strategies and payoff function are common knowledge among all the players a Assumptions on the players Rationality Players aim to maximize their payoffs Players are perfect calculators Each player knows that other players are rational
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 Si without knowledge of others'choices Then each player i receives his/her payoff 152· The game ends
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 a The normal-form( or strategic-form representation of a game Spe ecoles s a finite set of players (1, 2,.,n), >players' strategy spaces Ss,.Sn and their payoff functions uy u where u1:S1XS2×.XSn→R maxi S;∈S St,S;∈S,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 ■ Examp|e: S1={s11 12 13}S2={S21,S22} Player 2 s21 22 u,(s 11/S21) s11r521 )|a(s11s22),u2(s1122) Player 1 512 u(s 1221 12=21 1(S 1222r 1222 s13(s13,S2),u2(s1321)a2(s13,52),2(53,S2) 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