Static (or Simultaneous- Move) Games of Complete Information

Transcription

1 Statc (or Smultaneous- Move) Games of Complete Informaton Nash Equlbrum Best Response Functon F. Valognes - Game Theory - Chp 3

2 Outlne of Statc Games of Complete Informaton Introducton to games Normal-form (or strategc-form) representaton Iterated elmnaton of strctly domnated strateges Nash equlbrum Revew of concave functons, optmzaton Applcatons of Nash equlbrum Mxed strategy Nash equlbrum F. Valognes - Game Theory - Chp 3 2

3 Today s Agenda Revew of prevous classes Nash equlbrum Best response functon Use best response functon to fnd Nash equlbra Examples F. Valognes - Game Theory - Chp 3 3

4 Revew The normal-form (or strategc-form) representaton of a game G specfes: A fnte set of players {, 2,..., n}, players strategy spaces S S 2... S n and ther payoff functons u u 2... u n where u : S S 2... S n R All combnatons of the strateges. A combnaton of the strateges s a set of strateges, one for each player Mum Prsoner 2 Confess Prsoner Mum -, - -9, 0 Confess 0, -9-6, -6 F. Valognes - Game Theory - Chp 3 4

5 Revew Statc (or smultaneous-move) game of complete nformaton Each player s strateges and payoff functon are common knowledge among all the players. Each player chooses hs/her strategy s wthout knowledge of others choces. Then each player receves hs/her payoff u (s, s 2,..., s n ). The game ends. F. Valognes - Game Theory - Chp 3 5

6 Defnton: strctly domnated strategy In the normal -form game { S, S 2,..., S n, u, u 2,..., u n }, let s ', s " S be feasble strateges for player. Strategy s ' s strctly domnated by strategy s " f u (s, s 2,... s -, s ', s +,..., s n ) < u (s, s 2,... s -, s ", s +,..., s n ) for all s S, s 2 S 2,..., s - S -, s + S +,..., s n S n. s s strctly better than s regardless of other players choces Prsoner Mum Prsoner 2 Confess Mum -, - -9, 0 Confess 0, -9-6, -6 F. Valognes - Game Theory - Chp 3 6

7 Revew: terated elmnaton of strctly domnated strateges If a strategy s strctly domnated, elmnate t The sze and complexty of the game s reduced Elmnate any strctly domnated strateges from the reduced game Contnue dong so successvely F. Valognes - Game Theory - Chp 3 7

8 Iterated elmnaton of strctly domnated strateges: an example Player Up Down Player 2 Left Mddle Rght, 0, 2 0, 0, 3 0, 2, 0 Player Up Down Player 2 Left Mddle, 0, 2 0, 3 0, F. Valognes - Game Theory - Chp 3 8

9 New soluton concept: Nash equlbrum Player 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 The combnaton of strateges (B, R ) has the followng property: Player CANNOT do better by choosng a strategy dfferent from B, gven that player 2 chooses R. Player 2 CANNOT do better by choosng a strategy dfferent from R, gven that player chooses B. (B, R ) s called a Nash equlbrum F. Valognes - Game Theory - Chp 3 9

10 Nash Equlbrum: dea Nash equlbrum A set of strateges, one for each player, such that each player s strategy s best for her, gven that all other players are playng ther correspondng strateges, or A stable stuaton that no player would lke to devate f others stck to t (Confess, Confess) s a Nash equlbrum. Prsoner Mum Prsoner 2 Confess Mum -, - -9, 0 Confess 0, -9-6, -6 F. Valognes - Game Theory - Chp 3 0

11 2-player game wth fnte strateges S ={s, s 2, s 3 } S 2 ={s 2, s 22 } (s, s 2 )s a Nash equlbrum f u (s,s 2 ) u (s 2,s 2 ), u (s,s 2 ) u (s 3,s 2 ) and u 2 (s,s 2 ) u 2 (s,s 22 ). Player 2 s 2 s 22 s u (s,s 2 ), u 2 (s,s 2 ) u (s,s 22 ), u 2 (s,s 22 ) Player s 2 u (s 2,s 2 ), u 2 (s 2,s 2 ) u (s 2,s 22 ), u 2 (s 2,s 22 ) s 3 u (s 3,s 2 ), u 2 (s 3,s 2 ) u (s 3,s 22 ), u 2 (s 3,s 22 ) F. Valognes - Game Theory - Chp 3

12 Defnton: Nash Equlbrum In the normal-form game {S, S 2,..., S n, u, u 2,..., * * u n }, a combnaton of strateges ( s,..., s n) s a Nash equlbrum f, for every player, * * * u ( s,..., s, s, s,..., s ) u ( s * *,..., s *, s * +, s * + n,..., s for all s S. That s, s * solves * * * * Maxmze u ( s,..., s, s, s+,..., sn) Subject to s S Prsoner * n ) Mum Gven others choces, player cannot be betteroff f she devates from s * Prsoner 2 Confess Mum -, - -9, 0 Confess 0, -9-6, -6 F. Valognes - Game Theory - Chp 3 2

13 Cell-by-cell nspecton Player Up Down Player 2 Left Mddle Rght, 0, 2 0, 0, 3 0, 2, 0 Player Up Down Player 2 Left Mddle, 0, 2 0, 3 0, F. Valognes - Game Theory - Chp 3 3

14 Example: Toursts & Natves Bar Bar 2 $2 $4 $5 $2 0, 0 4, 2 4, 5 $4 2, 4 20, 20 28, 5 $5 5, 4 5, 28 25, 25 Payoffs are n thousands of dollars Bar Bar 2 $4 $5 $4 20, 20 28, 5 $5 5, 28 25, 25 F. Valognes - Game Theory - Chp 3 4

15 Best response functon: example Player 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 If Player 2 chooses L then Player s best strategy s M If Player 2 chooses C then Player s best strategy s T If Player 2 chooses R then Player s best strategy s B If Player chooses T then Player 2 s best strategy s L If Player chooses M then Player 2 s best strategy s C If Player chooses B then Player 2 s best strategy s R Best response: the best strategy one player can play, gven the strateges chosen by all other players F. Valognes - Game Theory - Chp 3 5

16 Example: Toursts & Natves Bar 2 $2 $4 $5 $2 0, 0 4, 2 4, 5 Bar $4 2, 4 20, 20 28, 5 $5 5, 4 5, 28 25, 25 Payoffs are n thousands of dollars what s Bar s best response to Bar 2 s strategy of $2, $4 or $5? what s Bar 2 s best response to Bar s strategy of $2, $4 or $5? F. Valognes - Game Theory - Chp 3 6

17 2-player game wth fnte strateges S ={s, s 2, s 3 } S 2 ={s 2, s 22 } Player s strategy s s her best response to Player 2 s strategy s 2 f u (s,s 2 ) u (s 2,s 2 ) and u (s,s 2 ) u (s 3,s 2 ). Player 2 s 2 s 22 s u (s,s 2 ), u 2 (s,s 2 ) u (s,s 22 ), u 2 (s,s 22 ) Player s 2 u (s 2,s 2 ), u 2 (s 2,s 2 ) u (s 2,s 22 ), u 2 (s 2,s 22 ) s 3 u (s 3,s 2 ), u 2 (s 3,s 2 ) u (s 3,s 22 ), u 2 (s 3,s 22 ) F. Valognes - Game Theory - Chp 3 7

18 Usng best response functon to fnd Nash equlbrum In a 2-player game, ( s, s 2 ) s a Nash equlbrum f and only f player s strategy s s her best response to player 2 s strategy s 2, and player 2 s strategy s 2 s her best response to player s strategy s. Mum Prsoner 2 Confess Prsoner Mum Confess -, - -9, 0 0, -9-6, -6 F. Valognes - Game Theory - Chp 3 8

19 Usng best response functon to fnd Nash equlbrum: example Player 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 s Player s best response to Player 2 s strategy L T s Player s best response to Player 2 s strategy C B s Player s best response to Player 2 s strategy R L s Player 2 s best response to Player s strategy T C s Player 2 s best response to Player s strategy M R s Player 2 s best response to Player s strategy B F. Valognes - Game Theory - Chp 3 9

20 Example: Toursts & Natves Bar Bar 2 $2 $4 $5 $2 0, 0 4, 2 4, 5 $4 2, 4 20, 20 28, 5 $5 5, 4 5, 28 25, 25 Payoffs are n thousands of dollars Use best response functon to fnd the Nash equlbrum. F. Valognes - Game Theory - Chp 3 20

21 Example: The battle of the sexes Chrs Opera Prze Fght Pat Opera Prze Fght 2, 0, 0 0, 0, 2 Opera s Player s best response to Player 2 s strategy Opera Opera s Player 2 s best response to Player s strategy Opera Hence, (Opera, Opera) s a Nash equlbrum Fght s Player s best response to Player 2 s strategy Fght Fght s Player 2 s best response to Player s strategy Fght Hence, (Fght, Fght) s a Nash equlbrum F. Valognes - Game Theory - Chp 3 2

22 Example: Matchng pennes Player Head Tal Player 2 Head Tal -,, -, - -, Head s Player s best response to Player 2 s strategy Tal Tal s Player 2 s best response to Player s strategy Tal Tal s Player s best response to Player 2 s strategy Head Head s Player 2 s best response to Player s strategy Head Hence, NO Nash equlbrum F. Valognes - Game Theory - Chp 3 22

23 Defnton: best response functon In the normal-form game {S, S 2,..., S n, u, u 2,..., u n }, f player, 2,..., -, +,..., n choose strateges s,..., s, s+,..., sn, respectvely, then player 's best response functon s defned by B ( s,..., s, s,..., s ) = { s S Player s best response : u ( s u + ( s,..., s,..., s n, s, s +, s, s,..., s + n ),..., s n ), for all s S Gven the strateges chosen by other players } F. Valognes - Game Theory - Chp 3 23

24 Defnton: best response functon An alternatve defnton: Player 's strategy s B ( s,..., s, s+,... sn) f and only f t solves (or t s an optmal soluton to) Maxmze u ( s,..., s, s, s+,..., sn) Subject to s S where s,..., s, s+,..., sn are gven. Player s best response to other players strateges s an optmal soluton to F. Valognes - Game Theory - Chp 3 24

25 Usng best response functon to defne Nash equlbrum In the normal-form game {S,..., S n, u,..., u n }, * * a combnaton of strateges ( s,..., s n) s a Nash equlbrum f for every player, s * B ( s *,..., s *, s * * +,..., s n ) A set of strateges, one for each player, such that each player s strategy s best for her, gven that all other players are playng ther strateges, or A stable stuaton that no player would lke to devate f others stck to t F. Valognes - Game Theory - Chp 3 25

26 Summary Nash equlbrum Best response functon Usng best response functon to defne Nash equlbrum Usng best response functon to fnd Nash equlbrum Next tme Concave functon and maxmzaton Applcatons Readng lsts Sec.2.A and.2.b of Gbbons F. Valognes - Game Theory - Chp 3 26