Static games: Coordination and Nash equilibrium

Transcription

1 Statc game: Coordato ad Nah equlbrum Lecture Game Theory Fall 204, Lecture Dael Spro, ECON3200/4200, Lecture 2

2 Ratoalzablty about Narrowg dow the belef I have ad the other player may have by dcardg thoe belef (me ad other ) that are ot ratoal to have, by keepg thoe belef that could be ratoal to have. The oly requremet that my trategy terally cotet wth the belef I have of the other trategy ad belef. No guaratee that the other player actually have the trategy I beleve they have Dael Spro, ECON3200/4200, Lecture 2 2

3 Ratoalzablty approprate whe player are trategcally ophtcated a tuato whch doe ot recur ofte. wth o commucato or outde coordato..e. whe I do ot kow what the other player beleve ad trategze Dael Spro, ECON3200/4200, Lecture 2 3

4 Two cocept profle Coder a et of where ) Bet repoe property (weak cogruty): each, trategy for all. The et cota oly bet repoe to tratege ha the bet, repoe there 2) Bet repoe completee: The et cota all bet repoe to tratege, but may cota ome o-br too. each bet repoe t. S property f, for each, BR () BR. ad ( complete f, for each ad Looely: ) My trategy to oly do what bet gve what I thk other pla ad 2) my trategy cota all the poble bet repoe gve what I thk other pla. Dael Spro, ECON3200/4200, Lecture 2 ).

5 Up utl ow, t all bee about belef, but wthout a guaratee that the belef are coordated/rght. What f the game recur ofte (eve though the oppoet chage from tme to tme). or the player ca commucate. or there outde coordato Dael Spro, ECON3200/4200, Lecture 2 5

6 Nah equlbrum Are there tratege for the two player o that o player wll regret h ow choce whe beg told of the other player choce? If ye, the uch a trategy profle a Nah equlbrum. Some example: Defto each player : (, hold I.e., for each player { } BR( ),,, the ) that, ( a u,, Nah (, BR( ) a equlbrum ) u ). If, trct (, f t, for ) for all for each player Nah., equlbrum Dael Spro, ECON3200/4200, Lecture 2 6

7 Obervato Reult : If the ( (,,,, ) a Nah equlbrum, ) urvve terated trct elmato. Reult : If ) the oly trategy profle whch urvve terated trct elmato, the (,, (,, ) a Nah equlbrum Dael Spro, ECON3200/4200, Lecture 2 7

8 Some game have o Nah equlbrum Some example: Such game have a Nah equlbrum mxed tratege. Iterpretato a a teady tate. Defto where S a player, mxed Reult - hold : Coder a trategy profle trategy (Nah, for each player that u 950) Nah (, ). equlbrum mxed - trategy Nah equlbrum. : u Every fte The profle f ( t,, (,, ), for each game ha ) for all (,, ) a Dael Spro, ECON3200/4200, Lecture 2 8

9 Example: Voluteer dlemma Suppoe two dvdual are obervg (each through her houe wdow) a robbery at the eghborg bak. Each oe would lke the robbery to be reported but there a cot of callg the polce. What the probablty that at leat oe wll call the polce? Dael Spro, ECON3200/4200, Lecture 2 9

10 Steady-tate terpretato of Nah equlbr. The game a model deged to expla ome regularty a famly of mlar tuato. Each partcpat "kow" the equlbrum ad evaluate whether t worthwhle to ue aother trategy. The terpretato requre that the player meet dfferet oppoet each tme. I game wth multple equlbra, wll the player coordate, ad f ye, o whch equlbrum? Some example: Geerally game theory doe ot ay how coordato ha emerged. Thrd teo: Coordato o a effcet NE Dael Spro, ECON3200/4200, Lecture 2 0

11 Defto : A trategy profle (Pareto) effcet f there o other trategy profle (,,,, ) uch that The cocept of effcecy every player ad u j ( ) u j u (,, ( ) u,, ( ) for ) ( ) for ome player j. Eglh: The combed tratege of the player the game are effcet f there t aother combato of tratege whch make omeoe better off wthout makg the other wore off. Implcato: Wthout effcecy there room for coordato/egotato/cotract to mprove for all Dael Spro, ECON3200/4200, Lecture 2

12 Ca Nah equlbrum be ued a a oluto cocept f the game oly played oce? Ye, f each player ca predct what each oppoet wll do. For each player, oly oe trategy urvve teratve elmato of trctly domated tratege. Through commucato before the game tart, the player make a elf-eforcg agreemet (coordate o a equlbrum). Gve a commo backgroud, the player are able to co-ordate o a equlbrum wthout commucato before the game tart (Schellg, 960, focal pot). A uque Nah equlbrum ot uffcet Dael Spro, ECON3200/4200, Lecture 2 2

13 The dfferece betwee cogruou et ad Nah equlbra Nah: My belef of the other trategy rght ad the other belef o my tratege are rght. Cogruty ad ratoalzablty: Our belef are ratoal but they may be wrog Dael Spro, ECON3200/4200, Lecture 2 3

14 Behavoral game theory Stadard game theory provde dcple for our aaly of the relato betwee the outcome of trategc teracto ad our aumpto about behavor. But doe the theory accurately decrbe ad predct real behavor? To tet game theory, oe ca Gather data about behavor real trategc tuato. Perform laboratory expermet wth moetary payoff. Behavoral game theory eek to lear about real behavor through laboratory expermet. Problem: Lab. ettg may ot reemble real trategc ettg. May be dffcult to cotrol the ubject payoff Dael Spro, ECON3200/4200, Lecture 2 4