Strategic games (Normal form games)

Transcription

1 Strategc Normal form ECON5200 Advanced mcroeconomc Lecture n game theory Fall 2010, Part G.B. Ahem, ECON Game theory tude mult-peron decon problem, and analyze agent that are ratonal have well-defned preference reaon trategcally take nto account ther knowledge and belef about what other do Clafcaton of non-cooperatve v. cooperatve trategc v. extenve Game wth perfect and mperfect nformaton G.B. Ahem, ECON

2 no communcaton before the game communcaton before the game Stude the outcome of ndvdual of ont acton acton when there when there no external external enforcement. enforcement. trct ene non-cooperatve game theory wde ene cooperatve game theory G.B. Ahem, ECON Statc Dyna- mc Incomplete Complete nformaton nformaton Perfect nformaton Imperfect nformaton Almot perf. nfo. Strategc Lecture 2 Lecture 3 Extenve mult-tage Bayean Extenve the general cae G.B. Ahem, ECON

3 Soluton concept A ytematc decrpton of outcome that may emerge n clae of Game theory ugget reaonable oluton for clae of and examne ther properte Interpretaton of oluton concept The evolutve teady tate nterpretaton The deductve educ- tve nterpretaton Bounded ratonalty wll not be treated n thee lecture G.B. Ahem, ECON Ratonal behavor Decon-maker chooe from a et S of tratege. Decon are made under uncertanty, where a fnte et of uncertan tate. Example The uncertanty may be trategc uncertanty relatng to the tratege of other. exogenou uncertanty relatng to the envronment. A tate-trategy trategy par, S lead to a conequence c C. A conequence functon g: S C agn a conequence to each tate-trategy par G.B. Ahem, ECON

4 Ratonal behavor cont Let the decon-maker be endowed wth a vnm utlty functon : C. Example a ubectve prob. dtr. over. A pure trategy preferred to f and only f ω g ω, ω g ω, ω ω Ancombe & Aumann decon-theoretc framework. It requre the avalablty of mxed tratege. A mxed trategy a ob. randomzaton over S. The exp. utlty of : ω g ω, S ω G.B. Ahem, ECON A trategc game Defnton 11.1 A trategc game cont of a fnte et N of player Statc Dynamc Incomplete Complete nformaton nformaton Perfect nformaton Imperfect nformaton Strategc Lecture 2 Extenve mult-tage Bayean Lecture 3 Extenve the general cae for each N, a non-empty et S of tratege for each N, a payoff functon u on S N S :, S, u A trategc game fnte f, for each N, S fnte. Interpretaton: the game played once?, g, Conequence or outcome multaneou acton? G.B. Ahem, ECON

5 Nah equlbrum for a trategc game Defnton 14.1 A Nah equlbrum of a trategc game a trategy profle S wth the property that, for each player N, Alternatve formulaton: S, u, u, Defne a et-valued functon B bet-repone fn: B { S S, u, u, } The trategy profle S a Nah equlbrum f and only f, for each player N, B G.B. Ahem, ECON Can Nah equlbrum be ued a a oluton concept f the game only played once? Ye, f each player can predct what each opponent wll do. For each player, only one trategy urvve teratve elmnaton of trctly domnated tratege. Through communcaton before the game tart, the player make a elf-enforcng agreement coordnate on an equlbrum. Gven a common background, the player are able to co-ordnate on an equlbrum wthout communcaton before the game tart Schellng, 1960, focal pont. A unque Nah equlbrum not uffcent G.B. Ahem, ECON

6 N {1,, n} Example: An aucton Player valuaton v, where v 1 v n 0. The player ubmt bd multaneouly. The obect gven to the player ubmttng the hghet bd f there are everal player wth the hghet bd, then the wnner the one wth the lowet ndex. Frt prce aucton: The wnner pay h bd. Second prce aucton: The wnner pay py the hghet bd among the non-wnner. Equlbra n a frt prce aucton? Equlbra n a econd prce aucton? G.B. Ahem, ECON Extence of Nah equlbrum Lemma 20.1 Kukutan fxed pont theorem Let X be a compact convex ubet of n, and let f : X X be a et-valued functon for whch for all x X the et fx non-empty and convex. the graph of f cloed Then there ext x X uch that x fx. Propoton 20.3 A trategc game ha a Nah equlbrum f for all N, the trategy et S non-empty and compact. the payoff functon u contnuou the payoff functon u qua-concave on S. Proof of Pro 203 poton G.B. Ahem, ECON

7 A Bayean game Defnton 25.1 A Bayean game cont of a fnte et N of player Statc Dynamc Incomplete Complete nformaton nformaton Perfect nformaton Imperfect nformaton Strategc Lecture 2 Extenve mult-tage Lecture 2 Bayean Lecture 3 Extenve the general cae for each N, a et A of acton for each N, a et T of type for each N, a prob. dtr. p on T N T that atfe, for all t T, p t, t 0 t T for each N, a vnm utlty functon u on the et A T, where A N A, and where u a, t player payoff f a, t realzed G.B. Ahem, ECON pot perpectv ve. An ex Nah equlbrum for a Bayean game A Nah equlbrum of a Bayean game a Nah equlbrum for the trategc game defned a follow: The et of player the et of all par, t for N and t T. For each player, t, the et of tratege A. For each player, t, the payoff functon defned by p t, t u, t a u a1, t,, a n t, t 1, n t T p t T t, t G.B. Ahem, ECON

8 Example B B B B t1, t2 ; p t1, t 1/ 4 B S B 2, 2 0, 0 S 0, 0 1, 1 2 B S B S t1, t2 ; p t1, t 1/ 4 B S B 2, 1 0, 0 S 0, 0 1, 2 2 S B S B S S S S t1, t2 ; p t1, t 1/ 4 t1, t2 ; p t1, t2 1/ 4 B S B S B 1, 2 0, 0 B 1, 1 0, 0 S 00 0, , S 00 0, 0 22 I a B B a S, t and t S, for both a Nah equlbrum for th Bayean game? G.B. Ahem, ECON Example: A econd prce aucton N {1,, n} A T V n p v,, v v for ome probablty 1 n 1 dtrbuton over V. The payoff u equal the expectaton of the, v a random varable whoe value gven v 1,, v n v max N \{ } a v f player wn. 0 otherwe. Equlbra? G.B. Ahem, ECON

9 The mxed extenon Defnton 32.1 A mxed extenon of a trategc game cont of a fnte et N of player Statc Dynamc Incomplete Complete nformaton nformaton Perfect nformaton Imperfect nformaton Strategc Lecture 2 Extenve mult-tage Bayean Lecture 3 Extenve the general cae for each N, a et S of mxed tratege, where S the et of prob. dtrbuton on S. for each N, a payoff functon U on N S agnng to each N S the expected value: S U u N G.B. Ahem, ECON Mxed trategy Nah equlbrum Defnton 32.3 A mxed trategy Nah equlbrum of a trategc game a Nah equlbrum of t mxed extenon. Propoton 33.1 Every fnte trategy game ha a mxed trategy Nah equlbrum. Lemma 33.2 Let G be a fnte trategc game. Then N S a mxed trategy Nah equlbrum f and only f, for each player, every pure trategy agned potve probablty by a bet repone to. Can be ued a prmtve defnton cf. Def Proof of Propoton 33.1 Proof of Lemma G.B. Ahem, ECON

10 Interpretaton of mxed tratege Mxed tratege a an obect of choce Mxed trategy Nah equlbrum a a teady tate. Mxed tratege a pure tratege of an extended game. Mxedtrategeapuretrategeof a a perturbed game. Mxed tratege a belef. Mxed trategy Nah equlbrum a a teady tate Mxed tratege a pure trat. n a perturbed game G.B. Ahem, ECON Ratonalzablty Defnton A et of trategy profle Z ha N Z the bet repone property f, for all N and Z, there ext a Z uch that. B Defnton 55.1 A trategy ratonalzable S f and only f there ext a et of trategy profle wth the bet rep. prop., uch that Z Z,. N Z G.B. Ahem, ECON

11 Further reult on ratonalzablty Lemma Every trategy ued wth potve probablty by ome player n a mxed trategy Nah equlbrum, ratonalzable. Propoton Let G be a fnte trategc game. Then there ext, for any N, a ratonalzable trategy for. Obervaton Even n a trategc game wth a unque Nah equlbrum, common knowledge of the player' ratonalty doe not mply that the player wll play th Nah equlbrum G.B. Ahem, ECON Never-bet repone and trct domnaton Defnton 59.1 A trategy S of player n the fnte trategc game G a never-bet repone f there no S uch that B. Defnton 59.2 A trategy S of player n the fnte trategc game G trctly domnated f there a mxed trategy S uch that, S u, u,, S Lemma 60.1 A trategy S of player n the fnte trategc game G a never-bet repone f and only f t trctly domnated G.B. Ahem, ECON

12 Iterated elmnaton of trctly domnated tratege Defnton 60.2 The et of X S of trategy profle n the fnte trategc game G urvve terated elmnaton of trctly domnated tratege f X t T N X and there a collecton of et X N t0 that atfe the followng condton for each N. 0 T X and S X X t1 t X X for each t 0,..., T 1. For each t 0,..., T 1, every trategy of player t t1 n X trctly domnated n the game G t \ X, where, t for each N, trategy et retrcted to X. T No trategy n trctly domnated n the game G T. X G.B. Ahem, ECON Iterated elmnaton of trctly domnated tratege cont. Lemma If Z N Z ha the bet repone property, then Z X. X N X ha the bet repone property. Propoton 61.2 If X N X urvve terated elmnaton of trctly domnated tratege n the fnte trategc game G, then, for each N, X the et of player ratonalzable tratege G.B. Ahem, ECON

13 13 Weak domnaton Defnton 62 1 Atrategy S of player n the 1 1, 0 1, 0 1, 1 0, T B L R 1 1, 0 1, 0 1, 1 0, T B L R Defnton 62.1 A trategy S of player n the fnte trategc game G weakly domnated f there a mxed trategy S uch that,,,, S u u S,,, S u u S G.B. Ahem, ECON Iterated elm. of weakly dom. tr. dubou becaue the reult depend on the the order of elmnaton, hard to gve the procedure an eptemc foundaton.