Introduction to game theory

Transcription

1 Introducton to game theory Lectures n game theory ECON5210, Sprng 2009, Part G.B. Ashem, ECON Overvew over lectures 1. Introducton to game theory 2. Modelng nteractve knowledge; equlbrum and ratonalzablty. 3. Extensve wth perfect nformaton theory 4. Extensve wth perfect nformaton applcatons Guest lecture: Andrés Perea Research semnar: Andrés Perea 5. Extensve wth mperfect nformaton 6. Behavoral game theory Later: Contract theory Tapas Kundu & Tore Nlssen G.B. Ashem, ECON Game theory studes mult-person decson problems, and analyzes agents that are ratonal have well-defned preferences reason strategcally take nto account ther knowledge and belefs about what others do Classfcaton of non-cooperatve vs. cooperatve strategc vs. extensve Games wth perfect and mperfect nformaton G.B. Ashem, ECON

2 no communcaton before the game communcaton before the game Studes the outcome of ndvdual actons when there s no external enforcement. strct sense non-cooperatve game theory wde sense ofont actons when there s external enforcement. cooperatve game theory G.B. Ashem, ECON Statc Complete nformaton nformaton Perfect nformaton Imperfect nformaton Almost perf. nfo. Strategc Bayesan Dynamc I Extensve mult-stage II Extensve the general case G.B. Ashem, ECON Soluton concepts A systematc descrpton of outcomes that may emerge n classes of Game theory suggests reasonable solutons for classes of and examnes ther propertes Interpretaton of soluton concepts The evolutve steady state nterpretaton The deductve eductve nterpretaton Bounded ratonalty wll not be treated n lectures G.B. Ashem, ECON

3 Mutual knowledge of an event all players know the event Common knowledge of an event all players know the event all players know that all players know the event all players know that all players know that all players know the event and so forth G.B. Ashem, ECON Ratonal behavor Decson-maker chooses from a set S of strateges. Decsons are made under uncertanty, where Ω s a fnte set of uncertan states. Example The uncertanty may be strategc uncertanty relatng to the strateges of others. exogenous uncertanty relatng to the envronment. A state-strategy par ω, s Ω S leads to a consequence c C. A consequence functon g: Ω S C assgns a consequence to each state-strategy par G.B. Ashem, ECON Ratonal behavor cont Let the decson-maker be endowed wth a vnm utlty functon υ : C R. a subectve prob. dstr. μ over Ω. Example A pure strategy s s preferred to s f and only f μ ω g ω, s > ω g ω, s υ μ υ ω Ω ω Ω Anscombe & Aumann s decson-theoretc framework. It requres the avalablty of mxed strateges. A mxed strategy σ s a ob. randomzaton over S. The exp. utlty of σ : σ s μ ω υ g ω, s s S ω Ω G.B. Ashem, ECON

4 A strategc game Defnton 11.1 A strategc game conssts of a fnte set N of players Statc Dynamc Complete nformaton Perfect nformaton Strategc I Extensve mult-stage nformaton Imperfect nformaton Bayesan II Extensve the general case for each N, a non-empty set S of strateges for each N, a payoff functon u on S = N S : s, s S, u s,, s = υ g s s Consequence A strategc game s fnte f, or outcome for each N, S s fnte. Interpretatons: the game s played once? smultaneous actons? G.B. Ashem, ECON Nash equlbrum for a strategc game Defnton 14.1 A Nash equlbrum of a strategc game s a strategy profle s S wth the property that, for each player N, Alternatve formulaton: s S, u s, s u s, s Defne a set-valued functon B best-response fn: B s { s S s S, u s, s u s, s } = The strategy profle s S s a Nash equlbrum f and only f, for each player N, s B s G.B. Ashem, ECON Can Nash equlbrum be used as a soluton concept f the game s only played once? Yes, f each player can predct what each opponent wll do. For each player, only one strategy survves teratve elmnaton of strctly domnated strateges. Through communcaton before the game starts, the players make a self-enforcng agreement coordnate on an equlbrum. Gven a common background, the players are able to co-ordnate on an equlbrum wthout communcaton before the game starts Schellng, 1960, focal pont. A unque Nash equlbrum s not suffcent G.B. Ashem, ECON

5 N = {1,, n} Example: An aucton Player s valuaton v, where v 1 > > v n > 0. The players submts bds smultaneously. The obect s gven to the players submttng the hghest bd f there are several players wth the hghest bd, then the wnner s the one wth the lowest ndex. Frst prce aucton: The wnner pays hs bd. Second prce aucton: The wnner pays the hghest bd among the non-wnners. Equlbra n a frst Equlbra n a second prce aucton? prce aucton? G.B. Ashem, ECON Exstence of Nash equlbrum Lemma 20.1 Kukutan s fxed pont theorem Let X be a compact convex subset of R n, and let f : X X be a set-valued functon for whch for all x X the set fx s non-empty and convex. the graph of f s closed Then there exsts x X such that x fx. Proposton 20.3 A strategc game has a Nash equlbrum f for all N, the strategy set S s non-empty and compact. the payoff functon u s contnuous the payoff functon u s quas-concave on S. Proof of Proposton G.B. Ashem, ECON A Bayesan game Defnton 25.1 A Bayesan game conssts of Statc Dynamc a fnte set N of players for each N, a set A of actons Complete nformaton Perfect nformaton Strategc I Extensve mult-stage for each N, a vnm utlty functon u on the set A T, where A = N A, and where u a, t s player s payoff f a, t s realzed. nformaton Imperfect nformaton Bayesan II Extensve the general case for each N, a set T of types for each N, a prob. dstr. p on T = N T that satsfes, for all t T, p t, t > 0 t T G.B. Ashem, ECON

6 An ex post perspectve. Nash equlbrum for a Bayesan game A Nash equlbrum of a Bayesan game s a Nash equlbrum for the strategc game defned as follows: The set of players s the set of all pars, t for N and t T. For each player, t, the set of strateges s A. For each player, t, the payoff functon s 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. Ashem, ECON Example B B B B ; p t1, t = 1/ 4 B 2, 2 0, 0 S 0, 0 1, 1 2 ; p t1, t = 1/ 4 B 2, 1 0, 0 S 0, 0 1, 2 2 S B ; p t1, t = 1/ 4 B 1, 2 0, 0 S 0, 0 2, 1 2 S S S S ; p t1, t = 1/ 4 B 1, 1 0, 0 S 0, 0 2, 2 Is a B = B a S, t and t S =, for both a Nash equlbrum for ths Bayesan game? G.B. Ashem, ECON Example: A second prce aucton N = {1,, n} A =R + T = V n p v1,, vn = Π = 1π v for some probablty dstrbuton π over V. The payoff u equals the expectaton of the, v a random varable whose value gven v 1,, v n s v max N \{ } a v f player wns. 0 otherwse. Equlbra? G.B. Ashem, ECON

7 The mxed extenson Defnton 32.1 A mxed extenson of a strategc game conssts of a fnte set N of players Statc Dynamc Complete nformaton nformaton Perfect nformaton Imperfect nformaton Strategc Bayesan I Extensve II mult-stage Extensve the general case for each N, a set ΔS of mxed strateges, where ΔS s the set of prob. dstrbutons on S. for each N, a payoff functon U on N ΔS assgnng to each σ N ΔS the expected value: = s S Π s U σ σ u s N G.B. Ashem, ECON Mxed strategy Nash equlbrum Defnton 32.3 A mxed strategy Nash equlbrum of a strategc game s a Nash equlbrum of ts mxed extenson. Proposton 33.1 Every fnte strategy game has a mxed strategy Nash equlbrum. Proof of Proposton 33.1 Proof of Lemma 33.2 Lemma 33.2 Let G be a fnte strategc game. Then σ N ΔS s a mxed strategy Nash equlbrum f and only f, for each player, every pure strategy assgned postve probablty by σ s a best response to σ. Can be used as prmtve defnton cf. Def G.B. Ashem, ECON Interpretatons of mxed strateges Mxed strateges as an obect of choce Mxed strategy Nash equlbrum as a steady state. Mxed strateges as pure strateges of an extended game. Mxed strateges as pure strateges of a perturbed game. Mxed strateges as belefs. Mxed strategy Nash equlbrum as a steady state Mxed strateges as pure strat. n a perturbed game G.B. Ashem, ECON