Extensive games with imperfect information

Transcription

1 Extensive games with impefect infomation ECON500 Advanced micoeconomics ectues in game theoy Fall 0, Pat G.. Asheim, ECON500-3 Incomplete Complete infomation infomation Pefect infomation Impefect infomation Static Dynamic ectue Stategic games ectue Extensive games (multi-stage games) ectue ayesian games ectue 3 Extensive games (the geneal case) c ulti-stage game ("Almost pefect" infomation) Not multi-stage game (Not "almost pefect" infomation) G.. Asheim, ECON500-3 Thee easons why a playe when making a move has only patial infomation about pevious actions Not pefect ecall Pefect ecall implies that the playe a playe does emembes not emembe what what he once he once knew. did the playe does not emembe what he once knew Not multi-stage Not OK OK only patial a playe infomation emembes about what an he opponent s once did. pevious action Not complete infomation asymmetic infomation about what natue has done Not OK OK G.. Asheim, ECON

2 Definition 00. An extensive game consists of A finite set N (the set of playes) A set H of sequences (the set of histoies) A set Z, whee Z H (the set of teminal histoies) A function P fom H\Z to N{c} (the playe function) A pobability distibution f c that fo each h with P(h) c assigns pobability to each action following h. Fo each i N, a patition I i of {h H P(h) i} s.t. A(h) A(h) if h and h ae in the same pat of the patition (I i : infomation patition; I i I i : infomation set). Fo each i N, a vn utility function on the set of pob. dist. ove Z, whee i (z) is playe i s utility if z Z is ealized (payoff function) G.. Asheim, ECON Definition 03. A pue stategy of playe i N in an extensive game is a function that assigns an action in A(I i ) to each infomation set I i I i. Stategy et X i (h) be the sequence of infomation sets that i uns into unde the histoy h,, and the actions that he takes at these infomation sets. Definition 03.3 An extensive game has pefect ecall if, fo each playe i N, we have that X i (h) X i (h) if the histoies h and h ae in the same infomation set of playe i G.. Asheim, ECON Example A, 0, 0,, 0, 0 I {{}} X () {} I {{}, {(, A), (, )}} X () {} X ((, A)) ({},, {(, A), (, )}) X ((, )) ({},, {(, A), (, )}) G.. Asheim, ECON

3 Definition. A mixed stategy of playe i in an extensive game is a pobability measue ove the set of i s pue stategies. A behavioal stategy of playe i is a collection (i ( Ii )) I i I of independent i pobability measues, whee i (I i ) is a pobability measue ove A i (I i ). Ab behavioal stategy t is a book whee the numbe of pages equals the numbe of infomation sets fo the playe and each page specifies a mixed action. A mixed stategy is a pobability distibution ove such books whee, fo each book, the pages specify pue actions G.. Asheim, ECON et ( i ) in be a pofile of mixed stategies o a pofile of behavioal stategies. The outcome O() of is the pobability distibution ove teminal histoies that ae ealized if each playe plays accoding to i. Two stategies fo a playe ae outcome-equivalent if, fo evey pue stategy pofile fo the opponents, they geneate the same outcome. Poposition 4. Fo each mixed stategy of a playe in a finite extensive game with pefect ecall, thee is an outcome-equivalent behavioal stategy. Definition A Nash equil. in mixed stat. of an ext. game is a stategy pofile of mixed stategies with the popety that, fo each playe i N and fo evey mixed stategy i of playe i, O(, )) ( O(, )) i ( i i i i i G.. Asheim, ECON Examples: Sequential equilibium, N, 3, 0, 0 0,, 3, 0, 0 0,,,, Fo 3, each 0, infomation 0 set a playe must choose a best esponse given his beliefs. 3, 0, 0 0,, 0,, G.. Asheim, ECON

4 oe examples: Sequential equilibium (cont), N, 3,, 0 0, 0 0, 3,, 0 0, 0 0, eliefs must be consistent with the stategy,, pofile. 3,, 0 What 3,, 0 equiements 0, 0 0, should be imposed on beliefs at infomation sets that ae eached with 0, pobability 0 0, 0? G.. Asheim, ECON Stuctual consistency? N Common beliefs? G.. Asheim, ECON500-3 Definition. An assessment in an ext. game is a pai (, ), whee is a pofile of behavioal stategies and is a funct. (a belief system) that assigns to each info. set a pobability measue on the set of histoies in the info. set. Definition 4. Conside an ext. game w/pefect ecall. An assessment is sequentially ational if we have that, fo each playe i and fo each info. set I i I i, playe i's stategy is a best esp. to the opp.s s stategies given i's s beliefs at I i. Definition 4. Cons. a finite ext. game w/pef. ecall. An ass. (, ) is consistent if thee is a sequence (( n, n )) n of ass.s that conveges to (, ), whee each n is compl. mixed and whee n is deived fom n using ayes' ule. Definition 5. An ass. is a sequential equilibium of a finite ext. game w/pefect ecall if it is sequentially ational and consistent G.. Asheim, ECON

5 efinements: Pefect equil. in stategic games i : mixed stategy of playe i, : pofile of mixed stategies A C Definition 48. Conside a finite stategic game. A 0, 0 0, 0 0, 0 (,, n ) is a pefect equilibium if thee is a k sequence ( ) of 0, completely 0,, mixed 0 stategy pofiles k that conveges to such that fo each i N, C 0, 0 0,, i is a best k esponse s to ii fo all k.. Poposition 48. A stategy pofile in a finite -playe stategic If game the playes is a pefect ae cautious, equilibium would if and not only if it is a mixed the stategy Nash Nash equilibia equilibium (A, A) and and (C, the C) stategy of neithe playe be is weakly stategically dominated. unstable? Poposition 49. Evey finite stategic game has a pefect equilibium G.. Asheim, ECON efinements: Pefect equil. in stategic games (cont) T,,,0, T,,0 0,0,0 T,, 0,0, 0,,0,0,0 Stuctual consistency?,,,,0 T 0,, 0,0,,0, 0,,0,0,0 Common beliefs? 0,0, G.. Asheim, ECON efinements: Pefect equil. in extensive games T T t 3, 0 3, 0 3, 0 T' ' T l 3, 0 3, 0 0, t l t 0,, 0 0, 03 0, l l 0, 0, 3 The agent stategic fom T' ' T' ' T 3,0,3 3,0,3 T 3,0,3 3,0,3 T' ' 4 3 0,,0,0, 0,,0 0,3,0 t l G.. Asheim, ECON

6 efinements: Pefect equil. in extensive games (cont) A, 0, a b 0, 33 3, 3 Aa Ab a,, 0,,,, 0 b 0, 3, 3 The agent stategic fom A,,,, A,,,, 0,,0,0, 0,,0 3,3,3 a b G.. Asheim, ECON efinements: Pefect equil. in extensive games (cont) Definition 5. A pefect equilibium in a finite extensive game w/pefect ecall is the pofile of behavioal stategies that coesponds to a pefect equilibium of the agent stategic fom of the game. Poposition 5. Fo evey pefect equilibium of a finite extensive game w/pefect ecall thee is a belief system such that (, ) is a sequential equilibium. Coollay 53. Evey finite extensive game w/pefect ecall has a pefect equilibium and thus also a sequential equilibium G.. Asheim, ECON Thee ae two pefect equilibia in pue stategies (i.e. the agent stategic fom has two pefect equilibia in pue stategies). The stategic fom has only one pefect equilibium., 0, 0, 0, 0, 0, 0, 0, Thee is one pefect equilibium (i.e. the agent stategic fom has one pefect equil.). The stategic fom has two pefect equilibia in pue stategies G.. Asheim, ECON