Syllabus. Prelmnares. Role of game theory n economcs. Normal and extensve form of a game. Game-tree. Informaton partton. Perfect recall. Perfect and mperfect nformaton. Strategy.. Statc games of complete nformaton. Domnant strateges. Nash eulbrum. 3. Games wth contnuous-space strateges 4. Dynamc games of complete nformaton. Backward nducton. Subgame perfect eulbrum. Forward nducton. 5. Repeated games. 6. Statc and dynamc games of ncomplete nformaton. 7. Behavoral game theory 8. Cooperatve games.
Robert Gbbons, A prmer n game theory references Chapters on game theory n Mas-Colell, Whnston and Green Other references Game theory classes by Yldz at Sloan BS (MIT), by Brandenburger at Stern BS (NYU) Other books on game theory: Myerson, Gardner, Fudengerg&Trole Applcatons to busness strategy: Branderburger and Nalebuff, Coopetton
prelmnares Game theory s about decson wth nteracton: more than one agent decdes. Each agent s payoff depends on the behavor of other agents Compare: a small consumer and a monopoly (decson theory) wth a duopoly (nteracton) Interacton leads to some dffcultes for agents:. Even f they agree on cooperaton, coordnaton ssues. Conflct 3
Gbbons Model buldng : an nformal descrpton s translated nto a formal problem Model buldng : nstead of addng all the relevant ssues of a problem, the formal representaton emphaszes a few aspects of the problem to analyze them Then evdence (through emprcal nvestgaton) must be found about ts relevance Smlar ssues arse n dfferent areas of economcs, and the same game-theoretc tools can be appled n dfferent settngs 4
Myerson, Journal of economc lterature, 999 Economc theory was transformed by Nash s deas What s economcs?. Before Nash, a specalzed socal scence concerned wth the producton and allocaton of materal goods.. Now, economcs s about the analyss of ncentves n all socal nsttutons. (Obvously, Nash s contrbutons s just one very mportant- among others) To analyze any socal nteracton, we need:.a descrpton of the knd of nteracton we want to analyze.a predcton of ndvduals lkely behavor 5
Normal-form game wth n players G = { S,, S n, u, u n } Strategy sets S,, S n (a concrete strategy of player s denoted s ). Payoffs functons u ( s s ),..., u ( s,..., s ),..., n n n : u : S... Sn R 6
remarks: Players know the strategy set and the payoff functons of the other players; they are aware the other players also know ther own set of strateges and payoff functons, etc - games of complete nformaton and - common knowledge, assumptons on ratonalty, etc. (see papers by Brandenburger) We should dstngush outcomes and payoffs: - f u ( s,..., ) s an outcome (money), then we usually assume players are rskneutral s n - f u ( s,..., ) functons s n s a payoff (utlty, player's preferences), then we use expected utlty 7
How to solve a game (n normal-form)?. Iterated elmnaton of strctly domnated strateges. Nash-eulbrum 8
. Iterated elmnaton of strctly domnated strateges Defnton: In a normal-form game, for player the strategy s " strctly domnates strategy ' s f for each combnaton of strateges of other players ( ' ) ( " ) s s, s, s,..., s u s,... s, s, s s u,...,,... + n < + n a "ratonal" player never plays domnated strateges (t could be a defnton of ratonalty; see Brandenburger) 9
An example of terated elmnaton of strctly domnated strateges (Gbbons, fgure..) Player Left Mddle Rght Player Up.0. 0. Down 0.3 0..0 0
Delete strategy rght for player Player Left Mddle Player Up.0. Down 0.3 0. Delete strategy down for player Player Left Mddle Player Up.0.
Delete strategy left for player Player Mddle Player Up. Caveats about solvng a game ths way:. It's not enough for player, say, to be ratonal; player must be ratonal and player must expect player to be ratonal and so on.. For most of the games we do not obtan a unue predcton, we at best dscard some outcomes (see for nstance the game n fgure..4 n Gbbons).
Best response Defnton: In a normal-form game, for player the strategy s ' s a best response to strateges ( s s, s,..., s ),... + n by ts rvals f ( ' ) s s, s, s,..., s u ( s,... s, s, s s ) u,...,,... + n + n for any other feasble strategy s S. ' The followng notaton s used often: s R ( s s, s,..., s ),... + n 3
Best responses n the prevous game? ( left) R ( mddle) top R = = R ( rght) = bottom R ( p left + p mddle + p rght)... 3 = R (). =... 4
The Prsoner's dlemma Player confess don't confess Player confess. 0.0 don't confess 0.0 6.6 Expected outcome? Ths s a clear example of conflct. 5
. Nash eulbrum ( * * ) Defnton: In a normal-form game, the strateges s for each player, the strategy * any s on S, s,..., n are a Nash eulbrum f ( * * * * ) s s a best response to s, s s s +,...,,..., n, that s, for ( * * * * ) ( * * * * * ) s s, s, s,..., s u s,..., s, s, s s u + n +,...,,..., n * ( * * * * ) In term of the best response correspondence, R s s, s,..., s s +,..., n 6
An example (from Gbbons) Player Left Mddle Rght Top 0.4 4.0 5.3 Player Mddle 4.0 0.4 5.3 Bottom 3.5 3.5 6.6 Whch s the Nash-eulbrum n pure strateges? 7
The prevous defnton s not very accurate. The Nash eulbrum also consders mxed strateges: Mxed strateges (Gbbons.3.A) Suppose the space of strateges of player s S { s,..., sk } for player s a probablty dstrbuton p ( p,..., p ) K k= p =. k A pure strategy s, for nstance, = (,0,...,0) p. =. Then a mxed strategy =, where 0 p and K k 8
How to justfy the Nash-eulbrum (NE) concept as the soluton of a game? From Mas-Colell et al.:. Ratonal nference?. If there s only one soluton of the game, t must be a NE. 3. Focal ponts 4. It s a self-enforcng agreement. Compellng argument when there s only one NE. Wth more than one NE, see below. 5. Stable socal conventon (convergence to NE, evolutonary economcs, John Maynard Smth n bology) 9
Battle of sexes Player opera football Player opera 3.5 0.0 football 0.0 5.3 Nash-eulbra n pure strateges (what about mxed strateges?) There s (maybe) a problem of coordnaton. 0
Stag-hunt game Player left rght Player top 9.9 0.8 bottom 8.0 7.7 Nash-eulbra n pure strateges There s a problem of coordnaton as before and moreover: - If the cells denote outcomes, not payoffs, (bottom, rght) seems less rsky. - Is the Nash-eulbrum (top, left) self-enforcng? Aumann, Farrell dscuss t.
An example wthout Nash eulbrum n pure strateges (Gbbons, exercse.) Player Left Rght Player Top. 0. Bottom. 3.0 Best responses of player to left and to rght? Best responses of player to top and to bottom? Best responses of player to = ( µ, µ ) p or ( p p ) Nash eulbrum when we consder mxed strateges: p left, rght =? * * 3 p =,, p =, 3 3 4 4
When players have fnte sets of strateges, a Nash eulbrum always exsts (t s not necessarly unue, and maybe t s an eulbrum n mxed strateges). The proof amounts to show that there s always a fxed pont n the correspondence ( R ( s ) R n ( s )),..., n : ( * ) ( * * ) R s ),..., R ( s ) = s s * ( n n,..., n 3
An example of a game n normal form wth 3 players. Player 3 chooses boxes. Box Player Left Mddle Player Up.0... Down 0.3. 0.. Box Player Left Mddle Player Up.0. 3.. Down.3. 0.. 4
The space of strateges was fnte untl now. It could be nfnte. examples:. Second prce aucton. players have valuatons { v,v }. Each player submts smultaneously a bd {,b } The player that submts the hgher bd gets the object and pays the lower bd: b. u ( b, b ) = v v b b 0 f f f b b b > = < b b b Domnant strategy for player? 5
. A duopoly that competes n uanttes: the Cournot game (Gbbons, applcaton..a). There s a lnear demand, D( p) A p = ; frms have constant margnal costs of producton c, and choose,. Assume the market clears (demand = supply): Q = +, and as a conseuence p ( + ) = A 0 whenever whenever + + < < 0 0 Strategy space S of frm : 0 (I wrte nstead of s ; for "uanttes"). 6
7 Frm solves: ( ) ( ) j j c A =, max π subject to 0. Frst order condton (FOC): Fnd that solves ( ) 0, = j π (Ths s ndeed a soluton: Second order condton ( ) 0, < = j π : the proft functon s a concave functon on ). From ( ) 0, = j π we obtan the best response functon ( ) ( ) < = otherwse c A whenever c A R j j j 0
The Nash eulbrum amounts to fndng the fxed pont n the correspondence ( ( ) R ( )) R :, ( ( * ) ( * ) ( * * ) R, R =, 8