Lecture Note 1: Foundations 1

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1 Economcs 703 Advanced Mcroeconomcs Prof. Peter Cramton ecture Note : Foundatons Outlne A. Introducton and Examples B. Formal Treatment. Exstence of Nash Equlbrum. Exstence wthout uas-concavty 3. Perfect ecall 4. Correlated Equlbra A. Introducton and Examples Defnton (oger Myerson [986]): Game theory s the study of mathematcal models of conflct and cooperaton between ntellgent and ratonal decson makers. atonal means that each ndvdual's decson-makng behavor s consstent wth the maxmzaton of subjectve expected utlty. Intellgent means that each ndvdual understands everythng about the structure of the stuaton, ncludng the fact that others are ntellgent ratonal decson makers. Game : Each of three players smultaneously pcks a number from [0,]. A dollar goes to the player whose number s closest to the average of the three numbers. In case of tes, the dollar s splt equally. Descrpton of a Game n Normal Form player N = {,...,n} strategy s S strategy vector (profle) s = (s,...,s n ) S = S... S n payoff functon u (s):s, whch maps strateges nto real numbers game n normal form Γ = {S,...,S n ;u,...,u n } Game (both pay aucton): Ten dollars s auctoned to the hghest of two bdders. The players alternate bddng. At each stage, the bddng player must decde ether to rase the bd by $ or to qut. The game ends when one of the two bdders quts n whch case the other bdder gets the ten dollars, and both bdders pay the auctoneer ther bds. Descrpton of a Game n Extensve Form (See Kreps and Wlson, Econometrca 98) These notes are based wthout restrant on notes by obert Gbbons, MIT.

2 An extensve form answers the questons: Who plays when, what they can do, what they know, what are the payoffs? $ 0, 0 $ 9, 0 $3 -, 8 $4 7, - $ , 6 Game 3: Game 4: l r l r l r l r : 0 3 : : 0 3 : Strategy: a complete plan of acton (what to do n every contngency). Informaton Set: for player s a collecton of decson nodes satsfyng two condtons: player has the move at every node n the collecton, and doesn't know whch node n the collecton has been reached. Perfect Informaton: each nformaton set s a sngle node. (Chess, checkers, go,...) Fnte games of perfect nformaton can be "solved" by backward nducton n the extensve form or elmnaton of weakly domnated strateges n the normal form. Imperfect Informaton: at some pont n the tree some player s not sure of the complete hstory of the game so far. Any extensve-form game can be represented n normal form. Two-person games wth fnte strategy sets are represented as bmatrx games:

3 Game 3 ll lr rl rr Game 4 l r 0, 0, 3, 3, -,-, 0 -, -, 0 3, 3 -, 4,- 0, 0 Extensve forms contan more nformaton than normal forms (e.g., two dfferent extensve forms can have the same normal form): l r l r l r : 0 3 : : 0 3 : M ~ M Game 5 (Prsoners' Dlemma): Two suspects are arrested and charged wth a crme. They are held n separate cells. The Dstrct Attorney separately offers each the chance to turn state's evdence (.e., to fnk on the other prsoner). A jal sentence of x years has utlty -x. The payoffs to the prsoners as a functon of ther decsons are gven by the bmatrx: Mum Fnk Mum, 5, 0 Fnk 0, 5 4, 4 3

4 What s ratonal? Domnated Strategy: x strctly domnates y f the player gets a hgher payoff from playng x than playng y, regardless of what the other players do. x weakly domnates y f the player's payoff s at least as great by playng x than y, regardless of what the other players do. What s ratonal n game 4? Nash equlbrum: For an n-person game n Normal form, a strategy profle s S s a Nash equlbrum n pure strateges f for all u (s ) u ( s, s ) for all s S - - ) - + n where s = ( s, K, s, s, K, s ). (Each player's strategy s a best response to the others' strateges.) Game 6 (Matchng Pennes): Each of two players smultaneously show a penny. If the pennes match (both heads or both tals), player gets 's penny. Otherwse, player gets 's penny. H T H T H, - -, T -,,- H T H T What s ratonal? Zero-Sum Game: Sum of the players' payoffs s zero, regardless of outcome. Mxed strategy: a randomzaton over pure strateges. B. Formal Treatment. Exstence of Nash Equlbrum (Fnte Games) Normal form game: Γ = (S,...,S n ; u,...,u n ) Pure Strategy Profle: s = {s,...,s n } S = S... S n Mxed Strategy Profle: σ = {σ,...,σ n } (S )... (S n ) where σ : S [0,] and σ (s ) = Pr( plays pure strategy s ). 4

5 Expected payoff: v ( σ ) = u (s) s (s ) n s S j= Other's strategy: σ = { σ, K, σ, σ, K, σ } j j + n n ( σ, σ$ + ) = { σ, K, σ, σ$, σ, K, σ } Def: An n-tuple of mxed strateges σ = (σ,...,σ n ) s a Nash Equlbrum f for every, v ( σ) v ( σ -, σ$ ) for every σ$ ( S ). Theorem (Nash): Every fnte game has a Nash equlbrum n mxed strateges. Theorem : Consder an n-person game Γ = {D,...,D n ;v,...,v n }, where D s the set of pure strateges avalable to and v : D... D n s 's payoff functon. If each D s a compact convex subset of a Eucldean space, and each v s contnuous and quasconcave n d, then Γ has a Nash equlbrum n pure strateges. uasconcave: v ( d, αd + ( α) d $ ) mn{ v ( d,d ), v ( d, d $ )},.e., the payoff from a convex combnaton of two strateges s at least as great as the payoff from the worst of the two strateges. Def: A correspondence ϕ from a subset T of Eucldean space to a compact subset V of Eucldean space s upper hemcontnuous at a pont x T f x r x and y r y, where y r ϕ(x r ) for every r, mples y ϕ(x). ϕ s upper hemcontnuous f t s upper hemcontnuous at every x T. Theorem 3 (Kakutan): If T s a nonempty compact convex subset of a Eucldean space, and ϕ s an upper hemcontnuous nonempty convex-valued correspondence from T to T, then ϕ has a fxed pont,.e., there s an x T such that x ϕ(x). Proof of Theorem : For each, defne a "best response" correspondence ϕ from D = D... D n to D as follows. For any d D, let ϕ (d) be the set of strateges whch maxmze 's payoff gven the others strateges are d -,.e., ϕ (d) = {d D v (d -,d ) v (d -, d $ ) for every d $ D }. ϕ s nonempty snce D s compact and v s contnuous. ϕ s convex-valued snce v s quasconcave n d. ϕ s upper hemcontnuous, snce v s contnuous. (Consder a sequence d r d D and a sequence d r d D, where d r ϕ (d r ) for every r. For any d $ D, v (d - r,d r ) v (d - r, d $ ), so snce v s contnuous v (d -,d ) v (d -, d $ ),.e., d ϕ (d).) Defne ϕ: D D by ϕ(d) = ϕ (d)... ϕ n (d). D s a compact subset of Eucldean space snce each D s, and ϕ s an upper hemcontnuous nonempty convex-valued correspondence snce each ϕ s. So by Kakutan's theorem, ϕ has a fxed pont. But a fxed pont of ϕ s just a Nash equlbrum of Γ..E.D. Proof of Theorem : et D = (S ). Each D s then a compact convex subset of Eucldean space, and each v s contnuous and quasconcave n d (ndeed lnear). Hence, by Theorem, there s a Nash 5

6 equlbrum n whch each player chooses a "pure" strategy from (S )..E.D.. Exstence wthout uasconcavty (Dasgupta and Maskn, ES 986) What f the strategy set s contnuous, but the payoff functons are not quasconcave? Then we can't use Kakutan's fxed pont theorem, so look at equlbra n mxed strateges. Theorem (Glcksberg). et each D be a non-empty and compact subset of Eucldean space. And let each v be contnuous. Then Γ has a Nash equlbrum n mxed strateges. [Snce any functon on a fnte set s contnuous, Nash's theorem s an mmedate corollary.] Dasgupta and Maskn extend ths result by relaxng the requrement that the payoffs be contnuous. 3. Perfect ecall A game has perfect recall f each player knows whatever he knew prevously, ncludng hs prevous actons. Example: brdge ( player or 4 player game). A behavoral strategy specfes a probablty dstrbuton over feasble actons at each nformaton set. W E S = {l, r, } Mxed: (p, p, - p - p ) Behavor: p = Pr() q = Pr(l,W) Nature + - H H l r S = {HH, H, H, } Mxed: (p, p, p, p ) 3 4 Behavor: p = Pr(H +) q = Pr(H -) Kuhn (953) showed that n games of perfect recall, mxed strateges can be descrbed by behavor strateges: they nduce dentcal dstrbutons over termnal nodes. 4. Correlated Equlbra (Aumann, Journal of Mathematcal Economcs 974) Example (Battle of the Sexes): U, 0, 0 D 0, 0, Nash equlbra: Payoffs: (U,) (,) (D,) (,) (/3U,/3) (/3,/3) 6

7 But by correlatng strateges on a mutually verfable con toss then players can get (3/,3/). Need communcaton! Example (Coordnaton Game): U 6, 6, 7 D 7, 0, 0 Nash equlbra: Payoffs: (U,) (,7) (D,) (7,) (/3U,/3) (4/3,4/3) Can they do better? et a random devce pck A, B, or C wth probablty /3 each. s told whether A s chosen; s told whether C s chosen. plays D f A and U otherwse; plays f C and otherwse. Yelds payoff (5,5). For a normal form game, a correlated strategy s a probablty dstrbuton p(s) over the set of pure-strategy n-tuples S. A medator recommends strategy accordng to randomzaton devce p( ) that s common knowledge among the players. Gven a recommended strategy s, player holds belefs about other's strateges p(s - s ) derved from the correlated strategy by Bayes' rule. Def: The correlated strategy p(s) s a correlated equlbrum of the medated game f for every and for all s such that p(s ) > 0, u (s,s )p(s s ) u (s,s )p(s s ) for all s S s S s S Theorem: Every pont n the convex hull of the Nash-equlbrum payoffs s a correlated-equlbrum payoff. Proof: Use a mutually observable randomzng devce. Theorem: The correlated-equlbrum payoffs are a convex polyhedron defned by lnear nequaltes, unlke the n- st degree equatons that determne Nash equlbra. Proof: The lnear nequaltes n the defnton of a correlated equlbrum determne a convex polyhedron n the space of correlated strateges, whch determnes a convex polyhedron of correlatedequlbrum payoffs. 7