CS294 Topcs n Algorthmc Game Theory October 11, 2011 Lecture 7 Lecturer: Chrstos Papadmtrou Scrbe: Wald Krchene, Vjay Kamble 1 Exchange economy We consder an exchange market wth m agents and n goods. Agent has utlty u and ntal wealth (endowment) w. We seek an allocaton of goods x such that x = w. We want the allocaton to be Pareto optmal,.e. the vector of utltes [u 1 (x 1 ),...,u n (x n )] s not domnated by any other vector of utltes. The frst theorem of welfare economcs states that any prce equlbrum p (a prce that satsfes no excess demand f j (p )=[ x (p ) w ] j 0 wth equalty f p j > 0) s Pareto-optmal. In 1960, Negsh shows that under certan condtons, a prce equlbrum maxmzes a lnear combnaton of utltes: Theorem 1. Negsh 1960 If w,j > 0, j then α 1,...,α m such that s at prce equlbrum. 2 Gross Substtutablty arg max α u (x ) x 1,...,x n Defnton 1. A utlty functon s sad to have the Gross Substtutablty property f ts excess demand functon satsfes f p j > 0 = j f the excess demand functon s non dfferentable, ths corresponds to the followng dfference equaton f (p j +, p j ) f (p j, p j ) > 0 = j, > 0 ths property descrbes a market where the demand for good ncreases when the prce of good j rses. Note that ths s not the case for complementary goods: f coffee and sugar are complementary, then the demand for coffee decreases when the prce of coffee rses. GS s addtve Note that f f 1 and f 2 are two excess demand functons that satsfy the GS property, then so does f 1 + f 2. Therefore, to check the GS property for a market, t suffce to check t for every agent. 2.1 Unqueness of the prce equlbrum n a GS exchange economy Theorem 2. Arrow-Block-Hurwcz 1959 If the utltes u are ncreasng, strctly concave, and satsfy the GS property, then there exsts a unque prce equlbrum p. 7-1
Proof: Assume there are two dfferent prce equlbra p, p > 0. Wlog (by rescalng and rendexng), assume that p =[1,p 2,...,p n ] p =[1,p 2 2,...,p n n ] then by the GS property, the excess demand for good 1 strctly ncreases when we ncrease the prces, one by one, to go from p to p 0=f 1 (p )=f 1 (1,p 2 2,...,p n n ) <f 1 (1,p 2,...,p n n ) 2 f 1 (p) therefore f 1 (p) > 0 and p s not a prce equlbrum. 2.2 Weak Axom of Revealed Preference (WARP) Theorem 3. Arrow-Block-Hurwcz 1959 Consder a market that satsfes the GS property. Suppose p s not a prce equlbrum. Then the prce equlbrum p les n the half-space gven by p.f(p) > 0 Ths shows that one can solve effcently (n polynomal tme) for the prce equlbrum usng the ellpsod algorthm. Proof: Wlog, assume that the prce equlbrum s p =[1,...,1] (otherwse, change the unts of the goods) and p =[1,p 2,...,p n ] (otherwse scale t). We want to show that 1.f(p) = f (p) > 0. Defne prces (p 1,...,p n )by p 1 =(1,...,1) = p p 2 =(1,p 2,...,p 2 ). p n =(1,p 2,...,p n )=p Let x k = f(p k ) be the excess demand at prce p k. Then x 1 = 0 and we want to show that x n = f(p) s such that 1.x n > 0. We have x n =(x n x n 1 )+(x n 1 x n 2 )+ +(x 2 x 1 ) Lemma 1. (Proof omtted.), 1.(x +1 x ) 0, and 0 such that 1.(x 0+1 x 0 ) > 0 Usng the lemma, the result follows mmedately 1.x n = 1(x n x n 1 )+ + 1.(x 2 x 1 ) > 0. 2.3 Tâtonnement n a GS exchange market Consder the tâtonnement dfferentl equaton n a GS market dp j dt = f j(p) p(0) = (1, 0,...,0) Assume the excess demand functons f j are contnuous. Then Theorem 4. Prce Adjustment mechansm (Arrow 1960) The tâtonnement DE has a soluton that converges to the prce equlbrum p. 7-2
Proof: Contnuty of f j guarantees the exstence of a soluton. Then gven the ntal condton p(0) = (1, 0,...,0), the soluton remans bounded p(t) [01] n t 0. To prove the convergence of the soluton to the prce equlbrum p, consder the potental functon We have V (p) = p(t) p 2 dv dt =2 j (p j (t) p j ) dp j(t) dt =2 j (p (t) p )f j (p) =2p.f(p) 2p.f(p) = 2p.f(p) If for some t 0 0, p(t 0 )saprceequlbrum,thenf(p(t 0 )) = 0 and p(t) =p(t 0 ) t t 0. Otherwse we have by WARP dv dt = 2p.f(p) < 0 t 0 Therefore V (t) converges to 0: assume by contradcton that V (t) = p(t) p 2 2 t 0 for dv some > 0. Then by contnuty of f, dt < η for some η > 0. Then lm t V (t) =, whch contradcts the fact that V (t) 0. Therefore t must be that lm t V (t) =0.e. lm t p(t) =p Defnton 2. Approxmate demand A bundle x s an approxmate demand for agent at prce p f u (x ) u,and 1 p.x p, where u = max x {u (x) p.x p.w } Defnton 3. Approxmate prce equlbrum A prce vector p s a strong approxmate equlbrum f x 1,...,x n such that x = x (p) (x s the demand of agent at prce p) x 1 w It s a weak approxmate equlbrum f x 1,...,x n such that x s an approxmate demand of agent at prce p. x 1 w Theorem 5. Papadmtrou and Yannakaks 10 Any prce adjustment mechansm takes (n worst case) exponentally many steps n 1/ to come to an approxmate equlbrum. 3 Demand functons for specal utltes Consder an exchange economy, a fxed agent wth utlty u(x), endowment w, budget B = p.w. Subscrpts j denote goods. 7-3
Lnear utltes u(x) = α j x j. Let j 0 = arg max j α j /p j. The demand s then gven by x j0 = B/p j0 x j =0 j = j 0 Cobb-Douglas utltes u(x) = j xλj j. The demand s x j = Bλ j /p j B Leontef utltes u(x) =mn j x j /α j. Let κ = be the consumpton rate. Then the demand j pj/αj s x j = κα j Pecewse lnear utltes u(x) = j α jγ(x j ) 1/ρ. CES utltes Constant Elastcty utltes. u(x) = j α jx ρ j The demand s x j = α p j 1/(1 ρ) B/ 3.1 CES exchange economes for ρ < 1 k 1/(1 ρ) αk The demand s strctly concave and ncreasng, therefore prce equlbra exst (but not necessarly unque). 3.2 CES exchange economes for 0 < ρ < 1 Theorem 6. A CES exchange economy wth 0 < ρ < 1 s GS As a consequence, we have unqueness of the prce equlbrum, and t can be found usng the ellpsod algorthm. Note that fndng prce equlbra for ρ = 0 (Cobb-Douglas) and ρ = 1 (Lnear) s also easy snce t can be solved by a convex program. 3.3 CES exchange economes for 1 ρ < 0 For 1 ρ < 0, the optmzaton problem max p subject to p ρ k u (x (p)) (1) x,j (p) w,j (2) s non convex, but under the change of varable y,j = p j x,j = s j x,j, the optmzaton problem becomes max u (y (s)) (3) s subject to y,j (s) s j w,j (4) Lemma 2. y,j (s) s convex. As a consequence, the optmzaton problem s convex and can be solved by ellpsod. 7-4
3.4 CES exchange economes for ρ < 1 Consder a market wth m = 2 agents and n = 2 goods, ntal endowments w 1 =(1, 0) and w 2 =(0, 1), and utltes Then u 1 (x 1,x 2 )=((ax 1 ) ρ + x ρ 2 )1/ρ u 2 (x 1,x 2 )=((x 1 /a) ρ + x ρ 2 )1/ρ Theorem 7. ρ < 1, a ρ such that (1/2, 1/2) and (1/3, 2/3) are the only prce equlbra for the above market. For such a market (gven by ρ,a ρ ), the set of prce equlbra s non convex, whch shows that fndng prce equlbra cannot be solved by a convex program for CES exchange economes wth ρ < 1. 3.5 Complexty of fndng equlbra n Leontef exchange economes A Leontef market s a specal case of CES market for ρ =. The Leontef utlty functons u (x 1,...,x n )=mnx,j /α,j j are not strctly concave, and prce equlbra may not exst. Fndng prce equlbra s at least as hard as PPAD. Ths can be shown by reducng Nash to computng a prce equlbrum of a Leontef exchange market. Ths uses a one-to-one correspondence between 2 player Nash games and Leontef markets. Theorem 8. Codenott-Saber-Varadarajan-Ye 06 Fndng a market equlbrum n a Leontef exchange economy s at least as hard as fndng a Nash equlbrum n a two player game. Proof: Consder a 2-person game gvenby utlty matrces (A, B). Ths game can be reduced to a 0 A symmetrc game (M,M T )wherem = 1+ B T. Fndng a Nash equlbrum for ths game s equvalent (by the fundamental theorem of Nash equlbra) to solvng the followng Lnear Complementarty 0 Program (LCP) MX + Y = 1 X.Y =0 X, Y 0 X represents the mxed strategy of the frst player, and Y represents ts complement: Y s how much below best response strategy s. Thus f s n the support of X (X > 0) then Y = 0 (by the constrant X.Y = 0).e. s a best response. Now consder a Leontef market wth n ntal endowments W =[w 1,...,w n ]=I (w, =1, w,j =0 j =..e. agent comes n wth one unt of good ) and utltes u (x 1,...,x n )=mn j x,j /M,j A (quas) prce equlbrum for ths Leontef economy s a prce vector p such that, κ = p / j M jp j s well defned. κ represents the utlty value of the optmal bundle of agent. The optmal bundle s x (p) =[M,1 κ,...,m,n κ ]. j, f j = M jκ 1 0.e. no excess demand of good j, wth equalty f p j > 0. Ths can be wrtten as f = M T κ 1 0 and f j p j =0 j. 7-5
Note that snce κ = p / j M jp j f j p j =0 j κ j p j =0 j Therefore, fndng a (quas) prce equlbrum for ths market s equvalent to solvng the LCP Mκ + f = 1 κ.f =0 κ, f 0 Ths completes the proof. Note that ths reducton holds approxmately for near Leontef markets (ρ = η 2 close to ). Thus n the regon where ρ s close to, fndng prce equlbra s stll PPAD-complete. 4 Producton markets Consder a market wth n goods, m agents (consumers), and q frms (producers). The j th frm has technology T j R n that descrbes the producton capabltes of the frm as follows: t T j,lett = t + + t. Then let I = { t < 0} (the support of t,whchrepresentsthenput goods) and O = {o t + o > 0} (the support of t +, whch represents the output goods). Then the frm can produce t + gven t as nput. We assume that consumer has an ntal endowment w R n, a utlty functon u, and that she owns fracton λ,j of frm j. 4.1 Prce mechansm After prces p are announced consumer buys x (p) that maxmzes ther utlty x (p) = arg max x {u (x), p.x p.w }. frm j chooses t j (p) that maxmzes ts proft t j (p) = arg max t Tj p.t the excess demand s gven by f(p) = x (p) w + j t j 4.2 Prce equlbra n economes of scale Theorem 9. Exstence of prce equlbra for economes of scale If the utltes u are ncreasng and strctly concave, and T j are closed convex compact sets, then prce equlbra for the producton market exst. The convexty of T j descrbes economes of scale where the output (goods produced by the frm) s a concave functon of the nputs (goods consumed by the frm durng the producton process). Ths s a far assumpton for some producton markets (e.g. ol extracton, mnng) but does not hold n general producton markets. Note If the producton market s GS, the prce equlbrum s not necessarly unque (as opposed to exchange markets) 7-6
4.3 Non-convex economes Now we consder a non-convex producton market, where the technology sets T j are non-convex. Theorem 10. Prce equlbra n non-convex producton markets and Complexty equlbra. Papadmtrou- Wlkns 11 In a non-convex producton market prce equlbra may not exst, and t s NP-hard to tell It s worse than NP-hard to fnd a Pareto allocaton There are complexty equlbra Complexty equlbra Roughly speakng, complexty equlbra are allocatons x (p) and producton decsons t j (p) that are not equlbra and non Pareto (n fact very far from beng Pareto, n the sense that there exsts allocatons and producton decsons where everyone s better off), yet everyone s stuck to ths default because t s computatonally mpossble to mprove. References [1] B. Codenott, A. Saber, K. Varadarajan, Y. Ye. Leontef Economes Encode Nonzero Sum Two-Player Games. ACM-SIAM Symposum on Dscrete Algorthms, 2006. [2] Kenneth J. Arraow, H. D. Block and Leond Hurwcz On The Stablty of the Compettve Equlbrum. Econometrca 27, 82 109, 1959. [3] Chrstos. H. Papadmtrou and Chrstopher. A. Wlkens. Economes Wth Non-Convex Productons and Complexty Equlbra. 7-7