Market Equilibrium: The Quest for the “Right” Model. Algorithmic Game Theory and Internet Computing. Vijay V. Vazirani Georgia Tech. Arrow-Debreu Theorem, 1954. Established existence of market equilibrium under very general conditions using a deep theorem from
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Market Equilibrium: The Quest for the “Right” Model Algorithmic Game Theoryand Internet Computing Vijay V. Vazirani Georgia Tech
Arrow-Debreu Theorem, 1954 • Established existence of market equilibrium under very general conditions using a deep theorem from topology - Kakutani fixed point theorem. • Provides a mathematical ratification of Adam Smith’s “invisible hand of the market”.
Today’s reality • New markets defined by Internet companies, e.g., • Microsoft • Google • eBay • Yahoo! • Amazon • Massive computing power available. • Need an inherently-algorithmic theory of markets and market equilibria.
Standard sufficient conditions on utility functions (in Arrow-Debreu Theorem): • Continuous, quasiconcave, satisfying non-satiation.
Complexity-theoretic question • For “reasonable” utility fns., can market equilibrium be computed in P? • If not, what is its complexity?
Irving Fisher, 1891 • Defined a fundamental market model • Special case of Walras’ model
Linear Fisher Market • DPSV, 2002: First polynomial time algorithm • Assume: • Buyer i’s total utility, • mi: money of buyer i. • One unit of each good j.
Why remarkable? • Equilibrium simultaneously optimizes for all agents. • How is this done via a single objective function?
Rational convex program • Always has a rational solution, using polynomially many bits, if all parameters are rational. • Eisenberg-Gale program is rational.
Auction for Google’s TV ads N. Nisan et. al, 2009: • Used market equilibrium based approach. • Combinatorial algorithms for linear case provided “inspiration”.
Build on combinatorial insights • V. & Yannakakis, 2007: Equilibrium is rational for Fisher and Arrow-Debreu models under separable, plc utilities. • Given prices p, are they equilibrium prices?
Case 1 partially allocated utility fully allocated amount ofj
p full & partial segments Network N(p) • Theorem:p equilibrium prices iff max-flow in N(p) = unspent money.
Network N(p) p’(1) m’(1) p’(2) m’(2) t s p’(3) m’(3) m’(4) p’(4) partially allocated segments
LP for max-flow in N(p); variables = fe’s • Next, let p be variables! • “Guess” full & partial segments – gives N(p) • Write max-flow LP -- it is still linear! • variables = fe’s& pj’s
Rationality proof • If “guess” is correct, atoptimality, pj’s are equilibrium prices. • Hence rational!
Rationality proof • If “guess” is correct, atoptimality, pj’s are equilibrium prices. • Hence rational! • In P??
Markets with piecewise-linear, concave utilities • Chen, Dai, Du, Teng, 2009: • PPAD-hardness for Arrow-Debreu model • Chen & Teng, 2009: • PPAD-hardness for Fisher’s model • V. & Yannakakis, 2009: • PPAD-hardness for Fisher’s model
Markets with piecewise-linear, concave utilities V, & Yannakakis, 2009: • Membership in PPAD for both models, Sufficient condition: non-satiation • Otherwise, for both models, deciding existence of equilibrium is NP-hard!
Markets with piecewise-linear, concave utilities – same for approx. eq. • Chen, Dai, Du, Teng, 2009: • PPAD-hardness for Arrow-Debreu model • Chen, Teng, 2009: • PPAD-hardness for Fisher’s model • V, & Yannakakis, 2009: • PPAD-hardness for Fisher’s model • Membership in PPAD for both models, Sufficient condition: non-satiation • Otherwise, deciding existence of eq. is NP-hard
NP-hardness does not apply • Megiddo, 1988: • Equilibrium NP-hard => NP = co-NP • Papadimitriou, 1991: PPAD • 2-player Nash equilibrium is PPAD-complete • Rational • Etessami & Yannakakis, 2007: FIXP • 3-player Nash equilibrium is FIXP-complete • Irrational
FIXP = Problems whose solutions are fixed points of Brouwer functions. (Given as polynomially computable algebraic circuits over the basis {+,-,*,/,max,min}.) • Etessami & Yannakakis, 2007: PPAD = Restriction of FIXP to piecewise-linear Brouwer functions.
M: market instance • Prove: PPAD alg. for finding equilibrium for M • Geanakoplos, 2003: Brouwer function-based proof of existence of equilibrium • Piecewise-linear Brouwer function??
Instead … • F: correspondence in Kakutani-basedproof. • G: piecewise-linear Brouwer approximation of F • (p*, x*): fixed point of G. Can be found in PPAD.
Instead … • F: correspondence in Kakutani-basedproof. • G: piecewise-linear Brouwer approximation of F • (p*, x*): fixed point of G. Can be found in PPAD Yields “guess” • Solve rationality LP to find equilibrium for M!
Linear Fisher Market • DPSV, 2002: First polynomial time algorithm • Extend to separable, plc utilities??
What makes linear utilities easy? • Weak gross substitutability: Increasing price of one good cannot decrease demand of another. • Piecewise-linear, concave utilities do not satisfy this.
