Algorithmic Game Theory and Internet Computing Vijay V. Vazirani Georgia Tech Market Equilibrium: The Quest for the “Right” Model

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”.

Need algorithmic ratification!!

The new face of computing

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. Today’s reality

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

Several buyers with different utility functions and moneys.

Several buyers with different utility functions and moneys. Find equilibrium prices.

Linear Fisher Market DPSV, 2002: First polynomial time algorithm Assume:  Buyer i’s total utility,  m i : money of buyer i.  One unit of each good j.

Eisenberg-Gale Program, 1959

prices p 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.

Combinatorial Algorithm for Linear Case of Fisher’s Model Devanur, Papadimitriou, Saberi & V., 2002 By extending the primal-dual paradigm to the setting of convex programs & KKT conditions

Auction for Google’s TV ads N. Nisan et. al, 2009: Used market equilibrium based approach. Combinatorial algorithms for linear case provided “inspiration”.

utility Piecewise linear, concave amount of j Additively separable over goods

Long-standing open problem Complexity of finding an equilibrium for Fisher and Arrow-Debreu models under separable, plc utilities?

How do we build on solution to the linear case?

utility amount of j Generalize EG program to piecewise-linear, concave utilities? utility/unit of j

Generalization of EG program

V. & Yannakakis, 2007: Equilibrium is rational for Fisher and Arrow-Debreu models under separable, plc utilities. Given prices p, are they equilibrium prices? Build on combinatorial insights

utility amount of j Case 1 fully allocated partially allocated

utility amount of j Case 2: no p.a. segment fully allocated

p full & partial segments Network N(p) Theorem: p equilibrium prices iff max-flow in N(p) = unspent money.

Network N(p) m’(1) m’(2) m’(3) m’(4) p’(1) p’(2) p’(3) p’(4) partially allocated segments s t

LP for max-flow in N(p); variables = f e ’s Next, let p be variables! “Guess” full & partial segments – gives N(p) Write max-flow LP -- it is still linear!  variables = f e ’s & p j ’s

Rationality proof If “guess” is correct, at optimality, p j ’s are equilibrium prices. Hence rational!

Rationality proof If “guess” is correct, at optimality, p j ’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

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.

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

Markets with piecewise-linear, concave utilities V, & Yannakakis, 2009: Membership in PPAD for both models. Path-following algorithm??

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-based proof. G: piecewise-linear Brouwer approximation of F (p*, x*): fixed point of G. Can be found in PPAD.

Instead … F: correspondence in Kakutani-based proof. 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!

How do we salvage the situation?? Algorithmic ratification of the “invisible hand of the market”

Is PPAD really hard?? What is the “right” model??

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.

utility Piecewise linear, concave amount of j

rate rate = utility/unit amount of j amount of j Differentiate

rate amount of j rate = utility/unit amount of j money spent on j

rate rate = utility/unit amount of j money spent on j Spending constraint utility function $20$40 $60

Theorem (V., 2002): Spending constraint utilities: 1). Satisfy weak gross substitutability 2). Polynomial time algorithm for computing equilibrium.

An unexpected fallout!! Has applications to Google’s AdWords Market!

rate rate = utility/click money spent on keyword j Application to Adwords market $20$40 $60

Is there a convex program for this model? “We believe the answer to this question should be ‘yes’. In our experience, non-trivial polynomial time algorithms for problems are rare and happen for a good reason – a deep mathematical structure intimately connected to the problem.”

Devanur’s program for linear Fisher

C. P. for spending constraint!

EG convex program = Devanur’s program Fisher market with plc utilities Spending constraint market

Price discrimination markets Business charges different prices from different customers for essentially same goods or services. Goel & V., 2009: Perfect price discrimination market. Business charges each consumer what they are willing and able to pay.

plc utilities

Middleman buys all goods and sells to buyers, charging according to utility accrued.  Given p, there is a well defined rate for each buyer.

Middleman buys all goods and sells to buyers, charging according to utility accrued.  Given p, there is a well defined rate for each buyer. Equilibrium is captured by a convex program  Efficient algorithm for equilibrium

Middleman buys all goods and sells to buyers, charging according to utility accrued.  Given p, there is a well defined rate for each buyer. Equilibrium is captured by a convex program  Efficient algorithm for equilibrium Market satisfies both welfare theorems!

Generalization of EG program works!

V., 2010: Generalize to Continuously differentiable, quasiconcave (non-separable) utilities, satisfying non-satiation.

V., 2010: Generalize to Continuously differentiable, quasiconcave (non-separable) utilities, satisfying non-satiation. Compare with Arrow-Debreu utilities!! continuous, quasiconcave, satisfying non-satiation.

EG convex program = Devanur’s program Price discrimination market (plc utilities) Spending constraint market

EG convex program = Devanur’s program Price disc. market Spending constraint market Nash Bargaining V., 2008 Eisenberg-Gale Markets Jain & V., 2007 (Proportional Fairness) (Kelly, 1997)

Triumph of Combinatorial Approach to Solving Convex Programs!!

A new development Orlin, 2009: Strongly polynomial algorithm for Fisher’s linear case. Open: For rest.

AGT&E’s gift to theory of algorithms! New complexity classes: PPAD, FIXP  Study complexity of total problems A new algorithmic direction  Combinatorial algorithms for convex programs