Algorithmic Game Theory and Internet Computing Vijay V. Vazirani Joint work with Jugal Garg & Ruta Mehta Dichotomies in Equilibrium Computation: Market Provide a Surprise

Equilibrium computation Has its own character, quite distinct from computability of optimization problems

Dichotomies Natural equilibrium computation problems exhibit striking

2-Nashk-Nash, k>2 Nature of solution Rational Algebraic; Irrational eg : Nash, 1950 ComplexityPPAD - completeFIXP - complete Practical algorithm Lemke - Howson ??

Arrow-Debreu Model n agents and g divisible goods. Agent i: has initial endowment of goods and a concave utility function (models satiation!)

Arrow-Debreu Model n agents and g divisible goods. Agent i: has initial endowment of goods and a concave utility function (piecewise-linear, concave)

Agent i comes with an initial endowment

At given prices, agent i sells initial endowment

… and buys optimal bundle of goods, i.e,

Several agents with own endowments and utility functions. Currently, no goods in the market.

Agents sell endowments at current prices.

Each agent wants an optimal bundle.

Equilibrium Prices p s.t. market clears, i.e., there is no deficiency or surplus of any good.

Arrow-Debreu Theorem, 1954 Celebrated theorem in Mathematical Economics Established existence of market equilibrium under very general conditions using a deep theorem from topology - Kakutani fixed point theorem.

Kenneth Arrow Nobel Prize, 1972

Gerard Debreu Nobel Prize, 1983

Arrow-Debreu Theorem, 1954 Celebrated theorem in Mathematical Economics Established existence of market equilibrium under very general conditions using a theorem from topology - Kakutani fixed point theorem. Highly non-constructive!

Computability of equilibria Walras, 1874: Tanonnement process Scarf, Debreu-Mantel-Sonnenschein, 1960s: Won’t converge! Scarf, Smale, …, 1970s: Many approaches

Open problem (2002) Separable, piecewise-linear, concave (SPLC) utility functions

Separable utility function

utility : piecewise-linear, concave amount of j

utility : piecewise-linear, concave amount of j SPLC utility: Additively separable over goods

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

Markets with separable, piecewise-linear, concave utilities V. & Yannakakis, 2009: Membership in PPAD for both models. Indirect proof, using the class FIXP of Etessami & Yannakakis, 2008.

Open The definition of PPAD was designed to capture problems that allow for path-following algorithms, in the style of Lemke-Howson and Scarf … It will be interesting to obtain a natural, direct algorithm for this task (hence leading to a more direct proof of membership in PPAD), which may be useful for computing equilibria in practice.

Complementary Pivot Algorithms (path-following) Lemke-Howson, 1964: 2-Nash Equilibrium Eaves, 1974: Equilibrium for linear Arrow-Debreu markets (using Lemke’s algorithm)

Eaves, 1975 Technical Report: Also under study are extensions of the overall method to include piecewise-linear utilities, production, etc., if successful, this avenue could prove important in real economic modeling. Eaves, 1976 Journal Paper:... Now suppose each trader has a piecewise-linear, concave utility function. Does there exist a rational equilibrium? Andreu Mas-Colell generated a negative example, using Leontief utilities. Consequently, one can conclude that Lemke’s algorithm cannot be used to solve this class of exchange problems.

Leontief utility (complementary goods) Utility = min{#bread, 2 #butter} Piecewise-linear, concave: Equilibrium may be irrational!

Leontief utility: is non-separable PLC Utility = min{#bread, 2 #butter} Only bread or only butter gives 0 utility

Devanur & Kannan, 2007, V. & Yannakakis, 2007: Equilibrium is rational for Fisher and Arrow-Debreu models under SPLC utilities. Using flow-based structure from DPSV. Rationality for SPLC Utilities

Theorem (Garg, Mehta, Sohoni & V., 2012): Complementary pivot (path-following) algorithm for Arrow-Debreu markets under SPLC utilities. Membership in PPAD, using Todd, Algorithm is simple, no stability issues, & is practical.

Experimental Results Inputs are drawn uniformly at random. |A|x|G|x#Seg#InstancesMin ItersAvg ItersMax Iters 10 x 5 x x 5 x x 10 x x 10 x x 15 x x 15 x x 20 x x 20 x

log(total no. of segments) log(no. of iterations)

SPLC Utilities Rest Nature of solution Rational Algebraic; Irrational e.g.: Mas-Colell, 1976 ComplexityPPAD - completeIn FIXP FIXP-hardness ? Practical algorithm GMSV, 2012 (Lemke-based) ??

Eaves, 1975 Technical Report: Also under study are extensions of the overall method to include piecewise-linear utilities, production, etc., if successful, this avenue could prove important in real economic modeling. Main general class of markets still unresolved.

Eaves, 1975 Technical Report: Also under study are extensions of the overall method to include piecewise-linear utilities, production, etc., if successful, this avenue could prove important in real economic modeling. Main general class of markets still unresolved. So far, no works exploring rationality …

Garg & V., 2013: SPLC Production: 1 finished good from 1 raw good, e.g., bread from wheat or corn. Total is additive.  LCP and complementary pivot algorithm  PPAD-complete  Rationality Non-separable PLC Production: From 2 or more raw goods: irrational!

SPLC Production Rest Nature of solution Rational Algebraic; Irrational e.g.: GV, 2013 ComplexityPPAD - completeIn FIXP FIXP-hardness ? Practical algorithm GV, 2013 (Lemke-based) ??

Separable vs non-separable This dichotomy arises in several places, e.g., concave flows.

Leontief-Free Utilities and Production!!

utility amount of crème brulee

utility amount of chocolate cake

Utilities of Crème Brulee and Chocolate Cake are not additively separable -- both satiate desire for dessert! Joint utility should be sub-additive

Capture via non-separable utilities! This non-separability is very different from that of Leontief utilities. Leontief: goods are complements Here: goods are substitutes

Garg, Mehta & V, 2013: Leontief-free utility functions Both are subclasses of PLC utilities, with no known generalization to differentiable, concave utilities.

Leontief utility Every bit of utility comes from all goods in appropriate proportions.

Leontief-free utility Every bit of utility comes from exactly one good. A specified set of goods competes for it, via mechanism of segments.

Segment An upper bound on utility allowed. A set of goods, with rate (utility/unit) for each good.

utility SPLC utilities are Leontief-free amount of j

Segments of an SPLC utility function Segment = a piece of some Set of goods = { j } Upper bound on utility = upper bound of piece Rate = slope of piece

Segments of a Leontief-free utility fn.

Leontief-free utility function

SPLC Utilities Rest Nature of solution Rational G MV, 2013 Algebraic; Irrational e.g.: Mas-Colell, 1976 ComplexityPPAD - complete GMV 2013 In FIXP FIXP-hardness ? Practical algorithm GMV, 2013 (Lemke-based) ?? Leontief-Free

Practical applications In pricing a new good, need to compare to substitutes.

Adding one Leontief-type constraint … … can lead to irrationality! Utility = min{#bread, 2 #butter}

Leontief-free production Garg, Mehta & Vazirani, 2013: full-fat milk + 2% milk yogurt + ice cream

SPLC Production Rest Nature of solution Rational GMV, 2013 Algebraic; Irrational e.g.: GMV, 2013 ComplexityPPAD - complete GMV, 2013 In FIXP FIXP-hardness ? Practical algorithm GMV, 2013 (Modified Lemke) ?? Leontief-Free

SPLC  Leontief-free In other contexts, e.g., algorithms for concave flows.

Are Leontief-free utilities sound from the viewpoint of economics?

Satiation (extreme cases) Intra-good:  No satiation: is linear  Max satiation: is 1-piece PLC

amount of good j

Satiation (extreme cases) Intra-good:  No satiation: is linear  Max satiation: is 1-piece PLC Inter-good: What does joint utility “look like” in extreme cases?

Consistent function

3 easy situations

1).

The best substitute for Crème Brulee is …

1).

a crème brulee b chocolate cake 2).

a crème brulee b chocolate cake 2).

a crème brulee b chocolate cake 2).

utility For a fixed bundle with > 1 good amount of good j 3).

utility For a fixed bundle amount of good j 3).

utility For a fixed bundle amount of good j 3).

amount of good

Total amount of good corres. to partitions If this holds for all goods, then t is feasible for x.

Hicks, 1932: Constant elasticity of substitution (CES) Utility functions and production sets.

Hicks, 1932: Constant elasticity of substitution (CES) Utility functions and production sets. Differences:  Highly specialized  Constant returns to scale  Differentiable

Complements vs Substitutes

Why found so late? Why in TCS?

Linear Complementarity Problem & Complementary Pivot Algorithms