Combinatorial Algorithms for Convex Programs Algorithmic Game Theory

Combinatorial Algorithms for Convex Programs Algorithmic Game Theory (Capturing Market Equilibria and Internet Computing Nash Bargaining Solutions) Vijay V. Vazirani Georgia Tech

What is Economics? ‘‘Economics is the study of the use of scarce resources which have alternative uses. ’’ Lionel Robbins (1898 – 1984)

How are scarce resources assigned to alternative uses?

How are scarce resources assigned to alternative uses? Prices!

How are scarce resources assigned to alternative uses? Prices Parity between demand supply

How are scarce resources assigned to alternative uses? Prices Parity between demand supply equilibrium prices

Do markets even admit equilibrium prices?

Do markets even admit equilibrium prices? General Equilibrium Theory Occupied center stage in Mathematical Economics for over a century

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

Do markets even admit equilibrium prices?

Do markets even admit equilibrium prices? Easy if only one good!

Supply-demand curves

Do markets even admit equilibrium prices? What if there are multiple goods and multiple buyers with diverse desires and different buying power?

Irving Fisher, 1891 n Defined a fundamental market model

linear utilities

For given prices, find optimal bundle of goods

Several buyers with different utility functions and moneys.

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

“Stock prices have reached what looks like a permanently high plateau”

“Stock prices have reached what looks like a permanently high plateau” Irving Fisher, October 1929

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

General Equilibrium Theory An almost entirely non-algorithmic theory!

The new face of computing

Today’s reality n New markets defined by Internet companies, e. g. , ¨ Google ¨ e. Bay ¨ Yahoo! ¨ Amazon n Massive computing power available. n Need an inherenltly-algorithmic theory of markets and market equilibria.

Combinatorial Algorithm for Linear Case of Fisher’s Model n Devanur, Papadimitriou, Saberi & V. , 2002 Using the primal-dual paradigm

Combinatorial algorithm n Conducts an efficient search over a discrete space. simplex algorithm vs ellipsoid algorithm or interior point algorithms. n E. g. , for LP:

Combinatorial algorithm n Conducts an efficient search over a discrete space. n E. g. , for LP: simplex algorithm vs ellipsoid algorithm or interior point algorithms. n Yields deep insights into structure.

n No LP’s known for capturing equilibrium allocations for Fisher’s model

n No LP’s known for capturing equilibrium allocations for Fisher’s model n Eisenberg-Gale convex program, 1959

n No LP’s known for capturing equilibrium allocations for Fisher’s model n Eisenberg-Gale convex program, 1959 n Extended primal-dual paradigm to solving a nonlinear convex program

Linear Fisher Market n B = n buyers, money mi for buyer i G = g goods, w. l. o. g. unit amount of each good : utility derived by i on obtaining one unit of j Total utility of i, n Find market clearing prices. n n n

Eisenberg-Gale Program, 1959

Eisenberg-Gale Program, 1959 prices pj

Why remarkable? n Equilibrium simultaneously optimizes for all agents. n How is this done via a single objective function?

Theorem n If all parameters are rational, Eisenberg-Gale convex program has a rational solution! ¨ Polynomially many bits in size of instance

Theorem n If all parameters are rational, Eisenberg-Gale convex program has a rational solution! ¨ n Polynomially many bits in size of instance Combinatorial polynomial time algorithm for finding it.

Theorem n If all parameters are rational, Eisenberg-Gale convex program has a rational solution! ¨ n Polynomially many bits in size of instance Combinatorial polynomial time algorithm for finding it. Discrete space

Idea of algorithm primal variables: allocations n dual variables: prices of goods n iterations: execute primal & dual improvements n Allocations Prices (Money)

How are scarce resources assigned to alternative uses? Prices Parity between demand supply

Yin & Yang

Nash bargaining game, 1950 n Captures the main idea that both players gain if they agree on a solution. Else, they go back to status quo. n Complete information game.

Example n n Two players, 1 and 2, have vacation homes: ¨ 1: in the mountains ¨ 2: on the beach Consider all possible ways of sharing.

Utilities derived jointly : convex + compact feasible set

Disagreement point = status quo utilities Disagreement point =

Nash bargaining problem = (S, c) Disagreement point =

Nash bargaining Q: Which solution is the “right” one?

Solution must satisfy 4 axioms: n Paretto optimality n Invariance under affine transforms n Symmetry n Independence of irrelevant alternatives

Thm: Unique solution satisfying 4 axioms

Generalizes to n-players n Theorem: Unique solution

Linear Nash Bargaining (LNB) n Feasible set is a polytope defined by linear constraints n Nash bargaining solution is optimal solution to convex program:

Q: Compute solution combinatorially in polynomial time?

Game-theoretic properties of LNB games n Chakrabarty, Goel, V. , Wang & Yu, 2008: ¨ Fairness ¨ Efficiency (Price of bargaining) ¨ Monotonicity

Insights into markets n V. , 2005: spending constraint utilities (Adwords market) n Megiddo & V. , 2007: continuity properties V. & Yannakakis, 2009: piecewise-linear, concave utilities n Nisan, 2009: Google’s auction for TV ads n

How should they exchange their goods?

State as a Nash bargaining game S = utility vectors obtained by distributing goods among players

Special case: linear utility functions S = utility vectors obtained by distributing goods among players

Convex program for ADNB

Theorem (V. , 2008) n If all parameters are rational, solution to ADNB is rational! ¨ Polynomially many bits in size of instance

Theorem (V. , 2008) n If all parameters are rational, solution to ADNB is rational! ¨ n Polynomially many bits in size of instance Combinatorial polynomial time algorithm for finding it.

Flexible budget markets n n n Natural variant of linear Fisher markets ADNB flexible budget markets Primal-dual algorithm for finding an equilibrium

How is primal-dual paradigm adapted to nonlinear setting?

Fundamental difference between LP’s and convex programs n Complementary slackness conditions: involve primal or dual variables, not both. n KKT conditions: involve primal and dual variables simultaneously.

KKT conditions

Primal-dual algorithms so far (i. e. , LP-based) n Raise dual variables greedily. (Lot of effort spent on designing more sophisticated dual processes. )

Primal-dual algorithms so far n Raise dual variables greedily. (Lot of effort spent on designing more sophisticated dual processes. ) ¨ Only exception: Edmonds, 1965: algorithm for max weight matching.

Primal-dual algorithms so far n Raise dual variables greedily. (Lot of effort spent on designing more sophisticated dual processes. ) ¨ Only n exception: Edmonds, 1965: algorithm for max weight matching. Otherwise primal objects go tight and loose. Difficult to account for these reversals -in the running time.

Our algorithm n Dual variables (prices) are raised greedily n Yet, primal objects go tight and loose ¨ Because of enhanced KKT conditions

Our algorithm n Dual variables (prices) are raised greedily n Yet, primal objects go tight and loose ¨ Because n of enhanced KKT conditions New algorithmic ideas needed!

Open Nonlinear programs with rational solutions!

Open Nonlinear programs with rational solutions! Solvable combinatorially!!

Primal-Dual Paradigm n Combinatorial Optimization (1960’s & 70’s): Integral optimal solutions to LP’s

Exact Algorithms for Cornerstone Problems in P n n n Matching (general graph) Network flow Shortest paths Minimum spanning tree Minimum branching

Primal-Dual Paradigm n Combinatorial Optimization (1960’s & 70’s): Integral optimal solutions to LP’s n Approximation Algorithms (1990’s): Near-optimal integral solutions to LP’s

Primal-Dual Paradigm n Combinatorial Optimization (1960’s & 70’s): Integral optimal solutions to LP’s n Approximation Algorithms (1990’s): Near-optimal integral solutions to LP’s WGMV 1992

Approximation Algorithms set cover Steiner tree Steiner network k-MST scheduling. . . facility location k-median multicut feedback vertex set

Primal-Dual Paradigm n Combinatorial Optimization (1960’s & 70’s): Integral optimal solutions to LP’s n Approximation Algorithms (1990’s): Near-optimal integral solutions to LP’s n Algorithmic Game Theory (New Millennium): Rational solutions to nonlinear convex programs

Primal-Dual Paradigm n Combinatorial Optimization (1960’s & 70’s): Integral optimal solutions to LP’s n Approximation Algorithms (1990’s): Near-optimal integral solutions to LP’s n Algorithmic Game Theory (New Millennium): Rational solutions to nonlinear convex programs n Approximation algorithms for convex programs? !