Auction Algorithms for Market Equilibrium

Rahul Garg

IBM India Research

Sanjiv Kapoor

Illionis Institute of Technology

Overview

The market equilibrium problem History and recent developments A parameterized linear programming

formulation The auction algorithm Analysis and proof outline Conclusions and future work

A (Fisher) Market

There are n buyers and m sellers Each seller has exactly one commodity

(seller j has aj amount of commodity j) Buyers have only money

(buyer i has ei units of money) Sellers want only money, buyers want only

commodities

A (Fisher) Market

Buyers have utilities on commodity bundlesui: R+

m R+ The utility function ui of buyer i maps an endowment of commodities to a “happiness” index

The buyers and sellers come to the market and exchange commodities to maximize happiness

Each buyer and seller acts independently to maximize its own happiness

There are n traders and m commodities Each trader has initial endowments of commodities

aij = amount of commodity j with trader i Traders have utilities on commodity bundles

ui: R+m R+

The utility function ui of trader i maps an endowment of commodities to a “happiness” index

The traders come to the market and exchange commodities to maximize happiness

Each trader acts independently and acts to maximize its own happiness

The General Market Model (Walras)

Market Equilibrium

Commodities are divisible xij: the amount of commodity

j with trader i after the trade Commodity j is tagged with

a price pj

xij is a solution to the optimization problem

No excess or deficiency of any commodity

m

j

jij

m

j

jij

n

i

n

iij

papxi

axjij

11

11

:

:

j

m

j

ij

m

j

jij

ij

papx

xu

11

i

:Subject to

)(:Maximize

Market Equilibrium

No incentive for a trade No deficiency or surplus of any commodity p1, p2, …, pm are equilibrium prices Prices are in terms of an abstract currency Prices invariant to scaling Real money is a commodity (say m) Real price of commodity j is pj / pm

Not an optimization problem

Market Equilibrium History

Posed by 1891 Fisher 1894 Walras (Walrasian Equilibrium)

Existence 1954 Arrow and Debreu

Computation Hydraulic apparatus by Fisher Walrasian tatonnement Convergence? Polynomial time algorithms?

Computation of Market Equilibrium Arrow et al. 1959

Stability of a local greedy price adjustment method for “Gross Substitute” utility functions

Eisenberg and Gale, 1959 Fisher model, additive linear utilities Optimization problem

Eaves, 1976 Linear complementarity problem Lemke’s algorithm

Newman and Primak, 1992 Ellipsoid method – provably polynomial-time method

Computation of Market Equilibrium

Devanur et al. 2002 Fisher model, separable additive and linear utilities Combinatorial algorithm based on max flows Complexity: n4/ max-flow computations ~ n7/

Jain et al 2003, Devanur and Vazirani 2003 Approximation algorithm for Walrasian model, linear utilities

Jain, 2004 General Walrasian model, additive linear utilities Ellipsoid method (similar to Eisenberg and Gale)

Ye 2004 Fisher and Walrasian model, linear utilities Complexity: n4 L

Algorithms for Market Equilibrium Centralized Slow Very difficult to define and report utility functions Impractical

Auction Algorithms for Market Equilibrium

Fisher and General Walrasian model Additive linear utilities Approximation algorithm Decentralized and distributed Very simple Natural auction interpretation Complexity: 1/ (n m2 + m n2) log vmax steps

The Market Equilibrium Problem Linear, additive utilities

m

j

jij

m

j

jij

j

n

i

n

iij

papxi

aaxjij

11

11

:

:

j

m

j

ij

m

j

jij

m

jijij

papx

xv

11

1

:Subject to

:Maximize

m

jijiji xvxu

1

)(

A Parameterized LP Formulation

A family of LPs pj are market clearing

prices iff there is a dual optimal with = 0

ijjjiij

ijjji

m

jjj

n

i

m

jijij

m

j

m

jjijjij

n

ijij

n

i

m

jijij

vpx

vp

apa

papxi

axj

xv

0

:Subject to

:Minimize

:

: :Subject to

:Maximize

11 1

1 1

1

1 1

maximized is /0

and

0

jijij

ijji

ijjiij

pvx

vp

vpx

Search for p such that optimal dual has = 0

The Auction Algorithm

Fix a bid increment factor (1 + ) Start with low prices A trader with “sufficient” surplus money finds its best

commodity a commodity that maximizes vij / pj

Acquires a best item by outbidding the current winning trader

Raises the price of the acquired commodity by (1 + )

Stop when all the traders have small surplus

Outbidding

yij

hij

pj

(1 + ) pj

hij

pj

(1 + ) pj

Trader i Item j

ykjTrader k

Trader i

Trader k

pj

ykj

Divisible Items

Every item may be sold two prices: pj and pj (1 + )

hij amount sold at pj (1 + ) yij amount sold at pj

Price at which item is available pj (1 + )yij

hij

Some Details

Initialize with pj = 1 for all j

Xij = hij = yij = 0 Demand set of a trader

Di = { j: vij / pj = max vik / pk }

Surplus of a trader ri = aij pj - yij pj - hij pj (1 + )

The Auction Algorithm

while i such that ri > aij pjpick j Diif j is unassigned then

get j at price 1else iff ykj > 0 for some k then

outbid k on item jupdate ri and rk

else increase pj by factor (1 + )i: yij = hij; hij = 0;recompute Di’s

endifendwhile

ri

rk

yij

hijpj

(1 + ) pj

yij

hijpj

(1 + ) pj

hij yij pj

(1 + ) pj

hij

pj = (1 + ) pj yij

(1 + ) pj

ri = aij pj - yij pj

- hij (1 + ) pj

Bidding process:• Transfers surplus• Reduces it by (1 + )

Price raise:• Increases surplus

A Primal-Dual Interpretation

Maintains dual feasibility Satisfies complementary

slackness Successively improves

primal feasibility Stops when primal

infeasibility is sufficiently small decreasesnever

0 becomes after only increases

/ ifonly increased

/max Define

0Set

j

ijj

ijijij

jijji

p

xp

pvx

pv

Analysis

Terminates and achieves approximate market clearing

If order of bidding is fixed, then time complexity is good

Analysis

Number of prices raises < O(m/ log (pmax)) Bidding in rounds

every bidders bids once in a round either a price is raised or r reduces by factor (1 + ) Gives a bound of

O(1 / 2 nm log (a pmax / ( amin)) log pmax)

Aggressive Bidders

Bidder i bidding on item j Raises prices: m / log (pmax) Knocks out a bidder (say k)

Can a bidder get the same item again? Yes: if j Dk

Exhausts surplus Can surplus come again? How many times?

A modification If i is getting j at pj then it upgrades to (1 + ) pj

If j Dk then k bids back on item immediately and gets it a price (1 + ) pj

Analysis

Define a bipartite graph G (i, j) G iff j Di

(j, i) G iff ykj > 0 and j Di

G is acyclic Three types of assignments

hij > 0

yij > 0 and j Di

yij > 0 and j Di

Bidders Items

i

j

k

j Di

ykj > 0 and j Di

Amortization

Define Y = (j, i) : yij > 0 Bidding

raises prices atmost m / log (pmax) times adds atmost n new edges in Y

exhausts an edge in Y: n m / log (pmax) times reduces surplus to zero

atmost n2 times for every price rise Demand set computation

atmost m / log (pmax) times requires atmost nm steps

Requires 1/ (n m2 + m n2) log vmax) steps

Conclusions

Fast polynomial time algorithm for approximate market equilibrium – linear additive utilities

Decentralized auction algorithm No need to reveal private information Natural and practical Conceivable implementation in grid

economies using software agents

Future Work

Fast algorithm for exact market equilibrium (linear utilities)

Strongly polynomial time exact equilibrium algorithms

Greedy monotone price mechanisms Separable additive gross substitute utilities General gross substitute utilities

General (concave) utilities