Algorithmic Game Theory and Internet Computing Vijay V. Vazirani Polynomial Time Algorithms For Market Equilibria

1) History and Basic Notions

Markets

Stock Markets

Internet

Revolution in definition of markets

Revolution in definition of markets New markets defined by  Google  Amazon  Yahoo!  Ebay

Revolution in definition of markets Massive computational power available

Revolution in definition of markets Massive computational power available Important to find good models and algorithms for these markets

Adwords Market Created by search engine companies  Google  Yahoo!  MSN Multi-billion dollar market Totally revolutionized advertising, especially by small companies.

New algorithmic and game-theoretic questions Queries are coming on-line. Instantaneously decide which bidder gets it. Monika Henzinger, 2004: Find on-line alg. to maximize Google’s revenue.

New algorithmic and game-theoretic questions Queries are coming on-line. Instantaneously decide which bidder gets it. Monika Henzinger, 2004: Find on-line alg. to maximize Google’s revenue. Mehta, Saberi, Vazirani & Vazirani, 2005: 1-1/e algorithm. Optimal.

How will this market evolve??

The study of market equilibria has occupied center stage within Mathematical Economics for over a century.

The study of market equilibria has occupied center stage within Mathematical Economics for over a century. This talk: Historical perspective & key notions from this theory.

2). Algorithmic Game Theory Combinatorial algorithms for traditional market models

3). New Market Models Resource Allocation Model of Kelly, 1997

3). New Market Models Resource Allocation Model of Kelly, 1997 For mathematically modeling TCP congestion control Highly successful theory

A Capitalistic Economy Depends crucially on pricing mechanisms to ensure: Stability Efficiency Fairness

Adam Smith The Wealth of Nations 2 volumes, 1776.

Adam Smith The Wealth of Nations 2 volumes, ‘invisible hand’ of the market

Supply-demand curves

Leon Walras, 1874 Pioneered general equilibrium theory

Irving Fisher, 1891 First fundamental market model

Fisher’s Model, 1891 milk cheese wine bread ¢ $$$$$$$$$ $ $$$$ People want to maximize happiness – assume linear utilities. Find prices s.t. market clears

Fisher’s Model n buyers, with specified money, m(i) for buyer i k goods (unit amount of each good) Linear utilities: is utility derived by i on obtaining one unit of j Total utility of i,

Fisher’s Model n buyers, with specified money, m(i) k goods (each unit amount, w.l.o.g.) Linear utilities: is utility derived by i on obtaining one unit of j Total utility of i, Find prices s.t. market clears, i.e., all goods sold, all money spent.

Arrow-Debreu Model, 1954 Exchange Economy Second fundamental market model Celebrated theorem in Mathematical Economics

Kenneth Arrow Nobel Prize, 1972

Gerard Debreu Nobel Prize, 1983

Arrow-Debreu Model n agents, k goods

Arrow-Debreu Model n agents, k goods Each agent has: initial endowment of goods, & a utility function

Arrow-Debreu Model n agents, k goods Each agent has: initial endowment of goods, & a utility function Find market clearing prices, i.e., prices s.t. if  Each agent sells all her goods  Buys optimal bundle using this money  No surplus or deficiency of any good

Utility function of agent i Continuous, monotonic and strictly concave For any given prices and money m, there is a unique utility maximizing bundle for agent i.

Agents: Buyers/sellers Arrow-Debreu Model

Initial endowment of goods Agents Goods

Agents Prices Goods = $25 = $15 = $10

Incomes Goods Agents =$25 =$15 =$10 $50 $40 $60 $40 Prices

Goods Agents Maximize utility $50 $40 $60 $40 =$25 =$15 =$10 Prices

Find prices s.t. market clears Goods Agents $50 $40 $60 $40 =$25 =$15 =$10 Prices Maximize utility

Observe: If p is market clearing prices, then so is any scaling of p Assume w.l.o.g. that sum of prices of k goods is 1. k-1 dimensional unit simplex

Arrow-Debreu Theorem For continuous, monotonic, strictly concave utility functions, market clearing prices exist.

Proof Uses Kakutani’s Fixed Point Theorem.  Deep theorem in topology

Proof Uses Kakutani’s Fixed Point Theorem.  Deep theorem in topology Will illustrate main idea via Brouwer’s Fixed Point Theorem (buggy proof!!)

Brouwer’s Fixed Point Theorem Let be a non-empty, compact, convex set Continuous function Then

Brouwer’s Fixed Point Theorem

Idea of proof Will define continuous function If p is not market clearing, f(p) tries to ‘correct’ this. Therefore fixed points of f must be equilibrium prices.

Use Brouwer’s Theorem

When is p an equilibrium price? s(j): total supply of good j. B(i): unique optimal bundle which agent i wants to buy after selling her initial endowment at prices p. d(j): total demand of good j.

When is p an equilibrium price? s(j): total supply of good j. B(i): unique optimal bundle which agent i wants to buy after selling her initial endowment at prices p. d(j): total demand of good j. For each good j: s(j) = d(j).

What if p is not an equilibrium price? s(j) p(j) s(j) > d(j) => p(j) Also ensure

Let S(j) S(j) > d(j) => N is s.t.

is a cts. fn. => is a cts. fn. of p => f is a cts. fn. of p

is a cts. fn. => is a cts. fn. of p => f is a cts. fn. of p By Brouwer’s Theorem, equilibrium prices exist.

is a cts. fn. => is a cts. fn. of p => f is a cts. fn. of p By Brouwer’s Theorem, equilibrium prices exist. q.e.d.!

Bug??

Boundaries of

B(i) is not defined at boundaries!!

Kakutani’s Fixed Point Theorem convex, compact set non-empty, convex, upper hemi-continuous correspondence s.t.

Fisher reduces to Arrow-Debreu Fisher: n buyers, k goods AD: n+1 agents  first n have money, utility for goods  last agent has all goods, utility for money only.

Money