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Algorithmic Game Theory Polynomial Time Algorithms and Internet Computing For Market Equilibria Vijay V. Vazirani 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, 1776. ‘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 $ $$$$$$$$$ ¢ wine bread cheese milk $$$$ People want to maximize happiness – assume Findutilities. prices s.t. market clears linear Fisher’s Model n buyers, with specified money, m(i) for buyer i k goods (unit amount of each good) U u x Linear utilities: uij is utility derived by i on obtaining one unit of j Total utility of i, i u u x x [0,1] i ij j ij ij j ij ij Fisher’s Model n buyers, with specified money, m(i) k goods (each unit amount, w.l.o.g.) U u x Linear utilities: uij is utility derived by i on obtaining one unit of j Total utility of i, i u u x i j Find prices s.t. market clears, i.e., all goods sold, all money spent. ij ij j ij ij 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 ui : R R k Continuous, monotonic and strictly concave For any given prices and money m, there is a unique utility maximizing bundle for agent i. Arrow-Debreu Model Agents: Buyers/sellers Initial endowment of goods Agents Goods Prices = $25 = $15 = $10 Agents Goods Incomes Agents $50 $60 Goods Prices =$25 $40 $40 =$15 =$10 Maximize utility U i : ( x1 , x2 , xn ) R Agents $50 $60 Goods Prices =$25 $40 $40 =$15 =$10 Find prices s.t. market clears Agents $50 $60 Goods Prices =$25 $40 $40 =$15 =$10 Maximize utility U i : ( x1 , xn ) R 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 : 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 S R be a non-empty, compact, convex set Continuous function Then n f :S S x S : f ( x) x Brouwer’s Fixed Point Theorem Idea of proof f : k k 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) < d(j) => p(j) s(j) > d(j) => p(j) Also ensure p k Let f ( p) p ' S(j) < d(j) => p( j ) [d ( j ) s( j )] p '( j ) N S(j) > d(j) => p( j ) p '( j ) N N is s.t. p '( j ) 1 j i : ui is a cts. fn. => i : B(i ) is a cts. fn. of p => j : d ( j ) is a cts. fn. of p => f is a cts. fn. of p i : ui is a cts. fn. => i : B(i ) is a cts. fn. of p => j : d ( j ) is a cts. fn. of p => f is a cts. fn. of p By Brouwer’s Theorem, equilibrium prices exist. i : ui is a cts. fn. => i : B(i ) is a cts. fn. of p => j : d ( j ) 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 k Boundaries of k B(i) is not defined at boundaries!! Kakutani’s Fixed Point Theorem S Rn convex, compact set non-empty, convex, f : S 2S upper hemi-continuous correspondence x S s.t. x f ( x) Fisher reduces to Arrow-Debreu Fisher: n buyers, k goods AD: first n+1 agents n have money, utility for goods last agent has all goods, utility for money only. Money