CSCI 3210: Computational Game Theory Market Equilibria: An - PDF document

11/16/20 CSCI 3210: Computational Game Theory Market Equilibria: An Algorithmic Perspective Ref: Ch 5 [AGT] Mohammad T . Irfan Email: mirfan@bowdoin.edu Web: www.bowdoin.edu/~mirfan Many of the slides are adapted from Vazirani's and Kleinberg-Tardos' textbooks. Market 1

11/16/20 Study of markets u General equilibrium (GE) theory u Seeks to explain the behavior of supply, demand and prices in an economy u Partial equilibrium vs GE Competitive equilibrium (CE) u AKA Walrasian equilibrium u Formal mathematical modeling of markets by Leon Walras (1874) u CE consists of prices and allocations u Equilibrium pricing: demand = supply u GE ⟹ CE, but CE ⇏ GE 2

11/16/20 Background u Good news u CE exists in Walrasian economy u Proved by Arrow and Debreu (1954) u Bad news u Existence proof is not algorithmic Arrow Debreu Background: why important? u 1 st Welfare Theorem u Any CE (Walrasian equilibrium) leads to a “ Pareto optimal ” allocation of resources Nobody can be better off without making somebody else worse off u 2 nd Welfare Theorem u Any Pareto opt outcome can be supported as a CE u Social justification u Let the competitive market do the work (everybody pursuing self-interest) u It will lead to Pareto optimality (socially optimal benefit) 3

11/16/20 Timeline u 1954 – 2001 u We are happy. Equilibrium exists. Why bother about computation? u Sporadic computational results u Eisenberg-Gail convex program, 1959 u Scarf’s computation of approximate fixed point, 1973 u Nenakov-Primak convex program, 1983 Today’s markets 4

11/16/20 Electronic marketplaces Need for algorithms u New types of markets u The internet market u Massive computational power available u Need to “compute” equilibrium prices u Effects of u Technological advances u New goods u Changes in the tax structure u Deng, Papadimitriou and Safra (2002)– Complexity of finding an equilibrium; polynomial time algorithm for linear utility case u Devanur, Papadimitriou, Saberi, Vazirani (2002) – polynomial time algorithm for Fisher’s linear case 5

11/16/20 Fisher economy u Irving Fisher (1891) u Mathematical model of a market Fisher's apparatus to compute equilibrium prices Fisher economy 6

11/16/20 Utility function utility amount of milk Utility function utility amount of bread 7

11/16/20 Utility function utility amount of cheese Total utility u Total utility of a “bundle” of goods = Sum of the utilities of individual goods 8

11/16/20 Easy problem u Prices given u What would be the optimal bundle of goods for a buyer? Bang-per-buck (BPB) Example: u 2 / p 2 > u 1 / p 1 > u 3 / p 3 p p p 1 2 3 Fisher market – setup u Multiple buyers, with individual budgets and utilities u Multiple goods, fixed amount of each good u Equilibrium/market-clearing prices u Each buyer maximizes utility at these prices u Buyers will exhaust their budgets u No excess demand or supply 9

11/16/20 Fisher’s linear case u Model parameters (what's given) u n divisible goods (1 unit each wlog) and n' buyers u e i = buyer i 's budget (integral wlog) u u ij = buyer i 's utility per unit of good j (integral wlog) u Linear utility functions u Want (not given): equilibrium allocations u x ij = amount of good j that i buys to maximize n ∑ utility u i ( x ) = u ij x ij j = 1 u No excess demand or supply Dual (proof later) u Want (not given): equilibrium/market- clearing prices u Prices: p 1 , p 2 , …, p n u After each buyer is assigned an optimal basket of goods ( x ij ’s) w.r.t. these prices, there's no excess demand or supply u x ij ’s at these prices: equilibrium/market-clearing allocations 10

11/16/20 Can we formulate an optimization routine? u Does LP work? u Anything else? Can we formulate this as an LP? u Just think about equilibrium allocations å åå = max u x ( ) max u x i ij ij i i j u Such that å " £ j x 1 ij i " ³ i j , x 0 ij 11

11/16/20 That LP doesn’t work ´ u x ( ) 2 u x ( ) u Utility functions and are i i equivalent ( x maximizes one iff x max other) u But… å + Maximize ( ) u x u x ( ) does not necessarily 1 i ¹ i 1 å + maximize 2 ( ) u x u x ( ) 1 i ¹ i 1 å " £ j x 1 ij i No LP formulation " ³ i j , x 0 ij is known! Main challenge Optimize Optimize Optimize buyer 1's utility buyer 2's utility buyer n's utility Global constraint: ∑ ∀ j x ij = 1 i Convert to a single optimization 12

11/16/20 Eisenberg-Gale Formulation of Fisher Market How to devise duals of nonlinear programs? Lagrange function KKT conditions 13

11/16/20 Review: Fisher’s linear case u Model parameters (what's given) u n divisible goods (1 unit each wlog) and n' buyers u e i = buyer i 's budget (integral wlog) u u ij = buyer i 's utility per unit of good j (integral wlog) u Linear utility functions u Want (not given): equilibrium allocations u x ij = amount of good j that i buys to maximize n ∑ utility u i ( x ) = u ij x ij j = 1 u No excess demand or supply Eisenberg-Gale convex program (1959) u Equilibrium allocations captured as u Optimal solutions to the Eisenberg-Gale convex program u Objective function u Money weighted geometric mean of buyers' utilities ∑ 1/ e i ∏ e i ∏ e i ∑ max( u i ) ⇔ max( u i ) ⇔ max e i log u i i i i i 14

11/16/20 Eisenberg-Gale convex program max $ & % log $ + %* , %* % * -+./&01 12 $ , %* ≤ 1, ∀/ % , %* ≥ 0, ∀9, / u Lagrange function : ,, ;, < = − $ & % log $ + %* , %* + $ ; * $ , %* − 1 + $ $ < %* −, %* % * * % % * KKT conditions ! " # "$ ∗ − + "$ u Stationary condition ∗ ∗ = ) $ 1 ∑ $ # "$ & "$ ∗ # "$ ∑ $ # "$ & "$ ∗ ≤ ) $ ! " ∗ ≤ 1, ∀1 u Primal feasibility . & "$ " ∗ ≥ 0, ∀4, 1 & "$ u Dual feasibility ∗ ≥ 0, ∀4, 1 ∗ , + "$ ) " u Complementary slackness ∗ . ∗ − 1 ∗ > 0 ⇒ . ∗ = 1 ) $ & "$ = 0 ⇔ ) $ & "$ " " ∗ −& "$ ∗ > 0 => + "$ ∗ = 0 ∗ + "$ = 0 ⇔ & "$ => BPB = total utility/budget 15

11/16/20 Does Eisenberg-Gail convex program work for Fisher market? u Prove: There exist market-clearing allocations (or prices) iff each good has an interested buyer (someone who gets positive utility for that good) u Theorem 5.1 (AGT pg. 107) u Prove that if each good has an interested buyer then 1. All goods are sold out 2. Each buyer spends all of their money while maximizing their utility u Prove reverse direction (iff) u Next assignment Example u 2 buyers, 1 good (1 unit of milk) Buyer 1 Buyer 2 Budget, e 1 = $100 Budget, e 2 = $50 u 11 = 10/unit of milk u 21 = 1/unit of milk utility utility amount of milk amount of milk x* 11 = ? x* 21 = ? 16

11/16/20 Solution u x 11 = 2/3, x 21 = 1/3 x represents x 11 obj. fun. x Solution u Why x 11 = 2/3, x 21 = 1/3? u Set price of milk = $150/unit x represents x 11 17

11/16/20 Primal-dual u p j = The price of good j at an equilibrium = Dual variable corresponding to the primal constraint for good j: ∑ x ij ≤ 1 i p j ’s: dual variables x ij ’s: primal variables Interesting properties u The set of equilibria is convex u Equilibrium prices are unique! u All entries rational => equilibrium allocations and prices rational 18