Market Com- munication in Production Economies Market Communication Christopher Wilkens in Communication Complexity Production Economies Market Communication Markets Arrow-Debreu Markets Christopher Wilkens Related Work DPS 2002 Large Endowments UC Berkeley Main Result Conclusion WINE ’10
A Communication Bound for Market Equilibrium Market Com- munication in Production Economies Christopher Wilkens In a market with m goods, m prices are sufficient to Communication communicate an equilibrium... Complexity Market Communication Markets Arrow-Debreu Markets Related Work DPS 2002 Large Endowments Main Result Conclusion
A Communication Bound for Market Equilibrium Market Com- munication in Production Economies Christopher Wilkens In a market with m goods, m prices are sufficient to Communication communicate an equilibrium... Complexity BUT we show the number of bits of information must be Market Communication polynomial in Markets Arrow-Debreu m , Markets Related Work n , the number of agents, and DPS 2002 Large l , the number of production firms. Endowments Main Result Conclusion
Communication Complexity Market Com- munication in Production Economies Christopher Yao’s 2-party model: Wilkens Alice has a private, n -bit string a Communication Complexity Bob has a private, n -bit string b Market Alice and Bob want to compute f ( a , b ) Communication Markets How many bits of communication are required to compute Arrow-Debreu f ( a , b )? Markets Related Work DPS 2002 Large Endowments Main Result Conclusion
Communication Complexity Market Com- munication in Production Economies Christopher Yao’s 2-party model: Wilkens Alice has a private, n -bit string a Communication Complexity Bob has a private, n -bit string b Market Alice and Bob want to compute f ( a , b ) Communication Markets How many bits of communication are required to compute Arrow-Debreu f ( a , b )? Markets Related Work DPS 2002 Example: Alice and Bob have strings a and b . Are a and b Large Endowments different? Main Result Conclusion
The Market Equilibrium Story Market Com- munication in Production Economies Christopher Wilkens ...but market equilibrium doesn’t fit this pattern: Communication Complexity Market Communication Markets Arrow-Debreu Markets Related Work DPS 2002 Large Endowments Main Result Conclusion
The Market Equilibrium Story Market Com- munication in Production Economies Christopher Wilkens ...but market equilibrium doesn’t fit this pattern: 1 The “Invisible hand” publishes prices. Communication Complexity Market Communication Markets Arrow-Debreu Markets Related Work DPS 2002 Large Endowments Main Result Conclusion
The Market Equilibrium Story Market Com- munication in Production Economies Christopher Wilkens ...but market equilibrium doesn’t fit this pattern: 1 The “Invisible hand” publishes prices. Communication Complexity 2 Agents sell endowment to the market. Market Communication Markets Arrow-Debreu Markets Related Work DPS 2002 Large Endowments Main Result Conclusion
The Market Equilibrium Story Market Com- munication in Production Economies Christopher Wilkens ...but market equilibrium doesn’t fit this pattern: 1 The “Invisible hand” publishes prices. Communication Complexity 2 Agents sell endowment to the market. Market Communication Markets 3 Firms maximize profit. Arrow-Debreu Markets Related Work DPS 2002 Large Endowments Main Result Conclusion
The Market Equilibrium Story Market Com- munication in Production Economies Christopher Wilkens ...but market equilibrium doesn’t fit this pattern: 1 The “Invisible hand” publishes prices. Communication Complexity 2 Agents sell endowment to the market. Market Communication Markets 3 Firms maximize profit. Arrow-Debreu Markets 4 Agents buy optimal bundle given market prices. Related Work DPS 2002 Large Endowments Main Result Conclusion
The Market Equilibrium Story Market Com- munication in Production Economies Christopher Wilkens ...but market equilibrium doesn’t fit this pattern: 1 The “Invisible hand” publishes prices. Communication Complexity 2 Agents sell endowment to the market. Market Communication Markets 3 Firms maximize profit. Arrow-Debreu Markets 4 Agents buy optimal bundle given market prices. Related Work DPS 2002 Large 5 The market clears. Endowments Main Result Conclusion
Market Communication Market Com- munication in Production Deng, Papadimitriou, and Safra (STOC 2002) Economies Setting: Christopher Wilkens n agents, each agent i ∈ [ n ]... Communication has private information x i ∈ X i , and Complexity must compute f i ( x 1 , . . . x n ). Market Communication Markets Agent 0, the“invisible hand,” knows ( x 1 , . . . x n ). Arrow-Debreu Markets Related Work DPS 2002 Large Endowments Main Result Conclusion
Market Communication Market Com- munication in Production Deng, Papadimitriou, and Safra (STOC 2002) Economies Setting: Christopher Wilkens n agents, each agent i ∈ [ n ]... Communication has private information x i ∈ X i , and Complexity must compute f i ( x 1 , . . . x n ). Market Communication Markets Agent 0, the“invisible hand,” knows ( x 1 , . . . x n ). Arrow-Debreu Markets Protocol: Related Work DPS 2002 1 Agent 0 computes x 0 = g 0 ( x 1 , . . . x n ) Large Endowments Main Result 2 Agent 0 broadcasts x 0 Conclusion 3 Each agent i computes f i ( x 1 , . . . , x n ) = g i ( x 0 , x i )
Market Communication Market Com- munication in Production Economies Christopher Wilkens Definition Communication A market communication protocol is a set of functions Complexity Market ( g 0 ( · ) , g 1 ( · ) , . . . g n ( · )) where g 0 : X 1 × . . . X n → X 0 , and Communication Markets g i ( g 0 ( x 1 , . . . x n ) , x i ) = f i ( x 1 , . . . x n ). Arrow-Debreu Markets The amount of market communication is the size of Related Work DPS 2002 x 0 = g 0 ( x 1 , . . . x n ). Large Endowments Main Result Conclusion
An Arrow-Debreu Market Market Com- munication in Production Economies m tradable, divisible goods Christopher n agents with... Wilkens utility functions Communication u i : R m + → R Complexity Market Communication endowment e i ∈ R m + Markets Arrow-Debreu Markets Related Work DPS 2002 Large Endowments Main Result Conclusion
An Arrow-Debreu Market Market Com- munication in Production Economies m tradable, divisible goods Christopher n agents with... Wilkens utility functions Communication u i : R m + → R Complexity Market Communication endowment e i ∈ R m + Markets l production firms... Arrow-Debreu Markets Related Work with a set of production possibilities DPS 2002 Large Endowments Y k ⊂ R m Main Result Conclusion agent i owns a σ i , k share of firm k
Market Equilibrium Market Com- munication in Production Equilibrium is tuple ( x , y , π ) such that Economies x i is an optimal consumption bundle: Christopher Wilkens x i ∈ argmax x | 0 ≤ π · x ≤ M i u i ( x ) Communication Complexity where M i = π · e i + � k σ i , k ( π · y k ) Market Communication y k is an optimal production plan: Markets Arrow-Debreu Markets y k ∈ argmax y ∈ Y k π · y Related Work DPS 2002 Large Endowments The market clears: Main Result Conclusion � � � 0 ≤ x i ≤ e i + y k i i k
Related Work Market Com- munication in Production Economies Dimensionality of message spaces Christopher Wilkens Economists’ analog to communication complexity m − 1 real numbers (prices) are optimal for representing a Communication Complexity market equilibrium Market Calsamiglia (1988) — parametric communication Communication Markets (analogous to market communication) does not help. Arrow-Debreu Markets Nisan and Segal (2006) — “prices” are required to Related Work DPS 2002 communicate an allocation Large Endowments Main Result Deng, Papadimitriou, and Safra (2002) — Market Conclusion communication
DPS 2002 Market Com- munication in Production Theorem (Deng, Papadimitriou, and Safra, STOC 2002) Economies Christopher If the non-satiation requirement for utilities is relaxed, Wilkens communicating an exchange equilibrium in the market Communication communication model requires Complexity Market Communication Ω ( n log( m + n )) Markets Arrow-Debreu Markets Related Work bits of communication. DPS 2002 Large Endowments Main Result Conclusion
DPS 2002 Market Com- munication in Production Theorem (Deng, Papadimitriou, and Safra, STOC 2002) Economies Christopher If the non-satiation requirement for utilities is relaxed, Wilkens communicating an exchange equilibrium in the market Communication communication model requires Complexity Market Communication Ω ( n log( m + n )) Markets Arrow-Debreu Markets Related Work bits of communication. DPS 2002 Large Endowments Weaknesses: Main Result Conclusion Relaxes non-satiation requirement. Proof uses Ω( n m )-bit numbers in players’ endowments.
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