# Comp325 Algorithmic and Game Theoretic Foundation for Internet Economics/Xiaotie Deng Lecture 19 Matching Market Equilibrium

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Comp325 Algorithmic and Game Theoretic Foundation for Internet Economics/Xiaotie Deng Lecture 19 Matching Market Equilibrium http://www.csc.liv.ac.uk/~deng/COMP325.html

Comp325 Algorithmic and Game Theoretic Foundation for Internet Economics/Xiaotie Deng Model

Comp325 Algorithmic and Game Theoretic Foundation for Internet Economics/Xiaotie Deng Utilities

Comp325 Algorithmic and Game Theoretic Foundation for Internet Economics/Xiaotie Deng Individual Rationality

Comp325 Algorithmic and Game Theoretic Foundation for Internet Economics/Xiaotie Deng Define Market Equilibrium Equilibrium corresponds to an allocation (a matching assignment x*) and a price vector p* such that – Individual rationality: each buyer gets the item achieves its highest utility: its own value minus price. – Market clearance: all items priced at more than reserved price are sold.

Comp325 Algorithmic and Game Theoretic Foundation for Internet Economics/Xiaotie Deng Example b1 b2 b3b4 C1=5 C2=7 C4=8 h12=9 h22=8 h14=7 h43=9 h41=8 h11=7 h21=5 h13=6 h24=8 h44=9 C3=6 h32=8 h34=8

Comp325 Algorithmic and Game Theoretic Foundation for Internet Economics/Xiaotie Deng Maximum Total Social Surplus

Comp325 Algorithmic and Game Theoretic Foundation for Internet Economics/Xiaotie Deng Social Welfare

Comp325 Algorithmic and Game Theoretic Foundation for Internet Economics/Xiaotie Deng Example b1 b2 b3b4 C1=5 C2=7 C4=8 a21=4 a22=1 a41=2 a34=1 a13=0 a11=2 a12=0 a31=1 a42=1 a44=1 C3=6 a23=2 a43=2

Comp325 Algorithmic and Game Theoretic Foundation for Internet Economics/Xiaotie Deng Example Simplied b1 b2 b3b4 C1=5 C2=7 C4=8 a12=4 a22=1 a14=2 a43=1 a11=2 a13=1 a24=1 a44=1 C3=6 a32=2 a34=2

Comp325 Algorithmic and Game Theoretic Foundation for Internet Economics/Xiaotie Deng An Integer Linear Program Model

Comp325 Algorithmic and Game Theoretic Foundation for Internet Economics/Xiaotie Deng Relaxation as a Linear Program

Comp325 Algorithmic and Game Theoretic Foundation for Internet Economics/Xiaotie Deng Example

Comp325 Algorithmic and Game Theoretic Foundation for Internet Economics/Xiaotie Deng Polynomial time solvability Theorem: There is a polynomial time algorithm for finding the social optimum. Proof: Two different ways to solve it. 1.The problem is a weighted bipartite matching problem, which can be solved in polynomial time. 2.The relaxed problem is a linear program which can be solved in polynomial time. Since the linear program is unimodular, it always has an integer solution. Unimodular matrix has its determinant equal to zeor or plus or minus one.

Comp325 Algorithmic and Game Theoretic Foundation for Internet Economics/Xiaotie Deng Dual LP

Comp325 Algorithmic and Game Theoretic Foundation for Internet Economics/Xiaotie Deng Example Simplied v1 v2 v3v4 C1=5 C2=7 C4=8 a12=4 a22=1 a14=2 a43=1 a11=2 a13=1 a24=1 a44=1 C3=6 a32=2 a34=2 u1u2u3u4

Comp325 Algorithmic and Game Theoretic Foundation for Internet Economics/Xiaotie Deng Example

Comp325 Algorithmic and Game Theoretic Foundation for Internet Economics/Xiaotie Deng Duality Conditions

Comp325 Algorithmic and Game Theoretic Foundation for Internet Economics/Xiaotie Deng Market Equilibrium Satisfies Duality Condition

Comp325 Algorithmic and Game Theoretic Foundation for Internet Economics/Xiaotie Deng Derive Duality Conditions

Comp325 Algorithmic and Game Theoretic Foundation for Internet Economics/Xiaotie Deng Optimal Primal Dual Solution Implies Equilibrium

Comp325 Algorithmic and Game Theoretic Foundation for Internet Economics/Xiaotie Deng Decide on Pricing and Allocation

Comp325 Algorithmic and Game Theoretic Foundation for Internet Economics/Xiaotie Deng Equivalence with Market Equilibrium Buyers rationality: – Each winner gets one of the maximum utility – Each loser has value less than every sellers value. Market clearance: – Unsold items priced at the sellers value.

Comp325 Algorithmic and Game Theoretic Foundation for Internet Economics/Xiaotie Deng Buyers rationality at Equilibrium

Comp325 Algorithmic and Game Theoretic Foundation for Internet Economics/Xiaotie Deng Market clearance

Comp325 Algorithmic and Game Theoretic Foundation for Internet Economics/Xiaotie Deng Theorem

Comp325 Algorithmic and Game Theoretic Foundation for Internet Economics/Xiaotie Deng The Core

Comp325 Algorithmic and Game Theoretic Foundation for Internet Economics/Xiaotie Deng The core

Comp325 Algorithmic and Game Theoretic Foundation for Internet Economics/Xiaotie Deng An Example Core F={f1,f2,f3,f4} v({i})=0, v({i,j})=1, v({I,j,k})=1, v(F)=2. Define x: x(i)=1/2. Then x is a member of the core for F.

Comp325 Algorithmic and Game Theoretic Foundation for Internet Economics/Xiaotie Deng In class exercises

Comp325 Algorithmic and Game Theoretic Foundation for Internet Economics/Xiaotie Deng Exercise: Prove Market equilibrium conditions Prices are not unique – Profits are obtained from the core – Dependent on the solutions for the core, we have different profits for the members in the maximum matching. – Price is equal to reserved price plus profit, which is different if the profit is different.

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