comp325 algorithmic and game theoretic foundation for internet economics/xiaotie deng lecture 19...
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Comp325 Algorithmic and Game Theoretic Foundation for
Internet Economics/Xiaotie Deng
Lecture 19Matching Market Equilibrium
http://www.csc.liv.ac.uk/~deng/COMP325.html
Comp325 Algorithmic and Game Theoretic Foundation for
Internet Economics/Xiaotie DengModel
• Input:• Two sided Market– Sellers: M={1,2,…, m} each with one house to sell– Purchasers: N={1,2,…,n} each wants to buy one house
• iM values its house at • j N values house i at h(i,j)
Comp325 Algorithmic and Game Theoretic Foundation for
Internet Economics/Xiaotie DengUtilities
• Private Value of House , price and profit – , i=1,2,…, m– Seller’s utility:
• Buyer j’s utility on item i:– , for iM
Comp325 Algorithmic and Game Theoretic Foundation for
Internet Economics/Xiaotie DengIndividual Rationality
• A buyer j is proactive, and wants an item i that maximizes – h(i,j)- , for all iM
• The seller i is passive and may not sell to a buyer who offers the highest price. When it sells for , the utility is:– , i=1,2,…, m
Comp325 Algorithmic and Game Theoretic Foundation for
Internet Economics/Xiaotie DengDefine 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 DengExample
b1 b2 b3 b4
C1=5 C2=7 C4=8
h12=9
h22=8
h14=7
h43=9h41=8h11=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 DengSocial Welfare
• Surplus of j on i: a(i,j)=max{0, h(i,j)- )}• For each subset S of M∪N– v(S) = Maximum matching in S with weight a• the maximum value of the sum of values of houses by
matched pairs of sellers & buyers for a matching in S.
• It measures the maximum social surplus the sellers and buyers in S can achieve.
Comp325 Algorithmic and Game Theoretic Foundation for
Internet Economics/Xiaotie DengExample
b1 b2 b3 b4
C1=5 C2=7 C4=8
a21=4
a22=1
a41=2
a34=1a13=0a11=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 DengExample Simplied
b1 b2 b3 b4
C1=5 C2=7 C4=8
a12=4
a22=1
a14=2
a43=1a11=2
a13=1
a24=1
a44=1
C3=6
a32=2
a34=2
Comp325 Algorithmic and Game Theoretic Foundation for
Internet Economics/Xiaotie DengAn Integer Linear Program Model
• Model: Let x(i,j) be one if house i is purchased by buyer j, zero otherwise• //max social surplus• Subject to constraints:– ; each house sells to 1 buyer– ; each buyer gets one house
Comp325 Algorithmic and Game Theoretic Foundation for
Internet Economics/Xiaotie DengRelaxation as a Linear Program
• //max social surplus• Subject to constraints:– ; each house sells to one buyer– ; each buyer gets one house
Comp325 Algorithmic and Game Theoretic Foundation for
Internet Economics/Xiaotie DengExample
• Max:
• Subject to constraints:• ; ;• ;
– And• ; ;• ;
– All
Comp325 Algorithmic and Game Theoretic Foundation for
Internet Economics/Xiaotie DengPolynomial 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 DengDual LP• Min • Subject to constraints:– ; //stable solution//u: profit of seller, r: profit of buyer
Comp325 Algorithmic and Game Theoretic Foundation for
Internet Economics/Xiaotie Deng
Example Simplied
v1 v2 v3 v4
C1=5 C2=7 C4=8
a12=4
a22=1
a14=2
a43=1a11=2
a13=1
a24=1
a44=1
C3=6
a32=2
a34=2
u1 u2 u3 u4
Comp325 Algorithmic and Game Theoretic Foundation for
Internet Economics/Xiaotie DengExample• Min • Subject to constraints:– ;– ;
– All
Comp325 Algorithmic and Game Theoretic Foundation for
Internet Economics/Xiaotie DengDuality Conditions• LP duality: Let (x*;u*,r*) be primal and dual
optimal solutions.1. implies //pairwise stable2. implies // profit sharing3. Given i: 4. Given j:
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 DengDerive Duality Conditions• Given a matching equilibrium, set x*(i*,j*) for an
edge (i*,j*) in the matching. Set u* be p*-c*, and Set x*(i,j)=0, and otherwise.
• Therefore, x* is feasible for the primal LP• u* and r* are feasible for the dual LP
• =• Therefore, the duality conditions hold.
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 DengDecide on Pricing and Allocation• Let (x*;u*,r*) be primal and dual optimal solutions.• Corresponding equilibrium – Allocation x*• Item i is assigned to buyer j if x(i,j)=1
– pricing vector: c+u*• j pays if x(i,j)=1.
Comp325 Algorithmic and Game Theoretic Foundation for
Internet Economics/Xiaotie DengEquivalence with Market Equilibrium• Buyer’s rationality:– Each winner gets one of the maximum utility– Each loser has value less than every seller’s value.
• Market clearance:– Unsold items priced at the seller’s value.
Comp325 Algorithmic and Game Theoretic Foundation for
Internet Economics/Xiaotie Deng Buyer’s rationality at Equilibrium• If j* get i*, then x*(i*,j*)=1 and x(i,j*)=0 for .– The utility of is by duality.– The utility of j* on other items will be • Which is less than by dual LP feasibility condition
• If buyer j* does not get any, then – – Therefore, – By Duality Condition 4,
Comp325 Algorithmic and Game Theoretic Foundation for
Internet Economics/Xiaotie DengMarket clearance• If item i* is not sold, then • Therefore, • By Duality Condition 3, . Therefore,
Comp325 Algorithmic and Game Theoretic Foundation for
Internet Economics/Xiaotie DengTheorem• The optimal primal solution gives the allocation
and the dual solution gives the price vector for the items which is a market equilibrium; and vice versa.
Comp325 Algorithmic and Game Theoretic Foundation for
Internet Economics/Xiaotie Deng
The Core
Comp325 Algorithmic and Game Theoretic Foundation for
Internet Economics/Xiaotie DengThe core
• Given a member set F, and a value function (a function maps a subset to a real number)
• A set of solutions (: i F) is in core if – Sum(: iS)≥ )– and
– Sum(: iF)= )
Comp325 Algorithmic and Game Theoretic Foundation for
Internet Economics/Xiaotie DengAn 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 DengIn class exercises
• F={1,2,…n}
• Find a member of the core for the above game or prove the core is empty (no member in core).
Comp325 Algorithmic and Game Theoretic Foundation for
Internet Economics/Xiaotie DengExercise: 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.