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.