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ECON0123 Week 5
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Week 5Matching markets and trading networks
EK Ch 10, 11
Matching markets
Bipartite matching
Preferences are represented as a link in a Bipartite graph
Agents
I i in w binary over
preferences objects
Objects
K ki km
Assignment
Milk tuksit
FielMileKolo
trekMIKEv0
alilkistulkki
set
Constricted
Perfect matching
When m=n, an assignment is a perfect matching find2objects
• Each node is connected by a link to the node it is assigned to acceptable
• No two nodes on the left are assigned to the same node on the right
Constricted sets:
Set
ofneighborsfromKofanodexfromI
µ set S Nls
Constricted
Assume n=m, if a Bipartite graph has no perfect matching, then it must contain a constricted set
Valuations and optimal assignments
The quality of the matching is the same of realised valuations
1265
7man
Quality
assignment
quality optimal
Bipartite matching is almost a special case of nding an optimal assignment, with valuations of 0 or
1
However, 0 is ambiguous and means that agents can be linked to unacceptable objects
Price- based decentralisation
• A planner who knows agents preferences may search for a perfect matching by making a
centralised computation
• A marker decentralises choices to agents, using prices to coordinate them
• A matching market adds valuations and prices to the assignment problem
Sellers
S s sn onehousepersale
Buyers Bibi bn wantsonehouse
, Value
gi's
housetoj ripO
i's sellsitat
theirhouseOand piso
p rice
valuation
of
j'spayoffVijpi
Preferred sellers:
j'spreferredseller maxvijpioplvii
i
izs
z z
zo z
ss s
t es
oo o
7s a
If each buyer gets the house from their preferred seller, a perfect matching is obtained with market-
clearing prices
to3,2
i If
S
san
thereisa seti.e
constricted
I
demand
excess for aso
foraperfectmatchingrequires
Tpa
Market clearing prices
Existence: for any set of buyer valuations, there exists a set of market-clearing prices
Optimality: for any set of market-clearing prices, a perfect matching in the preferred-seller graph
gives the maximum total valuation of any assignment of buyers to sellers
Proof:
◦ In the preferred-seller graph every buyer j is maximising their individual pay o , so the total
buyers’ payo is maximum
◦ The buyer’s total payo is equal to their total valuation minus the ∑ of prices, and the prices
do not depend on matching
Auction-like procedure to nd market-clearing prices:
• Sellers initially set prices at 0
• Buyers choose their preferred sellers
◦ If the preferred-seller graph has a perfect matching set we are done
◦ If not there is a constricted set K of buyers, then the number of sellers in N(K) is strictly
smaller than the number of buyers in K
• All the “rare” sellers in N(K) raise their prices by 1
• Iterate until a perfect matching is found
• If at some point all prices are positive, we apply a price reduction and subtractmin from each
pi
price and bring the lowest price down to zero
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