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Lecture 17\ Multi Item auction
Lecture 17
Multi Item auction
October 09, 2002 transcribed by Dhan Mahesh
In this class we would be discussing Multi Item Auction.
There are n buyers and m sellers each buyer wants to buy only one item
out of the many available. Each seller has a single item to sell.
Figure
One of the scenarios is a market of secondhand cars where each seller
brings a car for sale. Buyers are interested in at most one car
each. Buyers have preferences over cars. Also depending on the price
of the car. So each buyer will rank car and assigns some value
to it.
For example:
Car Model  Value (in Lakhs)  Price (in Lakhs) 
Maruti 800  1.0  0.5 
Lancer  5.0  1.0 
Which car would a buyer prefer? How would his/her preference be represented?
v_{i,j} stands for the maximum a buyer i is willing to pay for item j.
p_{j} stands for the price at which item j can be bought.
Buyers may want to maximize their surplus : v_{i,j}  p_{j} or the
profit ratio v_{i,j}/p_{j}
Every seller has a reserve price r_{j} on item j. The reserve
price represents the minimum price that a seller expects for the
item and he/she would like to maximize his/her surplus that is given
by p_{j}r_{j}
If the utilities are quasi linear in money then the social optimal
for the economy is to maximize the total surplus.
Define variable x_{i,j} which is 1 when buyer i gets
item j and 0 otherwise since each buyer can buy only one item.
Also an item can be bought by atmost one buyer. Therefore
Surplus for buyer i can be given by (v_{i,j}p_{j})x_{i,j} .The total buyer surplus is given by å_{j} x_{i,j}(v_{i,j}p_{j})
Similarly , total seller surplus is given by å_{i} x_{i,j} (p_{j}r_{j})
Therefore the total surplus given by
å_{i} X_{i,j}(v_{i,j}p_{j}) + å_{j} å_{i} X_{i,j}(p_{j}r_{j}) 
= å_{i} å_{j} X_{i,j} (v_{i,j}r_{j}) 
Note that the total surplus is independent of the prices. Therefore the problem of effecient allocation is same as finding maximum
weight matching in a bi partite graph matching with edge weights
equal to v_{i,j}r_{j}. This is formally stated as follows:

max
x_{i,j}

(v_{i,j}  r_{j})x_{i,j} 

such that x_{i,j} Î {0,1} & 
å
i

x_{i,j} £ 1 & 
å
j

x_{i,j} £ 1 

Maximum weighted bipartite matching gives allocation but the problem
of how much buyer has to pay to buy that item remains unsolved. Even
to get allocation also, one has to get teh maximum price a buyer is
willing to pay for an item and its reserve price from seller. Would
they give their current prices?
VCG procedure will give solution but let us see an alternative solution.
In the special case when m=1 ( there is a single item) ascending english auction leads to efficient allocation.
Final price = Second highest value.
If two items and many buyers:
For example, two items and three buyers as below:
Figure
Let us assume reserve prices are zero.
Assume open cry ascending auction.
buyer 1 and buyer 3 competes with eachother for item 1 till Rs 5
and then buyer 1 can outbid buyer 3 by quoting some price u such
that 5 < u £ 10. Similarly in case of item 2, buyer 1 and
buyer 2 would be competing for item 2 till Rs 15 and then if
buyer 1 gets chance he/she has to decide which item to choose. It is
obvious that he/she would choose item 1 and gets it at some u (5 < u £ 10). So buyer 2 would get item 2 at some price y such that
10 < y £ 15.
In general what procedure will give us stable and efficient
allocation? That procedure should lead to stable matching ( Self enforcing)
Bidding Procedure/strategy
General Case: At every step buyer i places a bid on item j
such that v_{i,j}p_{j} is maximized and then rests until somebody
outbids him/her.This procedure goes on like this. What will happen if
we go like this (sooner or later, bidding stops as price will come to v_{i,j}
So, we are going to get with a matching and price. This procedure converges to (M,P) (Matching, Price).
Theorem 1 Ascending Price simultaneous auctions lead to nearly
efficient allocations.[ Bertsekas ]
Stability:
A situation where when an auction terminates no buyer is interested in switching.
An allocation X of a price [p\vec] to be stable if
"i,j Sb_{i}+ Ss_{j} ³ v_{i,j}  r_{j} where the Sb_{i}
stand for the surplus of buyer i in the allocation X and Ss_{j} is
the surplus of seller of item j that is
v_{i,k}  p_{k} + p_{j}  r_{j} ³ v_{i,j}  r_{j} 

If this condition is not satisfied then i & j can deviate from
current allocation & be both better of , destabilizing X.
Claim: (M,P) is stable matching.
Figure
seller y tries to break up only if he gets more surplus. Buyer A tries
to break up only if he gets more surplus.
(A,y) would destabilize the matching iff
v_{1,2}  (p_{2} +e) ³ v_{1,1}  p_{1} 

Þ v_{1,2}  p_{2} > v_{1,1}  p_{1} 

Let us see under what conditions we would get stable matching?
After A places bid on x, price of y can increase (or stay
unchanged ). So his/her surplus on y can't be more than on x at a
future time. He/she placed a bid on y of p_{i} which is less that p_{2}
v_{1,1}  p_{1} = v_{1,2}  p_{i} 

Þ v_{1,2}  p_{2} > v_{1,2}  p_{i} as p_{2} > p_{i} 

Þ v_{1,1}  p_{1} > v_{1,2}  p_{2} 

Lets see whether this leads to maximum surplus / efficient allocation ?
Assume number of buyers = number of items.
We would be getting a unique stable matching but lots of prices
possible.
Lets look at efficient allocation
Figure  Figure 
Fig 5(i): Our algo  Fig 5(ii): Optimal (given by some other algo)

So, each buyer will have 2 edges and each seller has 2 edges.
So, we would get some disjoint cycles. As there are equal number of
buyers and sellers, if we start from any buyer we will come back to
that buyer.
Now we have to prove that the sum of the weights picked by our algo is
atleast equal to that of Optimal.
Figure
v_{1,1}  p_{1} ³ v_{1,2}  p_{2} 

v_{2,2}  p_{2} ³ v_{2,3}  p_{3} 

v_{3,3}  p_{3} ³ v_{3,1}  p_{1} 

add them up.
v_{1,1} + v_{2,2} + v_{3,3} ³ v_{1,2} + v_{2,3} + v_{3,1} 

This is true for all the cycles. Therefore our algo will give the
correct maximum weighted bipartite matching.
If not all items are matched.
(assume for 3 buyers and 3 items)
Figure
v_{1,1}  p_{1} ³ v_{1,3}  p_{3} 

v_{2,2}  p_{2} ³ v_{2,1}  p_{1} 

v_{3,3}  p_{3} ³ v_{1,3}  p_{3} 

add them up
v_{1,1} + v_{2,2} + v_{3,3} ³ v_{3,2} + v_{2,1} + v_{1,3} 

but v_{1,3} = 0;
another case:
Figure
v_{2,2}  p_{2} ³ v_{2,1}  p_{1} 

v_{3,3}  p_{3} ³ v_{3,2}  p_{2} 

v_{2,2} + v_{3,3} ³ v_{2,1} + v_{3,2} + v_{4,3}  p_{1} 

(v_{2,2}  r_{2}) + (v_{3,3}  r_{3}) ³ (v_{2,1}  r_{1}) + (v_{3,2}  r_{2}) + (v_{4,3}  r_{3}) since p_{1} = r_{1} 

Therefore our allocation gives better results.
So we can say that this procedure gives better results which are
competitive with respect to buyers and are stable
By VCG:
Take for every item each buyer is willing to pay and all the sellers
will sit and give to one who bids the maximum. We exclude that
bidder and find the price.
There can be different prices but same stable matching.
This method would be applicable for recruitment (employers and
employees).
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On 26 Nov 2002, 12:38.