Transcript Auctions -1

Auctions -2
Debasis Mishra
QIP Short-Term Course on Electronic Commerce
Indian Institute of Science, Bangalore
February 17, 2006
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Outline
Combinatorial Auctions
 The VCG Auction
 Iterative VCG Auctions

– One-to-one assignment
– General case: iBundle auction

Winner Determination Problem
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QIP Course on E-Commerce : Auctions- 2
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Combinatorial Auctions
Sale of multiple items simultaneously.
 Bidders can have non-additive values on items.
 Allows for bids on bundles of items.
 An example: a seller wants to sell two items:

– (a) a shopping complex in Goa
– (b) a shopping complex in Mumbai.

Two buyers with values:
– 5 for a, 6 for b, and 15 for a+b,
– 10 for a, 8 for b, and 10 for a+b.
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Applications in Practice
Spectrum wave auctions in different
countries: US, European nations, and
India too.
 Transportation lane auctions: London bus
routes, Home Depot in US.
 Airport time-slots by FAA (US).
 Sponsored search auctions by Google,
Yahoo!, and Microsoft.

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Efficient Mechanism Design
An economic and algorithmic view
 Input: “Bids” from buyers, Output: an
allocation (assignment of items to buyers)
and payments of buyers.
 Allocation – efficient allocation: one that
maximizes the total value of buyers or
total payoff/welfare of the system.
 Payment – to make money AND to give an
incentive to participate.
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Why are Incentives Important
Bidders often indulge in strategizing in “badly”
designed auctions – a fact from spectrum
auctions in US and Europe: leads to low revenue
and loss in efficiency.
 Strategizing is not easy in combinatorial
auctions.
 Incentive schemes makes strategies
straightforward – no need to strategize, just bid
in a straightforward manner. Leads to savings
for bidders in terms of “auction preparation
costs”.

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The Model
Set of items N={1,…,n}.
 Set of buyers M={1,…,m}.
 Bundles – any subset of items of N.
 Buyers have values on bundles: v(i,S)

– A1- S is a subset of T means v(i,S) <= v(i,T)
– A2- v(i,φ)=0
– A3 - if i pays amount p and gets bundle S, his
payoff is v(i,S) - p
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The Efficient Allocation (1 of 3)
An allocation is X: a partition of items in N to
bundles and assignment of these bundles to
buyers (buyers can get the empty bundle) – Xi is
bundle of i.
 Efficient allocation: one which maximizes
Σi v(i,Xi).
 An economy consists of buyers and their
valuations – E(B) is economy with buyers in B
(subset of M).
 E(M) is main economy and E(M-i) is a marginal
economy.

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The Efficient Allocation (2 of 3)
If X is an efficient allocation of E(B), it is a
disjoint partition of A but assigns bundles
to buyers in B only.
 If X is an efficient allocation of E(B),
denote V(B)= Σi v(i,Xi).
 For an integer programming formulation:

– y(i,S) = 1 if i is assigned bundle S and zero
otherwise.
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The Efficient Allocation (3 of 3)
Formulation (IP) for main economy
V(M)
= max ∑i,S v(i,S)y(i,S)
s.t.
∑S y(i,S)
=1
for all i in M,
∑i ∑S: j in S y(i,S) = 1
for all j in N,
y(i,S)
in {0,1}
for all i, S.

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The VCG Mechanism (1 of 2)
The Vickrey-Clarke-Groves (VCG) mechanism is
a sealed-bid auction.
 Allocation – Efficient allocation of main
economy.
 Payment - such that payoff of buyer i is V(M)V(M-i): marginal contribution of i

– Payment is: v(i,Xi)-[V(M)-V(M-i)], where X is efficient
allocation of main economy.

Bidding one’s true value as bid is a dominant
strategy.
– In many ways, it is a unique mechanism which is
efficient and has a dominant strategy.
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The VCG Mechanism (2 of 2)

Requires solving (IP) (m+1)
times in the worst case.
– (IP) is difficult to solve – NP Hard
(Rothkopf et al., 1998,
Management Science).
Requires every buyer to submit
an exponential-sized valuation
function.
 Example: 3 buyers, 2 items
 V(1,2)=8+8=16;
V(1)=12,V(2)=14.

1
2
a
b
a+b
8
6
9
8
12
14
– p1=8-[16-14]=6; p2=8-[16-12]=4
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An Easy Instance:
The Assignment Problem (1 of 2)

Every buyer is interested in at most one item.
– Values of buyers can be written as: v(i,j) value of
buyer i on item j.
(IP) reduces to following (LPA)
V(M)
= max ∑i,j v(i,j)y(i,j)
s.t.
∑j y(i,j) <= 1
for all i in M,
∑i y(i,j) <= 1
for all j in N,
y(i,j)
>= 0
for all i, j.

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An Easy Instance:
The Assignment Problem (2 of 2)
Dual of (LPA) is (DPA)
V(M)
= min ∑i qi + ∑j pj
s.t.
pj + qi >= v(i,j)
for all i, j,
pj >= 0 for all i, qj >= 0 for all j.
 Leonard (1983) showed that there exists an
optimal solution of (DPA) such that pj =VCG
payoff of buyer who is assigned item j in optimal
solution of (LPA) .

– This solution is obtained by maximizing ∑i qi over all
possible solutions of (DPA).
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Iterative Auctions
Iterative auctions – decentralized
implementation of sealed-bid auctions.
 Better preference elicitation – buyers can work
on bounds on valuations and no need to submit
the entire valuation function.
 More transparent – shown to generate more
revenue and efficiency (Cramton 1998, Eur.
Econ. Rev.).
 Popular in practice – English auction more
popular than the Vickrey auction.

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Linear Price Iterative Auction




For the assignment problem, pj denotes price on item j.
Price vector p – prices on items.
Demand set of buyer i at p: D(i,p)={j in N:v(i,j)- pj >=
v(i,k) – pk for all k in N} – payoff maximizing items.
An iterative auction (Demange et al., 1986):
– Start from zero price vector (or a low price). Initially no buyer is
assigned.
– A buyer bids when he is not assigned and his maximum payoff is
more than zero. A buyer bids (truthfully) by increasing the price
of an item in his demand set at the current price by ε.
– The auction stops when there is no bidding.

As ε approaches zero, this auction implements the VCG
outcome and truthful bidding is a (ex post) Nash
equilibrium for buyers.
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Example
Three buyers, two items:
v(1,1)=3, v(1,2)=4;
v(2,1)=2, v(2,2)=5;
v(3,1)=4,v(3,2)=4.
 At price (0,0) buyer 1 bids on item 2 to
make it (0,1). Now, buyer 3 bids on item 1
to make it (1,1). Now, buyer 2 bids on item
2 to make it (1,2) … will converge
approximately to (3,4).

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Complex Price Auctions

For general combinatorial auction settings, iterative
auctions require complex prices.
– Non-linear (every bundle has a price) and non-anonymous
(personalized prices for every buyer): p(i,S).
– The underlying theory for such complex prices can be found in
Bikhchandani and Ostroy (2002, J. of Econ. Theory).

Sometimes, such complex prices can be represented in a
simple way – when items are homogeneous and
marginal values on units/items are non-increasing
Ausubel (2004) designs an iterative auction that
maintains a single price but implicitly maintains nonlinear and non-anonymous prices (Bikhchandani and
Ostroy, 2005, Games and Econ. Behavior, Forthcoming)
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iBundle Auction (1 of 4)

First appeared in Parkes (1999, EC’99).
– Maintains non-linear and non-anonymous prices.

Given such a price vector p, demand set of buyer i is
D(i,p) = {all bundles S: v(i,S)-p(i,S) >= v(i,T) – p(i,T) for
all bundles T}
– payoff maximizing bundles at price p.

Supply set of seller at price p as L(p) = {all allocations X:
∑i p(i,Xi) >= ∑i p(i,Yi) for all allocations Y}
– revenue maximizing allocations.

Define L*(p)={all allocations X: X in L(p) and Xi in D(i,p)
or Xi = φ}: buyer compatible allocations in supply set.
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iBundle Auction (2 of 4)


Start from zero prices (or low prices).
At every iteration with price p:
– Collect demand sets of buyers at p.
– Find an allocation X (provisional allocation) from
L*(p)
 The auction ensures that L*(p) is not empty.
– Define losers in X as: O(X,p)={i: Xi notin D(i,p)}.
– If O(X,p) is empty go to last step. Else for every i in
O(X,p) and S in D(i,p) set p(i,S):=p(i,S) + 1 (this bid
increment is for convenience) and repeat.

The final allocation is final provisional allocation
X and payment of buyer i is p(i, Xi).
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iBundle Auction (3 of 4)

An example:
– v(1,a)= 4, v(1,b)=3, v(1,a+b)=6;
– v(2,a)=2, v(2,b) = 5, v(2,a+b)=8.

At price (0,0,0;0,0,0):
– D(1,p)=D(2,p)= {a+b}. L*(p)={1 gets a+b, 2 gets φ}.

Next price (0,0,0;0,0,1) where D( ) are unchanged.
– L*(p)={2 gets a+b, 1 gets φ}. Next price (0,0,1;0,0,1). This goes on …

Price reaches (0,0,2;0,0,2):
– D(1,p)={a,a+b}, D(2,p)={a+b}, L*(p)={1 gets a+b}.

Next price (0,0,2;0,0,3}:
– D(1,p)={a,a+b}, D(2,p)={b,a+b}, L*(p)={2 gets a+b}
– Next price (1,0,3;0,0,3} …

Finally, price reaches (3,2,5;0,2,5):
– D(1,p)={a,b,a+b}, D(2,p)={b,a+b}, L*(p)={1 gets a, 2 gets b}.
– Auction ends with 1 paying 3 and 2 paying 2.
– Note that these are VCG payments.
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iBundle Auction (4 of 4)



The fact that final payments in the iBundle auction is
VCG payment is no coincidence in the example.
Ausubel and Milgrom (2002, Frontiers of Theo. Econ.)
show that under a submodularity condition on V( ),
satisfied in the example, final payments in iBundle will
always be VCG payments.
For valuations that do not satisfy the submodular
condition, we have to apply iBundle for marginal
economies too and give discounts to buyers at the end
(Mishra and Parkes, 2005, J. of Econ. Theory,
Forthcoming).
– These discounts are marginal contribution of a buyer to the
revenue of the seller.
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Winner Determination Problem (1 of 2)
Computing L*(p) is called the winner
determination problem (WDP).
max ∑i ∑S in D(i,p) or φ p(i,S)y(i,S)
s.t.
∑S in D(i,p) or φ y(i,S)
=1
for all i in M,
∑S: j in S ∑i:S in D(i,p) y(i,S) = 1
for all j in
N,
y(i,S) in {0,1}
for all i, S in D(i,p) or φ.

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Winner Determination Problem (2 of 2)

(WDP) is NP-Hard
– Lot of research on quickly solving WDP
(Rothkopf et al. 1998, Tuomas Sandholm’s
papers).

Observe that (WDP) is a smaller version of
(IP).
– In iterative auctions, we solve smaller
instances of multiple (IP) instead of solving
one huge instance of (IP).
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Linear Programs and
Iterative Auctions
Close connection exists.
 Almost all iterative auctions can be
interpreted as a linear programming
algorithm to solve an appropriate linear
program (de Vries et al., 2005, J. of Econ.
Theory, Forthcoming).
 In particular, (almost) every auction in
literature is either a primal-dual algorithm
or a subgradient algorithm.

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Concluding Thoughts

Similar to ascending auctions, possible to design
descending auctions that implement the VCG outcome
(Mishra and Parkes, 2004a, 2004b).
– Descending auctions have better preference elicitation properties
than ascending ones.



Combinatorial auction design is difficult due to
complexity of the input.
Carefully handling the WDP stage should help
implement practical combinatorial auctions.
Besides incentives, stress should be given on simplicity
(unfortunately, simplicity and incentives are not
compatible) – simplicity in terms of prices, bidding
languages (how to submit bids).
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