Combinatorial Auctions
(Bidding and Allocation)
Adapted from Noam Nisan
What is a Combinatorial
Auction?
• Set of Products:
• Each customer can bid:
$700 for {
AND
$1200 for {
$6 for {
}
} OR $8 for {
} XOR $30 for {
$3 for {ANY 3}
}
}
The Model
• m items for sale: X = {x1,…,xm}
• n bidders: v1,…,vn
Every vi is a valuation of subsets of X:
vi : 2X  R+
vi(S ) = how much would I pay for S  X
• Bids are handed in sealed envelops.
• Auctioneer allocates the items among bidders
trying to maximize its revenue:
Find n disjoint sets - S1,…,Sn  X s.t.
vi(Si) is maximal.
Issues
• Bidding
– Expressiveness
– Simplicity
• Allocation
– Hardness Results
– Approximation Algorithms
• Payment
• Strategy
Bidding
• No Externalities
v (S ) depends only on S.
• Free Disposal
S T  v (S )  v (T )
• Normalization
v () = 0
Bidding
• Let S and T be disjoint item subsets.
We say S and T are:
– Complementary, if:
v (S T ) > v (S ) + v (T )
– Substitutes, if:
v (S T ) < v (S ) + v (T )
Bidding - Examples
• The additive valuation
– v (S ) = |S |
– No substitutabilities and no complementarities.
• The single item valuation
– Want to buy just one item.
– v (S ) = 1
(iff S is not empty)
– All items are substitutes of each other.
• The K-budget valuation
– Want to buy up to K items.
– v (S ) = min {K, |S |}
Bidding - Examples
• The majority valuation
– Want to buy most of the items.
– v (S ) = 0
for |S | < m / 2
1
otherwise
• Symmetric valuation
– Let p1,…,pn be non-negative numbers.
v (S ) = j=1…|S| pj
– Additive: pj=1
– K-budget: pj= 0
for j > K
1
otherwise
– Majority: pj= 1
for j = m/2
0
otherwise
Bidding - Examples
• Downward sloping symmetric valuation
– A symmetric valuation with
p1  p2  …  pn  0
– Is viewed as the “normal” economic case.
Bidding - Asymmetric Examples
• The monochromatic valuation
– m/2 red items, m/2 blue items.
– Want to buy items of just one color.
– For S having k reds and l blues,
v (S ) = max{k,l }
• One-of-each-kind valuation
– m/2 pairs of items.
– Want to buy just one item from each pair.
– For S having k pairs and l singletons (|S |=2k+l),
v (S ) = k +l
Bidding Languages
• A simple language:
– Specify v explicitly as a 2m vector.
– Impractical - bids are too big.
• Language must allow:
– To express any “reasonable” valuation with
polynomial (in m) size expressions.
– To be computationally easy: given v and S,
compute v (S ) in polynomial time.
• The applet-language:
– Specify v as a computer program.
– Doesn’t allow efficient allocation algorithms.
Basic Bidding Languages
• Atomic bids
– v = (S,p)
v (T ) = p
if T S
0
otherwise
– Can’t represent the additive valuation.
• OR bids
– v = (S1,p1) OR (S2,p2) OR … OR (Sk,pk)
– If Si and Sj are disjoint, v (Si Sj) = pi + pj
– Can express all bids with no substitutabilities
and only them.
Basic Bidding Languages
• XOR bids
– v = (S1,p1) XOR (S2,p2) XOR … XOR (Sk,pk)
– v (S ) = max pi s.t. S  Si
– Can express all bids.
– The additive valuation requires 2m atoms in
XOR language but only m atoms in OR language.
OR-of-XOR
• v = u1 OR u2 OR … OR uk
each ui is a XOR bid.
• The bidder is willing to obtain any number of u-s
for the sum of their prices.
• Downward sloping bid:
[({x1},p1) XOR … XOR ({xm},p1)] OR
[({x1},p2) XOR … XOR ({xm},p2)] OR
…
[({x1},pm) XOR … XOR ({xm},pm)]
OR-of-XOR
• Theorem: The monochromatic valuation requires
an (2m/2) size OR-of-XOR expression.
• Proof:
– W.L.O.G. every (S,p) is monochromatic.
– p = |S |
– Can’t have a blue atom in one clause and a red
atom in another.
– All atoms must be in one XOR clause.
– The additive valuation on m/2 red items
requires XOR of 2m/2 atoms.
XOR-of-OR
• v = u1 XOR u2 XOR … XOR uk
each ui is an OR bid.
• The bidder is willing to obtain the maximal u.
• Monochromatic bid:
(OR over all reds) XOR (OR over all blues)
• Theorem: Fix K = m/2. The K-budget valuation
1/4
requires an (2m ) size XOR-of-OR expression.
OR/XOR Formulae
• Definition: Let v and u be valuations. Then,
– (v XOR u)(S) = max{v(S),u(S)}
– (v OR u)(S) = max{v(R)+u(T)|RT=, RT=S}
• Stronger than OR-of-XOR  XOR-of-OR. E.g.:
v = (monochromatic on m/2 items) OR
(K-budget on m/2 items)
OR Bids with Phantom Items
• OR* bids (Fujishima et al.): Each bidder submits
an OR bid whose atoms (S,p) may introduce new
(phantom) items.
• Phantom items are used to express constraints,
e.g.:
(S1,p1) XOR (S2,p2) =
(S1{g},p1) OR (S2{g},p2)
• Theorem: Any OR/XOR formula of size s can be
rewritten as an OR* formula of size s and O(s 2)
phantom items.
The OR* Language
• Theorem: The majority valuation requires at least
(mm/2) atoms in the OR* language.
• Proof:
– (S,p) with p > 0 must have at least m/2 real
items.
– Every subset of m/2 real items must appear as
an atom (possibly with phantom items).
• Open problem: Is OR* strictly stronger than
OR/XOR?
• OR* can express externalities.
Bidding and Computability
• Definition 1: A bidding language is polynomially
interpretable if there exists a polynomial
algorithm receiving a bid b in the language and a
subset S as input, and outputs b (S ).
• Only Atomic and XOR are polynomially
interpretable.
• Definition 2: A bidding language is polynomially
verifiable if there exists an NP algorithm
receiving a bid b in the language, a subset S and a
proof w of a lower bound on b (S ).
Allocation
• Bids are given in OR*.
• Auctioneer can treat them as one OR* bid:
{Bi}i=1..n , where Bi=(Si,pi) is an atomic bid.
• Algorithmically – no difference between real and
phantom items.
SPP - Hardness
• SPP – Set Packing Problem
• Is equivalent to Max-Clique and MaxIndependent-Set with weighted vertices.
• Is approximable within O(n/log2n)
• Not approximable within n1/2- for any >0.
• Not approximable within n1- for any >0, unless
NP=ZPP.
Integer Programming
• Formalization of the allocation problem:
Maximize:
i=1…n xipi
Subject to:
i|jSj xi  1, for each j=1…m
xi{0,1} , for each i=1…n
• Relaxation to linear programming:
xi  0, for each i=1…n
Fractional Auctions
• Example: communication lines for sale
B1 = ({TA-Paris, Paris-NY, P}, 10)
B2 = ({TA-London, London-NY, P}, 10)
B1 is 1/3 winning and B2 is 2/3 winning.
Can use B1 for a 1/3 of its bandwidth, and
B2 – for 2/3 of its bandwidth.
Single Item Prices
• An allocation x1,…,xn is supported by single item
prices y1,…,yn if:
– For every winning bid (xi=1), pi  jSi yj
– For every losing bid (xi=0), pi  jSi yj
• The allocation is exactly supported if for every
winning bid, pi = jSi yj
• If every item belongs to some winning bid, it is a
full allocation.
Single Item Prices
• An auction admits single item prices if it has a full
allocation supported by single item prices.
• Theorem:
An auction admits single item prices
 The linear program produces {0,1} solutions.
Then the supporting prices are the solutions to
the dual linear program.
Single Item Prices - Example
• Bidder #1:
({A,P},5)
({A},5) XOR
OR ({B,P},6)
({B},6)
• Bidder #2:
({B},3)
• Bidder #1 wins A for 5$.
Bidder #2 wins B for 3$.
• Supporting prices: A = 2$, B = 3$, P = 3$
Cases Where LP Relaxation Is
Optimal
• Linear Order Bids
The items can be linearly ordered, G = {g1,…,gm},
such that all bids are for sub-ranges, S = {gk,…,gl}.
– Hierarchical Bids
All sets form a nested hierarchy.
 Every two bids S, T are either disjoint or
one contains the other.
• OR-of-XORs of Singleton Bids
– Single Item Bids
– Downward Sloping Symmetric Bids