Lecturer: Moni Naor

1
Algorithmic Game Theory Uri Feige Robi
Krauthgamer Moni NaorLecture 11
Combinatorial Auctions

• Lecturer Moni Naor

2
Announcements

• Course resumes to 1400-1600

3
The setting

• Set of alternatives A
• Who wins the auction
• Which path is chosen
• Who is matched to whom
• Each participant a value function viA ? R
• Can pay participants valuation of choice a with
  payment pi is vi(a)pi

Quasi linear preferences
4
Mechanism Design

• Mechanisms
• Recall We want to implement a social choice
  function
• Need to know agents preferences
• They may not reveal them to us truthfully
• Example
• One item to allocate
• Want to give it to the participant who values it
  the most
• If we just ask participants to tell us their
  preferences may lie
• Can use payments result is also a payment vector
  p(p1,p2, pn)

5
The Revelation Principle
Theorem if there exists an arbitrary mechanism
implementing a social choice function f in
dominant strategies, then there exists an
incentive compatible mechanism that implements
f The payments of the players in the incentive
compatible mechanism are identical to those
obtained at equilibrium in the original
mechanism Proof by simulation
6
Direct Characterization

• A mechanism is incentive compatible iff the
  following hold for all i and all vi
• The payment pi does not depend on vi but only on
  the alternative chosen f(vi, v-i)
• the payment of alternative a is pa
• The mechanism optimizes for each player
• f(vi, v-i) 2 argmaxa (vi(a)-pa)

7
Expected Revenues

• Theorem Revenue Equivalence under very general
  conditions, every two Bayesian Nash
  implementations of the same social choice
  function
• if for some player and some type they have the
  same expected payment then
• All types have the same expected payment to the
  player
• If all player have the same expected payment the
  expected revenues are the same

Not true when there are reservation prices!
8
Range Voting

• Each voter ranks the candidates in a certain
  range (say 0-99)
• The votes for all candidates are summed up and
  the one with highest total score wins
• Can be considered as a generalization of approval
  voting from the range 0-1
• No incentive for voter to rate a candidate lower
  than a candidate they like less.

9
Vickrey Clarke Grove Mechanism

• f(v1, v2, vn) maximizes ?i vi(a) over A
• Maximizes welfare
• There are functions h1, h2, hn where
• hi V1 ? V2 ? ? Vn ? R does not depend on vi
• we have that
• pi(v1, v2, vn) hi(v-i) - ?j ? i vj(f(v1, v2,
  vn))
• Clark Pivot rule Choosing hi(v-i) maxb 2 A ?j
  ? i vj(b)
• Payment of i when af(v1, v2,, vn)
• pi(v1, v2, vn) maxb 2 A ?j ? i vj(b) - ?j ? i
  vj(a)

Depends only on chosen alternative
Does not depend on vi
Social welfare when he does not participate
Social welfare (of others) when he participates
10
Examples of Auctions

• Sponsored Search
• Buying keywords (Google, Yahoo, MS)
• Spectrum Auctions
• Ebay
• Government procurement
• Privatization

FCC Auction 2008
11
????? ???
???? ????? ???'? ???? ?-3.8 ???????, ???? ????
?-4.11 ???????, ?????.

• ???? "????", 29 ???? 2005
• ??? ?????? ??????? ????? ????? ???????-???,
  ?????? ???'?.
• ?????? ????????? - ???'? 3.2 ???????, ??? 3.6
  ??????? ?"?
• ?????? ?????? ?? ???? ????? ????? ?? ??????,
  ????? ????? ?????
• ????? ???? ????? ????.
• ???? ???? ????? ??? ??? ?????? ????? ???? ??????
  ??? ?????? ??????, ??? ???'? ?????? ??????.
• ????? ???????, ????' ???? ??? ????????, ?????
  ?????? ??? ?????? "??? ?????? ???, ??? ???? ???
  ??????? ????? ??????? ?????? ?????. ?? ????? ???
  ??? ????????? ?? ?????? ???? ??????... ?????
  ??????? ??? ????? ????? ????? ????? ?????
  ???????, ??? ?? ??? ??????? ???? ????? ?? ????
  ????? ?? ?? ?????. ??? ???????? ?? ???? ???, ???
  ?????? ???? ?????, ????? ?????? ????? ????, ???
  ?????? ?????? ?????? ?????? ????".

From Ron Lavis slides
12
VCG turning mechanisms truthfulUsing exact
optimization
Exact optimization algorithms for social welfare f
VCG prices
truthful mechanism
Strategic players
13
VCG and Computational Complexity

• Several optimization must be performed in VCG
• f(v1, v2, vn) maximizes ?i vi(a) over A
• Clark Pivot rule hi(v-i) maxb 2 A ?j ? i vj(b)
• These may be hard optimization problems.
• What happens when we replace then with
  approximations?
• Incentive compatibility not guaranteed!

14
Combinatorial Auctions

• Set M of m indivisible items
• Set N of n bidders
• Preferences are on subsets S µ M bundles of
  items
• Valuation function vi 2M ? R
• vi(S) - value bidder i receives on bundle S
• Must satisfy monotonicity v(s) v(T) for S µ T
• Normalized v(?) 0
• Allocation subsets S1, S2, Sn where Si Å Sj
  ? for i?j
• Social welfare of allocation ?i1n vi(Si)

15
Combinatorial Auctions Example

• Allocation subsets S1, S2, Sn where Si Å Sj
  ? for i?j
• Social welfare of allocation ?i1n vi(Si)

Interested in any one of red subsets
items 1,, m
players
indivisible item
16
Issues

• Computational Complexity of finding the
  allocation maximizing the social welfare
• Representation and communication of the valuation
  functions vi
• May be large since defined for every subset
• How to analyze the strategy of the bidder in
  light of the obstacles

17
Today

• Complexity of Approximation
• Tight bounds for the single minded bidder
• Connection with communication complexity
• Demand queries and the relaxed LP formulation

18
Single Minded Auctions

• A valuation v is single minded if there is a
  bundle of items S and value v 2 R such that
• v(S) v if S µ S
• v(S) 0 for all other S
• Very simple to represent (Si, vi )
• Allocation problem for single minded bidders
• Given (Si, vi )i for bidders i1..n
• Find a subset W of winning bids such that Si Å
  Sj ? with maximum social welfare ?j 2 W vj

19
Single Minded Auctions Example

• Allocation problem for single minded bidders
• Given (Si, vi )i for bidders i1..n
• Find a subset W of winning bids such that Si Å
  Sj ? with maximum social welfare ?j 2 W vj
• Example communication links in a tree
• bidders want a path between some two nodes in the
  tree
• Since there is only one path between any pair of
  nodes, the bidders are single-minded
• Bidder i wants to connect si to ti
• Items edges of tree
• Si set of edges connecting (si, ti)
• vi 1

20
Complexity of Single Minded Auctions

• Allocation problem for single minded bidders
• Given (Si, vi )I for bidders i1..n
• Find a subset W of winning bids such that Si Å
  Sj ? with maximum social welfare ?j 2 W vj
• Claim Allocation problem for single minded
  bidders is NP-hard
• Proof Reduction from imaximum independent set.
• For a Graph G(V,E)
• each node ? bidder
• each edge ? item
• Si e 2 EI 2 e and vi1.
• Winning set W must correspond to an independent
  set
• Welfare of W is its size W
• Size of max W is exactly size of max independent
  set!

Size m
Size n
21
Computationally easy cases

• When each Si is of size 2
• Allocation problem corresponds to maximum
  weighted matching in the corresponding graph
• Item ? node
• Bidder ? edge, vi is the weight
• Can be solved in polytime
• When the items are on a line. Each Si is a
  contiguous segment
• Finding max weight independent set in in interval
  graphs possible in poly time

vi
item
vi
22
Approximation of Single Minded Auctions

• Allocation problem for single minded bidders
• Given (Si, vi )I for bidders i1..n
• Find a subset W of winning bids such that Si Å
  Sj ? with maximum social welfare ?j 2 W vj
• Size of max W is exactly size of max independent
  set
• What about approximating the best allocation?
• W is a c-approximation if for any other
  allocation W
• ?j 2 W vj / ?j 2 W vj c
• Limited by the approximation ratio of independent
  set
• Cannot be better than n1-? Hastad
• In terms of the number of items cannot be better
  than m1/2-?

23
Incentive Compatible Approximation Mechanism

• Want to satisfy both incentive compatibility and
  computational efficiency
• Lemma a mechanism for single-minded bidders
  where losers pay 0 is incentive compatible iff
• Monotonicity a bidder who wins with (vi , Si)
  also wins with (vi, Si) for any vi vi and
  Si µ Si
• Critical payment a bidder who wins with pays the
  minimum value needed for winning
• infimum of all vi, where (vi, Si) wins.

24
Incentive Compatible Approximation Mechanism

• Want to satisfy both incentive compatibility and
  computational efficiency
• Given (Si, vi )I for bidders i1..n
• Rank bidders by vi / v Si
• v1 / v S1 v2 / v S2
• Run a greedy algorithm, starting from large to
  small
• Add to W if not adjacent to any current member of
  W
• Allocation the set W
• Payments for i 2 W pi vj/ v (Sj / Si)
  where j is smallest index that
• Si Å Sj ? ?
• j wins without i
• and 0 if none exists

The value that would have made j appear before i
25
Approximation Ratio of the Mechanism

• Theorem Let OPT be allocation maximizing ?j 2
  OPT vj and W the output of the greedy algorithm.
  Then
• ?j 2 OPT vj vm ?j 2 W vj
• Proof
• For each i 2 W let OPTij 2 OPT, j i SiÅ Sj
  ? ?
• The set of elements that did not enter W
  because of i
• We know that OPT µ i 2 W OPTi
• Will show ?j 2 OPTi vj vm vi
• For all j 2 OPTi we know that vj vi/ v
  (Sj/Si)
• ?j 2 OPTi vj vi/ v Si ?j 2 OPTi v Sj

26
Approximation Ratio of the Mechanism

• Will show ?j 2 OPTi vj vm vi
• For all j 2 OPTi we know that vj vi/ v
  (Sj/Si)
• ?j 2 OPTi vj vi/ v Si ?j 2 OPTi v Sj
• By Caushy-Schwarz
• ?j 2 OPTi v Sj v OPTi v ?j 2 OPTi Sj
• For j 2 OPTi SiÅ Sj ? ?
• Since OPT is an allocation
• these intersections are disjoint and OPTi
  Si
• ?j 2 OPTi Sj m
• ?j 2 OPTi v Sj vSi vm

27
Today

• Complexity of Approximation
• Tight bounds for the single minded bidder
• Connection with communication complexity
• Demand queries and the relaxed LP formulation

28
Communication Complexity and Bidding

• Two (or more) parties
• Each party i can compute valuation function
  vi 2M ? R by an oracle call
• No succinct description
• Want to find optimal allocation by sending as few
  bits to each other.
• Suppose vi 2M ? 0,1
• Claim need to exchange an exponential number of
  bits to find optimal allocation

29
Communication Complexity
x2X
y2Y
Let fX x Y? Z Input is split between two
participants Want to compute outcome zf(x,y)
while exchanging as few bits as possible
30
A protocol is defined by the communication tree
z5
z0 z1 z2 z3 z4 z5 z6 z7 ...
31
A Protocol

• A protocol P over domain X x Y with range Z is a
  binary tree where
• Each internal node v is labeled with either
• avX? 0,1 or
• bvY? 0,1
• Each leaf is labeled with an element z 2 Z
• The value of protocol P on input (x,y) is the
  label of the leaf reached by starting from the
  root and walking down the tree.
• At each internal node labeled av walk
• left if av(x)0
• right if av(x)1
• At each internal node labeled bv walk
• left if bv(y)0
• right if bv(y)1
• The cost of protocol P on input (x,y) is the
  length of the path taken on input (x,y)
• The cost of protocol P is the maximum path length

32
Motivation for studying communication complexity

• Originally for studying VLSI questions
• Connection with Turing Machines
• Data structures and the cell probe model
• Boolean circuit depth
• Combinatorial Auctions

33
Communication Complexity of a function

• For a function fX x Y? Z the (deterministic)
  communication complexity of f (D(f)) is the
  minimum cost of protocol P over all protocols
  that compute f
• Observation For any function fX x Y? Z
• D(f) log X log Z
• Example
• let x,y µ 1,,n and let f(x,y)maxx y
• Then D(f) 2 log n

34
Combinatorial Rectangles

• A combinatorial rectangle in X x Y is a subset R
  µ X x Y such that R A x B for some A µ X and B
  µ Y
• Proposition R µ X x Y is a combinatorial
  rectangle iff (x1,y1) 2 R and (x2,y2) 2 R implies
  that (x1,y2) 2 R
• For Protocol P and node v let Rv be the set of
  inputs (x,y) reaching v
• Claim For any protocol P and node v the set Rv
  is a combinatorial rectangle
• Claim given the transcript of an exchange
  between Alice an Bob (but not x and y) possible
  to determine zf(x,y)

35
Fooling Sets

• For fX x Y? Z a subset R µ X x Y is
  f-monochromatic if f is fixed on R
• Observation any protocol P induces a partition
  of X x Y into f-monochromatic rectangles.
• The number of rectangles is the number of leaves
  in P
• A set Sµ X x Y is a fooling set for f if there
  exists a z 2 Z where
• For every (x,y) 2 S, f(x,y)z
• For every distinct (x1,y1), (x2,y2) 2 S either
• f(x1,y2)?z or
• f(x2,y1)?z
• Property no two elements of a fooling set S can
  be in the same monochromatic rectangle
• Lemma if f has a fooling set of size t, then
  D(f) log2 t

36
Examples

• Equality Alice and Bob each hold x,y 2 0,1n
• want to decide whether xy or not.
• Fooling set for Equality
• S(w,w)w 2 0,1n
• Conclusion D(Equality) n
• Disjointness let x,y µ 1,,n and let
• DISJ(x,y)1 if x ? y 1 and
• DISJ(x,y)0 otherwise
• Fooling set for Disjointness
• S(A, comp(A))A µ 1,,n
• Conclusion D(DISJ) n

37
Bidding as a function

• In bidding the parties are interested in finding
  a good allocation
• Can be viewed as computing a relation
• Can view the

38
Communication Complexity and Bidding

• Theorem every protocol that find the optimal
  allocation for every pair of 0,1 valuation
  v1,v2 must use
• m choose m/2 ¼ 2m/v (?/2 m)
• bits.
• Proof by a fooling set argument
• For every valuation v define dual v
  v(S)1-v(Sc)
• dual v is monotone
• Claim Let v ? u then the sequence of bits
  transmitted on (v,v) cannot be the same as that
  transmitted on (u,u)
• Claim ? Theorem there are at least m choose
  m/2 different valuations

1
m/2
0
Number of 1s
39

• Claim Let v ? u then the sequence of bits
  transmitted on (v,v) cannot be the same as that
  transmitted on (u,u)
• Proof if v ? u but same transcript on (v,v) and
  (u,u) then same transcript on (v,u) and
  (u,v).
• Same allocation, S, Sc, is produced in all 4
  cases
• For some T we have that v(T) ? u(T).
• WLOG v(T) 1 and u(T)0, u(Tc )1.
• Hence
• v(T) u(Tc )2. Therefore v(S) u(Sc ) 2.
• However v(S) v(Sc) u(S) u(Sc )2
• We get that u(S) v(Sc) 0 which is suboptimal

• Allocation produced
• Should be optimal

40
Approximation?

• Approximating the social welfare with a factor
  strictly smaller than minn,m1/2-? requires
  exponential communication

41
Today

• Complexity of Approximation
• Tight bounds for the single minded bidder
• Connection with communication complexity
• Demand queries and the relaxed LP formulation

42
Demand Queries
v(S) never explicitly represented!

• Demand query auctioneer presents a vector of
  item prices p1, p2, , pn
• the bidder reports a demand bundle for the
  prices a set S maximizing
• v(S) ?j 2 S pi
• Value query auctioneer presents a bundle S,
• the bidder reports his value v(S) for this bundle.

43
Linear Programming Formulation
For single minded bidder just xi,S

• Linear Programming Relaxation
• Variable xi,S for each bidder i 2 N and subset S
• Max ?i,S vi(S) xi,S
• s.t.
• ?i,S s.t. j 2 S xi,S 1 for all j 2 M
• ?S µ M xi,S 1 for all i 2 N
• xi,S 0 for all S µ M, i 2 N
• The integer Program xi,S 20,1

Exponential number of variables
xi,S 1 iff i receives bundle S
44
Dual Linear Programming Relaxation

• Variable ui for each bidder i 2 N and pj for each
  item j 2 M
• Minimize ?i 2 N ui ?j 2 M pi
• s.t.
• ui ?j 2 S pj vi(S) for all S µ M, i 2 N
• ui 0, pj 0 for all i 2 N, j2 M

Interpretation pj price of item j Ui demand
of user i
Exponential number of constraints
45
Demand Queries
v(S) never explicitly represented!

• Demand query auctioneer presents a vector of
  item prices p1, p2, , pn
• the bidder reports a demand bundle for the
  prices a set S maximizing
• v(S) ?j 2 S pi
• Theorem Linear Programming Relaxation can be
  solved in poly time (in n,m and number of bits of
  precision) using only demand queries with item
  prices.

46
Back to the RLP

• Use the dual problem (nm variables, exp
  constraints)
• Ellipsoid Method
• Uses Separation oracle for when given a
  candidate solution, either confirms that it is
  feasible or respond with a violated constraint
• Given (u,p) need to check ui vi(S) -?j 2 S pj
• query user i with demand query p response Di
• See whether ui ?j 2 Di pj vi(D_i)

ui ?j 2 S pj vi(S) for S µ M, i 2 N ui
0, pj 0 for i 2 N, j2 M
Value query
47
From the Dual to the RLP

• The Ellipsoid Algorithm makes a polynomial number
  of calls to the separation oracle
• Each time a constraint is returned poly
  altogether
• Remove all other constraints and obtain a
  reduced dual LP
• Ellipsoid Algorithm still returns the same result
• Use the dual of the reduced dual problem to get a
  solution to the original primal
• With a polynomial number of variable
• It is also a solution to the original (with 0s
  on all non-variables)
• What to do with the relaxed solution?
• Many approximation algorithms use it
• Randomized rounding is the obvious choice

Encountered in Separation
ui ?j 2 S pj vi(S) for S µ M, i 2 N ui
0, pj 0 for i 2 N, j2 M