COMBINATORIAL AUCTIONS
Reloaded
Review
Review Optimal allocation
Special case: Single minded bidders Incentive compatibility
Review Greedy algorithm (single minded) LPR & DLPR Walrasian Equilibrium
Every bidder receives his demand“demand” = maximum utility bundlePoly-time optimal allocation
ReviewLinear program relaxation (LPR)
Social welfareItem cannot be more than once
Social welfare
Bidder has at most one bundle
ReviewDual linear program relaxation (DLPR)
Lecture outline “The curse of dimensionality” Bidding languages Iterative auctions
QueriesCommunication complexityAscending auctions
The curse of dimensionality High-dimensional input data
for all and LPR
High dimensional solution vector for all and
DLPREnormous amount of constraints
Bidding languages
Iterative auction
Bidding languages
Goals Expressiveness
What kind of valuations can we express? Compactness
Use less bits for “interesting” kinds of valuations
OR bids Any subset can be fulfilled Example
({a,b}, 12) OR ({c,d}, 8}) Valuations
v({a}) = 0v({a,b}) = 12v({a,b,c}) = 12v({a,b,c,d}) = 12 + 8 = 20
OR bids Formal definition
are called atomic bids.
IntuitivelyTake all “valid” collections of itemsChoose the one that has maximum value
XOR bids Only one subset can be fulfilled Choose maximum-value subset Example
({a,b}, 12)XOR({c,d}, 8) Valuations
v({a,b}) = 12v{c,d}) = 8v({a,b,c,d}) = 12
Expression power XOR bids can represent any valuation
Just XOR all possible values for all subsetsCan be very inefficient
OR bids – super additive valuations implies
Expression power Additive valuation
Naturally represented by OR bid Unit-demand valuation
Naturally represented by XOR bid
OR/XOR combinations Defined inductively Let be valuations
Have more representation power
OR/XOR expression power Symmetric valuation
depends only on Downward sloping
Can be represented as with Downward sloping with cannot be
represented by OR bids and needs exponential size XOR bids.
OR/XOR expression power Theorem
OR of XOR bids can express any downward-sloping symmetric valuation of items in size
Proof – next slide
OR/XOR expression power For each define a clause that offers for
any single item
Define the final expression as
Since are non-increasing – first item taken from the second from and so on
OR/XOR expression power Example
We have 3 items and we have , and The expression is
Dummy items Goal – reuse single-minded allocation
algorithm for OR/XOR bids Method – represent everything as OR
bids Key observation
OR bids look like a collection of single-minded bids from different players.
Dummy items How? Use dummy items! Example
We have Add a dummy item dRewrite as
Apply recursively for OR/XOR bids We call this OR* bids
Dummy items Theorem: Any valuation that can be
represented by OR/XOR formula of size can be represented by OR* formula of size using at most dummy items
RemarksA valuation in terms of the original formula
is translated to in terms of OR* formula, where is the set of all dummy items.
The “size” of a formula is the amount of atomic bids it contains.
Dummy items Two stage proof
(1) Prove that we can construct an OR* formula of size s
(2) Prove that we need at-most dummy items in the OR* formula.
Dummy items Proof of (1) by induction Definition
to be the OR* translation of v. Base
A single pair is also an OR* bid Step for
Let and Define to be the union of atomic bids in and .We got a formula of the same size as .
Dummy items Example for XOR
Define dummy items
Translates to
Dummy items Step for
Define and Create dummy items for each pair of atomic
bids in and in Create
○ Transform each in to become ○ Similarly, transform each in
We got to be of the same size as .
Dummy items Proof of (2) Dummy item’s purpose is to disallow two
atomic bids to be taken concurrently. Thus we need dummy items – one for
each pair of atomic bids that cannot be taken concurrently.
Dummy items Conclusion: Every algorithm that can
handle single-minded bids in polynomial time can handle any OR/XOR combinations in polynomial time.
Iterative auctions
Iterative auctionQuery
Response
Allocation?
Motivation Reduce the amount of information transfer
Query mechanism that transmits less bits than OR/XOR?
Expressive power Preserve some privacy Bidder limitations
Bidders don’t know their valuationNeed effort to determine valuationsGuide bidders to the data relevant to the
mechanism
Goals Computational efficiency
How much information is transferred?How long does it take to determine an
allocation? Incentive compatibility
Why should the bidders answer the queries truthfully?
Queries The method of “asking for preferences” Value query
What is the value of a bundle S? Demand query
What would you like to buy for those prices?Formally: Given a set of prices , what is the
bundle S that maximizes
Expressive power Demand queries are more powerful than
value queries Lemma:
A value query may be simulated using demand queries, where is the number of bits in the representation of bundle’s value.
Exponential number of value queries may be needed to simulate a single demand query.
Demand queries allow solving the LPR problem efficiently
Solve the linear program Solve the DLPR Use a method that doesn’t need all
constraints at onceexponential amount of constraints!
Ellipsoid method to the rescue! Use the solution of DLPR to solve LPR
Ellipsoid method Solved LP problems by shrinking an
ellipsoid High level overview
Start with an ellipsoid that contains the solution.
Iteratively create a sequence such that:○ contains the solution
○ results from constraint violation test
Solve the linear program Ellipsoid method requirements
Given report weather is feasible or find a single constraint that violates.
DLPR constraints
Treat as utility and as prices Constraint violation checking – using
demand queries.
Solve the linear program Demand-query all bidders using the
prices . The results are . Calculate
Using demand queriesUsing a value query
If for all have a feasible point Otherwise – we have a violated
constraint.
Solving the primal LP
Maximize Subject to
DualMinimize Subject to
Solving the primal
A b
c
Solving the primal
A b
c
Solving the primal
A b
c
Solving the primal Use violated constraints from the
Ellipsoid algorithm Remove all other constraints still the
same solutionEllipsoid algorithm will produce the same
final ellipsoid
Solving the primal
A b
c
Solving the primal
A b
c
Solving the primal Solve the reduced-primal with
polynomial number of variables! Assign 0 to the unused ones.
Conclusions Facts
Walrasian Equillibrium LPR has integer solution LPR solved optimal allocation
LPR can be solved in polynomial timeLPR solution requires polynomial number of
demand queries. Conclusion
Walrasian Equillibrium Can use demand queries to find optimal allocation in poly-time using polynomial amount of queries.
Communication Complexity
There is no such thing as a free
lunch!
Proof Two bidders Valuations are binary functions
Optimal allocationA set such that is maximal
Theorem: Every protocol that finds optimal allocation for any pair must use bits of communication.
Proof outline
World domination auction!
Proof outline
Both parties simulate the aucioneer
Proof outline Let Prove that there are pairs of valuations
that need different bit-sequences. This implies: There are pairs of
valuations that require bits of communication.
Proof Definition
For a valuation we define a dual valuation Lemma:
Let be two 0/1 valuations. Then, in a welfare-maximizing combinatorial auction the sequence of bits transmitted on is not identical to the sequence of bits transmitted on .
Lemma visualized𝑢 𝑣≠𝑢∗ 𝑣∗
𝑏𝑖𝑡𝑠 (𝑢 ,𝑢∗ )≠𝑏𝑖𝑡𝑠 (𝑣 ,𝑣∗)
Lemma proof Suppose not. Then for some partition :
But this means that
We will show that this is impossible
Lemma proof Let such that
W.L.O.G and Therefore Optimal allocation for satisfies . Since
We conclude that
Contradiction to the fact that the protocol produces the same output for as well.
Theorem proof We still need to count the number of
different 0/1 valuations . Count only valuations such that
for for
Theorem proof We have sets of size Therefore we have such valuations We have at-least zero/one valuations.
Ascending Auctions
Ascending auctionQuery
Response
Allocation?
Increase prices
Ascending auction Intuitiveness Guaranteed to terminate May increase seller’s revenue
Ascending auctions Item-price auctions
Each item has a price .Bidder utility:
Bundle-price auctionFor each bidder each bundle has a price Bidder utility:
Item-price auctions Start with initial item prices Increase prices gradually Maintain tentative allocation Terminate when no items tentatively
allocated to one bidder is held by another.
Item-price auctions Intuitively – reaches (near) Walrasian
equilibrium. Is it always guaranteed to reach it? Under which conditions?
Substitutes valuation Increasing prices causes the bidders to
drop ONLY them items for which prices have increased
FormallyFor every pair of item-price vectors the
demand at prices contains all the items in the demand at prices whose price remained constant.
Substitutes valuation Four items A,B,C,D , , , Bidder demanded Price increased: , , , Good example: Bidder demands Bad example: Bidder demands
“Near” Walrasian equilibrium Allocation and prices are -Walrasian
equilibrium ifFor each bidder , is a demand of at prices
for and for For each in we have
Looks ALMOST like Walrasian equilibrium
Intuitively: The prices are “a bit too low” to be a real Walrasian equillibrium
Item price auctions We have an item-price ascending action
thatGiven and substitutes-valuation Finds an -Walrasian equilibrium allocation
However:Prices are NOT VCGNo full incentive-compatibility
The algorithm
Why does it work? Key observation:
At every stage For itself, For the rest
Items taken from by makes smaller and doesn’t affect
Prices for items outside are increased use substitutes property: only items outside are dropped from .
Can we do better?Bundle-price
auctions to the rescue
Bundle price auction More complex pricing scheme Nice reward: We can always reach an
equilibrium! Competitive equilibrium
Every bidder gets his demandAllocation maximizes seller’s revenue
Always exists!
Competitive equilibrium Allocation and bundle prices are called
competitive equilibrium ifFor every bidder , is a demand bundle. That
is, for any bundle we have
The allocation maximizes seller’s revenue – for any other allocation we have that
Always existsWelfare maximizing
Why do it?
In any competitive equilibrium the
allocation maximizes social welfare
Competitive equilibrium Let be a competitive equilibrium Consider some allocation . Demand bundles
Competitive equilibrium Sum over all bidders
Maximizing bidder revenue Conclusion In words: maximizes social welfare
How do we find it?
We don’t! We get -close!
-competitive equilibrium A bundle is an -demand for player
under bundle prices if for any other bundle we have
An -competitive equilibrium is the same as competitive equilibrium, except that the demand requirement is replaced with -demand.
A bundle-price auction
Summary Bidding languages
Simple OR and XOR formulasOR/XOR formulas
Iterative auctionsValue vs demand queriesSolving LPR using demand queries
Comm. Complexity: No free lunch! Ascending auctions
Some results
Questions?