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?