Download The Fixed-Price Auction

Truthful Randomized Mechanisms for
Combinatorial Auctions
Speaker: Shahar Dobzinski
Joint work with Noam Nisan and Michael Schapira
Combinatorial Auctions


A set of indivisible different items is for sale
Items might be:
– Complements:
v(TV) + v(VCR) < v(TV+VCR)
– Substitutes:
v(TV Toshiba) + v(TV Sony) > v(both TVs)
Combinatorial Auctions

Example:
Two bidders: Alice, Bob
Two items: a, b
v(a)
Alice
Bob

v(b) v(a+b)
0
3
4
2
2
3
Note: we maximize “welfare”, not the seller’s revenue.
FCC Spectrum Auctions
Combinatorial Auctions

Abstract many important resource allocation
problems.

Examples:
– FCC spectrum auctions
– Truckload transportation
– Airport slots
Combinatorial Auctions Definition


m items for sale.
n bidders, each bidder i has a valuation function
vi:2MR+.
Common assumptions:
 Normalization: vi()=0
 Monotonicity: ST  vi(T) ≥ vi(S)

Goal: find a partition S1,…,Sn such that the total
welfare Svi(Si) is maximized.

Difficulty: valuation length is exponential in n and m.
A Black-Box Approach
Efficient
allocation
Challenges

Two main challenges:
–
–
Computer science: compute an efficient allocation
in polynomial time.
Game theory: take into account that the bidders
are strategic.
Computer Science: The Complexity of
Combinatorial Auctions

Computing the optimal solution of a combinatorial
auction is hard:
–
–

NP-hard even for simple valuations (“single-minded
bidders”).
Even ignoring computational aspects it requires exponential
amount of communication (Nisan-Segal).
We can overcome these problems by using:
–
–
–
Heuristics
Assume priors on the input
Approximations
Approximations

Definition: A c-approximation algorithm is a polynomial time algorithm that on any input
returns a solution with value that is a factor c away from the optimal solution.

More formally:
–
–
–
OPT(i) = the value of the optimal solution given input i.
ALG(i) = the value of the solution produced by the algorithm.
ALG is a deterministic c-approximation algorithm (for a maximization problem) if it runs in polynomial
time and:
i: c * ALG(i) ≥ OPT(i)
–
Similarly, a randomized algorithm is a c-approximation algorithm if:
i: c * E[ALG(i)] ≥ OPT(i)
where the expectation is taken over the random coins of the algorithm.
Example: A Simple n-Approximation
Algorithm

The Algorithm: Bundle all items together. Assign the
new bundle to bidder i that maximizes vi(M).
50
32
40
Example: A Simple n-Approximation
Algorithm


Proposition: The allocation produced by the
algorithm is an n-approximation to the
optimal welfare.
Proof: denote the optimal allocation by
OPT1,…,OPTn.
Sni=1vi(M) ≥ Sivi(OPTi) = OPT
 i: vi(M) ≥ OPT/n
The Complexity of Approximating
Combinatorial Auctions

For any constant e> 0, approximating the welfare to
within a factor better than
min(n, m½-e) is hard:
–
–

NP-hard even for simple valuations (“single-minded
bidders”).
Requires exponential amount of communication (Nisan-Segal).
Several O(m½)–approximation algorithms are known.
–
Later we will see another one.
Game Theory: Handling the Strategic
Behavior of the Bidders

Our solution concept: dominant strategy
equilibrium.
–

Due to the revelation principle we limit ourselves
to truthful mechanisms.
Implementable using VCG!
–
Each bidder i pays: Sk≠ivi(OPTk) - OPT-I
where OPT-i denotes the optimal allocation of the auction
without the i’th bidder.

Are we done?
Problems with Implementing VCG


VCG requires finding the optimal allocation,
but it is hard to calculate this allocation!
Naïve Attempt: use an approximation
algorithm for calculating (approximate) VCG
prices.
–
Unfortunately, incentive-compatibility is not
preserved (Nisan-Ronen).
A Clash between Computer Science
and Game Theory

Game theoretically speaking the problem is solved,
but the solution requires exponential amount of time.
From a computer science point of view we know
several O(m½)-approximation algorithms, but we do
not know how to handle strategic bidders.

Can we combine both?

Theorem (wanted): There exists a polynomial time
truthful O(m½)-approximation algorithm for
combinatorial auctions.
Example: A Simple n-Approximation
Mechanism

The “second-price” mechanism: Bundle all items together. Assign
the new bundle to bidder i that maximizes vi(M). Let the winner pay
the second highest price.
50
32
40
Winner
pays 40!
Special Case: Single-Parameter
Settings

We know how to design a truthful m½-approximation algorithm
for combinatorial auctions with single-minded bidders (LehmannO’callaghan-Shoham).
–

Again, this approximation ratio is tight.
In general, single-parameter settings are pretty well
understood:
A single-parameter mechanism is truthful if and only if it is
monotone

Is it possible to design efficient approximation mechanism for
multi-parameter settings, like combinatorial auctions?
Randomness and Mechanism Design

Randomization might help.
–
Nisan & Ronen show a randomized truthful 7/4approximation mechanism for the makespan
problem with two players. They also show that
any deterministic mechanism can not achieve an
approximation ratio better than 2.
More on Randomized Mechanisms

Two notions of randomization:
–
–
“The universal sense”: a distribution over
deterministic mechanisms (stronger)
“In expectation”: truthful behavior maximizes the
expectation of the profit (weaker)


Risk-averse bidders might benefit from untruthful
behavior.
The outcomes of the random coins must be kept secret.
Previous Results and Our Contribution

Lavi & Swamy show a randomized O(m½)truthful in expectation mechanism.

We prove the following theorem:
Theorem: There exists an O(m½)-truthful in the
universal sense mechanism.
–
Actually our result is stronger – details to follow.
Combinatorial Auctions Definition


m items for sale.
n bidders, each bidder i has a valuation
function vi: 2MR+.
Common assumptions:
 Normalization:

vi()=0
 Monotonicity: ST  vi(T) ≥ vi(S)
Goal: find a partition S1,…,Sn such that the
total welfare Svi(Si) is maximized.
Our Mechanism: First Attempt

We will gradually devise our mechanism, in
each iteration we will make it stronger.

First, assume that the value of the optimal
solution is known.
Two Possible Cases


Fix an optimal solution
(OPT1,…,OPTn).
Two possible cases:
–
–
Value
OPT/m½
There is a bidder i such
that vi(M) ≥ OPT / m½.
For all bidders
Vi(OPTi) < vi(M)
< OPT / m½
1
2
3
4
OPT1
OPT2
OPT3
OPT4
Value
OPT/m½
Note: We will provide a different
O(m½)-mechanism for each case. Later we
will see how to combine them.
Case 1: a “Dominant” Bidder



Winner
pays 40!
Assumption: There is a
bidder i such that
vi(M) ≥ OPT / m½.
Then assigning all items to
bidder i is a good
approximation.
Our mechanism: the
“second-price” mechanism
50
32
40
Case 2: No “Dominant” Bidder


Assumption: For all
bidders
vi(OPTi) < OPT / m½.
Our mechanism: a
fixed-price auction
where each item has a
price of p = OPT / (2m)
Everything costs p
My price
is 2*p
Take your
most
profitable
bundle
Too
I paid p
Expensive
!
The Fixed-Price Auction



The fixed-price auction is clearly truthful.
Lemma: If for each bidder i,
vi(OPTi) < OPT / m½, then we get an O(m½)-approximation.
Proof: We need the following claim:
–
Claim: Let I={i | vi(OPTi) – p * |OPTi| > 0}.
Then SiIvi(OPTi) > OPT/2.

–
Informally, this means that “most” bundles in OPT are profitable under
fixed price of p.
Proof (of claim):
SiN \ I vi(OPTi) ≤ SiN \ I p * |OPTi| ≤ p * SiN \ I |OPTi|
≤ (OPT / (2m) ) * SiN \ I |OPTi| ≤ (OPT / (2m) ) * m ≤ OPT / 2
The Approximation Ratio of the FixedPrice Auction (continued)




If the mechanism gets to bidder iI, and all items from OPTi are still
available then bidder i will buy at least one item.
Whenever we sell a bundle S to bidder i, we gain a revenue of |S|*p.
Clearly,
vi(S) > |S|*p = |S| * OPT / (2m).
In the worst case, each item jS “belongs” to a different bidder in I. By
our assumption our “lose” is at most |S|*OPT / (m½). We also lose a
value of at most OPT / (m½) by not assigning i the bundle OPTi.
Corollary: for each item we sell at price OPT / (2m), we “lose” a value
of at most OPT / O(m½) from bidders in I. Since SiIvi(OPTi) > OPT/2,
we have an O(m½)-approximation mechanism for this case.
Choosing between the Second-Price
Auction and the Fixed-Price Auction

To “know” in which case are we, we flip a
random coin.
–
–
With probability ½ we run the second-price
auction, and with probablity ½ we run the fixedprice auction.
Still incentive compatible!
Proving the Correctness of the
Mechanism


Theorem: The mechanism is truthful in the
universal sense. The expected value of the
solution produced by it is O(m½).
Proof:
–
If there is a “dominant” bidder then:
Pr[the second-price auction was conducted] *
E[value of the second-price auction | there is
a dominant bidder] = ½ * m½
–
OPT/m½
OPT1 OPT2
OPT3
OPT4
OPT1 OPT2
OPT3
OPT4
if there is no “dominant” bidder
Pr[the fixed-price auction was conducted] *
E[value of the fixed-price auction | there is a
dominant bidder] = ½ * O(m½)
–
Value
In both cases we get an approximation ratio
of O(m½).
Value
OPT/m½
Removing Assumptions: Guessing
OPT


Observation: the value of OPT was only needed if there is no “dominant”
bidder.
Instead of knowing OPT, randomly partition the bidders, estimate OPT using
the “statistics” group, use this value for performing the fixed price auction
using the bidders in the second group.
–
Similar to the main idea of auctioning “digital goods”.
I know OPT!
(approx.)
Everything
costs p
Statistics
Group
Pros and Cons of the New Mechanism


The mechanism is incentive compatible.
However, estimating OPT (using the statistics
group) is still hard.
–

Recall that any approximation better than m½
requires exponential communication.
Let’s use the optimal fractional solution
instead.
The Linear Relaxation
Maximize: Si,Sxi,Svi(S)
Subject To:
– For each item j: Si,S|jSxi,S ≤ 1
– For each bidder i: SSxi,S ≤ 1
– For each i,S: xi,S ≥ 0


Despite the exponential number of variables, the LP relaxation
may still be solved in polynomial time using demand oracles.(NisanSegal).
OPT*=Si,Sxi,Svi(S) is an upper bound for the value of the optimal
integral allocation.
Two Possible Cases


Fix an optimal fractional
solution.
Two possible cases:
–
–
Value
OPT*/m½
 bidder i such that
vi(M) ≥ OPT* / m½.
For all bidders
vi(M) < OPT*/m½.
OPT*1 OPT*2
OPT*3
OPT*4
Value
OPT*/m½
OPT*1 OPT*2 OPT*3
OPT*4
Back to the Mechanism


Run the same mechanism as before, but this time
calculate an estimation of optimal fractional solution
OPT*, using the bidders in the statistics group.
For the fixed-price auction, use p=OPTSTAT* / (2m).
I know OPT*!
(approx.)
Everything
costs p
Statistics
Group
A Formal Description of the
Mechanism


With probability ½ run the second-price mechanism.
With probability ½ do the following:
–
–
–
–

With equal probability add each bidder to STAT or to FIXED.
Calculate OPT*STAT: the optimal fractional solution restricted
to bidders in the statistics group.
Let p = OPT*STAT / (2m)
Run the fixed-price auction with price p with the participation
of only bidders from FIXED.
Claim: The mechanism is truthful.
Proving the Approximation Ratio of the
Mechanism (if there is no dominant bidder)

Claim: With probability 1-o(1) it holds that:
OPT*STAT ≥ OPT*/4 and
OPT*FIXED ≥ OPT*/4

Corollary: With good probability
p ≥ OPT* / (4m)
–
Reminder: p = OPT*STAT / (2m)
The Approximation Ratio of the FixedPrice Auction (continued)

Claim:
Let I={(i ,S)| iFIXED and vi(S) – p*|OPT*| > 0}.
Then S(i,S)Ixi,Svi(Si) > OPT* / 4.

Proof :
S(i,S)I xi,svi(S) ≤ S(i,S)I xi,sp*|S|
≤ S(i,S)I xi,s(OPT*/(4m)) * |S|
≤ (OPT* / (4m) ) * m ≤ OPT* / 4
The Approximation Ratio of the FixedPrice Auction (continued)




If the mechanism gets to bidder iFIXED, and there is a bundle
S such that all items from S are still available and xi,s > 0, then
bidder i will buy at least one item.
Whenever we sell a bundle S to bidder i, we gain a revenue of
|S|*p. Clearly,
vi(S) > |S|*p = |S| * OPT* / (4m).
In the worst case, each item jS “belongs” to a different bundle
in I. By our assumption our “lose” is at most |S|*OPT / (m½).
Corollary: for each item we sell at price OPT* / (4m), we “lose”
a value of at most OPT* / O(m½) from bundles in I. Since
S(i,S)Ivi(S) > OPT*/4, we have an O(m½)-approximation
mechanism for this case.
Final Improvement: Increasing the
Probability of Success




The expectation of the solution provided by the
mechanism is indeed O(m½).
But it only succeeds if it guesses the “correct” case:
with probability ½.
Success probability can be increased using
amplification. However, truthfulness is not preserved.
Theorem: For any e>0, there exists a truthful
mechanism that achieves an O(m½ / e3)approximation with probability 1-e.
The Final Mechanism




Select each bidder to exactly one of the following groups: to
STAT with probability e/2, to FIXED with probability e/2, and to
SEC_PRICE with probability 1-e.
Calculate OPT*STAT: The optimal fractional solution restricted to
bidders in the statistics group.
Run a second-price auction with a reserve price OPT*STAT / m½
with the participation of only bidders from SEC_PRICE.
If there is no winner in the second-price auction:
–
–

Let p = OPT*STAT / (2m)
Run the fixed price auction with price p with the participation of
only bidders from FIXED.
Claim: The mechanism is truthful.
Correctness of the Final Mechanism

If there is a “dominant” bidder i, then he will be
chosen to SEC_PRICE with probability 1-e.
–


With probability of at most e the mechanism fails.
Since OPT*STAT ≤ OPT* the reserve price is at most
OPT* / m½.
Therefore, we will have a winner in the second-price
auction. The value we achieved is at least vi(M) >
OPT* / m½.
Handling the Case when there is no
Dominant Bidder


If there is no dominant bidder, then we have the
following:
Claim: With probability 1-o(1) it holds that:
OPT*STAT ≥ OPT*/ 4e and OPT*FIXED ≥ OPT* / 4e
–


With probability of at most o(1) the mechanism fails
If there is a winner in the second-price auction then
we are done.
Otherwise, we have a good estimation of OPT* (up
to O(e)), and the fixed-price auction will provide a
good approximation of the welfare.
Open Question & Other Results

Main open question: Is there a truthful deterministic
O(m½)-approximation algorithm for combinatorial
auctions?

Other results in the paper:
–
An O(log2m)-mechanism for combinatorial auctions with
XOS bidders

–
The XOS class includes all submodular bidders.
A general framework for designing truthful mechanisms for
combinatorial auctions.