Transcript slides

Truthful Randomized
Mechanisms for Combinatorial
Auctions
Speaker: Michael Schapira
Joint work with Shahar Dobzinski and
Noam Nisan
Hebrew University
Algorithmic Mechanism Design

Algorithmic Mechanism Design deals with designing
efficient mechanisms for decentralized computerized
settings [Nisan-Ronen].

Takes into account both the strategic behavior of the
different participants and the usual computational
efficiency considerations.

Target applications: protocols for Internet
environments.
Combinatorial Auctions

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 social-welfare Svi(Si) is maximized.
Challenges

Computer science: compute an optimal
allocation in polynomial time.

Game-theory: take into account that the
bidders are strategic.
Computer Science: The Complexity of
Combinatorial Auctions

For any constant e > 0, obtaining an
approximation ratio of min(n1-e, m½-e) is
hard:
– NP-hard even for simple valuations
(“single-minded bidders”).
– Requires exponential communication
(Nisan-Segal).

Several O(m½)–approximation algorithms
are known.
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!

Are we done?
A Clash between Computer Science
and Game Theory

VCG requires finding the optimal allocation,
but it is hard to calculate this allocation!

Why not use an approximation algorithm for
calculating (approximate) VCG prices?
–

Unfortunately, incentive-compatibility is not
preserved (Nisan-Ronen).
We need other techniques!
Deterministic Mechanisms

We know how to design a truthful m½-approximation
algorithm only for combinatorial auctions with single-minded
bidders (Lehmann-O’callaghan-Shoham).
–

This approximation ratio is tight.
Only two results are known for the multi-parameter case:
–
–
A pair of VCG-based algorithms: for the general case [Holzman-Kfir
Dahav-Monderer-Tennenholtz] and for the ”complement-free” case
[Dobzinski-Nisan-Schapira]. Both are far from what is computationally
possible.
A non-VCG mechanism for auctions with many duplicates of each
good [Bartal-Gonen-Nisan].
Theorem (wanted): There exists a polynomial time
truthful O(m½)-approximation algorithm for
combinatorial auctions.
Randomness and Mechanism Design

Randomness 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.
On Randomized Mechanisms
Two notions for the truthfulness of randomized
mechanisms:
– “universal truthfulness”: a distribution over
truthful deterministic mechanisms (stronger)
– “Truthfulness in expectation”: truthful
behavior maximizes the expected 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 presented a randomized O(m½)approximation mechanism that is truthful in
expectation. We prove the following theorem:
Theorem: There exists an O(m½)-approximation
mechanism that is truthful in the universal
sense.
–
Actually, our result is stronger (details to follow).
Our Mechanism: An Overview

We will describe our mechanism in several
steps.

First, assume that the value of the optimal
solution, OPT, is known.
Two Possible Cases

Fix an optimal solution
(OPT1,…,OPTn).
Value
OPT/m½

Two possible cases:
– There is a bidder i
such that
vi(M) ≥ OPT / m½.
– For all bidders
vi(M) < OPT / m½
1
2
3
4
OPT1
OPT2
OPT3
OPT4
Value
OPT/m½
We will provide a different
O(m½)-mechanism for each case. Later we
will see how to combine them.
The First Case (A “Dominant Bidder”)
is Easy

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!
The Second Case (There is no
“Dominant Bidder”):
The “fixed-price” mechanism:
1.Define a per-item price p=OPT / 2m
2.For every bidder i=1…n:
• Ask i for his most demanded bundle,
Si, given the per-item price p.
• Allocate Si to i, and charge him p|Si|.
The Second Case (No “Dominant
Bidder”):
A
B
C
D
E
p$ p$
p$
p$
p$
The Second Case (No “Dominant
Bidder”) :
Blue bidder takes
{A,D} and pays 2p.
A
B
C
D
E
p$ p$
p$
p$
p$
The Second Case (No “Dominant
Bidder”) :
Red bidder takes
{C} and pays p.
B
C
E
p$
p$
p$
The Second Case (No “Dominant
Bidder”) :
Green bidder takes
{B,E} and pays 2p.
B
E
p$
p$
Proving the Approximation Ratio of the
Fixed-Price Auction (if there is no dominant
bidder)

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:
Claim: Let PROFITABLE={i | vi(OPTi) – p * |OPTi| > 0}.
Then, S PROFITABLE vi(OPTi) > OPT/2.
i
–
Informally, this means that “most” bundles in OPT are
profitable given a fixed item-price of p.
Proving the Approximation Ratio of the
Fixed-Price Auction (if there is no dominant
bidder)
Proof (of claim):
SiN \ PROFITABLE vi(OPTi) < SiN \ PROFITABLE p * |OPTi| ≤
(OPT / (2m) ) * m = OPT / 2
Proving the Approximation Ratio of the
Fixed-Price Auction (if there is no dominant
bidder)

If the mechanism gets to bidder iPROFITABLE, and all
items in OPTi are unassigned then bidder i will purchase
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 is given to a different
bidder in OPT. Hence, we “lose” (compared to OPT) at
most |S|*OPT / (m½) by assigning the items in S to i.
We also lose a value of at most OPT / (m½) by not
assigning i the bundle OPTi.

This leads to a O(m½)-approximation to the social
welfare of the bidders in PROFITABLE (> OPT/2).
Choosing between the Second-Price
Auction and the Fixed-Price Auction

We flip a random coin.
–
With probability ½ we run the second-price
auction, and with probablity ½ we run the fixedprice auction.

Still truthful.

Still Guarantees the approximation ratio
(in expectation).
Getting Rid of the Assumption:

It is hard to estimate the value of OPT:
–
–
Recall that any approximation better than
m½ requires exponential communication.
Estimating OPT requires information from
the bidders.
We use the optimal fractional solution
instead.
 We get the information in a careful way.

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 can
still be solved in polynomial time using demand oracles (Nisan-Segal).
OPT*=Si,Sxi,Svi(S) is an upper bound on the value of the optimal
integral solution.
Two Possible Cases
Two possible cases:
–
 bidder i such that
vi(M) ≥ OPT* / m½.
Value
OPT*/m½
OPT*1 OPT*2
–
For all bidders
vi(M) < OPT*/m½.
OPT*3
OPT*4
Value
OPT*/m½
The mechanism for the first
case remains the same.
OPT*1 OPT*2 OPT*3
OPT*4
The Second Case (No “Dominant
Bidder”) :

The key observation: A randomly chosen set, that
consists of a constant fraction of the bidders, holds
(w.h.p.) a constant fraction of the total social
welfare.

This idea is similar to the main principle in randomsampling auctions for “digital goods”. [Fiat-GoldbergHartline-Karlin-Wright]

By partitioning the bidders into two sets of equal
size, we can use one set to gather statistics that will
determine the per-item price of the other.
The Second Case (No “Dominant
Bidder”) :

The mechanism:
–
Randomly partition the bidders into two sets of
size n/2: FIXED and STAT.
–
Calculate the optimal fractional solution for STAT,
OPT*STAT.
–
Conduct a fixed-price auction on the bidders in
FIXED with a per-item price of p=OPT*STAT/(2m).
Proving the Approximation Ratio of the
Fixed-Price Auction (if there is no dominant
bidder)

The mechanism is clearly universally truthful.

Theorem: If for each bidder i, vi(M)<(OPT*/m1/2)
then the fixed-price auction guarantees an O(m1/2)approximation.

Claim: With probability 1-o(1) it holds that:
OPT*STAT ≥ OPT*/4 and
OPT*FIXED ≥ OPT*/4
Proving the Approximation Ratio of the
Fixed-Price Auction (if there is no dominant
bidder)

Corollary: With high probability
p ≥ OPT* / (8m)
– Reminder: p = OPT*STAT / (2m) and
OPT*STAT > OPT*/4

Claim:
Let PROFITABLE={(i ,S)| iFIXED and vi(S) –
p*|OPT*| > 0}.
Then S(i,S)PROFITABLE xi,Svi(Si) > OPT* / 8.
Proving the Approximation Ratio of the
Fixed-Price Auction (if there is no dominant
bidder)

Claim:
For each item we sell at price OPT* / (8m), we
“lose” a value of at most OPT* / O(m½)
compared to the total social welfare of the
(fractional) bundles in PROFITABLE.
Since S(i,S)PROFITABLE vi(S) > OPT*/8, we obtain an
O(m½)-approximation mechanism for this
case (no dominant bidder).
Final Improvement: Increasing the
Probability of Success




The expected value of the solution provided by
the mechanism is indeed O(m½).
However, it only succeeds if it guesses the
“correct” case. This occurs with a probability of
½.
Success probability can be increased by running
both mechanisms and choosing the allocation
with the maximal value, or by 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.
A Truthful Mechanism for General Valuations:

Phase I: Partitioning the Bidders
Randomly partition the bidders into three sets: SEC-PRICE, FIXED,
and STAT, such that |SEC-PRICE|=(1-e)n, |FIXED|=(e/2)n, and
|STAT|=(e/2)n.

Phase II: Gathering Statistics
Calculate the value of the optimal fractional solution in the
combinatorial auction with all m items, but only with the bidders in
STAT. Denote this value by OPT*STAT.

Phase III: A Second-Price Auction
Conduct a second-price auction with a reserve price for selling the
bundle of all items to one of the bidders in SEC-PRICE. Set the
reserve price to be (OPT*STAT/m1/2). If there is a “winning bidder”
allocate all the items to him. Otherwise, proceed to the next phase.
A Truthful Mechanism for General Valuations:

Phase IV: A Fixed-Price Auction
Conduct a fixed-price auction with the bidders in FIXED and a peritem price of p=(eOPT*STAT/8m).
Correctness of the Final Mechanism

If there is a “dominant” bidder i, then he will be
in 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 secondprice auction. The social welfare value we
achieved is at least vi(M) > OPT* / m½.
Handling the Case when there is no
Dominant Bidder

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 to the total social welfare.
Other Results

Using the same general framework we
design a universally truthful O(log2m)approximation mechanism for combinatorial
auctions with XOS bidders.

The XOS class includes all submodular
valuations.
–
–
Submodular: v(ST) + v(S T) ≤ v(S) + v(T).
Semantic Characterization: Decreasing Marginal
Utilities.
Open Questions
Designing a truthful deterministic mechanism
for combinatorial auctions that obtains a
O(m1/2) approximation ratio.
truthful
approximations:
computationally
achievable:
Submodular valuations:
e/(e-1)-e
log2m
(Feige, Vondrak)
Complement Free valuations:
m1/2
(Dobzinski-Nisan-Schapira)
2
(Feige)