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:2MR+.Common assumptions:

Normalization: vi()=0Monotonicity: ST vi(T) ≥ vi(S)

Goal: find a partition S1,…,Sn such that the total social-welfare vi(Si) is maximized.

Challenges

Computer science: compute an optimal allocation in polynomial time.

Game-theory: take into account that the bidders are strategic.

For any constant > 0, obtaining an approximation ratio of min(n1-, m½-) is hard:– NP-hard even for simple valuations

(“single-minded bidders”).– Requires exponential communication

(Nisan-Segal).

Several O(m½)–approximation algorithms are known.

Computer Science: The Complexity of Combinatorial Auctions

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 7/4-

approximation 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).

Two possible cases:– There is a bidder i

such thatvi(M) ≥ OPT / m½.

– For all bidders vi(M) < OPT / m½

1 2 3 4

Value

OPT1 OPT2 OPT3 OPT4

Value

OPT/m½

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 / 2m2.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 DC EB

p$ p$ p$ p$ p$

A DC EB

p$ p$ p$ p$ p$

Blue bidder takes{A,D} and pays 2p.

The Second Case (No “Dominant Bidder”):

C EB

p$ p$ p$

Red bidder takes{C} and pays p.

The Second Case (No “Dominant Bidder”):

EB

p$ p$Green bidder takes{B,E} and pays 2p.

The Second Case (No “Dominant Bidder”):

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, iPROFITABLE vi(OPTi) > OPT/2.

– 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):

iN \ PROFITABLE vi(OPTi) < iN \ 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 iPROFITABLE, 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 jS 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 fixed-price 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: i,Sxi,Svi(S)

Subject To:– For each item j: i,S|jSxi,S ≤ 1 – For each bidder i: Sxi,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*=i,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.

– For all bidders vi(M) < OPT*/m½.

OPT*1 OPT*2 OPT*3 OPT*4

Value

OPT*1 OPT*2 OPT*3 OPT*4

Value

OPT*/m½

OPT*/m½

The mechanism for the first

case remains the same.

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 random-sampling auctions for “digital goods”. [Fiat-Goldberg-Hartline-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).

The Second Case (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 andOPT*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)| iFIXED and vi(S) – p*|OPT*| > 0}.Then 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 (i,S)PROFITABLE vi(S) > OPT*/8, we obtain an O(m½)-approximation mechanism for this case (no dominant bidder).

Proving the Approximation Ratio of the Fixed-Price Auction (if there is 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 >0, there exists a truthful mechanism that achieves an O(m½ / 3)-approximation with probability 1-.

Phase I: Partitioning the BiddersRandomly partition the bidders into three sets: SEC-PRICE, FIXED, and STAT, such that |SEC-PRICE|=(1-)n, |FIXED|=()n, and |STAT|=()n.

Phase II: Gathering StatisticsCalculate 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 per-item price of p=(OPT*STAT/8m).

A Truthful Mechanism for General Valuations:

Correctness of the Final Mechanism

If there is a “dominant” bidder i, then he will be in SEC_PRICE with probability 1-.– With probability of at most 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 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*/ 4 and OPT*FIXED ≥ OPT* / 4– 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(), 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(ST) + v(ST) ≤ v(S) + v(T).– Semantic Characterization: Decreasing Marginal Utilities.

m1/2

(Dobzinski-Nisan-Schapira)

Open Questions

Submodular valuations:

e/(e-1)-(Feige, Vondrak)

computationallyachievable:

truthful approximations:

Complement Free valuations:

2(Feige)

log2m

Designing a truthful deterministic mechanism for combinatorial auctions that obtains a

O(m1/2) approximation ratio.