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, BobTwo items: a, b

Note: we maximize “welfare”, not the seller’s revenue.

0 3 4

2 2 3

Alice

Bob

v(a) v(b) v(a+b)

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:2MR+.Common assumptions:

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

Goal: find a partition S1,…,Sn such that the total welfare vi(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

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.

Approximations

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.

ni=1vi(M) ≥ ivi(OPTi) = OPT

i: vi(M) ≥ OPT/n

The Complexity of Approximating Combinatorial Auctions

For any constant > 0, approximating the welfare to within a factor better than min(n, m½-) 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: k≠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 (Lehmann-

O’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 7/4-

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.

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: 2MR+.Common assumptions:

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

Goal: find a partition S1,…,Sn such that the total welfare vi(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:– There is a bidder i such

that vi(M) ≥ OPT / m½.

– For all bidders Vi(OPTi) < vi(M) < OPT / m½

1 2 3 4

Value

OPT1 OPT2 OPT3 OPT4

Value

OPT/m½

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

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

Winner pays 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 Take your most

profitable bundle

My price is 2*p I paid p

Too 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 iIvi(OPTi) > OPT/2.

Informally, this means that “most” bundles in OPT are profitable under fixed price of p.

– Proof (of claim):

iN \ I vi(OPTi) ≤ iN \ I p * |OPTi| ≤ p * iN \ I |OPTi|

≤ (OPT / (2m) ) * iN \ I |OPTi| ≤ (OPT / (2m) ) * m ≤ OPT / 2

The Approximation Ratio of the Fixed-Price Auction (continued)

If the mechanism gets to bidder iI, 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 jS “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 iIvi(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 fixed-price 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

– if there is no “dominant” bidderPr[the fixed-price auction was conducted] *

E[value of the fixed-price auction | there is a dominant bidder] = ½ * O(m½)

– In both cases we get an approximation ratio of O(m½).

OPT1 OPT2 OPT3 OPT4

ValueOPT/m½

OPT1 OPT2 OPT3 OPT4

ValueOPT/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”.

Statistics Group

I know OPT! (approx.)

Everything costs p

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: 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 may still be solved in polynomial time using demand oracles.(Nisan-Segal).

OPT*=i,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: 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

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).

Statistics Group

I know OPT*!(approx.)

Everythingcosts p

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 Fixed-Price Auction (continued)

Claim: Let I={(i ,S)| iFIXED and vi(S) – p*|OPT*| > 0}.Then i,S)Ixi,Svi(Si) > OPT* / 4.

Proof :

i,S)I xi,svi(S) ≤ ,S)I xi,sp*|S|

≤ i,S)I xi,s(OPT*/(4m)) * |S|

≤ (OPT* / (4m) ) * m ≤ OPT* / 4

The Approximation Ratio of the Fixed-Price Auction (continued)

If the mechanism gets to bidder iFIXED, 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 jS “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 (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 >0, there exists a truthful mechanism that achieves an O(m½ / 3)-approximation with probability 1-.

The Final Mechanism

Select each bidder to exactly one of the following groups: to STAT with probability /2, to FIXED with probability /2, and to SEC_PRICE with probability 1-.

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-.

– 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 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*/ 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 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.