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 Note: we maximize “welfare”, not the seller’s revenue 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 v i :2 M  R +. Common assumptions: Normalization: v i (  )=0 Monotonicity: S  T  v i (T) ≥ v i (S) Goal: find a partition S 1,…,S n such that the total welfare  v i (S i ) 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 v i (M)

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 OPT 1,…,OPT n.  n i=1 v i (M) ≥  i v i (OPT i ) = OPT   i: v i (M) ≥ OPT/n

The Complexity of Approximating Combinatorial Auctions For any constant  > 0, approximating the welfare to within a factor better than min(n, m ½-  ) is hard: – NP-hard even for simple valuations (“single-minded bidders”). – Requires exponential amount of communication (Nisan-Segal). Several O(m ½ )–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≠i v i (OPT k ) - 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(m ½ )-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(m ½ )-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 v i (M). Let the winner pay the second highest price Winner pays 40!

Special Case: Single-Parameter Settings We know how to design a truthful m ½ -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(m ½ )- truthful in expectation mechanism. We prove the following theorem: Theorem: There exists an O(m ½ )-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 v i : 2 M  R +. Common assumptions: Normalization: v i (  )=0 Monotonicity: S  T  v i (T) ≥ v i (S) Goal: find a partition S 1,…,S n such that the total welfare  v i (S i ) 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 (OPT 1,…,OPT n ). Two possible cases: – There is a bidder i such that v i (M) ≥ OPT / m ½. – For all bidders V i (OPT i ) < v i (M) < OPT / m ½ Value OPT 1 OPT 2 OPT 3 OPT 4 Value OPT/m ½ Note: We will provide a different O(m ½ )-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 v i (M) ≥ OPT / m ½. Then assigning all items to bidder i is a good approximation. Our mechanism: the “second-price” mechanism Winner pays 40!

Case 2: No “Dominant” Bidder Assumption: For all bidders v i (OPT i ) < OPT / m ½. 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, v i (OPT i ) < OPT / m ½, then we get an O(m ½ )-approximation. Proof: We need the following claim: – Claim: Let I={i | v i (OPT i ) – p * |OPT i | > 0}. Then  i  I v i (OPT i ) > OPT/2. Informally, this means that “most” bundles in OPT are profitable under fixed price of p. – Proof (of claim):  i  N \ I v i (OPT i ) ≤  i  N \ I p * |OPT i | ≤ p *  i  N \ I |OPT i | ≤ (OPT / (2m) ) *  i  N \ I |OPT i | ≤ (OPT / (2m) ) * m ≤ OPT / 2

The Approximation Ratio of the Fixed- Price Auction (continued) If the mechanism gets to bidder i  I, and all items from OPT i 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, v i (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 / (m ½ ). We also lose a value of at most OPT / (m ½ ) by not assigning i the bundle OPT i. Corollary: for each item we sell at price OPT / (2m), we “lose” a value of at most OPT / O(m ½ ) from bidders in I. Since  i  I v i (OPT i ) > OPT/2, we have an O(m ½ )-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(m ½ ). 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 * m ½ – 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(m ½ ) – In both cases we get an approximation ratio of O(m ½ ). OPT 1 OPT 2 OPT 3 OPT 4 Value OPT/m ½ OPT 1 OPT 2 OPT 3 OPT 4 Value OPT/m ½

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 m ½ requires exponential communication. Let’s use the optimal fractional solution instead.

The Linear Relaxation Maximize:  i,S x i,S v i (S) Subject To: – For each item j:  i,S|j  S x i,S ≤ 1 – For each bidder i:  S x i,S ≤ 1 – For each i,S: x i,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,S x i,S v i (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 v i (M) ≥ OPT* / m ½. – For all bidders v i (M) < OPT*/m ½. OPT* 1 OPT* 2 OPT* 3 OPT* 4 Value OPT* 1 OPT* 2 OPT* 3 OPT* 4 Value OPT*/m ½

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=OPT STAT * / (2m). Statistics Group I know OPT*! (approx.) Everything costs 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)| i  FIXED and v i (S) – p*|OPT * | > 0}. Then   i,S)  I x i,S v i (S i ) > OPT* / 4. Proof :   i,S)  I x i,s v i (S) ≤  ,S)  I x i,s p*|S| ≤   i,S)  I x i,s (OPT*/(4m)) * |S| ≤ (OPT* / (4m) ) * m ≤ OPT* / 4

The Approximation Ratio of the Fixed- Price 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 x i,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, v i (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 / (m ½ ). Corollary: for each item we sell at price OPT* / (4m), we “lose” a value of at most OPT* / O(m ½ ) from bundles in I. Since  (i,S)  I v i (S) > OPT*/4, we have an O(m ½ )-approximation mechanism for this case.

Final Improvement: Increasing the Probability of Success The expectation of the solution provided by the mechanism is indeed O(m ½ ). 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(m ½ /  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 / m ½ 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* / m ½. Therefore, we will have a winner in the second-price auction. The value we achieved is at least v i (M) > OPT* / m ½.

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(m ½ )-approximation algorithm for combinatorial auctions? Other results in the paper: – An O(log 2 m)-mechanism for combinatorial auctions with XOS bidders The XOS class includes all submodular bidders. – A general framework for designing truthful mechanisms for combinatorial auctions.