Approximations and Truthfulness: The Case of Multi-Unit Auctions

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Approximations and Truthfulness: The Case of Multi-Unit Auctions. Shahar Dobzinski Joint work with Noam Nisan and with Shaddin Dughmi. Auctions. Clean Air Auction. 1990’s: The US government decided to decrease the atmospheric levels of sulfur dioxide.

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ShaharDobzinski

Joint work with Noam Nisan and with ShaddinDughmi

Clean Air Auction
• 1990’s: The US government decided to decrease the atmospheric levels of sulfur dioxide.
• From 18.9 million tons to 9 million tons.
• Power plants tried to avoid legislation.
• Estimated cost of a decrease in a single ton: 700\$
• Solution: (annual) auction of emission allowances.
• Cost of a 1-ton allowance: ~70\$.
Definition of Multi-Unit Auctions
• n bidders, m (homogenous) items.
• vi(s) denotes the value of bidder i for a bundle of s items (represented by black boxes).
• Normalization: vi(0) = 0
• Monotonicity: vi(s+1) ≥ vi(s)

Goal: find an allocation (s1,…,sn), Ssi≤m, that maximizes the welfareSivi(si).

Fast running time: running time must be polynomial in n and log m.

A generalization of Knapsack.Input: n objects, each has size si and value vi , capacity m. Goal: Find a maximum-value subset of the objects with total size of at most m.

Computation: NP-hard to solve, but a (1-e)-approximation algorithm exists.

• Incentive Compatibility:
• Truthfulness: a player is never better off by misreporting his true value. For example, second price auction.
• Technically: dominant strategies, private values, quasi-linear utilities.

Incentives: The VCG mechanism is a truthful mechanism for multi-unit auctions.

Can we obtain an algorithm that is fast, truthful and approximates the welfare well?

Related Work

Mu’alem-Nisan, Kothari-Parkes-Suri, Lavi-Swamy, Briest-Krysta-Vocking, Lavi-Mu’alem-Nisan, Balcan-Blum-Mansour, …

Theorem: There exists a poly time deterministic truthful ½-approximation mechanism.

Some evidence that deterministic truthful mechanisms cannot achieve a better approximation ratio in polynomial time.

Theorem: There exists a truthful poly time randomized (1-e)-approximation mechanism.

A truthful mechanism (VCG):

• Find the optimal allocation (o1,…,on). Assign the bidders items accordingly.
• Pay each bidder i: Sj≠ivj(oj).

Proof of truthfulness: the profit of a bidder is the welfare of the allocation. E.g., bidder 1’s profit is v1(o1)+Sj>1vj(oj) = Sjvj(oj) = OPT

• Approximation algorithm + VCG payments?
• Truthful only if the algorithm is maximal-in-range.

Maximal in Range: limit the range and fully optimize over the restricted range.

The mechanism:

• Split the items into n2equi-sizedbundles each of size m/n2.
• Allocate these bundles optimally.

Lemma: the algorithm is truthful.

Using VCG payments, since the algorithm is maximal-in-range.

Lemma: the algorithm runs in polynomial time.

By using dynamic programming.

Lemma: the algorithm guarantees an approximation ratio of ½.

• WLOG, all items are allocated in the optimal solution (o1,…on). Let o1 ≥ m/n.
• Claim: There exists an allocation in the range that holds at least half of the value of the optimal solution.
• Proof case 1: If v1(o1) > Si>1vi(oi), allocate all items to bidder 1 and get a ½ approximation.
• Proof case 2: If v1(o1) ≤ Si>1vi(oi), round up each oi to the nearest multiple of m/n2, and set o1 to 0. We get a ½ approx.We added at most (m/n2)*n=m/n items, but removed at least m/n items, thus the allocation is valid and in the range.

Theorem: Every maximal-in-range (½+e)-approx algorithm requires at least mqueries to the black boxes.

The proof follows from the following two claims:

Claim: Let A be an MIR (½+e)-approximation algorithm for multi-unit auctions with 2 bidders. Then, A’s range must contain all allocations.

Claim [Nisan-Segal]: Optimally solving multi-unit auctions (even with only 2 bidders) requires at least m queries to the black boxes.

Claim: Let A be an MIR (½+e)-approximation algorithm for multi-unit auctions with 2 bidders. Then, A’s range must contain all allocations.

Proof: Otherwise, there is an allocation (k,m-k) that is not in A’s range.

Consider the following instance: Bidder 1 values a bundle of at least k items with 1 (and 0 o/w), and Bidder 2 values a bundle of at least m-k items with 1 (and 0 o/w).

The optimal welfare is 2, but A provides welfare of at most 1.

Proving Impossibilities:Characterize and Optimize

Step 1 (characterize): (essentially)every truthful mechanism with an approximation ratio better than ½ is maximal in range.

Step 2 (optimize): a maximal in range algorithm cannot provide an approximation ratio better than ½ in poly time

Characterize

+

Optimize

=

Impossibility

What’s Next?
• Truthfulness in Expectation: Bidding truthfully maximizes the expected profit of each bidder. Expectation is over the internal random coins of the algorithm.
• Good for risk-neutral bidders.
• Stronger than Bayesian incentive compatibility.

Theorem: there exists a (1-e)-approximation mechanism that is truthful in expectation.

Maximal In Distribution Range:

The range is a set of distributions over allocations.

Choose the distribution in the range that maximizes the expected social welfare

Randomly select an allocation according to the distribution.

• Pay each bidder the sum of the values of the others in the realized allocation.
• The expected profit of a bidder is the welfare of the best distribution, hence truthfulness in expectation.

The range contains only weighted allocations.

Each allocation s=(s1,…,sn) will have a weight ws. With probability ws allocate according to s and with probability 1-ws allocate nothing.

The range (“reward simplicity”):

Weights have the form wt=(1-e)(1+d)t, where d= log(1/1-e) / log m

The weight of the allocation s=(s1,…,sn) is wt, where t is the maximal integer s.t. all si’s are multiples of 2t.

Relaxation: the optimal distribution has weight w0.

At least one bidder is allocated an odd number of items.

We have good approx ratio and truthfulness. Poly time?

The Algorithm
• Each bidder computes a step function based on the valuation: round down each value to the nearest power of (1+d/2).Poly # of “breakpoints”.
• Each bidder transmits each breakpoint and n subsequent values (“neighbors”).I.e., if there is a breakpoint at v(5), transmit v(5),v(6),…,v(5+n)

Some valuation function

Value

Items

Select the best weighted allocation that consists of only breakpoints and neighbors.

Correctness

Lemma: Let WOPT= w0*(o1,…,on). Each oi is either a breakpoint or a neighbor of such.

Proof: Suppose not. Round down each oi to the nearest breakpoint o’i. We removed at least n items.

Add one item to each o’i that is odd: ai.

w1Svi(ai) = (1+d)w0Svi(ai) ≥ (1+d)w0Svi(o’i) ≥ (1+d)WOPT/(1+d/2) > WOPT

Summary

A deterministic truthful ½ approximation mechanism.

An impossibility for MIR mechanisms.

hardness for almost all truthful mechanisms.

Characterize and Optimize.

A truthful in expectation randomized (1-e)-approximation mechanism.

Open Questions

Are there any good MIDR mechanisms for other settings?

An O(1)-approximation mechanism for combinatorial auctions with submodular bidders?

See [Dughmi-Fu-Kleinberg] for initial results.

Basic Requirements from Auctions

Social goal implementation: maximize welfare, revenue, fairness, …

Fast running time: polynomial.

• Incentive Compatibility:
• Truthfulness: a player is never better off by misreporting his true value. For example, second price auction.

Technically: dominant strategies, private values, quasi-linear utilities.

This talk: auction design via multi-unit auctions.

Modern Auctions I: eBay

Many simultaneous auctions, many buyers and sellers.