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Truth Revelation in Approximately Efficient Combinatorial Auctions
Daniel Lehman, Liadan Ita O’Callaghan and Yoav Shoham
@ Journal of the ACM, 2002
Presenter – Muntasir Raihan Rahman
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Outline of the Presentation
Definition of Combinatorial Auctions Assumptions Main Contribution of the Paper Definition and Notations Single Minded Bidders Hardness of GVA for Single Minded Bidders Greedy Allocation Algorithms Conditions for Ensuring Truthfulness for Single Minded Bidders A Truthful Payment Scheme for the Greedy Allocation Algorithms Extension to Complex Bidders Revenue Considerations Strength and Weaknesses of the Paper Future Work and Open Problems
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Combinatorial Auction (CA)
An Auction where Multiple goods are available Bidders can bid for subsets of goods Single seller, multiple buyers Reverse auction (single buyer, multiple sellers) : similar One stage sealed bid CA
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Assumptions
Normalization: Zero valuation for empty set.
Free Disposal: Throw away excess items.
No Externalities: An agent’s valuation is independent of the other agents valuations.
Complementarity: Valuation for a set of items can be more than the sum of the valuations for the
individual items. Example: a left shoe and a right shoe.
Substitutability: Valuation for a set of items is less than the sum of the valuations for the individual
items. Example: two tickets to the same football game.
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Main Contribution of the Paper
Present simple approximate allocation algorithms for CA’s that posses the following two properties: The algorithm performs a reasonably effective optimization.
There is a performance guarantee. There is a payment scheme that converts the approximation algorithm
into a truthful mechanism. Not necessarily VCG payments.
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Definition and Notation
P : set of n bidders G : set of k goods A partial allocation function
A : G → P , P = P U {unallocated}′ ′ Partial since not all goods are required to be allocated. O = P ′ G is the set of all possible allocations.
Type : a bidders evaluation of goods A function v: 2G → R+ Θ is the set of bidder types: Θ = R+
2G
t is the true type of a bidder t ∊ Θ d is the declared type (maybe a lie), d ∊ Θ T=<t1, …, tn> : vector of true types. D=<d1, …, dn> : vector of declared types P is a vector of n payments. P ∊ R+
n
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Further Definitions
Utility: Assume quasi-linear utility. For a bidder of type t, the utility for any bundle s G
and payment x, u = t (s) – x.
Direct Mechanism for CA: For a declared type vector D, a direct mechanism is a
tuple <f,p> f is the allocation algorithm p is the payment vector: p = p(D)
pi(D) denotes the payment for bidder I gi(D) denotes the bundle obtained by i for the
allocation f.
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Recap: Truthful Mechanism
Consider a mechanism M = <f, p> M is truthful iff for every agent i, every type t
and any vector D t(gi(D′)) – pi(D′) ≥ t(gi(D)) – pi(D) D ′ is the vector obtained by replacing the ith
coordinate of D with t. Generalized Vickrey Auctions (GVA):
General method for designing truthful mechanisms. GVA chooses an optimal allocation that maximizes
social welfare. GVA payment: the opportunity cost that each agents
presence introduces to all other players. GVA ensures individual rationality
each agents utility is non-negative.
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Single Minded Bidders
A severe restriction on bidding types. A single minded bidder only cares about a single
bundle of items. Zero valuation for any other subset.
Formally, a bidder is single minded iff There is a set S and a value v such that
t(s′) = v if s s⊆ ′ t(s′) = 0 otherwise
A single minded bid is denoted as <s, v>. Also called an atomic bid.
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Real Life Examples of Single Minded Bids
Authors claim single minded bids not unrealistic and uncommon.
Examples: Used for selling timber in New Zealand: single minded bidders
A bidder interested in all locations in a specific geographical area. Contiguous bids.
Auctions for pollution rights Auto manufacturer selling auto parts
Buyers interested only in one set of parts that can be completed into a working vehicle.
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An Example from Computer Science
Consider a communication network where the underlying topology is a tree. So there is a unique path between any pair of nodes. The links are the items to be auctioned off. The players are source sink pairs <s, t>. Each player interested in building a path from source to sink. So each player is only interested in the set of links that constitute the unique path from
its source to sink. So the players are single minded.
Green player Red player
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GVA for Single Minded Bidders
Can we apply the GVA mechanism for single minded bidders to obtain an optimal allocation in polynomial time?
Unfortunately, NO! Theorem [Hastad, 1999; Sandholm, 2002]:
Finding an allocation that maximizes social welfare in the case of single minded bidders is NP-hard.
There is no polynomial time (polynomial in k+n) approximation algorithm with ratio better than k ε-1/2 unless NP=ZPP, where k is the number of items and n is the number of bidders.
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Proof Sketch of the Hardness Result
Reduction from the Weighted Independent Set (WIS) problem. Given a graph G=(V,E)
We will assign a weight wv to each vertex v V∊ .
An independent set is a set of vertices such that no two vertices are adjacent. The WIS problem asks for the independent set with maximum weighted sum.
Reduction: Consider the combinatorial auction with n players and k items. Construct a graph GCA(V, E)
V is the set of players E is the set of items The edges connected to a vertex v correspond to the single bundle of items v is interested
in. The weight of the vertex v is the valuation of the corresponding player for his bundle.
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Reduction (Continued)
Agent 1
Agent 3 Agent 4
Agent 2V1 : 7
V1 : 3V1 : 5
V1 : 4
Agent 1 bid: <7, {a,b}>
Agent 2 bid: <4, {a,d}>
Agent 3 bid: <5, {b,c}>
Agent 4 bid: <3, {c,d}>
Max weighted independent set: vertices corresponding to Agent 1 and Agent 4
Total weight 7+3=10
This corresponds to the optimal allocation
Giving a and b to Agent 1, c and d to Agent 2
c
db
a
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Greedy Allocation Algorithms
Finding optimal allocation for single minded bidders is computationally intractable.
So authors propose a family of simple greedy algorithms. The greedy algorithms consist of two phases:
Sorting Phase: The bids are sorted according to some criteria (norm). [Defined later] This phase runs in O(nlogn) time. [quick sort]
Allocation Phase: Let L be the list of bids obtained in the first phase. For each bid in L
If the current bid does not conflict with the previously allocated bids Then accept current bid.
Else Reject current bid.
This phase requires linear time.
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Efficiency of the Greedy Algorithm
The greedy algorithm is polynomial time computable. So it achieves computational efficiency.
What about economic efficiency? In order to increase economic efficiency,
The norm must be selected in such a way that bids that are more likely to increase efficiency are pushed forward in the sorted list.
A bid can increase economic efficiency if: It has high valuation Has a small number of items [this can help in the later stage by keeping
more items free for later bids] A norm that can satisfy these two properties :
For a bid b = <s, v>, norm(b) = (v / |s|), where |s| is the size of the subset.
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Efficiency (Continued)
So the bids should be sorted in decreasing order of the norm value. Infact any norm of the form norm(b) = (v / |s|x) will do. The case x = 0.5 is especially interesting, since in this case the
greedy algorithm can guarantee the best possible approximation ratio. [Recall the hardness result for single minded bidders !]
[Theorem]: The greedy allocation scheme with norm (v / |s|1/2) approximates the
optimal allocation within a factor of k1/2. This is the best possible approximation ratio that can be achieved!
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Greedy Algorithm in Action
Consider the case of two goods {a, b} and three bidders {R, G, B}. R: <10,{a}>, G: <19, {a,b}>, B: <8, {b}> norm(R)=10, norm(G)=9.5, norm(B)=8
give a to R Bid of G conflicts with R, so G gets nothing! Finally B gets b. Social welfare = 18.
Notice that this is not optimal, the optimal allocation would give both a and b to G with total social welfare = 19!
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How to Ensure Truthfulness
How can the greedy allocation be made truthful? Intuitive idea: use VCG payments. However a theorem by [Nisan and Ronen, 2000] shows that
A truthful mechanism with VCG payments must find optimal allocations.
But the greedy algorithms are not optimal! So an approximate allocation is incompatible with VCG payments. Need to search for alternate payment rules that can ensure
truthfulness for single minded bidders. Impose further restrictions on the bidders and allocation schemes.
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Sufficient Conditions for Truthfulness for Single Minded Bidders
Properties of allocation algorithms and payment schemes that will ensure truthfulness.
Exactness: A single minded bidder gets exactly the set that he desires, nothing
more, or he gets nothing. Bid <s, v>, bidder gets the set s if he wins, not any superset of s, if he
loses he gets nothing. A natural condition, since free disposal means the agent doesn’t care
about the additions items.
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Sufficient Conditions (Continued)
Monotonicity: If <s, v> is a winning bid, then for s′ s⊆ and v ′ ≥ v, <s′, v′ > is also a
winning bid. Proposing more money for fewer goods cannot cause a winning bidder
to lose. A very natural assumption. The GVA scheme also satisfies monotonicity.
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Sufficient Conditions (Continued)
Implications of Monotonicity: [Lemma] In a mechanism that satisfies exactness and monotonicity, for
any bidder j, there exists a critical value vc such that If j bids below vc, he wins nothing If j bids above vc, he wins the exact set of items he desired.
The proof follows immediately from the conditions of exactness and monotonicity.
The GVA mechanism also satisfies this property. Recall the Vickrey auction
Here the critical value is the second highest bid A bidder bidding below that will lose
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Sufficient Conditions (continued)
Critical: A satisfied bidder pays exactly the critical value. Compare with Vickrey auction.
The winner pays the second highest bid [critical value] The payment is independent of the agents own bid. Setting a payment below or above the critical value can encourage
lying. Participation:
An agent who wins nothing, pays nothing. Very natural restriction! Akin to Individual Rationality
All players utility are non-negative.
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Characterization of a Truthful Mechanism
[Theorem] If a mechanism satisfies the properties: exactness, monotonicity, participation and critical, then it is a truthful mechanism.
The proof follows easily from the definition of the properties!
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A truthful Mechanism with Greedy Allocation
A Greedy Payment scheme: [case 1]: If an agent loses, he pays nothing. Consider a winning bid for agent j. [case 2]: If the items allocated to j do not conflict with any other later
bids, then j pays nothing. This is because j is not hurting anyone. Example: j gets {a,b}, and no one else wants a or b.
[case 3]: Let k be the first agent in the sorted list whose bid conflicted with the bid of j. This means if j did not participate in the auction, k could have won! So j must have appeared before k in the sorted bid list of the obtained
in the first phase of the greedy allocation algorithm. Let <s,v> be j’s bid. j pays |s|.norm(k)
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Greedy Payment in Action
Consider two goods {a, b} and three players {R, G, B}. Bids:
R: <10, {a}>, G: <19, {a, b}>, B: <8, {b}> norm(R) = 10, norm(G)=9.5, norm(B)=8 R wins a and B wins b, G loses G pays 0 [did not win] R blocks G from winning
So R pays 1. norm(G)=9.5 B does not block anyone
So B pays 0
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The Final Result
[Theorem] The payment scheme and the greedy allocation compose a truthful mechanism for single minded bidders.
Proof Sketch: To prove truthfulness, we have to show that the four properties are
satisfied. Participation:
Trivial, since according to the payment rule, a losing bidder pays nothing.
Monotonicity: Let <s, v> be a winning bid with norm n. Consider <s′, v′> with norm n′, where s′ s⊆ and v >′ v. So n′> n, so the bid <s′, v′> will be pushed higher in the sorted list of
bids <s′, v′> remains a winning bid!
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Proof Continued
Critical: The payment is exactly equal to the critical value. Paying less will make a wining bid to lose. For a winner j, let p (j)= |s|. norm(k), where k is defined as before. If j bids v′ < p (j), then
v′ < |s|.norm(k) v′ / |s| < norm(k) norm(j) < norm(k) This means the bid of j will be kicked below the bid of k. j will no longer be a winner.
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Complex Bidders
Single minded bidder model too restrictive! Consider Complex bidders. A Complex bidder can be considered as a collection of single
minded bidders. Example: complex bid [<10,{a,b}>,<3,{a}>,<4,{b}>]
The complex bidder controls three single minded bidders <10,{a,b}> <3,{a}> <4,{b}> Will the single minded bidders act cooperatively or competitively?
The authors show that no payment scheme can make greedy allocation truthful for complex bidders. Proof by showing counter-examples [no formal proof]
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Revenue Maximization
Revenue maximization in combinatorial auctions is an active area of research!
The authors show by examples that the revenue generated by the greedy allocation and payment does not always coincide with revenue from GVA!
Recent approaches: Virtual Valuation Combinatorial Auctions (VVCA) [Sandholm, 2004]
Virtual Valuations: linear transformation of real valuations. Applied to VCG mechanism, can it be used with greedy allocation for single minded
bidders? Item Pricing for Revenue Maximization [Balcan, Blum, Mansour, 2008]
Pricing single items instead of bundle of items in VCG.
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Strengths and Limitations
(+) Well written paper, many illustrative examples to clarify important ideas.
(+) The greedy allocation algorithm and payment schemes are polynomial time computable and easy to understand.
(-) Model of single minded bidder too restrictive.
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Future Work Most computational results for Combinatorial Auctions focus on worst
case computational complexity. What about average case complexity? Some voting rules that are hard in the worst case are polynomial time
computable in the average case. This could inspire average case analysis of hard combinatorial auction
problems using some Junta Distributions. The greedy algorithm is very intuitive. What about other heuristics?
Approximation algorithms based on linear programming relaxations of the integer programming formulation of CA. Randomized mechanisms [truthful in expectation] [Lavi, Swamy 2005] Is it possible to get deterministic truthful mechanisms for single minded bidders
using relaxation? Other search heuristics like Genetic Algorithms, Simulated Annealing.
Complex Bidders: Find truthful mechanisms for other type of approximation schemes in case of
complex bidders.
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References
Truth Revelation in Approximately Efficient Combinatorial Auctions, Daniel Lehman, Liadan Ita O’Callaghan, Yoav Shoham, 2002.
Lectures on Combinatorial Auctions, Tim Roughgarden, 2006. Combinatorial Auctions, chapter 11, Algorithmic Game Theory, edited by
Noam Nisan, Tim Roughgarden, Eva Tardos, Vijay V Vajirani, 2007. Combinatorial Auctions, edited by Peter Cramton, Yoav Shoham, Richard
Steinberg, 2006.
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