Truth Revelation in Approximately Efficient Combinatorial Auctions

1. Multiagent Systems 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

2. Multiagent Systems 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

3. Multiagent Systems 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

4. Multiagent Systems 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.

5. Multiagent Systems 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.

6. 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+ • Θis 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 Multiagent Systems

7. 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. Multiagent Systems

8. 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. Multiagent Systems

9. 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. Multiagent Systems

10. Multiagent Systems 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.

11. Multiagent Systems 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 playerRed player

12. Multiagent Systems 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-1/2unless NP=ZPP, where k is the number of items and n is the number of bidders.

13. Multiagent Systems 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.

14. Multiagent Systems Reduction (Continued) V1 : 7 V1 : 4 a Agent 2 Agent 1 b d Agent 1 bid: <7, {a,b}> Agent 2 bid: <4, {a,d}> Agent 3 bid: <5, {b,c}> Agent 4 bid: <3, {c,d}> Agent 3 Agent 4 c V1 : 5 V1 : 3 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

15. Multiagent Systems 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.

16. Multiagent Systems 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.

17. Multiagent Systems 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 ofk1/2. • This is the best possible approximation ratio that can be achieved!

18. Multiagent Systems 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!

19. Multiagent Systems 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.

20. Multiagent Systems 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.

21. Multiagent Systems Sufficient Conditions (Continued) • Monotonicity: • If <s, v> is a winning bid, then for s′⊆ sandv′ ≥ 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.

22. Multiagent Systems Sufficient Conditions (Continued) • Implications of Monotonicity: • [Lemma] In a mechanism that satisfies exactness and monotonicity, for any bidder j, there exists a critical valuevc such that • If j bids belowvc, he wins nothing • If j bids abovevc, 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

23. Multiagent Systems 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.

24. Multiagent Systems 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!

25. Multiagent Systems 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)

26. Multiagent Systems 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

27. Multiagent Systems 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′⊆ sandv′ > 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!

28. Multiagent Systems 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.

29. Multiagent Systems 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]

30. Multiagent Systems 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.

31. Multiagent Systems 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.

32. Multiagent Systems 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 timecomputable 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 programmingrelaxations 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.

33. Multiagent Systems 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.

34. Multiagent Systems Questions? || /*Comments*/