Vincent Conitzer, Tuomas Sandholm, Carnegie Mellon University Paolo Santi, Pisa University

1. Elicitation in combinatorial auctions with restrictedpreferences and bounded interdependency between items Vincent Conitzer, Tuomas Sandholm, Carnegie Mellon University Paolo Santi, Pisa University Corresponding to papers: Santi, Conitzer, Sandholm, “Towards a Characterization of Polynomial Preference Elicitation in CAs” (COLT-04) Conitzer, Sandholm, Santi, “Combinatorial Auctions with k-wise Dependent Valuations” (Draft)

2. Introduction

3. Combinatorial auction • Can bid on combinations of items • Bidder’s perspective: • Allows bidder to express what she really wants • Avoids exposure problems • No need for lookahead / counterspeculation • Auctioneer’s perspective: • Automated optimal bundling • Winner determination problem: • Label bids as winning or losing so as to maximize sum of bid prices • Each item can be allocated to at most one bid • If approximating, watch incentives • => Better allocations of items than in noncombinatorial auctions

4. Another complex problem in combinatorial auctions: • In direct-revelation mechanisms (e.g. VCG), bidders bid on all 2m combinations • Need to compute the valuation for exponentially many combinations • Each valuation computation can be NP-complete local planning problem • E.g. carrier company bidding on trucking tasks: TRACONET [Sandholm AAAI-93] • Need to communicate the bids • Need to reveal the bids => Loss of privacy & strategic info • Bidding languages [Sandholm 98, 99; Nisan 00; Hoos & Boutilier 01] do not solve the problem

5. ? for $ 1,000 for $ 1,500 for What info is needed from an agent depends on what others have revealed Elicitor Clearing algorithm Elicitor decides what to ask next based on answers it has received so far Conen & Sandholm IJCAI-01 workshop on Econ. Agents, Models & Mechanisms, ACMEC-01

6. Related research • Nondeterministic (i.e., oracle) models • Bikhchandani & Ostroy JET-02 • Gul & Stacchetti JET-00 • Conitzer & Sandholm AAAI-02 • Parkes AMEC-02 • Nisan & Segal 03 • Segal 04 • Deterministic models • Ascending CAs, e.g. Parkes 99; Wurman & Wellman 00; Ausubel & Milgrom 02; Kwasnicka, Ledyard, Porter, DeMartini 04 • General elicitation framework • General preferences (no externalities, free disposal) • Conen & Sandholm IJCAI-01 workshop, ACMEC-01, AAAI-02, AMEC-02 • Hudson & Sandholm AMEC-02, AAMAS-03, AAMAS-04 • Restricted valuation classes [Techniques from computational learning theory] • Zinkevich, Blum, Sandholm ACMEC-03 • Blum, Jackson, Sandholm, Zinkevich COLT-03, JMLR-04 • Lahaie & Parkes ACMEC-04

7. Partial vs. full elicitation • In general, can achieve savings in elicitation by basing queries to one agent on answers from others • Here, will assume that auctioneer will want to know each agent’s entire preference function • So can focus on eliciting one agent’s function • Will assume that agent’s valuation function is drawn from a restricted class of functions

8. Model • Set of items I for sale • Bidder has true valuation function v: 2I  • Elicitor knows class of functions C with v  C • Elicitor’s goal is to identify v • Elicitor can ask bidder for v(B) for any bundle B • Counts as one (value) query • Distinguish between eliciting using • polynomial #queries • polynomial time • May take significant time to compute which query to ask

9. Some examples of polynomial-query elicitable classes

10. PLUS MAX ALL ALL 1000 500 400 100 200 150 Read-once valuations [Zinkevich, Blum, Sandholm 03] Valuations are represented by a tree Leaf nodes correspond to items and their values Nonleaf nodes (gates) perform operations including: SUM: computes the sum of its children MAXc: computes sum of the c highest inputs ATLEASTc: returns sum of inputs if at least c nonzero RO+M: only MAX and SUM allowed

11. A B C D 5 3 = + 2 Toolt (=ToolboxDNF)[Zinkevich, Blum, Sandholm 03] Valuation represented by polynomial with items as variables Using only t monomials 3A + 5AB + 2AC + 4D All coefficients must be nonnegative Can be elicited in O(mt) queries

12. A B C D 4 3 + = - 2 1 Tool-t (slight variation) Here, weights on monomials with  2 items must be negative 3A + 6B + 2C + D - 2AB - AC Thrm. Can be elicited in O(mt) queries Proof: First ask all singletons. Then, discover monomials one by one. Only need to find minimal subset of items that has value less than sum of contained monomials discovered so far. So, start by querying grand bundle and remove items one by one.

13. A B C v({B}) = 2 v({A}) = 1 v({C}) = 2 v({A, B}) = 4 v({B, C}) = 3 v({A, B, C}) = 5 Interval bids Items are ordered on a line Value of bundle = sum of values of disjoint components IMPLIED: v({A, C}) = v({A})+v({C}) = 3 Thrm. Can be elicited using m(m+1)/2 queries if ordering is known Thrm. Can be elicited using m2 – m + 1 queries if ordering is not known, but v({x, y}) > v({x}) + v({y}) iff x and y are adjacent Proof: Ask all singletons and pairs to find adjacencies (m(m+1)/2), then ask remaining components (m(m-1)/2 – (m-1)), for total of m2 – m + 1 queries

14. … … Tree bids require exponential queries Natural generalization: tree such that value of bundle = sum of values of disjoint components Requires exponentially many queries: There are 2m/2 such connected bids

15. Bounded interdependency

16. Node = item 0+1+2 = 3 3 0 -2 3 1 1 2 G2 = 2-wise dependent valuations Value of bundle = sum of values of nodes/edges in bundle

17. -2 3 0 Node = item 3 1 1 1 2 3-edge Gk = k-wise dependent valuations Value of bundle = sum of values of nodes/edges/multiedges in bundle For example, k=3:

18. Gk basic elicitation results • Thrm. Every valuation function has a unique Gm representation • Proof: Suppose we have found the unique weights for multiedges up to size j. Then weight of multiedge over S (with |S| = j+1) must be v(S) – S’Sw(S’) • Thrm. A function in Gk can be elicited in O(mk) queries • Proof: Query all bundles of size k or less. Again, weight of multiedge over S (with |S| = j+1  k) must be v(S) – S’Sw(S’), so can use dynamic programming

19. 1 1 1 1 1 1 Optimal clearing is still hard in G2 • Pf: reduces from EXACT-COVER-BY-3-SETS • Can get total value of 2m/3 if and only if an exact 3-cover exists

20. 2 1 9 7 1 3 6 Special case: union of graphs is forest • Thrm. Can solve clearing problem to optimality by dynamic programming in time O(mn)

21. Approximating with G2 or Gk • Thrm. Suppose there exists some v’ in Gk such that for any bundle S, |v(S) – v’(S)| ≤ δ. Then, using O(mk) queries, we can construct a function g in Gk such that for any bundle S, |v(S) – g(S)| ≤ δ(1+(|S| choose k)). • Bound is tight for G2 • Thrm.Suppose that all the weights in v’s Gm graph are positive. Then, using m(m+1)/2 queries, we can construct a function g in G2 such that for any S, |v(S)-g(S)| ≤ (M(v)/2) ((|S|(|S|+1)/2) (1+ |(|S|-1)/2) • here M(v) is a measure of the function’s disagreement with the same function without any multiedges

22. Unions of classes

23. Polynomial-query elicitable valuation classes closed under pairwise union • Let C1,C2 be valuation classes that can be elicited with polynomial #queries • Using algorithms A1,A2 with query bounds p1(n),p2(n) • Consider the following simple algorithm for C1 C2 • f1A1 , f2A2 • If f1= f2, return it • Otherwise, find bundle S such that f1(S)  f2(S) • Query v(S) • If f1(S) = v(S), return f1, otherwise f2 • At most p1(n) + p2(n) + 1 queries • Gives no bound on computation: checking identity of functions in steps 2, 3 may take lot of computation

24. 1 0 1 0 I-{b} I-{c} I-{a} {b} {c} {a} {} I 2 1 2 1 p1(n) + p2(n) + 1 bound is tight • Consider the following classes: • C1= {fs} where • fs(B) = 0 if B is empty or B = {s} • fs(B) = 2 if B = I • fs(B) = 1 otherwise • To elicit C1, simply ask v({s}) for every s • Need at most m-1 queries • C2 = {f-s} where • f-s(B) = 0 if B is empty • f-s(B) = 2 if B = I or B = I – {s} • f-s(B) = 1 otherwise • To elicit C2, simply ask v(I-{s}) for every s • Need at most m-1 queries • To elicit C1 C2 , need to find {s} or I-{s} with value different from 1 • Need 2m-1 = 2(m-1) + 1 queries

25. Does taking the union ever make computation harder? • Answer: yes. Consider following class: • G2U: valuation is given by graph from G2 (with positive edge weights) + upper bound u on value 2 A 3 u = 6 3 v({A, C}) = 6 C 2 1 B • Easy to elicit: • ask all singletons, all pairs to get graph • ask grand bundle to get u

26. Taking the union may make computation harder… • Now consider the following class: • G2UH: same as G2U except no more than half of bundle’s value can come from edges • require: no edge worth more than sum of endpoints 2 A 3 u = 20 3 v({A, B, C}) = 10 C 2 1 B • Again, easy to elicit: • ask all singletons, all pairs to get graph • ask grand bundle to get u

27. How computationally hard is it to elicit G2U G2UH? • Thrm. It is coNP-complete to determine whether a function from G2U and another from G2UH (represented by their graphs and u) are identical • That is, it is NP-complete to find a bundle whose query would distinguish them

28. Proof of hardness every vertex: weight 1 • Reduction from CLIQUE problem every edge: weight (k+) / (k choose 2) Required clique size: k (say, 3) u = 2k +  • Clique of size k would have k vertex weight and k+  edge weight • So, G2UH at-most-half-from-edges constraint would be binding • Cannot happen when there are fewer edges • For larger sets, the u-constraint is binding

29. Optimized polynomial-time elicitation algorithm forRO+M Tool-t  Toolt  G2  INT Thrm. Runs in polynomial time and uses at most O(m(m+t)) queries

30. Towards characterizing easily elicitable valuation functions

31. v(B) = 1 iff B contains all items corresponding to a color … Polynomial inferability • Inferring a bundle = ascertaining its value from queries on other bundles • Bundle is polynomially noninferable (strongly polynomially noninferable) wrt C if for some (any) function in C, the bundle’s valuation cannot be inferred using polynomially many queries • Thrm. There exists a class of functions where • exponentially many bundles are polynomially noninferable • no bundles are strongly polynomially noninferable • the class cannot be elicited using polynomially many queries. • Proof uses [Angluin 88] idea, functions of the form:

32. Conclusions • Focused on learning full valuation function in restricted classes • New easy-to-elicit classes of valuations • Tool-t, Interval, Gk • Clearing for G2 is NP-complete • But easy if union of graphs is forest • Approximation with functions from G2 or Gk • Polyquery elicitable classes closed under pairwise union • But computation required may go from polynomial to NP-hard • Efficient algorithm for union of most of the classes studied • Even classes without strongly polynomially noninferable bundles may require exponentially many queries for elicitation

33. Future research • Can Interval class be elicited with polynomially many queries without knowing the order? • Can we come up with a more general characterization of what makes valuation functions easy to elicit? • What if we have a restricted class of valuations and we only need to elicit enough to allocate (or compute VCG payments)?

34. Thank you for your attention!