Preference Elicitation in Combinatorial Auctions: An Overview Tuomas Sandholm [For an overview, see review article by Sandholm & Boutilier in the textbook Combinatorial Auctions, MIT Press 2006, posted on course home page] I collected some of our material below. Leave out, delete, change and add as you like. I hope that you can use a little bit of the stuff. Cheers, Wolfram.

Setting Combinatorial auction: m items for sale Private values auction, no allocative externalities So, each bidder i has value function, vi: 2m  R Free disposal Unique valuations (to ease presentation)

Another complex problem in combinatorial auctions: “Revelation problem” In direct-revelation mechanisms (e.g. VCG), bidders bid on all 2#items combinations Need to compute the valuation for exponentially many combinations Each valuation computation can be NP-complete local planning problem For example if a carrier company bids on trucking tasks: TRACONET [Sandholm AAAI-93, …] Need to communicate the bids Need to reveal the bids Loss of privacy & strategic info

Revelation problem … Agents need to decide what to bid on Waste effort on counter-speculation Waste effort making losing bids Fail to make bids that would have won Reduces economic efficiency & revenue

What info is needed from an agent depends on what others have revealed ? 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 SET UP AS QUESTION: CAN ANYTHING BE DONE TO AVOID THE NEED FOR ALL THE INFO ? Conen & Sandholm IJCAI-01 workshop on Econ. Agents, Models & Mechanisms, ACMEC-01

Conen & Sandholm IJCAI workshop-01, ACMEC-01 Elicitor skeleton Repeat: Decide what to ask (and from which bidder) Ask that and propagate the answer in data structures Check whether you know the optimal allocation of items to agents. If so, stop Conen & Sandholm IJCAI workshop-01, ACMEC-01

Incentive to answer elicitor’s queries truthfully Elicitor’s queries leak information across agents Thrm. Nevertheless, answering truthfully can be made an ex post equilibrium [Conen&Sandholm ACMEC-01] Elicit enough to determine optimal allocation overall, and for each agent removed in turn Use VCG pricing Push-pull mechanism If a bidder can endogenously decide which bundles for which bidders to evaluate, then no nontrivial mechanism – even a direct revelation mechanisms - can 1) be truth-promoting, and 2) avoid motivating an agent to compute on someone else’s valuation(s) [Larson&Sandholm AAMAS-05] (Even if the queries asked from an agent depend on others’ answers so far, answering truthfully is ex post equilibrium [Conen & Sandholm ACM-EC-01])

First generation of elicitors Rank lattice based elicitors [Conen & Sandholm IJCAI-01 workshop, ACMEC-01, AAAI-02, AMEC-02]

Rank Lattice [1,1] [1,2] [2,1] [1,3] [2,2] [3,1] [1,4] [2,3] [3,2] Rank of Bundle Ø A B AB for Agent 1 4 2 3 1 for Agent 2 4 3 2 1 [1,1] [1,2] [2,1] [1,3] [2,2] [3,1] [1,4] [2,3] [3,2] [4,1] [2,4] [3,3] [4,2] [3,4] [4,3] [4,4] Infeasible Feasible Dominated

A search algorithm for the rank lattice Algorithm PAR “PAReto optimal“ OPEN  [(1,...,1)] while OPEN  [] do Remove(c,OPEN); SUC  suc(c); if Feasible(c) then PAR  PAR  {c}; Remove(SUC,OPEN) else foreach node  SUC do if node  OPEN and Undominated(node,PAR) then Append(node,OPEN) Thrm. Finds all feasible Pareto-undominated allocations (if bidders’ utility functions are injective, i.e., no ties) Welfare maximizing solution(s) can be selected as a post-processor by evaluating those allocations Call this hybrid algorithm MPAR (for “maximizing” PAR) PAR stands for Pareto-optimal. the suc()-function determines the immediate successors of c as {d in C| exists d[j] ) c[j] + 1 for some j and d[i] = c[i] for all i not = j}

Value-Augmented Rank Lattice Value of Bundle Ø A B AB for Agent 1 0 4 3 8 for Agent 2 0 1 6 9 17 [1,1] 14 13 [1,2] [2,1] 9 10 12 [1,3] [2,2] [3,1] 8 9 [1,4] [2,3] [3,2] [4,1] [2,4] [3,3] [4,2] [3,4] [4,3] [4,4]

Search algorithm family for the value-augmented rank lattice Algorithm EBF “Efficient Best First“ OPEN  {(1,...,1)} loop if |OPEN| = 1 then c  combination in OPEN else M  {k  OPEN | v(k) = maxnode  OPEN v(node) } if |M|  1  node  M with Feasible(node) then return node else choose c  M such that c is not dominated by any node  M OPEN  OPEN \ {c} if Feasible(c) then return c else foreach node  suc(c) do if node  OPEN then OPEN  OPEN  {node} Thrm. Any EBF algorithm finds a welfare maximizing allocation Thrm. VCG payments can be determined from the information already elicited EBF = Efficient-Best-First

Best & worst case elicitation effort Best case: rank vector (1,...,1) is feasible One bundle query to each agent, no value queries VCG payments are all 0 Thrm. Any EBF algorithm requires at worst (2#items #bidders – #bidders#items)/2 + 1 value queries Proof idea. Upper part of the lattice is infeasible and not less in value than the solution Not surprising because in the worst case, finding a provably (even approximately) optimal allocation requires exponentially many bits to be communicated no matter what query types are used and what query policy is used [Nisan&Segal J. Economic Theory 2006] We will prove this later

EBF minimizes feasibility checks Def: An algorithm is admissible if it always finds a welfare maximizing allocation Def: An algorithm is admissibly equipped if it only has value queries, and a feasibility function on rank vectors, and a successor function on rank vectors Thrm: There is no admissible, admissibly equipped algorithm that requires fewer feasibility checks (for every problem instance) than an (arbitrary) EBF algorithm

MPAR minimizes value queries Thrm. No admissible, admissibly equipped algorithm (that calls the valuation function for bundles in feasible rank vectors only) will require fewer value queries than MPAR MPAR requires at most #bidders#items value queries

Differential-revelation Extension of EBF Information elicited: differences between valuations Hides sensitive value information Motivation: max ∑ vi(Xi)  min ∑ [vi(r-1(1)) – vi(Xi)] Maximizing sum of value  Minimizing difference between value of best ranked bundle and bundle in the allocation Thrm. Differences suffice for determining welfare maximizing allocations & VCG payments 2 low-revelation incremental ex post incentive compatible mechanisms ... with free disposal, v_i(r^-1(1) = v(grand_bundle) (r^-1 is the inverse of the rank function, giving the bundle at a rank). for Vickrey payments: payment of i = (sum of the differences of the best allocation without i) – (sum of differences of the best allocation without i and without Xi)

Differential elicitation ... Questions (start at rank 1) “tell me the bundle at the current rank” “tell me the difference in value of that bundle and the best bundle“ increment rank Natural sequence: from “good” to “bad” bundles

Policy-independent elicitor algorithms

What query should the elicitor ask next ? Simplest answer: value query Ask for the value of a bundle vi(b) How to pick b, i?

Hudson & Sandholm AMEC-02, AAMAS-04 Random elicitation Asks randomly chosen value queries whose answer cannot yet be inferred Thrm. If the full-revelation mechanism makes Q value queries and the best value-elicitation policy makes q queries, random elicitation makes on average value queries Proof idea: We have q red balls, and the remaining balls are blue; how many balls do we draw before removing all q red balls? Hudson & Sandholm AMEC-02, AAMAS-04

Random elicitation Not much better than theoretical bound queries 2 agents 4 items 80 1000 Full revelation 60 100 Queries 40 10 20 1 2 3 4 5 6 7 8 9 10 2 3 4 5 6 items agents

Querying random allocatable bundle-agent pairs only… Bundle-agent pair (b,i) is allocatable if some yet potentially optimal allocation allocates bundle b to agent i How to pick (b,i)? Pick a random allocatable one Asking only allocatable bundles means throwing out some queries Thrm. This restriction causes the policy to make at worst twice as many expected queries as the unrestricted random elicitor. (Tight) Proof idea: These ignored queries are either Not useful to ask, or Useful, but we would have had low probability of asking it, so no big difference in expectation

Querying random allocatable bundle-agent pairs only… Much better Almost (#items / 2) fewer queries than unrestricted random Vanishingly small fraction of all queries asked ! Subexponential number of queries queries queries 80 1000 Full revelation 60 100 Queries 40 10 20 1 2 3 4 5 6 7 8 9 10 2 3 4 5 6 items agents

Best value query elicitation policy so far Focus on allocations that have highest upper bound. Ask a (b,i) that is part of such an allocation and among them, pick the one that affects (via free disposal) the largest number of bundles in such allocations. Number of items for sale Fraction of values queried before provably optimal allocation found Omniscient elicitor “Most important slide of this segment of the talk” Optimal elicitor implementable, but utterly intractable. Hudson & Sandholm AMEC-02, AAMAS-04

Worst-case number of bits transmitted (nondeterministic model) Exponential (even to approximately optimally allocate the items within ratio better than 1/2) [Nisan & Segal JET-06; see also CS-friendly version from Nisan’s home page] L is the number of items Proof.

Restricted preferences Even worst-case number of queries is polynomial when agents’ valuation functions fall within certain natural classes…

Zinkevich, Blum & Sandholm ACMEC-03 Read-once valuations PLUS Returns sum of c highest-valued inputs if at least k inputs are positive, 0 otherwise GATEk,c MAX ALL ALL 1000 500 400 200 100 150 Thrm. If an agent has a read-once valuation function, the number of value queries needed to elicit the function is polynomial in items Thrm. If an agent’s valuation function is approximable by a read- once function (with only MAX and PLUS nodes), elicitor finds an approximation in a polynomial number of value queries Uses techniques from query learning. Zinkevich, Blum & Sandholm ACMEC-03

Zinkevich, Blum & Sandholm ACMEC-03 Toolbox valuations Items are viewed as tools Agent can accomplish multiple goals Each goal has a value & requires some subset of tools Agent’s valuation for a package of items is the sum of the values of the goals that those tools allow the agent to accomplish E.g. items = medical patents, goals = medicines Thrm. If an agent has a toolbox valuation function, it can be elicited in O(#items #goals) value queries Zinkevich, Blum & Sandholm ACMEC-03

Zinkevich, Blum & Sandholm ACMEC-03 Computational complexity of finding an optimal allocation after elicitation Thrm. Given one agent with an additive valuation fn and one agent with a read-once valuation fn, allocation requires only polynomial computation Thrm. With 2 agents with read-once valuations (even with just MAX, SUM, and ALL gates), it is NP-hard to find an allocation that is better than ½ optimal Thrm. Given 2 agents with toolbox valuations having s1 and s2 terms respectively, optimal allocation can be done in computation time poly(m, s1+s2) Uses techniques from query learning. Zinkevich, Blum & Sandholm ACMEC-03

2-wise dependent valuations 0+1+2 = 3 1 3 -2 2 Node = item m items Prop. If an agent has a 2-wise dependent valuation function, elicitor finds it in m(m+1)/2 queries Thrm. If an agent’s valuation function is approximately 2-wise dependent, elicitor finds an approximation in m(m+1)/2 queries Thrm. Every super-additive valuation function is approximately 2-wise dependent Thrm. These results generalize to k-wise dependent valuations using O(mk) queries Conitzer, Sandholm & Santi Draft-03, AAAI-05

Gk = k-wise dependent valuations G1  G2  …  Gm G1 = linear valuations: Easy to elicit & allocate Gk where k ≥ 2 is a constant: Easy to elicit, NP-hard to allocate if graph cycle free (i.e. forest), allocation polytime Gg(m) where g(m) is an arbitrary (sublinear) fn s.t. g(m) approaches infinity as m approaches infinity: Hard to elicit & NP-hard to allocate Gm contains all valuation fns Conitzer, Sandholm & Santi Draft-03

Combining polynomially elicitable classes Thrm. If class C1 (resp. C2) is elicitable using p1(m) (resp. p2(m)) queries, then C1 union C2 is elicitable in p1(m) + p2(m) + 1 queries. Tight Santi, Conitzer, Sandholm COLT-04

Blum, Jackson, Sandholm & Zinkevich JMLR-04 In some settings, learning only a tiny part of valuation fns suffices to allocate optimally Blum, Jackson, Sandholm & Zinkevich JMLR-04 Consider 2 agents with valuations f and g Each has some subsets of items that he likes Each such subset is of size log m Agent’s valuation is 1 if he gets a set of items that he likes, 0 otherwise Since there are bundles of size log m, some members of this class cannot be represented in poly(m) bits => can require super-polynomial number of queries to learn an agent’s valuation fn But… Thrm. Optimal allocation can be determined in poly(m) queries Proof: Try random partitions of items into two equal-sized sets Derandomization: A set of assignments to m boolean variables is (m,k)-universal if for every subset of k variables, the induced assignments to those variables cover all 2k settings. Naor and Naor (1990) give efficient constructions of such sets using only 2O(k) log m assignments. We can use k = O(log m), so the construction is polynomial time and space. Each of these assignments corresponds to a partition of items, and we ask f and g for their valuations on each one and take the best.

Blum, Jackson, Sandholm & Zinkevich JMLR-04 In some settings, learning only a tiny part of valuation fns suffices to allocate optimally… There can be super-polynomial power even when valuation fns have short descriptions Let each agent have some distinguished bundle S’ Agent’s valuation is 1 for all bundles of size ≥ |S’|, except for S’ itself 0 otherwise Prop. It can take value queries to learn such a valuation fn Thrm. With two agents with such valuation fns, the optimal allocation can be determined in 4 + log2 m value queries Proof. First find |S’| in log2 m + 1 queries using binary search. Then make 3 arbitrary queries of size |S’|. At most 1 of them can return 0. Call the other two sets T and T’. We then query the other agent for M-T; if it returns 1, then T, M-T is an optimal allocation. Otherwise, T’, M-T’ is optimal. Blum, Jackson, Sandholm & Zinkevich JMLR-04

Power of interleaving queries among agents Observation: In general (not just in combinatorial auctions), we can elicit without interleaving within a number of queries that is exponential in q where q is the number of queries used when eliciting with interleaving. Proof: Contingency plan tree is (merely) exponential in the number of queries

Other results on elicitation Interleaving value & order queries [Hudson & Sandholm AMEC-02, AAMAS-04] Bound-approximation queries [Hudson & Sandholm AMEC-02, AAMAS-04] Elicitation in exchanges (for multi-robot task allocation) [Smith, Sandholm & Simmons AAAI-02 workshop] Eliciting bid-taker’s non-price preferences in (combinatorial) reverse auctions [Boutilier, Sandholm, Shields AAAI-04]

Demand queries “If the prices (on items or some bundles) were p, which bundle would you buy?”

Value queries vs. demand queries A value query can be simulated by a polynomial number of (item-price) demand queries [Blumrosen&Nisan EC-05, see also their SIAM J. Computing 2010 paper] Proof. Elicit value of adding one item at a time into the bundle. The marginal value of each such addition is done via binary search on that item’s price. A demand query cannot be simulated in a polynomial number of value queries [Blumrosen&Nisan EC-05] There exists restricted CAs where optimal allocation can be found in poly bits, but exponential number of demand (and thus value) queries are needed [Nisan & Segal TARK-05]

Ascending combinatorial auctions Demand queries Per-item prices vs. bundle prices Discriminatory vs. nondiscriminatory prices Exponential communication complexity, but polynomial in special classes (e.g., when items are substitutes) [Nisan-Segal JET-06] To allocate optimally, enough info has to be elicited to determine the competitive equilibrium prices [Parkes chapter; Nisan-Segal JET-06] (more on this in the next slide deck) Could also use descending prices

Ascending combinatorial auctions… Thm [Blumrosen & Nisan JET-10]. To achieve efficiency, the number of trajectories in an ascending item-price CA may have to be exponential in the number of items even if the trajectories can be interleaved and what is done on a trajectory can depend on what happened on other trajectories Thm [Blumrosen & Nisan JET-10]. Any anonymous ascending (even bundle-price) CA may fail to find an efficient allocation

Recall the XOR-bidding language [Sandholm ICE-98, IJCAI-99] ({umbrella}, $4) XOR ({raincoat}, $5) XOR ({umbrella,raincoat}, $7) XOR … Bidder’s valuation is the highest-priced term, of the terms whose bundle the bidder receives

Power of bundle prices Thrm. [Lahaie & Parkes ACMEC-04] Using bundle-price demand queries (even when only poly(m) bundles are priced) and value queries, an XOR-valuation can be learned in O(m2 #terms) queries Thrm. [Blum, Jackson, Sandholm, Zinkevich COLT-03, JMLR-04] If the elicitor can use value queries and item-price demand queries only, then 2Ω(√m) queries are needed in the worst case even if each agent’s XOR-valuation has only O(√m) terms

Conclusions on preference elicitation in combinatorial auctions Reduces the number of local plans needed Capitalizes on multi-agent elicitation Truth-promoting push-pull mechanism Sequential-> atomic: SAY: each xi is evaluated in the context of all other xi’s.

Future research on preference elicitation Scalable general elicitors (in queries, CPU, RAM) New polynomially elicitable valuation classes More powerful queries, e.g. side constraints Using models of how costly it is to answer different queries [Hudson & Sandholm AMEC-02, AAMAS-04] Strategic deliberation [Larson & Sandholm] Other applications (e.g. voting [Conitzer & Sandholm AAAI-02, EC-04])

Tradeoffs between Agent’s evaluation complexity Amount revealed to the auctioneer (crypto) Amount revealed to other agents (vs. to elicitor) Bits communicated Elicitor’s computational complexity (knowing when to terminate, what to ask next) Elicitor’s memory usage (e.g., implicit candidate list) Designer’s objective Designing for specific prior & eliciting using the prior Terminating before optimal allocation, …