Generalizing Preference Elicitation In Combinatorial Auctions

1
Generalizing Preference Elicitation In
Combinatorial Auctions

• Benoît Hudson
• Tuomas Sandholm
• Carnegie Mellon University

2
Combinatorial Auctions

• We are selling k items to n agents.
• In series economically inefficient.
• Use a Generalized Vickrey Auction
• Economically efficient (under reasonable
  assumptions)
• NP-hard, inapproximable
• Exponential communication, inapproximable

3
Experiments

• Simulations
• Draw agents valuation functions from a random
  distribution.
• Distribution repeatedly pick a random value for
  a random unvalued bundle, observing free disposal
• Pick each agent's function independently
• Each point on plots is average of 50 runs.
• Every plot has the same sets of instances.

4
Rank queries

• Proposed by Conen Sandholm01
• Have each agent rank their preferences (total
  ordering over the bundles)
• This forms a lattice
• Go down the lattice in best-first order

5
Rank lattice Experiments
While the elicitation ratio with 2 agents falls
to 25, the ratio quickly goes to 1 as the number
of agents grows (4 or 5 agents). This is because
on average, agents will tend to get
lower-valued bundles as competition increases.
6
Elicitation Framework

• Goal minimize number of bids
• Approach
• Auctioneer elicits information about values of
  bundles of goods from agents
• C set of all possible allocations
• while C gt 1
• Decide what to elicit based on what it knows.
• Elicits one bid, stores the answer.
• Prunes allocations now known to be suboptimal.

7
Storing the bids for fast inference

• Interval constraint networks, 1 per agent
• Nodes store upper/lower bounds on value of
  bundle
• Edge (b,b) means vi(b) ? vi(b)
• At start create all nodes, add edges for free
  disposal.
• Could make other assumptions (like
  sub-modularity, etc.)

8
Pruning

• Need to determine whether information gathered so
  far proves v(c) ? v(c')
• First compute the difference in value to each
  agent
• di(c,c') is at least LBi(ci) - UBi(ci')
• but if there is a path in the constraint network
  from ci to ci', then di(c,c') is non-negative.
• Then sum up over all agents, and check
  that d(c,c') ? 0

9
Pruning quickly

• Pruning is the biggest computational costs in the
  algorithms must be done quickly.
• Reason for high cost need to search for paths in
  the constraint network.
• time linear in size of network, thus exponential
  in number of items

10
Pruning quickly

• Can prove if UB(c) lt UB(c') then c does not
  dominate c' and similarly with lower bounds.
  see paper
• Hence
• Dominates(c, c') if UB(c) lt UB(c') return
  false if LB(c) lt LB(c') return false if
  d(c,c') lt 0 return false return true

11
Bound-approximation queries

• Larson,Sandholm01 model bounded-rational
  agents as having on-line algorithms
• Get better bounds UBi(b) and LBi(b) with more
  time spent deliberating
• Query agent i, please spend t time units
  tightening the (upper or lower) bound on b
• How to choose i, b, t, U or L ?

12
Bound-approx query policy

• For simplicity, fix t 0.2 units (1 unit gives
  exact)
• Can choose randomly.
• More complicated policy does slightly better
• Choose query that will change the bounds on
  allocatable bundles the most
• Dont know exactly how much bounds will change
• Assume all legal answers equiprobable, sample to
  get expectation

13
Bound-approx query results
As the number of items grows, the elicitation
ratio falls. Note that the bound-approx
algorithm can get charged for up to cost 2 for
bundle, if it tightens both the lower and upper
bounds.
14
Bound-approximation a note

• We calculated the change but what is change from
  ? ?
• Can prove that we always need an upper bound on
  value of grand bundle.
• See below
• Policy actually is first, ask everyone for a UB
  on the grand bundle. Then elicit as described.
• After that, we neednt worry about ?.

15
Value and Order Queries

• Value queries the elicitor asks for the precise
  value of a bundle.
• clearly expensive
• Order queries the elicitor asks for the agent
  which of two bundles it prefers
• in many cases, extremely easy to answer
• if the values are highly divergent (a car vs. an
  eraser)
• if the items are very similar (2 computers,
  identical except that one has a slightly larger
  hard drive)

16
Value and Order Queries

• Order queries alone cannot find the globally
  optimal solution (only Pareto-optimal ones)
• Idea interleave value and order queries.
• Value policy pick a random agent-bundle pair of
  unknown value from a candidate allocation.
• Order policy pick a random agent-bundle-bundle
  triplet of unknown relation.
• For experiments order queries cost .1, value
  queries cost 1. Compare to asking all value
  queries.

17
ValueOrder results
About 10 fewer value queries total cost equal
to value only policy. Need (1) better order
query choice (2) better interleaving
18
The Grand Bundle Is Always Revealed

• To optimally clear an auction, the elicitor must
  elicit the grand bundle, unless one of the agents
  wins it.
• Proof the elicitor must place an upper bound on
  the value of the grand bundle to each agent i,
  otherwise allocating all the items to i could be
  worth an infinite amount.
• Furthermore If using value or bound-approx
  queries, the elicitor provably cannot do better
  by avoiding asking about the grand bundle.

19
Vickrey payments

• Above we solved winner determination.
• In a Vickrey auction, need to compute payments as
  well.
• In practice, very little extra is needed.
• No more at two agents (theoretical bound 4 extra
  queries in practice, 0 extra queries)
• About 20 more with 3-5 agents.

20
Reverse auctions

• Forward auction auctioneer selling items to
  agents.
• each item sold at most once
• Reverse auction auctioneer buying items from
  agents.
• each item bought at least once
• Reverse is easier to approximate (still NP-hard,
  exponential communication)

21
Reverse Auctions
Experiments on reverse auctions. The upper lines
are for the bound-approx algorithm, the
lower lines for the mixed policy. As the number
of agents grows, the elicitation ratio
falls, unlike in forward auctions.
22
Conclusion

• Elicitation can reduce amount of valuation
  function that needs to be revealed
• Save on communication
• Save on bidder-side computation
• Reduce revelation of private information
• Scale in items and maintain benefit in agents
• reverse auctions improve with more agents

23
Future work

• Theoretical results on these policies.
• Better policies, query types.
• Making a near-optimal policy
• Better data structures algorithms
• Current space and time exp in items, poly in
  agents
• Link with demand queries
• Auctioneer proposes prices, sees what agents
  would buy.