1. Ascending Combinatorial Auctions = a restricted form of preference elicitation in CAs
Tuomas Sandholm

2. Advantages of ascending CAs
Same motivation as other multiagent preference elicitation methods
Transparency
Dynamic exchange of information
With correlated values, can lead to increased revenue

3. Price hierarchy We consider several classes of pricing functions:
Linear: pj for each jÎG, p(S) = ΣjÎSpj
Non-linear: p(S) for each bundle S
Non-linear and non-anonymous: pi(S) for each bundle S and bidder i
3 generalizes 2 generalizes 1

4. Competitive equilibrium
Let agent i’s surplus πi(Si,p) = vi(Si) – pi(Si)
Let ΠS(S,p) = Σi pi(Si)
Prices p and allocation S* are in competitive equilibrium (CE) if:
πi(Si*, p) = maxS [vi(S) – pi(S), 0] (for all i)
ΠS(S*, p) = maxS Σi pi(Si) s.t. S feasible
So, a CE (S*,p) is such that S* maximizes the payoff of every bidder and the seller, given the prices
Allocation S* is said to be supported by p in CE
Theorem: Allocation S* is supported in CE iff S* is efficient
CE prices always exist (e.g. pi = vi)
Proof of Theorem: Chapter 8 of Combinatorial Auctions book. Proof technique: LP duality argument for suitably extended LP formulation of the CAP.

5. Existence of CE prices Some ascending CAs are designed to output a CE
We just saw that non-linear, non-anonymous prices always exist
But linear and non-linear anonymous prices do not always exist
Under what conditions do they exist? …

6. When do linear CE prices exist?
Theorem If each agent’s valuation function satisfies “goods are substitutes”, then linear CE prices exist
Special cases
Unit-demand valuations
Additive valuations
Downward-sloping valuations

7. When do non-linear anonymous prices exist?
Non-linear anonymous prices exist if
valuations are supermodular, i.e., increasing returns, or
bidders are single-minded, or
bidders have safe valuations (each pair of bundles with positive value share at least one item)

8. Minimal CE prices
Def. Minimal CE prices are CE prices where the seller’s revenue is minimized
For certain valuations, minimal CE prices correspond to VCG payments
Thus, truthful bidding is ex post equilibrium
Since minimal CE prices are a restriction of CE prices, a minimal CE allocation is efficient
Minimal CE prices always provide upper bound on VCG payments

9. Buyers are substitutes
Let w(L) for L Í I denote the value of the efficient allocation for CAP(L)
Def. A valuation v satisfies the buyers are substitutes (BAS) condition if: w(I) – w(I \ K) ≥ SiÎK [w(I) – w(I \ i)] for all K Ì I
Thm. BAS holds iff VCG payments are supported in minimal CE
Bikhchandani and Ostroy 2002

10. Buyer-submodular
Recall: Buyers are substitutes (BAS) if: w(I) – w(I \ K) ≥ SiÎK [w(I) – w(I \ i)] for all K Ì I
Slightly stronger version: Buyer-submodular (BSM): w(L) – w(L \ K) ≥ SiÎK [w(L) – w(L \ i)] for all K Ì L, L Í I
Some ascending CAs require the BSM condition to terminate in a minimal CE

11. Universal CE prices BAS does not hold in many practical cases
Then, by the previous theorem, VCG not reachable in minimal CE
We can reach a stronger condition by further restricting the price equilibrium concept
Defn Prices p are universal competitive equilibrium (UCE) prices if p are CE prices and p-i are CE prices for CAP(I \ i)
UCE prices (non-linear, non-anonymos) always exist (e.g. pi = vi)
Minimal CE prices are universal iff BAS holds
VCG outcome and payments determinable from UCE prices
Thm. Let p be UCE with efficient allocation S*. The VCG payment to bidder i is: qi = pi(Si*) – [PI*(p) – PI\i*(p)] where PL*(p) = maxS ∑ pi(Si) for bidders L Í I, S feasible

12. Communicational complexity lower bounds
Thm Any CA that implements an efficient allocation must compute CE prices
Thm Any CA that implements the VCG outcome must compute UCE prices

13. Designing ascending CAs
Timing
Continuous: faster propagation of info, difficult winner determination
Discrete: runs according to planned schedule
Feedback
Prices, bids, provisional allocation
Tradeoff between effective bid guidance and mitigating risk of collusion
Bidding rules
Bid improvement rule
Percentage improvement rule
Activity rules (to avoid sniping)
Termination conditions
Fixed vs. rolling
Bidding language
Proxy agents

14. Price-based ascending CAs
Each auction in this family has roughly the same structure
In each round, announce prices and allocation
Receive bids
Update prices and allocation
Stop if termination criterion met

15. Price-based ascending CAs
Name
Valuations
Price structure
Language
Price update method
Outcome KC
Substitutes
Non-anon items
OR-items
Greedy CE
SAA
Items GS
XOR
Minimal
Min CE
Aus
Single
VCG
iBundle
BSM
Non-anon bundles
General
dVSV
Clock-proxy
Items (+proxy)
RAD OR
LP-based
????
AkBA
Anon bundles
iBEA MP
Results assume truthful bidding

16. Price update methods
Greedy: Price is increased on some set of the over-demanded items/bundles
Minimal: Price is increased on a minimal set of over-demanded items
Or, on the bids from a set of minimally undersupplied bidders
LP (primal-dual)-based:
Formulate CA as an LP with integral optima. Dual should allow convergence to UCE prices (or minimal CE prices in the case of BAS)
Use bidding language that is expressive for straightforward bidding, and formulate a WDP to compute feasible primal solution that minimizes violation of complementary slackness conditions as represented by bids
Terminate when provisional allocation and ask prices satisfy complementary slackness conditions (and thus represent a CE), and also satisfy any additional conditions needed to compute VCG payments (e.g., UCE conditions or minimal CE conditions under BAS)
Otherwise, adjust prices to make progress toward an optimal dual solution that satisfies these conditions
A set of items is overdemanded if demand sets unsatisfiable
A set of bidders is undersupplied if some bidder not satisfied in allocation

17. Primal-dual auction design

18. Primal-dual example: iBundle(2)
Non-linear, anonymous prices
XOR bidding
Winning bids carried over from previous round
A bidder is competitive if she has at least one bid above current ask price
Prices are increased by e on bundles that receive a bid from a losing bidder
In general, could use primal-dual LP algorithms to “jump” the prices to the next vertex instead of incrementing them just a bit.
Prices and provisional allocation provided as feedback
Terminates when each competitive bidder wins a bundle
Thm Terminates with allocation within 3min{n,m}e of the efficient solution (under reasonable strategic assumptions)
Proof uses LP duality and complementary-slackness

19. Other CA designs used in practice
Clock-proxy auction [Chapter 5 of CA book]
Run a parallel clock auctions for the items until no item is over-demanded. Then run a last-and-final proxy round
Combines the simple and transparent price discovery of the clock auction with the efficiency of the proxy auction
Linear pricing maintained as long as possible, but is abandoned in the proxy round to improve efficiency and enhance revenue
Revealed preference consistency requirement
Other core-selecting CAs [e.g., Day & Milgrom]
(actually select a core for revealed valuations, assuming bidders act truthfully)
But bidders are not generally motivated to bid truthfully
If bidders use envy-reducing strategies, then these converge to an envy-free fixed point, and those points have revenue same or greater than VCG [Othman & Sandholm AAAI-10]
Can be supported by envy-quotes
Constraint generation is used to make this computationally feasible

20. Open problems
Design ex post truthful ascending CA that does not suffer from problems of VCG (collusion, low-revenue)
See two technical preference elicitation problems in our JMLR-04 paper