Ascending Combinatorial Auctions
= a restricted form of preference
elicitation in CAs
Tuomas Sandholm
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
Price hierarchy
•
We consider several classes of pricing
functions:
1. Linear: pj for each jG, p(S) = ΣjSpj
2. Non-linear: p(S) for each bundle S
3. Non-linear and non-anonymous: pi(S) for
each bundle S and bidder i
•
3 generalizes 2 generalizes 1
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:
1. πi(Si*, p) = maxS [vi(S) – pi(S), 0] (for all i)
2. 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)
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? …
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
When do non-linear anonymous
prices exist?
• Non-linear anonymous prices exist if
1. valuations are supermodular, i.e.,
increasing returns, or
2. bidders are single-minded, or
3. bidders have “safe” valuations (each pair
of bundles with positive value share at
least one item)
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
Buyers are substitutes
• Let w(L) for L  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) ≥ SiK [w(I) – w(I \ i)] for all K  I
• Thm. BAS holds iff VCG payments are
supported in minimal CE
Buyer-submodular
• Recall: Buyers are substitutes (BAS) if:
w(I) – w(I \ K) ≥ SiK [w(I) – w(I \ i)] for all K  I
• Slightly stronger version: Buyersubmodular (BSM):
w(L) – w(L \ K) ≥ SiK [w(L) – w(L \ i)] for all K  L, L I
• Some ascending CAs require the BSM
condition to terminate in a minimal CE
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
• Def. 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-anonymous) 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  I, S feasible
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
Design dimensions of 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 collusion risk
• Bidding rules
– Bid improvement rule / percentage improvement rule
– Activity rules (to help de-motivate sniping)
– Revealed preference rules
• Termination conditions
– Fixed vs. rolling
• Bidding language
• Proxy agents
• Price update rules: myopic vs. planned ahead [Nguyen & Sandholm
EC-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
Price-based ascending CAs
Name
Valuations
Price structure
Language
Price update
method
Outcome
KC
Substitutes
Non-anon items
OR-items
Greedy
CE
SAA
Substitutes
Items
OR-items
Greedy
CE
GS
Substitutes
Items
XOR
Minimal
Min CE
Aus
Substitutes
Items
Single
Greedy
VCG
iBundle &
Ascending
proxy
BSM
Non-anon bundles
XOR
Greedy
VCG
…
General
…
…
…
Min CE
dVSV
BSM
Non-anon bundles
XOR
Minimal
VCG
Clock-proxy
BSM
Items (+proxy)
XOR
Greedy
VCG
…
General
…
…
…
Min CE
RAD
General
Items
OR
LP-based
-
AkBA
General
Anon bundles
XOR
LP-based
-
iBEA
General
Non-anon bundles
XOR
Greedy
VCG
MP
General
Non-anon bundles
XOR
Minimal
VCG
Results assume
straightforward
bidding
Price update methods
• Greedy: Price is increased on some set of the over-demanded
items/bundles
• LP-based (Connection between auctions and optimization algorithms
goes back at least to Danzig (1963). For an excellent modern
presentation about this for CAs, see Mechanism Design: A Linear
Programming Approach by Vohra, Cambridge Univ. Press, 2011.)
– Subgradient algorithm (slower convergence, and convergence can
require price adjustments to become infinitesimally small)
– Primal-dual algorithm (faster converge)
Subgradient-algorithm-based CA framework
• Initialize prices (potentially on bundles and
nonanynymous) to zero
• Repeat
–
–
–
–
Each agent i choose a surplus-maximizing bundle Bi
If a bidder has zero surplus, he reports that he is inactive
Seller chooses revenue-maximizing allocation a*
If each active bidder k gets her most-preferred bundle
Bk, stop
– Otherwise, for each active bidder k, increase price pk(Bk)
by Δ>0
Primal-dual auction design
Primal-dual-algorithm-based CAs
• Algorithm:
– 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
• The primal-dual approach also tells how much each price can be changed
• Not all algorithms in this family are ascending
– Can choose an ascending variant that works correctly --- by ensuring that a
certain “overdemand property” is satisfied throughout the auction process.
For this, it suffices to start from zero prices and use a “minimal” price update:
• Prices are increased on the bids from a set of “minimally undersupplied bidders” in the
provisional allocation
Other CA designs used in practice
• Clock-proxy auction [Ausubel, Cramton & Milgrom, Ch. 5 of CA book]
– Run a parallel clock auctions for the items until no item is overdemanded. Then run a last-and-final proxy round
• In proxy round, bidders report values to their respective proxy agents. The proxy agents
iteratively submit package bids on behalf of the bidders, selecting the best profit
opportunity for a bidder given the bidder’s inputted values. The auctioneer then selects
the provisionally winning bids that maximize revenues. This process continues until the
proxy agents have no new bids to submit.
• XOR bids; all bids remain active; revealed preference consistency requirement
• 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
• Other core-selecting CAs [e.g., Day & Milgrom]
– Constraint generation is used to make this computationally feasible
– (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
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