Truthful Risk-Managed Combinatorial Auctions
Alan Holland
Alan Holland & Barry O'Sullivan
Cork Constraint Computation Centre,
Department of Computer Science,
University College Cork, Ireland
[a.holland|b.osullivan]@4c.ucc.ie
Barry O'Sullivan
This work was supported by Enterprise Ireland under
grant number PC/ 2005/0117.
Background: Combinatorial Auctions, Risk Management and the
Importance of being Truthful
ABSTRACT
Given a winning-bid withdrawal in a combinatorial auction, finding an alternative repair solution of adequate revenue without causing
undue disturbance to the remaining winning bids in the original solution may be difficult or even impossible. This bid-taker's exposure
problem may be preemptively addressed by finding a solution that is robust to winning-bid withdrawal. We introduce the concept of
monotonicity-in-expectation. We provide impossibility results concerning truthful mechanisms for robust solutions with bounded socialwelfare losses in which the bid-taker cannot rescind items from winning bidders to repair a solution. We also show that this result extends
to combinatorial auctions that include bid bonds. However, we present a positive result regarding truthfulness for combinatorial auctions
in a restricted setting that comprises a computationally efficient allocation algorithm that seeks to maximize expected social welfare
Truthful Risk-Managed Combinatorial Auctions
Can we develop a truthful mechanism whose allocations are
risk-managed to counteract potential winning-bid withdrawal?
Solution Robustness
A robust solution to a combinatorial auction is one
that can withstand winning-bid withdrawal by making
changes easily to form a repair solution of adequate revenue.
Robustness is a preventative measure that protects against future
uncertainty by trading-off revenue for solution stability. The weighted
super solutions (WSS) framework has been proposed as a means of
finding such solutions [Holland and O’Sullivan, AAAI'05].
Can we incorporate such an allocation scheme within a truthful
mechanism?
Let probabilities of withdrawal be exogenous (e.g. verifiable & unforeseeable
external events that enforce withdrawal)
1: Robust Allocations
(guarantees on max social welfare
losses following a bid withdrawal)
4: Maximize expected
social welfare
Maximizing Expected Social Welfare Polytime Approximation
We develop an LP-based approximation
scheme that maximizes expected social welfare
for known single-minded bidders in the case of
a single bid withdrawal.
1. Use LP-relaxation of integer program.
2. Set bids whose probability of being declared
a winner >0.5 as winners.
(No guarantees on max social
welfare loss)
Requirement: Monotonicity in Expectation
... whenever a bid wins with probability p, an
increase in the bid amount means that it wins
with at least probability p, assuming other
bids are fixed.
2: Irrevocable Winner
Determination
* Bidders bid upon desired combinations of items. Winner
determination is NP-complete.
* In the event of a winning bidder reneging upon a bid, the bid-taker
may incur a large loss in revenue because the remaining items have
already been assigned to other bidders. This can be addressed by
preempting such eventualities with careful allocation.
*The Revelation Principle implies that in a wide variety of settings, only
truthful revelation mechanisms in which agents truthfully announce
their valuations need to be considered when the maximization of a
social objective is required. Incentive compatibility poses computational
challenges. VCG-based mechanisms requires solving multiple winner
determination problems optimally.
Monotone-in-Expectation
See paper for proof
3: Leveled Commitment
(penalties attached)
Poly-time Risk Managed
Allocations
Impossible
(monotonicity still violated)
Impossible
(robustness violates
monotonicity requirement)
VCG-based Robust
Truthful Allocations:
Not possible
(m+1)-approximaion scheme
In the case of an auction with m items, we develop an
approximation scheme incorporating allocations based on
the max of a greedy heuristic and our LP-based scheme.
Payments are based on the critical value as described in
{Mu'alem and Nisan '02}
Future Work
Our Our positive results rely on quite
stringent requirements regarding bidder
preferences. An open question remains
whether we can develop a truthful
mechanism whose allocation scheme
incorporates risk-management whilst bidder
preferences are not so restricted.