Unformatted text preview: ovided in equilibrium solutions. The winner-determination complexity of Groves mechanisms in combinatorial domains also limits their applicability as problems get large; e.g.
winner-determination in the combinatorial allocation problem is NP-hard.
Approaches to resolve this tension between game-theoretic and computational properties include:
64 Approximation methods. Compute approximate outcomes based on agent strategies,
and make connections between the accuracy of approximation and game-theoretic
properties of the mechanism.
Distributed computation. Move away from a centralized model of mechanism implementation towards models of decentralized computation to compute the outcome of
a mechanism, based on information about agent preferences.
Special cases. Identify tractable special cases of more general problems, and restrict
the implementation space to those tractable special cases.
Compact preference representation languages. Provide agents with expressive and
compact methods to express their preferences, that avoid unnecessary details, make
structure explicit, perhaps introduce approximations, and make the problem of computing optimal outcomes more tractable.
Dynamic mechanisms. Instead of requiring single-shot direct-revelation, allow agents
to provide incremental information about their preferences for di erent outcomes and
solve easy problem instances without complete information revelation. The challenge is to make mechanisms computationally feasible without sacri cing useful
game-theoretic properties, such as e ciency and strategy-proofness. 3.2 Computation and the Generalized Vickrey Auction
The Generalized Vickrey Auction GVA is a classic mechanism with many important
applications in distributed computational systems NR01, WWWMM01 . As described in
Section 2.4, the GVA is a strategy-proof and e cient mechanism for the combinatorial
allocation problem, in which there are a set of items, G , and a set of agents, I , and the
goal is to compute a feasible allocation of items to maximize the total value across all
agents. Agents report values vi S for each bundle S G , and the GVA computes an
optimal allocation based on reported values and also solves one additional problem with
each agent taken out of the system to compute payments.
From a computational perspective the GVA presents a number of challenges: 65 Winner determination is NP-hard. Winner determination in the GVA is NP-hard,
equivalent to the maximum weighted set packing problem. The auctioneer must
solve the winner-determination problem once with all agents, and then once more
with each agent removed from the system to compute payments.
Agents must compute values for an exponential number of bundles of items. The
GVA requires complete information revelation from each agent. The valuation problem for a single bundle can be hard Mil00a , and in combinatorial domains there are
an exponential number of bundles to consider.
Agents must communicate values for an exponential number of bundles of items.
Once an agent has determined its preferences for all possible outcomes it must communicate that...
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- Fall '13