Computational Mechanism Design Chapter 3

The winner determination complexity of groves

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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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