Computational Mechanism Design Chapter 3

In these types of combinatorial problems a mechanism

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Unformatted text preview: d to solve hard local optimization problems to compute their costs to perform di erent bundles of jobs; each bundle may require that the agent computes an optimal schedule for its eet of vehicles. In these types of combinatorial problems a mechanism must not require an agent to report its complete valuation function. In addition, an agent must be able to compute its optimal strategy without computing its complete valuation function. It is not helpful to require less information if the agents must still compute values for all bundles to provide that information. In Chapter 8 I introduce a bounded-rational compatible characterization of auctions. The theory of bounded-rational compatibility precisely captures this idea that an agent can participate in an auction without performing unnecessary valuation work. In a bounded-rational compatible auction an agent can compute its equilibrium strategy with an approximate valuation function, at least in some problem instances. Two interesting approaches to reduce information revelation in mechanisms for combinatorial allocation problems are: 1 Retain the direct-revelation structure, but provide a high-level bidding language or bidding program" to allow an agent to represent and de ne" its local problem without explicitly solving its local problem in all possible scenarios. 2 Implement a dynamic mechanism, that requests information incrementally from agents and computes the optimal allocation and Vickrey payments without complete information revelation. The rst method may be helpful when speci cation is easier than valuation, i.e. it is easier for an agent to de ne how it determines its value for a bundle of items than it is to explicitly compute its value for all possible bundles. The second method may be 73 helpful when the iterative procedure terminates without complete information revelation by agents, and when an agent can provide incremental information without computing its complete valuation function. Let us consider each in turn. Bidding Programs and High Level Bidding Languages In choosing a bidding language for a mechanism there is a tradeo between the ease with which an agent can represent its local preferences, and the ease with which the mechanism can compute the outcome. Nisan Nis00 describes the expressiveness of a language, which is a measure of the size of a message for a particular family of valuation functions, and the simplicity of a language, which is a measure of the complexity involved in interpreting a language and computing values for di erent outcomes. A natural starting point in combinatorial auctions is the XOR bidding language, S1 ; p1  xor S2 ; p2 , which essentially allows an agent to enumerate its value for all possible sets of items. This bidding language is simple to interpret, in fact given a bid b in the XOR language, the auctioneer can compute the value bS  for any bundle in polynomial time Nis00 . However, this bidding language is not very expressive. An obvious example is P provided with a linear valuation function, vS  = x2S vx. XOR bids for this valuation function are exponential in size explicitly enumera...
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