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...ery set of other players’ strategies (t1, . . . , ti−1, ti+1, . . . , tn), the inequality ui(M(t1, . . . , ti−1, si, ti+1, . . . , tn)) + # ≥ ui(M(t1, . . . , ti−1, ti, ti+1, . . . , tn)) holds – so, si may not be a best possible strategy, but it is always within # of best possible, regardless of what other agents do. In the AMD context, one also insists that, for all i, the computation of si is tractable, given the information available to agent i, and that the computation of M(·) is tractable, given a strategy vector (s1, . . . , sn). 13One can consider the virtual implementation literature [39, 1], in which mechanisms produce lotteries over outcomes, to be a form of approximation. We don’t discuss virtual implementation here. Nisan and Ronen [53] were the first to address the question of approximate computation in AMD. They considered VCG mechanisms in which optimal outcomes are NP-hard to compute (as they are in combinatorial auctions). They pointed out that, if an optimal outcome is replaced by a computationally tractable, approximately optimal outcome, the resulting mechanism may no longer be strategyproof. The above discussion of how we should define “approximating the SH mechanism...

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...e. Since the efficient action profile can be approximated arbitrarily closely by a completely mixed action profile, the principal can implement an almost efficient mixed action profile (Abreu and Sen =-=[1]-=-, Matsushima [26]) and extract the full surplus. 2 This paper is hardly the first attempt to consider endogenous types in mechanism design. Probably the most famous class of models with endogenous typ...

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...e entire type space is common knowledge; this approach thus pins down the higher-order beliefs uniquely, a significantly stronger assumption. When one requires robust implementation, one should expect the conditions on SCFs to become stronger than in the classic approach. Indeed, for exact implementation, the robust analysis is known to turn conditions that are already somewhat restrictive into much stronger ones.5 In view of the strength of the conditions required for robust exact implementation, one might prefer to consider its approximate version: virtual implementation (Matsushima (1988), Abreu and Sen (1991)). For classic environments where the type spaces are common knowledge, Abreu and Matsushima (1992a, b) (henceforth, we use A&M to refer to these authors) characterize virtual implementation in iterative elimination of strictly dominated strategies. This solution concept places weak requirements on agents’ rationality and is closely related to rationalizability, which we use. A&M (1992b) find that incentive compatibility and a measurability condition, which we shall refer to as A&M measurability, are necessary conditions. In quasi-transferable environments, which allow small utility transfers ...