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Tuomas Sandholm Professor Computer Science Department Carnegie Mellon University

(More on) characterizing dominant-strategy implementation in quasilinear environments (see, e.g., Nisan’s review article). Tuomas Sandholm Professor Computer Science Department Carnegie Mellon University. Some characterization results. Prop. A mechanism is incentive compatible iff

chase-knox
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Tuomas Sandholm Professor Computer Science Department Carnegie Mellon University

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  1. (More on) characterizing dominant-strategy implementation in quasilinear environments(see, e.g., Nisan’s review article) Tuomas Sandholm Professor Computer Science Department Carnegie Mellon University

  2. Some characterization results • Prop. A mechanism is incentive compatible iff • Agent i’s payment does not depend on vi, and • The mechanism picks an outcome (within its range) that optimizes for each player: f  argmaxo{ vi(o) – pi(o)} • Can also characterize in the space of social choice functions only: • Def. f satisfies Weak Monotonicity (WMON) if f(vi ,v-i) = a  b = f(v’i ,v-i) implies vi(a) - vi(b)  v’i(a) – v’i(b) • In words: if the social choice changes when a single agent changes his valuation, then it must be because the agent increased his value of the new choice relative to his value of the old choice. • Thm. If a mechanism is incentive compatible, then f satisfies WMON. If all domains of preferences Vi are convex sets, then for every f that satisfies WMON, there exists a payment rule such that the mechanism is incentive compatible.

  3. Affine maximizers • Generalization of Groves mechanisms • f  argmaxo{ co + iwivi(o) } • Prop. If the payment for agent i is hi(v-i) - ji (wj/wi) vj(o) – co/wi, then the mechanism is incentive compatible • Thm (Roberts). If |O|  3, f is onto O, Vi = O for every i, and the mechanism is incentive compatible, then f is an affine maximizer

  4. Single-parameter domains • Setting: • Vi is one-dimensional • For each agent, there is a set of equally-preferred “winning” outcomes and equally preferred “losing” outcomes • Assume “normalized”, that is, losing agents pay 0 • Thm. Mechanism is incentive compatible iff • f is monotone in every vi, and • every winning agent pays his critical value

  5. (Essentially) uniqueness of prices • Thm. • Assume the domains of Vi are connected sets (in the usual metric in Euclidean space) • Let (f, p1,…pn) be an incentive compatible mechanism • The mechanism (f, p’1,…p’n) is incentive compatible iff p’i(v1,…vn) = pi (v1,…vn) + hi (v-i)

  6. Network interpretation of incentive compatibility constraints • See, e.g., the overview article by Rakesh Vohra that is posted on the course web page • Similar approach also available for Bayes-Nash implementation

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