More on characterizing dominant strategy implementation in quasilinear

(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 – 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.

Affine maximizers • Generalization of Groves mechanisms • f argmaxo{ co + i wi vi(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

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

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

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