1. (More on) characterizing dominant-strategy implementation in quasilinear environments (see, e.g., Nisan’s review: Chapter 9 of Algorithmic Game Theory book) Tuomas Sandholm Professor Computer Science Department Carnegie Mellon University

2. Some characterization results (see Nisan’s review chapter) Prop. A mechanism is incentive compatible iff –Agent i’s payment does not depend on his reported v i, but only on the alternative chosen, and –The mechanism picks an outcome (within its range) that optimizes for each player: f in argmax o { v i (o) – p i (o) } Can also characterize in the space of social choice functions only: Def. f satisfies Weak Monotonicity (WMON) if f(v i,v -i ) = a ≠ b = f(v’ i,v -i ) => v’ i (b) - v i (b) ≥ v’ i (a) - v i (a) –In words: if 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 domains of preferences V i are convex sets, then for every f that satisfies (even just local) WMON, there exists a payment rule such that the mechanism is incentive compatible. –First part is easy to prove, see page 227 of Algorithmic Game Theory book –Second part holds if outcome space is finite [Saks and Yu EC-05], or loop integral of f is zero around every sufficiently small triangle [Archer&Kleinberg EC-08] –They also show that the theorem applies to non-convex V i by studying functions f that apply to the convex hull –They also show how a truthful f can be stitched together from locally truthful f i ’s Somewhat unsatisfactory: it is not clear exactly what the WMON functions are. WMON is a local condition. Is there a global condition? Yes for unrestricted or very restricted V i. Largely open for practical problems that lie in between.

3. Unrestricted V i and affine maximizers Affine maximizers are a generalization of Groves mechanisms f in argmax o { c o +  i w i v i (o) } Prop. If the payment for agent i is h i (v -i ) -  j≠i (w j /w i ) v j (o) – c o /w i, then the mechanism is incentive compatible Thm (Roberts 1979). If |O| ≥ 3, f is onto O, V i = R O for every i, and the mechanism is incentive compatible, then f is an affine maximizer

4. Single-parameter domains Setting: –V i is one-dimensional, i.e., V i in R –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 v i, and –every winning agent pays his critical value

5. (Essentially) uniqueness of prices Thm. –Assume the domains of V i are connected sets (in the usual metric in Euclidean space) –Let (f, p 1,…p n ) be an incentive compatible mechanism –The mechanism (f, p’ 1,…p’ n ) is incentive compatible iff p’ i (v 1,…v n ) = p i (v 1,…v n ) + h i (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 [Weak monotonicity and Bayes–Nash incentive compatibility Games and Economic Behavior, Volume 61, Issue 2, November 2007, Pages 344-358 Rudolf Müller, Andrés Perea, Sascha Wolf]Weak monotonicity and Bayes–Nash incentive compatibility