Algorithmic Mechanism Design (1999; Nisan, Ronen)

Formulation The framework consists of a set A of alternatives, or outcomes, and n players, or agents. Each player i has a valuation function vi : A → < that assigns a value to each possible alternative. This valuation function belongs to a domain Vi of all possible valuation functions. Let V = V1 × · · · × Vn, and V−i = ∏ j 6=i Vj. Observe that this generalizes the shortest path example of above: A is all the possible s − t paths in the given graph, ve(a) for some path a ∈ A is either −we (if e ∈ a) or zero. A social choice function f : V → A assigns a socially desirable alternative to any given profile of players’ valuations. This parallels the notion of an algorithm. A mechanism is a tuple M = (f, p1, ..., pn), where f is a social choice function, and pi : V → < (for i = 1, ..., n) is the price charged from player i. The interpretation is that the social planner asks the players to reveal their true valuations, chooses the alternative according to f as if the players have indeed acted truthfully, and in addition rewards/punishes the players with the prices. These prices should induce “truthfulness” in the following strong sense: no matter what the other players declare, it is always in the best interest of player i to reveal her true valuation, as this will maximize her utility. Formally, this translates to: Definition 1 (Truthfulness). M is “truthful” (in dominant strategies) if, for any player i, any profile of valuations of the other players v−i ∈ V−i, and any two valuations of player i vi, v i ∈ Vi, vi(a)− pi(vi, v−i) ≥ vi(b)− pi(v i, v−i) where f(vi, v−i) = a and f(v ′ i, v−i) = b. Truthfulness is quite strong: a player need not know anything about the other players, even not that they are rational, and still determine the best strategy for her. Quite remarkably, there exists a truthful mechanism, even under the current level of abstraction. This mechanism suits all problem domains, where the social goal is to maximize the “social welfare”: Definition 2 (social welfare maximization). A social choice function f : V → A maximizes the social welfare if f(v) ∈ argmaxa∈A ∑ i vi(a), for any v ∈ V . Notice that the social goal in the shortest path domain is indeed welfare maximization, and, in general, this is a natural and important economic goal. Quite remarkably, there exists a general technique to construct truthful mechanisms that implement this goal: Theorem 1 (Vickrey-Clarke-Groves (VCG)). Fix any alternatives set A and any domain V , and suppose that f : V → A maximizes the social welfare. Then there exist prices p such that the mechanism (f, p) is truthful. This gives “for free” a solution to the shortest path problem, and to many other algorithmic problems. The great advantage of the VCG scheme is its generality: it suits all problem domains. The disadvantage, however, is that the method is tailored to social welfare maximization. This turns out to be restrictive, especially for algorithmic and computational settings, due to several reasons: (i) different algorithmic goals: the algorithmic literature considers a variety of goals, including many that cannot be translated to welfare maximization. VCG does not help us in such cases. (ii) computational complexity: even if the goal is welfare maximization, in many settings achieving exactly the optimum is computationally hard. The CS discipline usually overcomes this by using approximation algorithms, but VCG will not work with such algorithm – reaching exact optimality is a necessary requirement of VCG. (iii) different algorithmic models: common CS models change “the basic setup”, hence cause unexpected difficulties when one tries to use VCG (for example, an online model, where the input is revealed over time; this is common in CS, but changes the implicit setting that VCG requires). This is true even if welfare maximization is still the goal.