Dynamic mechanism design

AbstractIn this paper we address the question of designing truthful mechanisms for solving optimization problems on dynamic graphs with selfish edges. More precisely, we are given a graph G of n nodes, and we assume that each edge of G is owned by a selfish agent. The strategy of an agent consists in revealing to the system–at each time instant–the cost at the actual time for using its edge. Additionally, edges can enter into and exit from G. Among the various possible assumptions which can be made to model how this edge-cost modifications take place, we focus on two settings: (i) the dynamic, in which modifications can happen at any time, and for a given optimization problem on G, the mechanism has to maintain efficiently the output specification and the payment scheme for the agents; (ii) the time-sequenced, in which modifications happens at fixed time steps, and the mechanism has to minimize an objective function which takes into consideration both the quality and the set-up cost of a new solution. In both settings, we investigate the existence of exact and approximate truthful (w.r.t. to suitable equilibrium concepts) mechanisms. In particular, for the dynamic setting, we analyze the minimum spanning tree problem, and we show that if edge costs can only decrease and each agent adopts a myopic best response strategy (i.e., its utility is only measured instantaneously), then there exists an efficient dynamic truthful (in myopic best response equilibrium) mechanism for handling a sequence of k declarations of edge-cost reductions having runtime O((h+k)logn), where h is the overall number of payment changes

On the Existence of Truthful Mechanisms for the Minimum-Cost Approximate Shortest-Paths Tree Problem

Let a communication network be modeled by a graph G = (V,E) of n nodes and m edges, where with each edge is associated a pair of values, namely its cost and its length. Assume now that each edge is controlled by a selfish agent, which privately holds the cost of the edge. In this paper we analyze the problem of designing in this non-cooperative scenario a truthful mechanism for building a broadcasting tree aiming to balance costs and lengths. More precisely, given a root node r ∈V and a real value λ≥1, we want to find a minimum cost (as computed w.r.t. the edge costs) spanning tree of G rooted at r such that the maximum stretching factor on the distances from the root (as computed w.r.t. the edge lengths) is λ. We call such a tree the Minimum-cost λ -Approximate Shortest-paths Tree (λ-MAST). First, we prove that, already for the unit length case, the λ-MAST problem is hard to approximate within better than a logarithmic factor, unless NP admits slightly superpolynomial time algorithms. After, assuming that the graph G is directed, we provide a (1 + ε)(n – 1)-approximate truthful mechanism for solving the problem, for any ε> 0. Finally, we analyze a variant of the problem in which the edge lengths coincide with the private costs, and we provide: (i) a constant lower bound (depending on λ) to the approximation ratio that can be achieved by any truthful mechanism; (ii) a (1+ [(n-1)/(l)])(1+n−1)-approximate truthful mechanism