Sensitivity Analysis for Convex Separable Optimization Over Integral Polymatroids

Abstract

We study the sensitivity of optimal solutions of convex separable optimization problems over an integral polymatroid base polytope with respect to parameters determining both the cost of each element and the polytope. Under convexity and a regularity assumption on the functional dependency of the cost function with respect to the parameters, we show that reoptimization after a change in parameters can be done by elementary local operations. Applying this result, we derive that starting from any optimal solution there is a new optimal solution to new parameters such that the L 1 -norm of the difference of the two solutions is at most two times the L 1 -norm of the difference of the parameters.We apply these sensitivity results to a class of non-cooperative games with a finite set of players where a strategy of a player is to choose a vector in a player-specific integral polymatroid base polytope defined on a common set of elements. The players' private cost functions are regular, convex-separable and the cost of each element is a non-decreasing function of the own usage of that element and the overall usage of the other players. Under these assumptions, we establish the existence of a pure Nash equilibrium. The existence is proven by an algorithm computing a pure Nash equilibrium that runs in polynomial time whenever the rank of the polymatroid base-polytope is polynomially bounded. Both the existence result and the algorithm generalize and unify previous results appearing in the literature.We finally complement our results by showing that polymatroids are the maximal combinatorial structure enabling these results. For any non-polymatroid region, there is a corresponding optimization problem for which the sensitivity results do not hold. In addition, there is a game where the players' strategies are isomorphic to the non-polymatroid region and that does not admit a pure Nash equilibrium.

Uniqueness and computation of equilibria in resource allocation games

“…In particular, Harks et al devised an algorithm with running time n 2 mpδ{k 0 q 3 , where n is the number of players, m the number of resources, and δ is an upper bound on the maximum demand of the players (cf. Corollary 5.2 [32]). As δ is encoded in binary, however, the algorithm is only pseudopolynomial even for player-specific affine cost functions.…”

“…The class of integrally-splittable congestion games has been studied before by Tran-Thanh et al [72] for the case of player-independent convex cost functions and later by Harks et al [32,36] (for the more general case of polymatroid strategy spaces and player-specific convex cost functions). In particular, Harks et al devised an algorithm with running time n 2 mpδ{k 0 q 3 , where n is the number of players, m the number of resources, and δ is an upper bound on the maximum demand of the players (cf.…”

“…Again, the idea is to compute a pure Nash equilibrium for an associated integral splittable congestion game. It is known that one can compute pure Nash equilibria for integral splittable polymatroid congestion games with convex cost functions within a running time that is pseudo-polynomial in the aggregated demand of the players (see Harks et al [32,36]). In this paper we compute for each ą 0 a packet size k such that the k -integral equilibrium is guaranteed to be an -approximate equilibrium.…”

“…On the other hand, there are polynomial algorithms for symmetric network congestion games (cf. Fabrikant et al [25]), for matroid congestion games with player-specific cost functions (Ackermann et al [2,3]), for polymatroid congestion games with player-specific cost functions and polynomially bounded demands (Harks et al [32,36]) and for so-called total unimodular congestion games (see Del Pia et al [22]). Further results regarding the computation of approximate equilibria in congestion games can be found in Caragiannis et al [14,15], Chien and Sinclair [17] and Skopalik and Vöcking [68].…”

“…It is left to compute an equilibrium x k 0 for integral game G k 0 . Such integral games have been studied in the literature before, see Harks et al [32,36]. In particular, [32, Algorithm 1] has running time Opnmpδ{k 0 q 3 q.…”