Algorithms for pure Nash equilibria in weighted congestion games

Abstract

In large-scale or evolving networks, such as the Internet, there is no authority possible to enforce a centralized traffic management. In such situations, game theory, and especially the concepts of Nash equilibria and congestion games [Rosenthal 1973] are a suitable framework for analyzing the equilibrium effects of selfish routes selection to network delays. We focus here on single-commodity networks where selfish users select paths to route their loads (represented by arbitrary integer weights). We assume that individual link delays are equal to the total load of the link. We then focus on the algorithm suggested in Fotakis et al. [2005], i.e., a potential-based method for finding pure Nash equilibria in such networks. A superficial analysis of this algorithm gives an upper bound on its time, which is polynomial in n (the number of users) and the sum of their weights W . This bound can be exponential in n when some weights are exponential. We provide strong experimental evidence that this algorithm actually converges to a pure Nash equilibrium in polynomial time. More specifically, our experimental findings suggest that the running time is a polynomial function of n and log W . In addition, we propose an initial allocation of users to paths that dramatically accelerates this algorithm, compared to an arbitrary initial allocation. A by-product of our research is the discovery of a weighted potential function when link delays are exponential to their loads. This asserts the existence of pure Nash equilibria for these delay functions and extends the result of Fotakis et al. [2005].

On the Existence of Pure Nash Equilibria in Weighted Congestion Games

“…Also related is the early work of Rosenthal [33] who showed that in weighted congestion games where players are allowed to split their demand integrally, a PNE need not exist. On the positive side, Fotakis et al [16] and Panagopoulou and Spirakis [30] proved the existence of a PNE in games with affine and exponential costs, respectively. Dunkel and Schulz [13] showed that it is strongly NP-hard to decide whether or not a weighted congestion game with nonlinear cost functions possesses a PNE.…”

“…(iii) ⇒ (ii) follows because every weighted congestion games with such cost functions possesses a weighted potential; see Fotakis et al [16], Harks et al [22], and Panagopoulou and Spirakis [30].…”

“…We say that is consistent if every game in possesses a PNE. Using this terminology, the results of Fotakis et al [16,17], Harks et al [22], and Panagopoulou and Spirakis [30] yield that is consistent if contains either affine functions or certain exponential functions. A natural open question is to decide whether there are further consistent functions, that is, functions guaranteeing the existence of a PNE.…”

“…Harks et al [22] generalized this existence result to nonuniform exponential cost functions of the form c f x = a f e x + b f for some a f b f ∈ , where a f and b f may depend on the facility f , while must be equal for all facilities. It is worth noting that the positive results of Fotakis et al [16,17], Harks et al [22], and Panagopoulou and Spirakis [30] are particularly important as they establish existence of PNE for the respective sets of cost functions independent of the underlying game structure, that is, independent of the underlying strategy set, demand vector, and number of players, respectively. In this paper, we further explore the equilibrium existence problem in weighted congestion games.…”

“…For games with affine cost functions, Fotakis et al [16,17] proved that every weighted congestion game possesses a PNE. Later, Panagopoulou and Spirakis [30] proved that PNE always exist for instances with uniform exponential cost functions c f x = e x . Harks et al [22] generalized this existence result to nonuniform exponential cost functions of the form c f x = a f e x + b f for some a f b f ∈ , where a f and b f may depend on the facility f , while must be equal for all facilities.…”