Stackelberg Strategies and Collusion in Network Games with Splittable Flow

Abstract

We study the impact of collusion in network games with splittable flow and focus on the well established price of anarchy as a measure of this impact. We first investigate symmetric load balancing games and show that the price of anarchy is bounded from above by m, where m denotes the number of coalitions. For general networks, we present an instance showing that the price of anarchy is unbounded, even in the case of two coalitions. If latencies are restricted to polynomials, we prove upper bounds on the price of anarchy for general networks, which improve upon the current best ones except for affine latencies.In light of the negative results even for two coalitions, we analyze the effectiveness of Stackelberg strategies as a means to improve the quality of Nash equilibria. In this setting, an α fraction of the entire demand is first routed centrally by a Stackelberg leader according to a predefined Stackelberg strategy and the remaining demand is then routed selfishly by the coalitions (followers).For a single coalitional follower and parallel arcs, we develop an efficiently computable Stackelberg strategy that reduces the price of anarchy to one. For general networks and a single coalitional follower, we show that a simple strategy, called SCALE, reduces the price of anarchy to 1 + α . Finally, we investigate SCALE for multiple coalitional followers, general networks, and affine latencies. We present the first known upper bound on the price of anarchy in this case. Our bound smoothly varies between 1.5 when α = 0 and full efficiency when α = 1.

Efficiency of Restricted Tolls in Non-atomic Network Routing Games

“…Essentially, we show that these tolls achieve an efficiency that depends on the degree of the polynomial and the smallest ratio between the threshold value and the latency of an arc (see Section 4 for precise statements). The technique that we use to establish these bounds rests on a (λ, µ)-smoothness approach [16] that has previously been used successfully to bound the price of anarchy of network routing games [2,8] and in a more general context in [16]. We also prove that our bounds are tight, even for parallel-arc networks.…”

“…We next derive a bound on the ratio C( f )/C( f * ), where f * is an optimal flow. We adapt the (λ, µ)-smoothness approach [16] which was previously used successfully to bound the price of anarchy of network routing games [2,8] and in a more general context in [16].…”

“…Given an instance of the restricted network toll problem with polynomial latency functions of degree p, the efficiency of θ-restricted tolls as defined in (8) is no worse than…”

“…Define threshold functions θ a (x) = 0 and θā(x) = δℓā(x). Let the tolls (τ a ) a∈A be defined as in (8). Note thatε p = δ.…”

The Limits of Smoothness: A Primal-Dual Framework for Price of Anarchy Bounds

“…Roughgarden [11] showed that every POA bound proved using a "smoothness argument" (see Definition 1) -the most frequently employed method for establishing POA bounds (e.g. [5,6,9,10,12]) -applies automatically to (at least) all CCE of the game. A basic problem is to characterize the distributions over outcomes to which smoothness bounds always apply.…”