Equilibria of Atomic Flow Games are not Unique

Bhaskar, Umang; Fleischer, Lisa; Hoy, Darrell; Huang, Chien‐Chung

doi:10.1137/1.9781611973068.82

“…They also derived conditions under which Nash equilibria are unique. Uniqueness of Nash equilibria has been further studied by Fleischer et al [4] and Orda et al [26].…”

Section: Related Workmentioning

confidence: 99%

Stackelberg Strategies and Collusion in Network Games with Splittable Flow

Harks¹

2010

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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.

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“…Note that this definition of the price of anarchy is slightly different from the standard nonatomic selfish routing model ( [30]), since there may be qualitatively different equilibria, see [4].…”

Section: The Modelmentioning

confidence: 99%

Stackelberg Strategies and Collusion in Network Games with Splittable Flow

Harks¹

2009

Approximation and Online Algorithms

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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.

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“…Nonatomic equilibria are known to be essentially unique, but this is not the case for atomic splittable routing games, where uniqueness criteria were recently obtained by Bhaskar et al [3]. Equilibria in routing games are known to be inefficient, and considerable research in algorithmic game theory has focused on quantifying this inefficiency in terms of the price of anarchy (PoA) [18,20] of the game, which measures, for a given objective, the worst-case ratio between the objective values of an equilibrium and the optimal solution.…”

Section: Related Workmentioning

confidence: 99%

“…Our results extend to atomic splittable routing games if we assume that for all valid choices of parameters of the latency functions and tolls (as encountered during the ellipsoid method), the underlying atomic splittable routing game has a unique Nash equilibrium. Here, by uniqueness we mean that if f and g are two Nash equilibria, then f i e = g i e for all commodities i and edges e. This is not without loss of generality, but is known to hold, for example, if all latency functions are convex polynomials of degree at most 3, or if the graph is a generalized nearly-parallel graph and xl e (x) is strictly convex for all e (see [3]). When we say that tolls τ induce a flow f * = (f * i ) i≤k here, we mean that the flow of every commodity i on every edge e is f * i e in the resulting equilibrium.…”

Section: Extensionsmentioning

confidence: 99%

Achieving target equilibria in network routing games without knowing the latency functions

Bhaskar

¹

,

Ligett

²

,

Schulman

³

et al. 2019

Games and Economic Behavior

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The analysis of network routing games typically assumes, right at the onset, precise and detailed information about the latency functions. Such information may, however, be unavailable or difficult to obtain. Moreover, one is often primarily interested in enforcing a desired target flow as the equilibrium by suitably influencing player behavior in the routing game. We ask whether one can achieve target flows as equilibria without knowing the underlying latency functions.Our main result gives a crisp positive answer to this question. We show that, under fairly general settings, one can efficiently compute edge tolls that induce a given target multicommodity flow in a nonatomic routing game using a polynomial number of queries to an oracle that takes candidate tolls as input and returns the resulting equilibrium flow. This result is obtained via a novel application of the ellipsoid method. Our algorithm extends easily to many other settings, such as (i) when certain edges cannot be tolled or there is an upper bound on the total toll paid by a user, and (ii) general nonatomic congestion games. We obtain tighter bounds on the query complexity for series-parallel networks, and single-commodity routing games with linear latency functions, and complement these with a query-complexity lower bound. We also obtain strong positive results for Stackelberg routing to achieve target equilibria in series-parallel graphs.Our results build upon various new techniques that we develop pertaining to the computation of, and connections between, different notions of approximate equilibrium; properties of multicommodity flows and tolls in series-parallel graphs; and sensitivity of equilibrium flow with respect to tolls. Our results demonstrate that one can indeed circumvent the potentially-onerous task of modeling latency functions, and yet obtain meaningful results for the underlying routing game. *

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Equilibria of Atomic Flow Games are not Unique

Cited by 26 publications

References 13 publications

Stackelberg Strategies and Collusion in Network Games with Splittable Flow

Stackelberg Strategies and Collusion in Network Games with Splittable Flow

Stackelberg Strategies and Collusion in Network Games with Splittable Flow

Achieving target equilibria in network routing games without knowing the latency functions

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