In this work we study of competitive situations among users of a set of global resources. More precisely we study the effect of cost policies used by these resources in the convergence time to a pure Nash equilibrium. The work is divided in two parts. In the theoretical part we prove lower and upper bounds on the convergence time for various cost policies. We then implement all the models we study and provide some experimental results. These results follows the theoretical with one exception which is the most interesting among the experiments. In the case of coalitional users the theoretical upper bound is pseudo-polynomial to the number of users but the experimental results shows that the convergence time is polynomial.
Deep Dive into Study of the effect of cost policies in the convergence of selfish strategies in Pure Nash Equilibria in Congestion Games.
In this work we study of competitive situations among users of a set of global resources. More precisely we study the effect of cost policies used by these resources in the convergence time to a pure Nash equilibrium. The work is divided in two parts. In the theoretical part we prove lower and upper bounds on the convergence time for various cost policies. We then implement all the models we study and provide some experimental results. These results follows the theoretical with one exception which is the most interesting among the experiments. In the case of coalitional users the theoretical upper bound is pseudo-polynomial to the number of users but the experimental results shows that the convergence time is polynomial.
General goal of the current work is the study of competitive situations among users of a set of global resources. In order to analyze and model these situations we use as tools, game theoretic elements [OR94], such as Nash Equilibria, congestion games and coordination mechanisms. Every global resource debit a cost value to its users. We assume that the users are selfish; ie. their sole objective is the maximization of their personal benefit. An Nash equilibrium (NE) is a situation in which no user can increase his personal benefit by changing only his or her own strategy unilaterally.
More specific, we are interested in the KP-model or parallel links2 model with n users(jobs) and m edges(machines) and we study convergence methods to pure Nash Equilibria, in which all the strategies a user can select are deterministic. Generally, a game has not always a pure Nash equilibrium. Although we are going to study cases in which there is always at least one Nash equilibrium. We define as cost policy of an edge the function which computes the cost of each user of this edge.
One method of convergence in a pure Nash equilibrium is, starting from an initial configuration, to allow all users to selfishly change their strategies (one after the other) until they reach a pure Nash equilibrium. We are interested in the convergence time to pure Nash Equilibria, that is the number of these selfish moves. Firstly, we study the makespan cost policy, in which each edge debits its total load to everyone that use it. In the most simple case, the whole procedure is divided into several steps. At each step, the priority algorithm choose one user from the set of users that benefit by changing their current strategy. For this model, named ESS-model, the convergence time is at the worst case exponential to the number of users. We present the effect of several priority algorithms to the convergence time and results for the major different cases of edges (identical, related, unrelated) [EDKM03].
Another approach, with applications to distributed systems, is the concurrent change of strategies (rerouting) [EDM05] in which more than one user can change simultaneously his strategy. This model is more powerful than ESS because of its real life applications but we are not analyzing it in this work.
An extension to ESS-model is that of coalitions, in which the users can contract alliances. This model comes from cooperative game theory. In this case we have to deal with groups of users changing selfishly their group strategies. We restrict our attention to coalitions of at most 2 users in the identical machines introduced in [FKS06]. The pairs of coalitional users can exchange their machines making a 2-flip move as a kind of pair migration. There is a pseudo-polynomial upper bound to the convergence time to NE in this model.
Another model of convergence, a little different than the others stated above, is the construction of an algorithm that delegates strategies to the users unselfishly without increasing the social cost. Informally, social cost is a total metric of the system performance depending on the users strategies. This model is named nashification and the algorithm nashify provides convergence to a pure Nash equilibrium in polynomial number of steps without increasing the social cost [FGL + 03].
As far as the coordination mechanisms are concerned, they are a set of cost policies for the edges, that provides motives to the selfish users in order to converge to a pure Nash equilibrium with decreased social cost.
In this paper, we study the effect of coordination mechanisms in the convergence time. We study the following cases:
Cost Policies: (1) Makespan, (2) Shortest Job/User First (SJF), (3) Longest Job/User First (LJF), (4) First In First Out (FIFO).
Priority Algorithms: (a) max weight job/user (maw), (b) min weight job/user (miw), (c) fifo, (d) random.
Note that each machine uses a cost policy and the next for migration user is chosen using a priority algorithm and the above combinations can result in linear, polynomial or exponential convergence.
In the concept of coalitions there is a need of tight braking when we have to choose among pairs of users. We study two algorithms: Coalition Priority Algorithms: (i) max weight pair (map), (ii) min weight pair (mip).
Note that there is a arrangement of pairs comparing the subtraction of weights for each pair.
The paper’s results are divided in two categories: theoretical and experimental.
Theoretical Results. We study the convergence time for SJF, LJF and FIFO policies.
Especially for FIFO we prove in identical machines case a tight linear bound and a pseudopolynomial bound in unrelated machines case.
Lemma 1 Under the FIFO cost policy in the ESS-model, there is an upper bound of n 2 2 w max steps for convergence to NE in the unrelated machines case and n -1 steps in the identical machines case. This result is independent from the priority algorithm.
Proof Sketch. Note tha
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