Convergence time to Nash equilibrium in load balancing

We study the number of steps required to reach a pure Nash equilibrium in a load balancing scenario where each job behaves selfishly and attempts to migrate to a machine which will minimize its cost. We consider a variety of load balancing models, including identical, restricted, related, and unrelated machines. Our results have a crucial dependence on the weights assigned to jobs. We consider arbitrary weights, integer weights, k distinct weights, and identical (unit) weights. We look both at an arbitrary schedule (where the only restriction is that a job migrates to a machine which lowers its cost) and specific efficient schedulers (e.g., allowing the largest weight job to move first). A by-product of our results is establishing a connection between various scheduling models and the game-theoretic notion of potential games. We show that load balancing in unrelated machines is a generalized ordinal potential game, load balancing in related machines is a weighted potential game, and load balancing in related machines and unit weight jobs is an exact potential game.

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