Game Theory — Fall 2010 Instructor : Chaitanya Swamy

Q1: Recall the load balancing game from class: there are m machines and n jobs. Each job j is a player: it has a certain processing time pj and its strategy is to choose a machine to get assigned to. Given an assignment of jobs to machines, the load Li of machine i is the total processing time of the jobs assigned to it, and the cost incurred by a job j is the load of the machine to which it is assigned. In this question, we address the issue of how quickly the natural heuristic of letting players make improving-moves converges to a Nash equilibrium. (You might want to read the proof of Theorem 20.6 in the book before attempting this question.) Given an assignment {i(j)}, we say that a job j is unsatisfied if it can move to some machine (other than i(j)) and reduce its cost. A best-response move of job j is a move that minimizes its cost (given the current assignment).