Reference no: EM131389455
Problem -
Consider the following load balancing game: There are m machines F = {1, . . . , m} and n players. Each player i may choose a single machine on which to run his job: for all i, Ai = F. But now each player's job may take a different amount of time to run on each machine (the machines are not identical - one may have a faster CPU, one may have a faster graphics card, etc). For each machine j ∈ F and each player i, there is a corresponding weight wi,j describing how long it takes to run player i's job on machine j. The cost of machine j is the sum of the weights of the jobs assigned to it: lj(a) = ∑i:a_i = j wi,j. The cost of player i is the cost of the machine he selected: ci(a) = la_i (a).
The load balancing game we considered in class was the special case when each player had the same weight on every machine: wi,j = wi,j' ≡ wi for all j, j'.
1. Fix an action profile a and suppose player i makes a best-response move (i.e. one that strictly decreasing his cost) from playing on machine ai to playing on machine a'i. Write a' = (a'i, a-i). Show that for all j ∉ {ai, a'i}, lj(a)= lj(a') and that:
max(la_i(a'), la'_i(a')) < max(la_i (a), la'_i(a))
2. Show that the load balancing games on unrelated machines has an ordinal potential function (equivalently, a total ordering on all action profiles a ∈ A such that best response moves only move to states later in the ordering), and conclude that such games always have pure strategy Nash equilibria.
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