The Price of Optimum in Stackelberg Games

Consider a system M of parallel machines, each with a strictly increasing and differentiable load dependent latency function. The users of such a system are of infinite number and act selfishly, routing their infinitesimally small portion of the total flow r they control, to machines of currently minimum delay. It is well known that such selfishness if modeled by a noncooperative game may yield a Nash Equilibrium onM with cost unboundedly worst than the overallOptimum one. We model such a system as a Stackelberg or Leader-Followers game, and present a simple algorithm that computes the least flow βM (or “price of optimum”) that must be controlled by a Leader in order to impose the overall optimum cost on M , as well as Leader’s optimum strategy. The efficiency of our algorithm depends on the computation of the optimum and Nash assignment on such systems. Such assignments can be computed efficiently [4] for the classes of latency functions that we are interested in. Our motivation was [21] were the open question of computing βM on a arbitrary system M was posed. We were also greatly inspired from [7], in which the computation of βM was a major issue. In that article systems with M/M/1 latency functions were studied, which are widely met in real world applications. Furthermore, βM (n) was computed explicitly for Stackelberg games with either n = 1 or a finite number n of Followers. It was demonstrated that βM (n) is nondecreasing on n, and as n increases it becomes harder for the Leader to impose the overall optimum. Most notably, it was conjectured that if n → ∞ then it is not possible for the Leader to impose system’s overall optimum. This comes into contrast to our theoretical results. As a by-product, we present a simple algorithm that computes the Nash Equilibrium for a systemM with M/M/1 latency functions. We should stress here that the model of parallel machines, despite its simplicity, incurs the worst coordination ratio as proved in [19].