11,318 research outputs found
Smooth Inequalities and Equilibrium Inefficiency in Scheduling Games
We study coordination mechanisms for Scheduling Games (with unrelated
machines). In these games, each job represents a player, who needs to choose a
machine for its execution, and intends to complete earliest possible. Our goal
is to design scheduling policies that always admit a pure Nash equilibrium and
guarantee a small price of anarchy for the l_k-norm social cost --- the
objective balances overall quality of service and fairness. We consider
policies with different amount of knowledge about jobs: non-clairvoyant,
strongly-local and local. The analysis relies on the smooth argument together
with adequate inequalities, called smooth inequalities. With this unified
framework, we are able to prove the following results.
First, we study the inefficiency in l_k-norm social costs of a strongly-local
policy SPT and a non-clairvoyant policy EQUI. We show that the price of anarchy
of policy SPT is O(k). We also prove a lower bound of Omega(k/log k) for all
deterministic, non-preemptive, strongly-local and non-waiting policies
(non-waiting policies produce schedules without idle times). These results
ensure that SPT is close to optimal with respect to the class of l_k-norm
social costs. Moreover, we prove that the non-clairvoyant policy EQUI has price
of anarchy O(2^k).
Second, we consider the makespan (l_infty-norm) social cost by making
connection within the l_k-norm functions. We revisit some local policies and
provide simpler, unified proofs from the framework's point of view. With the
highlight of the approach, we derive a local policy Balance. This policy
guarantees a price of anarchy of O(log m), which makes it the currently best
known policy among the anonymous local policies that always admit a pure Nash
equilibrium.Comment: 25 pages, 1 figur
Improving the Price of Anarchy for Selfish Routing via Coordination Mechanisms
We reconsider the well-studied Selfish Routing game with affine latency
functions. The Price of Anarchy for this class of games takes maximum value
4/3; this maximum is attained already for a simple network of two parallel
links, known as Pigou's network. We improve upon the value 4/3 by means of
Coordination Mechanisms.
We increase the latency functions of the edges in the network, i.e., if
is the latency function of an edge , we replace it by
with for all . Then an
adversary fixes a demand rate as input. The engineered Price of Anarchy of the
mechanism is defined as the worst-case ratio of the Nash social cost in the
modified network over the optimal social cost in the original network.
Formally, if \CM(r) denotes the cost of the worst Nash flow in the modified
network for rate and \Copt(r) denotes the cost of the optimal flow in the
original network for the same rate then [\ePoA = \max_{r \ge 0}
\frac{\CM(r)}{\Copt(r)}.]
We first exhibit a simple coordination mechanism that achieves for any
network of parallel links an engineered Price of Anarchy strictly less than
4/3. For the case of two parallel links our basic mechanism gives 5/4 = 1.25.
Then, for the case of two parallel links, we describe an optimal mechanism; its
engineered Price of Anarchy lies between 1.191 and 1.192.Comment: 17 pages, 2 figures, preliminary version appeared at ESA 201
- …