Stackelberg Scheduling Strategies

1
Stackelberg Scheduling Strategies

• By
• Tim Roughgarden

Presented by Alex Kogan
2
Abstract

• We consider setting of scheduling jobs on a set
  of shared machines with load-dependent latency
  functions. The system performance is measured by
  the total latency of the system.
• We assume that the users selfishly wish only to
  minimize the latencies of their own jobs.
• In this case, the total latency is non-optimal.

3
Abstract (cont.)

• If theres a mix of selfishly controlled jobs
  and centrally controlled jobs, the assignment
  of centrally controlled jobs will influence the
  subsequent actions of the selfish users.
• Were interested in assigning the centrally
  controlled jobs in the best possible way (that
  maximizes the overall system performance ).

4
Coping with Selfishness

• In many large-scale systems (like the Internet),
  there is no central authority controlling the
  allocation of shared resources.
• The users act selfishly (non-cooperative game).
• Results in Nash eq. gt sub-optimal performance.
• Given a system with a mix of centrally and
  selfishly controlled jobs, how can centrally
  controlled jobs assignment to induce good
  behavior from the non-cooperative users?

5
Stackelberg Games

• The roles of different players are asymmetric.
• One player acts as a leader (according to some
  strategy).
• All other agents (the followers) react
  independently and selfishly to the leaders
  strategy, reaching a Nash equilibrium relative to
  the leaders strategy.
• The Stackelberg equilibrium is the minimum-cost
  equilibrium achieved by a Stackelberg strategy.

6
The Central Questions

• Given a set of m machines with load-dependent
  latencies and a large number of very small jobs
  to be scheduled, we can ask
• Among all leader strategies for a given set of
  machines and jobs, can we characterize and/or
  compute the strategy inducing the Stackelberg
  equilibrium - i.e., the eq. of minimal total
  latency?
• What is the worst-case ratio between the total
  latency of the Stackelberg eq. and that of the
  optimal assignment of jobs to the machines?

7
Results

• We give a simple polynomial-time algorithm
  algorithm for computing a leader strategy that
  induces an equilibrium with total latency no more
  then 1/? times the optimal (? - the fraction of
  centrally controlled jobs).
• We give an O(m2) algorithm for computing a
  strategy inducing total latency of at most
  4/(3?) of the optimal in special case of linear
  latency functions.

8
Results (cont.)

• Computing the strategy inducing the Stackelberg
  equilibrium is NP-hard, even when the latencies
  are linear!

9
The Model

• Set M of m machines 1, 2, , m
• li() is the latency of machine i (continuous and
  non-decreasing).
• (M, r) - an instance with machines M, rate r and
  no centrally controlled jobs.
• (M, r, ?) - a Stackelberg instance, where ??(0,1)
  indicates the fraction of the centrally
  controlled traffic.

10
Stackelberg Strategies and Induced Equilibria

• Definition A Stackelberg strategy for the
  Stackelberg instance (M, r, ?) is an assignment
  feasible for (M, ?r).
• Definition Let s be a strategy for Stackelberg
  instance (M, r, ?) where machine i has latency
  function li, and let li(x) li(si x) for each
  i?M. An equilibrium induced by s is an assignment
  t at Nash equilibrium for (M,(1-?)r) w.r.t.
  latency functions li.

11
The Aloof Strategy

• If x is the optimal assignment for (M, ?r), put
  s x.
• The minimum-cost strategy (ignoring the existence
  of jobs that are not centrally controlled).
• Poor performance.

12
The Scale Strategy

• If x is the optimal assignment for (M, r), put
  s ? x.
• The optimal assignment of the jobs, suitably
  scaled.
• Poor performance.

13
The LLF Strategy

• Both the Aloof and the Scale strategies suffer
  from the same flaw both dont consider the
  selfish users behavior.
• Its reasonable for a good strategy to give
  priority to the machines that are least appealing
  to selfish users - machines with relatively high
  latency.
• We consider the Largest Latency First strategy.

14
The LLF Strategy (cont.)

• The LLF steps
• Compute the optimal assignment x for (M, r)
• Index the machines of M so that l1(x1) ?
  ... ? lm(xm)
• Let k ? m be minimal with ?igtk xi ? ?r
• Put si xi if i gt k, sk ?r - ?igtk xi, si
  0 if i lt k
• A machine i is saturated by s if si xi. LLF
  saturates machines of the largest latency until
  therere no centrally-controlled jobs remaining.

15
The LLF Performance Guarantee

• For arbitrary latency functions, the LLF always
  induces an assignment of cost no more than 1/?
  times that of the optimal assignment.
• can be computed in polynomial time
• For linear latency functions, the LLF performance
  guarantee is 4/(3 ?).
• can be computed in O(m2)

16
The Complexity of Computing Optimal Strategies

• The LLF strategy not always provides the optimal
  result.
• The problem of computing the optimal Stackelberg
  strategy is NP-hard, even for instances with
  linear latency functions.