Price of Anarchy of Multi-Stage Machine Scheduling Games

TL;DR

The paper investigates how selfish, greedy agents affect efficiency in multi-stage (flow-shop-like) machine scheduling. It shows that greedy least-loaded decisions at each stage may diverge from subgame-perfect equilibria, establishing exact PoA for single-stage greedy scheduling as $2 - 1/m$ and a bounded PoA for multi-stage settings between $2 - 1/m_{max}$ and $3 - 1/m_{max}$. The analysis derives a recurrence that bounds how much completion times can grow when moving from one stage to the next, enabling a global PoA bound. These results illuminate the inefficiency of simple greedy policies in pipelines and distributed workflows and point to open questions about SPNE performance and more general machine types.

Abstract

In this paper, we extend the discussion of the price of anarchy of machine scheduling games to a multi-stage machine setting. The multi-stage setting arises naturally in manufacturing pipelines and distributed computing workflows, when each job must traverse a fixed sequence of processing stages. While the classical makespan price of anarchy of $2 - \frac{1}{m}$ has been established for sequential scheduling on identical machines, the efficiency loss in multi-stage scheduling has, to the best of our knowledge, not been previously analyzed. We assume that each task follows a greedy strategy and gets assigned to the least-loaded machine upon arrival at each stage. Notably, we observe that in multi-stage environments, greedy behavior generally does not coincide with a subgame perfect Nash equilibrium. We continue with analyzing the equilibrium under greedy choices, since it is logical for modeling selfish agents with limited computational power, and may also model a central scheduler performing the common least-load scheduling heuristics. Under this model, we first show that in single-stage scheduling, greedy choice again yields an exact price of anarchy of $2 - \frac{1}{m}$. In multi-stage scheduling, we show that the completion time from one stage to the next increases by at most two times the maximum job execution time. Using this relationship, we derived the price of anarchy of multistage scheduling under greedy choices to lie within $[2 - \frac{1}{m}, 3 - \frac{1}{m}]$, where $m$ denote the maximum number of machines in one stage.

Figures (4)

• Figure 1: Model of our multi-stage scheduling game. All tasks get released at $P_0$ at $t=0$ in a given order. Tasks are sequentially assigned to machines in the first stage. Once execution in the first stage is finished, it enters $P_1$ and gets assigned to machine in the second stage. The tasks are greedy and is always assigned to the machine with least load when it enters a new stage. Each task repeat the process of selecting machines, queuing, and execution until it reaches the end of the $k$ stage network $P_k$. This network models a procedure where tasks of different sizes require an identical $k$ step procedure, and the goal is to minimize the maximum completion time.
• Figure 2: An illustration of a M-machine stage, where jobs are released at source $s$ with release time $r_j$, processed through one of the machines from Machine 1 to m, then completed with completion time $c_j$
• Figure 3: Illustration of a network of Multi-Stage Identical Machine Scheduling. The network consists of $k$ stage. Each stage is a parallel of $m_i$ machines with identical speed $s_i$.
• Figure 4: A multi-stage machine game where greedy choice does not result in subgame perfect Nash equilibrium

Theorems & Definitions (14)

• Lemma 1
• proof
• Lemma 2
• proof
• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof