Bounding the Price of Anarchy of Weighted Shortest Processing Time Policy on Uniform Parallel Machines

Abstract

This article investigates the performance of the Weighted Shortest Processing Time (WSPT) rule as a local sequencing policy in a scheduling game for uniformly related parallel machines, where the social objective is the total weighted completion time. Our research aims to establish improved upper bounds for the price of anarchy in this game. We determine two bounds, incorporating parameters that characterize the instance family, such as the speed of the fastest machine ($s_m$) and the number of machines (m). One bound establishes a fixed upper bound for the price of anarchy, while the other outperforms the parametric upper bound found in the existing literature. These newly derived bounds provide better insights into the performance of the scheduling game under study, proving that the price of anarchy is upper bounded by $min\left\{s_m\left(1+1/2s_m-1/2m\right),m,4\right\}$.

1. Introduction

Job scheduling problems are recurring in several fields, including industrial production, project management, and logistics. The efficient allocation of resources and time optimization are critical factors for ensuring the optimal performance of complex systems. In this context, game theory emerges as an innovative approach to comprehending the dynamics of decision-making processes, offering insights that can significantly enhance real-time decision-making efficiency. The scheduling problems viewed from the game theory perspective are known as scheduling games. We recommend referring to Heydenreich et al. [1] for a more in-depth introduction to scheduling games. This article provides a comprehensive overview of this field of research, addressing key concepts.

The scheduling game approach has applications across diverse problem domains, particularly in computing and networks. Notable examples include load balancing in computational grids [2] and traffic routing in networks [3,4,5,6,7,8]. Additionally, this approach has been applied for modeling traffic flows and route choices in transport networks [9,10,11]; it has also been used to support decision-making for subcontracting production orders [12] among other applications.

1.1. Scheduling Game Description

This study considers the scheduling game in an environment of uniformly related parallel machines. In this environment, each machine operates at a different speed. Unlike the classic scheduling problem, where a single decision maker (central authority) must decide the allocation and sequence of jobs, each job now belongs to an agent who decides which machine will process its job. In this decentralized decision approach, each job (as an agent) makes a decision that maximizes its own utility (or minimizes its disutility) without regard to the performance of some central or social objective. Depending on the utility of the jobs, a distinction can be made between two models [1,13]: the congestion model (load balancing) and the sequencing model (scheduling).

In a congestion model [14], each job receives a utility indirectly related to the workload of the selected machine, that is, the total processing time related to all jobs assigned to the same machine. At the same time, the social objective is to minimize the makespan of the overall schedule. Note that the makespan is a minimax social choice objective. Some relevant works related to this model are presented in [5,14,15,16,17,18,19,20,21,22,23,24,25,26].

In the sequencing model, each job receives a utility indirectly related to its own completion time. In a decentralized way, each job selects a machine, trying to minimize its completion time. At the same time, the social objective is to minimize the total completion time (weighted or unweighted). Note that this objective is a minsum social choice objective. Then, to determine the sequence of jobs to be processed, each machine announces its local sequencing policies in advance, known as local policy or coordination mechanism [22]. If the policy depends solely on the processing requirements of the jobs assigned to the same machine, it is a strongly local policy rule.

Regarding the choice of jobs, their strategies can be considered in two ways: pure and mixed. A pure strategy implies that an agent must select a single machine. In contrast, a mixed strategy implies that the agent can choose multiple strategies over pure strategies, like a probability distribution.

The two most commonly used local policies for pure strategies in a minsum social choice objective are the Shortest Processing Time (SPT) rule, which sequences jobs in increasing order of their processing time, and the Weighted Shortest Processing Time (WSPT) rule, which sequences jobs in descending order of their weight-to-processing time ratio ($w_j/p_j$). Some relevant works related to the sequencing model are presented in [12,13,27,28,29,30,31,32,33,34,35,36,37]. Some of these works present results for other local policies [1,27,29,31,32,35,38]. On the other hand, for congestion games, the performance of SPT rules, such as local policy, has also been studied [21,25,26,39,40,41].

One of the fundamental aspects of decentralized decision-making involves achieving convergence to an equilibrium, commonly referred to as the Nash equilibrium. This equilibrium represents a state where no player is incentivized to change their strategy unilaterally. While Nash equilibria may not always be attainable for pure strategies, they are guaranteed to exist for mixed strategies. Studies such as [1,16,25,38] demonstrate that employing the WSPT or SPT rule as the local policy for pure strategies ensures convergence to a Nash equilibrium.

In general, the value of the social objective for some Nash equilibrium can be worse than that of the social objective for the optimal schedule (social optimum) obtained through centralized decision-making. This inefficiency of the game is quantified with the Price of Anarchy (PoA) proposed by Koutsoupias and Papadimitriou [14]. Given a game, the PoA for a game instance is the worst-case ratio of the social cost at a Nash equilibrium to the cost at the social optimum. The PoA for a game is the worst-case value of the PoA overall game instances.

According to the classification scheme proposed by [35], a scheduling game can be described by the three-field notation $\alpha \left|\beta \right|\gamma$. The $\alpha$ field describes the machine environment and the local sequencing policy used by each machine (in parentheses), the $\beta$ field contains entries relating to the job specifications and also indicates the egoistic utility ($ut$) criterion of the jobs, and the $\gamma$ field indicates the social objective. Therefore, the problem considered in this study is represented by $Q\left(WSPT\right)|ut=-C_j|\sum w_j{C}_j$. Meanwhile, the centralized problem, represented by $Q\left|\right|\sum w_j{C}_j$ in the scheduling notation [42,43], is a well-known NP-Hard problem.

1.2. Related Works

Table 1. Price of anarchy in sequencing games on parallel machine environments for minimizing total weighted completion time. Local sequencing policies: Shortest Processing Time (SPT) and Weighted Shortest Processing Time (WSPT).

Scheduling Game	Price of Anarchy ($\mathit{\rho }$)	Reference
$P\left(SPT\right)|ut=-C_j|\sum C_j$	$\rho =1$	[25,35]
$P\left(WSPT\right)|ut=-C_j|\sum w_j{C}_j$	$\rho =1.2071$	[1,44]
$Q_2\left(SPT\right)|ut=-C_j|\sum C_j$	$\rho \in \left[1.1830,1.6180\right]$	[35]
$Q_2\left(SPT\right)|ut=-C_j|\sum C_j$	$\rho \ge 1.1875$	[37]
$Q\left(SPT\right)|ut=-C_j|\sum C_j$	$\rho \in \left[1.5820,2\right]$	[34]
$Q_m\left(SPT\right)|ut=-C_j|\sum C_j$	$\rho \le 2-\frac{2}{(n+m)(n+1)}$a	[37]
$Q_2\left(WSPT\right)|ut=-C_j|\sum w_j{C}_j$	$\rho \le \frac{s_m}{s_1}+\frac{1}{4}$	[45]
$Q_m\left(WSPT\right)|ut=-C_j|\sum w_j{C}_j$	$\rho \le \frac{3}{2}\left(1-\frac{1}{m}\right)\frac{s_m}{s_1}+\frac{1}{m}$b	[45]
$Q\left(WSPT\right)|ut=-C_j|\sum w_j{C}_j$	$\rho \ge 2$	[45]
$P\left(WSPT\right)|ut=-C_j,{\mathcal{M}}_j|\sum C_j$c	$\rho =4$	[32]
$P\left(WSPT\right)|ut=-C_j,{\mathcal{M}}_j|\sum w_j{C}_j$	$\rho =4$	[32]
$Q\left(WSPT\right)|ut=-C_j,{\mathcal{M}}_j|\sum w_j{C}_j$	$\rho =4$	[33]
$R\left(WSPT\right)|ut=-C_j|\sum w_j{C}_j$	$\rho =4$	[32,33]

^a n and m represent the number of jobs and machines, respectively. ^b $s_m$ and $s_1$ represent the speed of the fastest and slowest machine, respectively. ^c ${\mathcal{M}}_j$ denotes machine eligibility restrictions.

presents the PoA for scheduling games in parallel machine environments that use the WSPT (or SPT) rule as the local sequencing policy, where the social objective is to minimize the total completion time, whether weighted or unweighted. In cases where a single value is provided, the PoA has already been proven to be tight. For instances where a range or inequality is given, these values represent the lower and upper bounds for the PoA. Note that this highlights a research opportunity to establish a tight PoA for the game or narrow the gap between the upper and lower bounds.

For the studied problem, Muñoz and Parra [45] establish that the PoA is at least two and propose the following parametric upper bounds:

$$\begin{array}{cc}\hfill & \frac{s_2}{s_1}+\frac{1}{4},\mathrm{for}m=2,\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill & \frac{3}{2}\left(1-\frac{1}{m}\right)\frac{s_m}{s_1}+\frac{1}{m},\mathrm{for}m>2.\hfill \end{array}$$

(2)

For the unweighted case of the studied problem, the literature establishes that the PoA ranges from $1.582$ to 2 (see Table 1 for more details) [34,37]. Meanwhile, for the particular case of $m=2$ machines, the PoA ranges from $1.1875$ to $1.618$ [35,37].

On the other hand, in [32,33], the PoA is examined for two generalizations related to the studied problem. In [32], the scheduling game on unrelated machines is analyzed, while [33] explores the scheduling game on uniformly related machines with the additional consideration of machine eligibility restrictions. These restrictions establish that each job j can be processed by a subset ${\mathcal{M}}_j$ of machines. Both studies demonstrate that the PoA of their respective problems is strictly 4.

According to the literature related to the study of the PoA in sequencing games [25,32,33,34,35,37,45], the typical methodology involves two stages: establishing lower bounds and upper bounds for the PoA. Upper bounds indicate the potential worst-case social performance, while lower bounds highlight instances where the Nash equilibrium demonstrates poor social performance. When upper and lower bounds coincide, a tight PoA is established, concluding the analysis for that sequencing game and the specific local policy. The establishment of upper bounds involves characterizing schedules, determining properties of the social optimum and the Nash equilibrium, bounding the total weighted completion time of these schedules, and establishing relationships among these values. This is typically achieved through the use of inequalities. To establish lower bounds, the usual method is to design problem instances that demonstrate the worst-case performance for PoA. Often, trial-and-error studies are necessary to achieve this objective. According to the literature, problem instances to illustrate worst-case scenarios can be classified into small, parametric, and large (asymptotic) instances.

A problem closely related to the problem discussed corresponds to assignment-sequencing models for multi-agent scheduling, also referred to as the Pareto-scheduling problem, e.g., in [46,47,48,49,50,51,52]. In this context, a finite set of customers (or agents) each possess a set of jobs. These jobs must be processed on parallel machines. Each customer has an objective function for their jobs, and manufacturers aim to find the best scheduling scheme considering the trade-off between the objectives. Another related work involves the fair allocation of divisible and indivisible chores, which can be seen as a scheduling problem on unrelated parallel machines. For example, in [53], the price of fairness with utilitarian diswelfare is studied.

1.3. Our Study

This study investigates the worst-case performance for the WSPT rule as the local sequencing policy for the $Q\left(WSPT\right)|ut=-C_j|\sum w_j{C}_j$ problem. Specifically, our objective is to establish an improved parametric upper bound for the PoA.

Our proof technique follows a similar approach to that used in [45], establishing a mathematical expression for the Nash equilibrium condition. Subsequently, we establish specific properties of the solutions to the problem, the Nash equilibria, and the social optimum. The main difference from the approach in [45] lies in determining the lower bound for the social optimum. While [45] derives a lower bound based on the inequality proposed by Eastman et al. [54] for a related problem, we employ a lower bound proved in [55] for the centralized decision problem, $Q\left|\right|\sum w_j{C}_j$. This allows us to establish a tighter parametric upper bound for the studied problem.

Our main result establishes that the PoA for any pure Nash equilibrium in the $Q\left(WSPT\right)|ut=-C_j|\sum w_j{C}_j$ problem, where ${\sum }_{i\in \mathcal{M}}{s}_i=m$, is at most

$$min\left\{s_m\left(1+\frac{1}{2s_m}-\frac{1}{2m}\right),m,4\right\}.$$

This implies an improved bound for the PoA compared to the parametric upper bounds proved in [45].

The remainder of this article is structured as follows. Section 2 formalizes the scheduling game under investigation and explains the notation utilized throughout this study. This section also includes an analysis of the properties of optimal (social) solutions and pure Nash equilibria. Section 3 is dedicated to studying the price of anarchy. In Section 4, we discuss our findings. Finally, we summarize the main conclusions of this study in Section 5.

2. Problem Statement and Notation

The scheduling game addressed in this study is set in an environment with m uniformly related parallel machines, where the local policy assumed by each machine is the WSPT rule, which is known to allow a set of jobs to be optimally sequenced on a single machine [56] (Thm. 3). Each machine may have a different speed in this setting and can process one job at a time without interruption. Let $\mathcal{M}$ represent the set of machines and $s_i$ the processing speed of machine $i\in \mathcal{M}$. Without loss of generality, we assume that the machines are indexed in increasing order based on their speed,

$$s_1\le s_2\le \cdots \le s_m,$$

(3)

and we rescale the machine speeds such that

$${\displaystyle \sum _{i\in \mathcal{M}}{s}_i=m.}$$

(4)

Let $\mathcal{J}$ be the set of n jobs (agents) that must choose a single machine to perform their processing. Each job $j\in \mathcal{J}$ has a processing requirement $p_j\ge 0$ and a weight or relative importance $w_j\ge 0$. Let $\mathit{v}$ be the vector representing the choices made by all the jobs. This vector, combined with the local policy, defines the schedule that will be implemented to process the entire set of jobs. In this vector, $v_j$ indicates the machine chosen by job j. More precisely, $v_j=i$ indicates that in the schedule $\mathit{v}$, job j selected the machine i. Then, its processing time will be $p_j/s_i$. Additionally, we define ${\mathcal{J}}_i\left(\mathit{v}\right)\subseteq \mathcal{J}$ as the set of jobs that chose machine i as their strategy.

Regarding the local job-sequencing policy, the WSPT rule determines that the jobs must be sequenced according to the $w_j/p_j$ ratios in descending order. In the remainder of the article, we will use the notation ≺ and ≻ to represent the sequence induced by the WSPT rule, considering that ties in the ratio are broken arbitrarily. We will also use the notation ⪯ and ⪰ to represent preceding and succeeding jobs, including the job itself.

For a schedule $\mathit{v}$, where $v_j=i$, the completion time of job j is determined as:

$$C_j\left(\mathit{v}\right)=\sum _{\begin{array}{c}k\in {\mathcal{J}}_i\left(\mathit{v}\right)\\ k⪯j\end{array}}\frac{p_k}{s_i}=\sum _{\begin{array}{c}k\in {\mathcal{J}}_{v_j}\left(\mathit{v}\right)\\ k⪯j\end{array}}\frac{p_k}{s_{v_j}}.$$

(5)

The total weighted completion time of schedule $\mathit{v}$ is defined by

$$Z\left(\mathit{v}\right)=\sum _{j\in \mathcal{J}}{w}_j{C}_j\left(\mathit{v}\right)=\sum _{i\in \mathcal{M}}\sum _{j\in {\mathcal{J}}_i\left(\mathit{v}\right)}\sum _{\begin{array}{c}k\in {\mathcal{J}}_i\left(\mathit{v}\right)\\ k⪯j\end{array}}\frac{w_j{p}_k}{s_i}=\eta \left(\mathit{v}\right)+\sum _{j\in \mathcal{J}}\sum _{\begin{array}{c}k\in {\mathcal{J}}_{v_j}\left(\mathit{v}\right)\\ k\prec j\end{array}}\frac{w_j{p}_k}{s_{v_j}}.$$

(6)

where,

$$\eta \left(\mathit{v}\right)=\sum _{i\in \mathcal{M}}\sum _{j\in {\mathcal{J}}_i\left(\mathit{v}\right)}\frac{w_j{p}_j}{s_i}=\sum _{j\in \mathcal{J}}\frac{w_j{p}_j}{s_{v_j}},$$

(7)

is the weighted sum of the processing times of schedule $\mathit{v}$. For this weighted sum and any schedule $\mathit{v}$, the following property was established [55] (Lem. 3):

$$\eta \left(\mathit{v}\right)\ge \sum _{j\in \mathcal{J}}\frac{w_j{p}_j}{s_m}.$$

(8)

Throughout the rest of this article, we denote the social optimum solution and the pure Nash equilibrium as ${\mathit{x}}^{*}$ and $\mathit{x}$, respectively.

2.1. Pure Nash Equilibrium

In decentralized decision-making, each job selects a machine that minimizes its disutility, modeled by its completion time. Then, the condition required for schedule $\mathit{x}$ to be a Nash equilibrium is that no job has incentives to change its choice. More precisely, any change in his choice deteriorates his utility. Thus, the condition for $\mathit{x}$ to be a Nash equilibrium is:

$$C_j\left(\mathit{x}\right)\le \frac{p_j}{s_h}+\sum _{\begin{array}{c}k\in {\mathcal{J}}_h\left(\mathit{x}\right)\\ k\prec j\end{array}}\frac{p_k}{s_h}\mathrm{for}\mathrm{all}j\in \mathcal{J},h\in \mathcal{M}.$$

(9)

The left-hand side of the Equation (9) indicates the completion time of the job j in the schedule $\mathit{x}$, while its right-hand side establishes the completion time that job j would have if it changed its choice from being processed by machine $x_j$ to machine h. In Section 3.1, we will utilize this Nash equilibrium condition to upper bound the PoA.

2.2. Social Optimum

In centralized decision-making, the social optimum is determined by a central authority. This optimum represents a schedule that minimizes the total weighted completion time. Recently, in Muñoz et al. [55] (Theor. 1), a lower bound was established for the optimal total weighted completion time. This inequality is presented in the following theorem.

Theorem 1.

([55] (Theor. 1)). For any optimal schedule ${\mathit{x}}^{*}$ of the $Q\left|\right|\sum w_j{C}_j$ problem, where ${\sum }_{i\in \mathcal{M}}{s}_i=m$, the following expression holds:

$$2Z\left({\mathit{x}}^1\right)-\eta \left({\mathit{x}}^1\right)\le 2Z\left({\mathit{x}}^{*}\right)-\eta \left({\mathit{x}}^{*}\right).$$

(10)

In Theorem 1, ${\mathit{x}}^1$ represents a schedule in which all jobs are assigned to a fictitious machine operating at speed m (the total number of machines). Note that by Equation (4), the number of machines is equivalent to the sum of the speeds of all the machines. For this schedule, we have the following expressions that allow us to determine its total weighted completion time and its weighted sum of the processing times:

$$Z\left({\mathit{x}}^1\right)=\sum _{j\in \mathcal{J}}\sum _{\begin{array}{c}k\in \mathcal{J}\\ k⪯j\end{array}}\frac{w_j{p}_k}{m},$$

(11)

$$\eta \left({\mathit{x}}^1\right)=\sum _{j\in \mathcal{J}}\frac{w_j{p}_j}{m}.$$

(12)

3. Results

This section introduces parametric and fixed upper bounds for the price of anarchy of the $Q\left(WSPT\right)|ut=-C_j|\sum w_j{C}_j$ problem. In Lemma 1, we establish a parametric expression that bounds the PoA for a family of instances defined by the number of machines (m) and the speed of the fastest machine ($s_m$). Additionally, Lemma 3 provides a fixed upper bound that uses the number of machines as input (a common practice for scheduling problems).

3.1. Parametric Bound for the Price of Anarchy

Let $\mathcal{I}(m,s_m)$ be the family of instances of the $Q\left|\right|\sum w_j{C}_j$ problem, where m uniformly related machines must process a given set of jobs, and the speed of the fastest machine is represented by $s_m$. In addition, it also considers the rescaling established in Equation (4).

The following lemma establishes the parametric upper bound of the studied PoA.

Lemma 1.

For any instance of the family $\mathcal{I}(m,s_m)$, where ${\sum }_{i\in \mathcal{M}}{s}_i=m$, the price of anarchy of the $Q\left(WSPT\right)|ut=-C_j|\sum w_j{C}_j$ problem for any pure Nash equilibrium is at most

$$s_m\left(1+\frac{1}{2s_m}-\frac{1}{2m}\right).$$

(13)

Proof. 

Starting from the pure Nash equilibrium condition, Equation (9), and multiplying both sides of the inequality by $w_j$,

$$w_j{C}_j\left(\mathit{x}\right)\le \frac{w_j{p}_j}{s_h}+\sum _{\begin{array}{c}k\in {\mathcal{J}}_h\left(\mathit{x}\right)\\ k\prec j\end{array}}\frac{w_j{p}_k}{s_h}\forall j\in \mathcal{J},h\in \mathcal{M}.$$

(14)

Multiplying both sides of Equation (14) by $s_h/m$ and summing over all $h\in \mathcal{M}$, leads to

$$\begin{array}{c}{w}_j{C}_j\left(\mathit{x}\right)\le w_j{p}_j+\sum _{h\in \mathcal{M}}\sum _{\begin{array}{c}k\in {\mathcal{J}}_h\left(\mathit{x}\right)\\ k\prec j\end{array}}\frac{w_j{p}_k}{m}=w_j{p}_j+\sum _{\begin{array}{c}k\in \mathcal{J}\\ k\prec j\end{array}}\frac{w_j{p}_k}{m}\hfill \\ \hfill =\left(1-\frac{1}{m}\right)w_j{p}_j+\sum _{\begin{array}{c}k\in \mathcal{J}\\ k⪯j\end{array}}\frac{w_j{p}_k}{m},\forall j\in \mathcal{J}.\end{array}$$

(15)

By summing over all $j\in \mathcal{J}$ in Equation (15) and using Equations (11) and (12), we have

$$Z\left(\mathit{x}\right)\le \left(1-\frac{1}{m}\right)\sum _{j\in \mathcal{J}}{w}_j{p}_j+Z\left({\mathit{x}}^1\right)=\left(1-\frac{1}{2m}\right)\sum _{j\in \mathcal{J}}{w}_j{p}_j+Z\left({\mathit{x}}^1\right)-\frac{\eta \left({\mathit{x}}^1\right)}{2}.$$

(16)

Utilizing Theorem 1 in Equation (16), we obtain the following expression:

$$Z\left(\mathit{x}\right)\le \left(1-\frac{1}{2m}\right)\sum _{j\in \mathcal{J}}{w}_j{p}_j+Z\left({\mathit{x}}^{*}\right)-\frac{\eta \left({\mathit{x}}^{*}\right)}{2}.$$

(17)

Note that, in Equation (17), the coefficient multiplying ${\sum }_j{w}_j{p}_j$ is always positive, and according to Equation (8), we have

$$\sum _{j\in \mathcal{J}}{w}_j{p}_j\le s_m\eta \left({\mathit{x}}^{*}\right).$$

(18)

From Equations (17) and (18), we have

$$Z\left(\mathit{x}\right)\le Z\left({\mathit{x}}^{*}\right)+\left(s_m-\frac{s_m}{2m}-\frac{1}{2}\right)\eta \left({\mathit{x}}^{*}\right).$$

(19)

Note that in Equation (19), the coefficient multiplying $\eta \left({\mathit{x}}^{*}\right)$ is always positive since $m\ge 2$ and $s_m\ge 1$. This is,

$$s_m-\frac{s_m}{2m}-\frac{1}{2}=s_m\left(1-\frac{1}{2m}-\frac{1}{2s_m}\right)\ge s_m\left(1-\frac{1}{4}-\frac{1}{2}\right)=\frac{s_m}{4}>0.$$

(20)

By Equation (6), we have that

$$\eta \left({\mathit{x}}^{*}\right)\le Z\left({\mathit{x}}^{*}\right).$$

(21)

Therefore, by Equations (19)–(21), we have

$$Z\left(\mathit{x}\right)\le Z\left({\mathit{x}}^{*}\right)\left(s_m+\frac{1}{2}-\frac{s_m}{2m}\right).$$

(22)

Finally, the proof is concluded by isolating $Z\left(\mathit{x}\right)/Z\left({\mathit{x}}^{*}\right)$. □

The subsequent lemma illustrates that the introduced parametric upper bound from Lemma 1 outperforms the parametric upper bound reported in [55].

Lemma 2.

The parametric upper bound for the price of anarchy introduced in this study for the $Q\left(WSPT\right)|ut=-C_j|\sum w_j{C}_j$ problem is tighter than the parametric upper bound previously reported in the literature.

Proof. 

To compare the parametric upper bounds, we define $D_i$ as the difference between the parametric upper bounds for i parallel machines. Specifically, let $D_2$ represent the difference between the parametric upper bounds given by (1) and (13), and $D_m$ denote the difference between the parametric upper bounds specified by (2) and (13). Note that proving the non-negativity of $D_2$ and $D_m$ is sufficient to establish the lemma.

For the case of $m=2$ machines, we have

$$D_2=\left(\frac{s_2}{s_1}+\frac{1}{4}\right)-s_2\left(1+\frac{1}{2s_2}-\frac{1}{4}\right)=\frac{s_2}{s_1}-\frac{3s_2}{4}-\frac{1}{4}.$$

However, according to the assumptions (3) and (4), it is stated that $s_1\le 1$ and $s_2\ge 1$, implying

$$D_2\ge s_2-\frac{3s_2}{4}-\frac{1}{4}=\frac{1}{4}\left(s_2-1\right)\ge 0.$$

(23)

On the other hand, for $m>2$ machines, we have

$$\begin{array}{cc}\hfill D_m& =\frac{3}{2}\left(1-\frac{1}{m}\right)\frac{s_m}{s_1}+\frac{1}{m}-s_m\left(1+\frac{1}{2s_m}-\frac{1}{2m}\right)\hfill \\ \hfill & =s_m\left(\frac{3}{2s_1}\left(1-\frac{1}{m}\right)-\frac{1}{s_m}\left(\frac{1}{2}-\frac{1}{m}\right)-1+\frac{1}{2m}\right).\hfill \end{array}$$

Note that, for $m\ge 2$, $1-1/m$ and $1/2-1/m$ are non-negative. Then, according to the assumptions (3) and (4), it is stated that $s_1\le 1$ and $s_m\ge 1$, implying

$$D_m\ge s_m\left(\frac{3}{2}\left(1-\frac{1}{m}\right)-\left(\frac{1}{2}-\frac{1}{m}\right)-1+\frac{1}{2m}\right)=0.$$

(24)

Equations (23) and (24) demonstrate that $D_i\ge 0$ for all $i\ge 2$, thereby concluding the proof. □

3.2. Fixed Bound for the Price of Anarchy

The following lemma establishes the fixed upper bound for the PoA.

Lemma 3.

The price of anarchy for any pure Nash equilibrium of the $Q\left(WSPT\right)|ut=-C_j|\sum w_j{C}_j$ problem, where ${\sum }_{i\in \mathcal{M}}{s}_i=m$, is at most m.

Proof. 

From Lemma 1 we have that

$$PoA\le s_m\left(1+\frac{1}{2s_m}-\frac{1}{2m}\right)=s_m\left(\frac{2m-1}{2m}+\frac{1}{2s_m}\right).$$

(25)

By Equations (3) and (4) we have that $s_m\in [1,m]$. Then, from Equation (25), the PoA can be upper-bounded as follows:

$$PoA\le s_m\left(\frac{2m-1}{2m}\right)+\frac{1}{2}\le m\left(\frac{2m-1}{2m}\right)+\frac{1}{2}=m.$$

□

3.3. Main Result

The following theorem establishes our main result, wherein the parametric upper bound for PoA from Lemma 1 and the fixed PoA from Lemma 3, is utilized in conjunction with the fixed PoA proved in [33] for the $Q\left(WSPT\right)|ut=-C_j,{\mathcal{M}}_j|\sum w_j{C}_j$ problem (refer to Table 1). The use of this result is based on the fact that the studied problem is a particular case of the problem with machine eligibility restrictions. More precisely, when $|{\mathcal{M}}_j|=m$, i.e., each job can be processed by any of the m machines.

Theorem 2

For the family of instances $\mathcal{I}(m,s_m)$ of the $Q\left(WSPT\right)|ut=-C_j|\sum w_j{C}_j$ problem, where ${\sum }_{i\in \mathcal{M}}{s}_i=m$, the price of anarchy for any pure Nash equilibrium is, at most,

$$min\left\{s_m\left(1+\frac{1}{2s_m}-\frac{1}{2m}\right),m,4\right\}.$$

4. Discussion

In the previous section, an analysis was conducted on families of instances of the $Q\left(WSPT\right)|ut=-C_j|\sum w_j{C}_j$ problem. The analysis showed that the pure PoA is upper-bounded by Equation (13). Additionally, Lemma 2 established that this upper bound is tighter than the previous upper bound documented in the literature [45].

For two machines, the upper bound on the PoA is set at

$$\frac{1}{2}+\frac{3s_2}{4},$$

converging monotonically to

$$\frac{1}{2}+s_m,$$

as the number of machines (m) increases.

Another feature of the proposed parametric upper bound is its complementarity with the fixed upper bound established in [33]. This complementarity is explained in Theorem 2. However, we are concerned about which of the other two bounds is tighter. The answer to this concern lies in the speed of the fastest machine. We will analyze the cases where $m\le 4$ and $m>4$.

For $m\le 4$, analyzing

$$s_m\left(1+\frac{1}{2s_m}-\frac{1}{2m}\right)\le m,$$

results in $s_m\le m$. Thus, by assumptions (3) and (4), the parametric upper bound is always tighter than the fixed upper bound.

On the other hand, for $m>4$, analyzing

$$s_m\left(1+\frac{1}{2s_m}-\frac{1}{2m}\right)\le 4,$$

leads to

$$s_m\le \frac{7m}{2m-1}.$$

Consequently, the parametric upper bound is tighter than the fixed upper bound for some specific values of $s_m$. More precisely, for $m=5$, the parametric upper bound will be better for $s_m\le 35/9\approx 3.89$. This value converges monotonically to $3.5$ as $m\to \infty$.

Finally, we extend the results of Lemma 1 to the environment of identical parallel machines. In this case, the studied scheduling game is a generalization of the problem in an identical parallel machine environment. This problem is represented by $P\left(WSPT\right)|ut=-C_j|\sum w_j{C}_j$. By setting $s_m=1$ in Equation (13), we obtain $(3m-1)/2m$. This expression is well-documented in the literature and coincides with the expression for the Price of Anarchy (PoA) of the scheduling game for mixed strategies [34] and the robust PoA in the case of unweighted jobs [36].

5. Conclusions

This study presents upper bounds for the price of anarchy of a sequencing game on a uniformly related parallel machine environment. In this setting, each job acts as an agent, selecting a machine for its processing without considering the impact of its decision on other agents (jobs). The social goal is to minimize the total weighted completion time, and the local sequencing policy executed by each machine is the Weighted Shortest Processing Time (WSPT) rule.

The research focused on establishing properties for the Nash equilibrium and the optimal solution of the problem, aiming to determine a parametric upper bound for the price of anarchy based on the parameters: the number of machines (m) and the speed of the fastest machine ($s_m$). The research has determined that the price of anarchy is upper-bounded by

$$s_m\left(1+\frac{1}{2s_m}-\frac{1}{2m}\right).$$

The notable findings of this study are that the proposed parametric upper bound outperforms the previously reported parametric upper bound and the complementarity of this new bounds with the fixed upper bounds. Upper bounding the PoA by the number of machines and a fixed value of 4.

The obtained results offer new insights into the efficiency of the scheduling game under study, particularly in environments with two or three parallel machines. The findings are especially noteworthy for instances where the parametric upper bound reports a value less than 4.