A New Time-Sensitive Graph Model for Conflict Resolution with Simultaneous Decision-Maker Moves

Abstract

The time dimension critically shapes decision-making and conflict evolution in real-world scenarios. This paper extends the Graph Model for Conflict Resolution (GMCR) framework by integrating time attributes, proposing a novel Time-Sensitive GMCR (TSGMCR) methodology that supports concurrent moves by multiple decision-makers (DMs). Within TSGMCR, we define new stability concepts and implement comparative analysis. The methodology is applied to the Jakarta–Bandung high-speed railway project conflict, demonstrating its effectiveness in resolving complex real-world conflicts and identifying beneficial coalition formations.

1. Introduction

Conflicts, as common phenomena in human society, emerge from confrontations among multiple decision-makers (DMs) due to competing interests, divergent values, or power struggles. The development of conflicts is significantly influenced by the strategic choices, preferences, and behavioral patterns of DMs, which collectively give rise to complex interactions and determine potential equilibrium outcomes. Within the paradigm of Graph Model for Conflict Resolution (GMCR), a novel Time-Sensitive GMCR (TSGMCR) framework incorporating temporal dimensions into conflict analysis and coalition analysis is proposed, which provides a more realistic representation of strategic interactions and enhances the explanatory power of conflict resolution methodologies.

As a nonquantitative game theoretical framework, GMCR provides a systematic methodology for analyzing strategic conflicts under limited information, high uncertainty, and complex interests [1,2], such as water resources conflict [3,4,5,6], technological transformation decisions [7,8], and social disputes [9,10], among others. Unlike classical game theory, GMCR operates without cardinal utility measurements, instead relying on ordinal rankings of potential conflict outcomes (or states) to capture DMs’ preferences. Developed from metagame analysis [11] and conflict resolution theory [12], GMCR introduces diverse stability concepts—such as Nash stability, sequential stability (SEQ), general metarationality (GMR), and symmetric metarationality (SMR)—to model how DMs navigate strategic interactions while anticipating counteractions from opponents. As a crucial component of the GMCR framework, coalition analysis provides formal mechanisms to examine how multiple DMs coordinate their actions to achieve collective advantages during conflict evolution [13,14,15]. Subsequent research has centered around stability analysis of coalitions, with Kilgour et al. [16] proposing coalitional improvements (CI) and coalitional Nash (CNash) stability by evaluating a coalition’s impact on conflict equilibria rather than preference similarity. Later, Inohara and Hipel [17,18] defined other stability concepts, including coalitional general metarationality (CGMR), symmetric metarationality (CSMR), and sequential stability (CSEQ). Recent studies have explored some novel definitions, such as those of Pareto stabilities [19] and the full coalition set [20]. Zhu et al. [19] extended CI to a broader concept in which at least one DM would benefit and nobody has a preference for declining, further discussing the inter-relationships within and among non-cooperative, classical, and Pareto coalitional equilibria [21]. Zhao et al. [20] considered all possible coalitional scenarios, proposing the concepts of a full coalition set and universal coalition set to compute coalition stabilities under unknown coalitional situations. Another research focus of GMCR is on preference [22], which orders states around the DMs’ options, thereby providing support for stability analysis. Li et al. [23] provided an uncertain preference structure by considering the uncertainty of preference. Hamouda et al. [24] proposed a strength preference structure to depict the strong or weak preferences of DMs for a given state. Xu et al. [25] incorporated the uncertainty and strength of preference, putting forward a hybrid preference structure. Believing that hesitant fuzzy preference shares similarities with three-way decisions, Chen et al. [26] proposed a novel three-way decision graph model for conflict resolution (3WD-GMCR) framework to obtain the intensity ranking of conflict states.

However, previous studies on GMCR have failed to incorporate time dimensions into the analytical framework of GMCR, consequently limiting its effective application to conflicts where time-sensitive factors are absent. A key contribution by Inohara [27] was the introduction of an asynchronous time framework, where DMs may require varying durations to complete unilateral moves. Under this model, state transitions that take less time are prioritized, while longer transitions are treated as infeasible moves. Later, He [28] put forward a new time-sensitive GMCR approach that sets the move rules of DMs so as to ensure that all DMs have opportunities to realize unilateral moves rather than considering only the move with the shortest state transition time. Meanwhile, the deadlines of conflict are considered in He [28]’s model. Existing literature maintains that DMs must move in a sequence, as in the classical graph model, failing to account for scenarios where multiple DMs move simultaneously. In real-world conflicts, multiple DMs frequently implement their strategies concurrently rather than sequentially, awaiting prior moves. Hence, when one DM is executing a move, it is more general and reasonable for other DMs to make their moves simultaneously.

In the time-sensitive graph model proposed by He [28], DMs execute their moves sequentially according to a predefined order. Specifically, only after one DM completes its decision-making process and achieves a state transition does the next eligible DM initiate its move. However, in most real-world conflict scenarios, DMs are unlikely to remain passive while waiting for opponents to complete their moves. Hence, to more accurately represent the dynamics of time-sensitive graph models, we formally redefine the following concepts: Time-Sensitive Unilateral Move (TSUM) and Time-Sensitive Unilateral Improvement (TSUI).

This paper establishes a novel time-sensitive GMCR framework that incorporates time attributes and accommodates concurrent moves by multiple DMs.

The remainder of this paper is organized as follows. In Section 2, the classical GMCR approach is briefly introduced. The proposed novel Time-Sensitive GMCR (TSGMCR) approach is put forward in Section 3. In Section 4, the proposed approach is applied to the Jakarta–Bandung high-speed railway project conflict to demonstrate its applicability and effectiveness. Finally, the conclusions and future work are summarized in Section 5.

2. Preliminaries

For complex real-world disputes and strategic conflicts, GMCR provides a robust methodological framework for systematic modeling, analysis, and understanding of conflict dynamics. This section presents the fundamental concepts and symbolic representations central to GMCR, along with standardized mathematical formulations and rigorous conceptual definitions essential for conflict analysis.

2.1. Key Concepts in GMCR

The application of the GMCR methodology requires focus on four fundamental elements: decision-makers (DMs), feasible states, preference structures, and possible state transitions. To illustrate these concepts, consider a scenario involving Countries A, B, and C, which are engaged in a trade dispute. Country A has the option to impose tariffs, while Countries B and C each possess two choices: raising tariffs or pursuing economic transition. Each country independently selects “Yes” or “No” for its respective options, and the combination of these choices defines distinct conflict states. For instance, one possible state occurs when Country A imposes tariffs, Country B raises tariffs, and Country C opts for economic transition. Driven by self-interest, DMs exhibit varying preferences over these states. Country A, for example, may favor a state where neither B nor C imposes tariffs, as this could enhance its trade surplus. State transitions occur when DMs modify their choices, reflecting the dynamic nature of conflicts within the theoretical framework of conflict analysis. For reference,

Table 1. Key variables and notation in this paper.

Notation	Description
N	The set of all decision-makers (DMs) in conflict
k	A specific DM in N
S	The set of all states in conflict
$A^k$	The set of state transitions by DM k
${\mathrm{\Omega }}_i$	The preference statement of DMs
${\mathrm{\Psi }}_i\left(s\right)$	The truth value of state s under the ith preference statement
$\mathrm{\Psi }\left(s\right)$	The preference score of state s
$R_k\left(s\right)$	The Unilateral Move (UM) list of DM k in state s
$R_k^{+}\left(s\right)$	The Unilateral Improvement (UI) list of DM k in state s
H	A nonempty coalition of $H\subseteq N$
$R_H\left(s\right)$	The reachable list of coalition H in state s
$R_H^{+}\left(s\right)$	The UI list of coalition H in state s
t	Time elapsed since the beginning of the conflict
T	The deadline of the conflict
$s^t$	A specific state at time t
${\tau }_k$	The time required to finish UMs by DM k
$O_k$	The strategy of DM k
$o_k^i$	The choice of option i controlled by DM k
$t{R}_k\left(s^t\right)$	The Time-Sensitive Unilateral Move (TSUM) list of DM k in state $s^t$
$t{R}_k^{+}\left(s^t\right)$	The Time-Sensitive Unilateral Improvement (TSUI) list of DM k in state $s^t$
${\sigma }_k(s^t,s^{t_1})$	The preference radio of a TSUM from $s^t$ to $s^{t_1}$ for DM k
$t{R}_H\left(s^t\right)$	the Time-Sensitive Unilateral Moves of a Coalition (CTSUM) (H) in state (s)
$t{R}_H^{+}\left(s^t\right)$	The Time-Sensitive Unilateral Improvements of a Coalition (CTSUM) (H) in state s

provides a comprehensive list of key variables and symbolic notations employed throughout this study.

2.2. Classical Graph Model for Conflict Resolution

A graph model can be represented by a four-tuple set ($G=(N,S,A_k,P_k)$, where $N=\{1,2,\dots ,n\}$ represents the finite set of DMs; $S=\{s_1,s_2,\dots ,s_z\}$ is the finite set of feasible states, which is composed of the options chosen by all DMs together; $A_k\in S\times S$ denotes the set of oriented arcs, each of which means indicates a movement from one state to another that is solely controlled by DM k ($k\in N$); and $P_k=\{{\succ }_k,{\sim }_k\}$ represents the relative preferences of DM k. $s_i{\succ }_k{s}_j$ means DM k prefers $s_i$ to $s_j$, while $s_i{\sim }_k{s}_j$ indicates state $s_i$ is indifferent to $s_j$ for DM k.

In GMCR, there are three primary methods for acquiring preference information: direct ranking, option weighting, and option prioritization. Direct ranking directly ranks all feasible states based on the DMs’ viewpoints, making it particularly suitable for smaller-scale conflict scenarios. Option weighting incorporates an analytic hierarchy process to determine the preference ranking of feasible states based on the weights of DMs and options. For large-scale conflicts, option prioritization provides a more scientific and accurate approach to ranking states, which formulates preference statements by combining DMs’ options with logical operators, such as “if”, “or (∣)”, “and (&)”, and “not (-)”. These statements are prioritized from highest to lowest, and the truth value of each state is determined based on whether it satisfies the preference statements. Supposed $\mathrm{\Omega }=\{{\mathrm{\Omega }}_1,{\mathrm{\Omega }}_2,\dots ,{\mathrm{\Omega }}_j\}$ is a set of preference statements (in descending order of priority); the truth value of each state ($s\in S$) under a given preference statement (${\mathrm{\Omega }}_q$($1\le q\le j$), ${\mathrm{\Psi }}_q\left(s\right)$) is calculated as follows:

$${\mathrm{\Psi }}_q\left(s\right)=\left\{\begin{array}{cc}{2}^{j-q}\hfill & ,if{\mathrm{\Omega }}_{f}istruefors\hfill \\ 0\hfill & ,otherwise\hfill \end{array}\right.$$

(1)

where j is the cardinality of $\mathrm{\Omega }$.

Then, the total truth value of state k can be applied to determined its preference ranking. For state $s\in S$, the total truth value of state $\mathrm{\Psi }\left(s\right)$ is defined as follows:

$$\mathrm{\Psi }\left(s\right)=\sum _{q=1}^j{\mathrm{\Psi }}_q\left(s\right)$$

(2)

The concepts of a unilateral move (UM) and unilateral improvement (UI) [1] are crucial in GMCR, defining possible strategic interactions. A UM occurs when a DM moves independently between states, while a UI refers to a UM leading to a more favorable state.

Definition 1 

(Unilateral Move, UM). For each $k\in N$ and $s\in S$, $R_k\left(s\right)=\{t\in S\mid (s,t)\in A_k\}$ is called the UM list of DM k from state s in one step.

Definition 2 

(Unilateral Improvement, UI). For each $k\in N$ and $s\in S$, $R_k^{+}\left(s\right)=\{t\in R_k\left(s\right)\mid t{\succ }_{k}s\}$ is called the UI list of DM k from state s in one step.

In a conflict, a nonempty subset ($H\subseteq N$) with $\left|H\right|\ge 2$ is defined as a coalition. For an n-DM ($n\ge 2$) graph model, for DM $k\in N$, an important coalition ($N\setminus \left\{k\right\}$) is composed of the opponents of DM k, where “ ∖” means set subtraction. In contrast to a single DM, the UM and UI of a coalition permit each DM withinthe coalition to make multiple moves while prohibiting any single DM from executing consecutive moves. These concepts are formally defined as follows.

Definition 3 

(Reachable List of a Coalition). Let $s\in S$ be the initial state and $H\subseteq N$ be a nonempty coalition. The reachable list of H from s ($R_H\left(s\right)$) can be determined by the following inductive rules:

(1) ${\mathrm{\Omega }}_H(s,s^{\prime })=\varnothing$ for all $s^{\prime }\in S$, where ${\mathrm{\Omega }}_H(s,s^{\prime })$ denotes the set of all last DMs in H by UMs from s to $s^{\prime }$;

(2) If DM $k\in H$ and $s^{\prime }\in R_k\left(s\right)$, then $s^{\prime }\in R_H\left(s\right)$ and ${\mathrm{\Omega }}_H(s,s^{\prime })={\mathrm{\Omega }}_H(s,s^{\prime })\cup \left\{k\right\}$;

(3) If $s^{\prime }\in R_H\left(s\right)$, $k\in H$ and $s^{″}\in R_k\left(s^{\prime }\right)$, then, provided ${\mathrm{\Omega }}_H(s,s^{\prime })\ne \left\{k\right\}$, $s^{″}\in R_H\left(s\right)$, and ${\mathrm{\Omega }}_H(s,s^{\prime })={\mathrm{\Omega }}_H(s,s^{\prime })\cup \left\{k\right\}$.

Definition 4 

(UIs of a Coalition). Let $s\in S$ be an initial state and $H\subseteq N$ be a nonempty coalition of DMs. The UIs of H from s ($R_H^{+}\left(s\right)$), can be defined inductively as follows:

(1) ${\mathrm{\Omega }}_H^{+}(s,s^{\prime })=\varnothing$ for all $s^{\prime }\in S$, where ${\mathrm{\Omega }}_H^{+}(s,s^{\prime })$ denotes the set of all last DMs in H by UIs from s to $s^{\prime }$;

(2) If DM $k\in H$ and $s^{\prime }\in R_k^{+}\left(s\right)$, then $s^{\prime }\in R_H^{+}\left(s\right)$ and ${\mathrm{\Omega }}_H^{+}(s,s^{\prime })={\mathrm{\Omega }}_H^{+}(s,s^{\prime })\cup \left\{k\right\}$;

(3) If $s^{\prime }\in R_H^{+}\left(s\right)$, $k\in H$, and $s^{″}\in R_k^{+}\left(s^{\prime }\right)$, then, provided ${\mathrm{\Omega }}_H^{+}(s,s^{\prime })\ne \left\{k\right\}$, $s^{″}\in R_H^{+}\left(s\right)$, and ${\mathrm{\Omega }}_H^{+}(s,s^{\prime })={\mathrm{\Omega }}_H^{+}(s,s^{\prime })\cup \left\{k\right\}$.

Several stability definitions are commonly used in stability analysis, including Nash, GMR, SMR, and SEQ. These definitions for the four stabilities are briefly reviewed as follows [29].

Definition 5 

(Nash stability, Nash). For each $k\in N$, a state ($s\in S$) is Nash stable if and only if $R_k^{+}\left(s\right)=\varnothing$.

Definition 6 

(General metarationality, GMR). For each $k\in N$, a state ($s\in S$) is GMR stable if and only if, for every $s_1\in R_k^{+}\left(s\right)$, there exists at least one $s_2\in R_{N\setminus k}\left(s_1\right)$ with $s{⪰}_k{s}_2$.

Definition 7 

(Symmetric metarationality, SMR). For each $k\in N$, a state ($s\in S$) is SMR stable if and only if, for every $s_1\in R_k^{+}\left(s\right)$, there exists at least one $s_2\in R_{N\setminus k}\left(s_1\right)$ with $s{⪰}_k{s}_2$ and $s{⪰}_k{s}_3$ for all $s_3\in R_k\left(s_2\right)$.

Definition 8 

(Sequential stability, SEQ). For each $k\in N$, a state ($s\in S$) is SEQ stable if and only if, for every, $s_1\in R_k^{+}\left(s\right)$, there exists at least one $s_2\in R_{N\setminus k}^{+}\left(s_1\right)$ with $s{⪰}_k{s}_2$.

3. Time-Sensitive Graph Model for Conflict Resolution

In real-world conflicts, the evolution of disputes and their potential resolutions can be influenced by the time required for DMs to evaluate, select, and implement available options. Additionally, DMs often act simultaneously rather than sequentially, further complicating the dynamics of conflict. Take the trade conflict mentioned in Section 2 as an example. For Country B, the time needed to raise tariffs would differ from that required for economic transition. Such temporal differences are difficult to represent within the classical graph model. Thus, the Time-Sensitive Graph Model for Conflict Resolution (TSGMCR) framework should be modeled more precisely. In this section, we proceed to formalize a new time-sensitive GMCR framework through rigorous definitions.

3.1. Structure of Time-Sensitive Graph Model for Conflict Resolution

Definition 9 

(Time-Sensitive Graph Model for Conflict Resolution). A time-sensitive graph model is represented as $G=\{N,S,{\left(A_k\right)}_{k\in N},{\left(P_k\right)}_{k\in N},{\left({\tau }_k\right)}_{k\in N}\}$, where N represents the finite set of DMs, S denotes the finite set of states, $A_k$ is the set of oriented arcs representing DM k’s UMs, $P_k=\{{\succ }_k,{\sim }_k\}$ represents the relative preferences of DM k, and ${\tau }_k$ is the set of time required to finish UMs by DM k.

Each DM $k\in N$ has a set of options, and for each option, there are two possible choices: choose (“Y”) or not choose (“N”). Let $O_k=\{o_k^1,o_k^2,\dots \}$ represent DM k’s strategy, which means the selection of options controlled by DM k. The union of all DMs’ strategies (${\cup }_{k\in N}{O}_k$) constitutes a state. Specifically, $O_k\left(s\right)$ is defined as the strategy of DM k in state s. Inspired by the definition of time in GMCR proposed by He et al. [28], T is defined as the duration of the conflict, indicating the time period from the beginning to the end of the conflict being analyzed in the GMCR framework. Let t denote the time elapsed from the initial moment of the conflict, where $t\le T$. The state ($s\in S$) at time t is represented as $s^t$.

Definition 10 

(Time of Unilateral Move for DMs, ToUM). For DM $k\in N$, the time taken to finish the UM from $s^t$ to $s^{t_1}$ is denoted as ${\tau }_k(s^t,s^{t1})$. This time parameter satisfies the following properties:

(1) ${\tau }_k(s^t$, $s^{t_1})\ne {\tau }_k(s^{t_1},s^t)$ is permitted;

(2) If DM k modifies the strategy expressed as $O_k=\{o_k^1,\dots ,o_k^m\}$ to $O_k^{\prime }=\{{o^{\prime }}_k^1,\dots ,{o^{\prime }}_k^m\}$ to transfer states from $s^t$ to $s^{t_1}$, the total ToUM is ${\tau }_k(s^t,s^{t_1})=\max \{{\tau }_k^{o_k^1},\dots ,{\tau }_k^{o_k^m}\}$, where $o_k^i$ and ${o^{\prime }}_k^i$ represent the selection of DM k (“Y” or “N”) regarding the ith option and ${\tau }_k^{o_{k_i}}$ denotes the time required for DM k to change the selection of the ith option ($o_k^i$). If $o_k^i={o^{\prime }}_k^i$, ${\tau }_k^{o_k^i}=0$.

Note that this paper considers a DM’s strategy modification process—including the decision-making, preparation, and implementation phases—as essentially representing the DM’s state transition process. When DM k initiates a UM, this action triggers a state transition process that lasts until the target state is achieved after a duration of time expressed as ${\tau }_k(s^t,s^{t_1})$.

Example 1. 

Taking the trade conflict mentioned at the beginning of this chapter as an example, we will introduce the concept of ToUM in actual conflicts. Let $o_1$ to $o_5$ represent the options of decision-makers A, B, and C. The detailed information on the options and states is shown in

Table 2. DMs, options, and states in Example 2.

DMs	Options	States
		${\mathit{s}}_{\mathbf{1}}$	${\mathit{s}}_{\mathbf{2}}$	${\mathit{s}}_{\mathbf{3}}$	${\mathit{s}}_{\mathbf{4}}$	${\mathit{s}}_{\mathbf{5}}$	${\mathit{s}}_{\mathbf{6}}$	${\mathit{s}}_{\mathbf{7}}$	${\mathit{s}}_{\mathbf{8}}$	${\mathit{s}}_{\mathbf{9}}$	${\mathit{s}}_{\mathbf{10}}$
A	$o_1$	N	Y	Y	Y	Y	Y	Y	Y	Y	Y
B	$o_2$	N	N	Y	N	N	N	Y	N	Y	N
	$o_3$	N	N	N	Y	N	N	N	Y	N	Y
C	$o_4$	N	N	N	N	Y	N	N	N	Y	Y
	$o_5$	N	N	N	N	N	Y	Y	Y	N	N

. Assuming that, for B, moving from N to Y takes 2 months for $o_2$ and 1.8 months for o2, then the ToUM for B’s unilateral move from $s_2$ to $s_3$ is ${\tau }_B(s_2,s_3)=\max (2,0)=2$, and the ToUM for the unilateral move from $s_2$ to $s_4$ is ${\tau }_B(s_2,s_4)=\max (0,1.8)=1.8$.

3.2. Unilateral Moves and Unilateral Improvements in TSGMCR

In TSGMCR, UMs and UIs are influenced by asymmetric ToUM and temporal constraints. Hence, to more accurately represent the dynamics of time-sensitive graph models, we formally redefine the following concepts: time-sensitive UM (TSUM) and time-sensitive UI (TSUI).

Definition 11 

(Time-Sensitive Unilateral Moves). For each $k\in N$ and $s^t\in S$, the set of TSUMs is defined as $t{R}_k\left(s^t\right)$, where $s^{t_1}\in t{R}_k\left(s^t\right)$ if and only if the following conditions hold:

(1) $(s^t,s^{t_1})\in A_k$;

(2) ${\tau }_k(s^t,s^{t_1})\le T-t$;

(3) $k\in U\left(t\right)$, where $U\left(t\right)\subseteq N$ is the set of DMs not engaged in transitions at time t.

The first condition means a TSUM must be a UM. The second condition denotes that the ToUM in the TSUM must not exceed the remaining time ($T-t$). The third condition indicates the rules of TSUMs, stating that a DM is only eligible to initiate a TSUM when the DM is not undergoing state transition. This is because in order to simplify the model, it is assumed in this paper that the DM is unable to change their strategy again during the state transfer process, which will also be reflected in the subsequent definitions.

Using the trade conflict case introduced above as an example, we conduct detailed modeling analysis to illustrate the concept of TSUMs proposed in this paper.

Example 2. 

In a trade conflict involving A, B, and C, the option assignments are as follows: option $o_1$ is controlled by A, and B has two options, written as $o_2$ and $o_3$, while $o_4$ and $o_5$ are two options controlled by C. The DMs, options, and states are listed in Table 2. Possible UMs and specific information are shown in Figure 1. The deadline of the conflict is 4, written as $T=4$.

As shown in Figure 1a, each arrow represents a state transition, with the DM for that transition labeled above it and the time consumed by each transition written above the target state. For example, $R_C\left(s^2\right)=\{s^5,s^6\}$ and ${\tau }_C(s^2,s^5)=1.5$, and ${\tau }_C(s^2,s^6)=1$. Note that in real-world conflicts, the scales of ToUMs can be years, months, or days, which is omitted in this example.

Assume that DM C initiates a TSUM from $s_2$ to $s_6$ at $t=0$, as illustrated in Figure 1b. Consequently, according to Figure 1a, there are two TSUMs to choose for B; one is to move to $s_7$ with ${\tau }_B(s^6,s^7)=2$, and the other is to move to $s_8$ with ${\tau }_B(s^6,s^8)=1.8$. Now, consider another scenario, as depicted in Figure 1c, in which C and B each initiate a TSUM at $t=0$. Their option changes are labeled below the arrows. At $t=1.5$, C completes the state transition to $s_5$. Based on Figure 1a, B can initiate a UM to $s_9$. However, at $t=1.5$, the move from $s_5$ to $s_9$ is not a TSUM for B because the transition initiated by B from $s_2$ to $s_4$ has not been completed, as explained in (3) of Definition 11. At $t=1.5$, B has a TSUM to $s_9$ with ${\tau }_B(s^{10},s^9)=2$. Note that $2\le T-t=4-1.8=2.2$, and the move complies with (2) in Definition 11.

Inspired by He [28], the preference ratio is defined as the ratio of the increase in preference score to the state transition time.

 Definition 12 

(Preference Ratio). For each $k\in N$, $s^t\in S$, and $s^{t_1}\in t{R}_k\left(s^t\right)$, the preference ratio of the TSUM of k (${\sigma }_k(s^t,s^{t_1})$) can be calculated as follows:

$${\sigma }_k(s^t,s^{t_1})={\displaystyle \frac{{\mathrm{\Psi }}_k\left(s^{t_1}\right)-{\mathrm{\Psi }}_k\left(s^t\right)}{{\tau }_k(s^t,s^{t_1})}}$$

(3)

where ${\mathrm{\Psi }}_k\left(s^{t_1}\right)$ and ${\mathrm{\Psi }}_k\left(s^t\right)$ represent the preference scores of $s^{t_1}$ and $s^t$ for DM k, respectively; ${\tau }_k(s^t,s^{t_1})$ denotes the oUM from $s^t$ to $s^{t_1}$; and ${\mathrm{\Psi }}_k\left(s^{t_1}\right)$ and ${\mathrm{\Psi }}_k\left(s^t\right)$ can be calculated based Equations (1) and (2).

As illustrated in Definition 12, a higher preference score and lower ToUM results in a higher preference ratio. Obviously, the TSUMs with higher preference ratios are most valuable to DMs, as they enable greater improvements with less time— which constitutes precisely the criterion DMs should consider when selecting TSUIs. A formal definition of a TSUI is introduced below.

Definition 13 

(Time-Sensitive Unilateral Improvements). For each $k\in N$ and $s^t\in S$, the set of TSUIs is defined as $t{R}_k^{+}\left(s^t\right)$, where $s^{t_1}\in t{R}_k^{+}\left(s^t\right)$ if and only if the following conditions hold:

(1) $(s^t,s^{t_1})\in A_k$ and $s^{t_1}{\succ }_k{s}^t$;

(2) ${\tau }_k(s^t,s^{t_1})\le T-t$;

(3) ${\sigma }_k(s^t,s^{t_1})\ge {\sigma }_k(s^t,s^{t_2})$, $\forall s^{t_2}\in R_k^{+}\left(s^t\right)$;

(4) $k\in U\left(t\right)$, where $U\left(t\right)\subseteq N$ is the set of DMs not engaged in transitions at time t;

According to Definitions 11 and 13, the primary distinctions between TSUIs and TSUMs lie in the first and third conditions in Definition 13. The first condition means a TSUI must be a UI, and the third condition represents the fact that a TSUI has the largest preference ratio of all possible UIs. Note that this does not necessarily mean that the TSUI is unique, as the same can exist for preference ratios.

He [28] imposed strict requirements on both the selection and ordering of DMs for the initiation of a TSUM or TSUI, while this paper considers the case where DMs move simultaneously, and it is no longer the DM with the shortest ToUM that is eligible to move. Therefore, in this paper, multiple DMs are allowed to move simultaneously, provided they are not in the process of state transition. Note that during a DM’s state transition, the other DMs will not know what move it has made until it completes those move. A comparative analysis between existing definitions of TSUMs and TSUIs and those proposed in this paper is presented in

Table 3. Comparative analyses between existing and proposed definitions of TSUMs and TSUIs.

Dimension	Classical GMCR [1,2]	Inohara [27]	He [28]	This Paper
DM’s Move Order in UM	N/A	The DM with the minimum state transition time is prioritized	Each DM is restricted to a single action per round, and the DM with the minimum state transition time is prioritized	The DMs not in state transition processes are eligible
DM’s Move Order in UI	N/A	Same as UM requirements	Based on the requirements of UM, prioritizes DM with minimum state transition time and maximum preference ratio	Same as UM requirements, but DMs must select state transitions with the maximum preference ratio
Deadline?	No	No	Yes	Yes
Time of UM/UI?	No	Yes	Yes	Yes
Multi-DM Simultaneous Move?	No	No	No	Yes

.

Example 3 is provided to demonstrate the concepts of TSUI proposed in this paper.

Example 3. 

Using the same scenario and notation as in Examples 1 and 2, the preference scores were calculated and are listed in

Table 4. Preference scores of states in Example 2.

DMs	States
	${\mathit{s}}_{\mathbf{1}}$	${\mathit{s}}_{\mathbf{2}}$	${\mathit{s}}_{\mathbf{3}}$	${\mathit{s}}_{\mathbf{4}}$	${\mathit{s}}_{\mathbf{5}}$	${\mathit{s}}_{\mathbf{6}}$	${\mathit{s}}_{\mathbf{7}}$	${\mathit{s}}_{\mathbf{8}}$	${\mathit{s}}_{\mathbf{9}}$	${\mathit{s}}_{\mathbf{10}}$
B	2	4	12	13	7	1	9	10	13	9
C	1	7	9	11	16	8	15	5	2	2

. Between the TSUMs for B at $s_2^0$, which means state $s_2$ at $t=0$, ${\sigma }_B(s_2,s_4)={\displaystyle \frac{13-4}{1.8}}>{\sigma }_B(s_2,s_3)={\displaystyle \frac{12-4}{2}}$, and between the TSUMs for C at $s_2^0$, ${\sigma }_C(s_2,s_5)={\displaystyle \frac{16-7}{1.5}}>{\sigma }_C(s_2,s_3)={\displaystyle \frac{9-7}{1}}$. Consequently, at $t=0$, B tends to initiate a TSUI to $s_4$, and C would move to $s_5$.

Let $t{R}_H\left(s^t\right)$ and $t{R}_H^{+}\left(s^t\right)$ represent the TSUM set and TSUI set of coalition H from state $s^t\in S$, in which any DM has exactly one move opportunity, unlike the classical definition that allows coalition members to make multiple moves. This is because the proposed TSGMCR framework permits simultaneous moves by DMs, thereby invalidating the requirement of alternating movement imposed by Definitions 3 and 4 of the classic graph model. The formal definitions are specified as follows.

Definition 14 

(Time-Sensitive Unilateral Moves of a Coalition, CTSUM). For $H\subseteq N$, $s^t,s^{t_1}\in S$ with $H\ne \varnothing$, $s^{t_1}$ is the CTSUM of H in state $s^t$, such that $s^{t_1}\in t{R}_H\left(s^t\right)$ and the following conditions hold:

(1) $H\subseteq U\left(t\right)$, where $U\left(t\right)\subseteq N$ denotes the set of DMs not in the state transition process at time t;

(2) For each $k\in H$, DM k initiates a TSUM to state $s^{t_k}\in t{R}_k\left(s^t\right)$, and K is the set of DMs initiating TSUMs;

(3) ${\tau }_H(s^t,s^{t_1})=ma{x}_{k\in K}{\tau }_k(s^t,s^{t_k})\le T-t$, where $s^{t_1}$ is the state when the CTSUM has completed. State $s^{t_1}$ can be determined by $s^{t_1}={\cup }_{k\in K}{O}_k\left(s^{t_k}\right){\cup }_{k^{\prime }\in N-H}{O}_{k^{\prime }}\left(s^t\right)$, where $O_k\left(s^{t_k}\right)$ represents the option of DMs in K in state $s^{t_k}$,and $O_{k^{\prime }}\left(s^t\right)$ denotes the option of DMs outside of coalition H. Note that K is the set of DMs initiating TSUMs.

Definition 15 

(Time Sensitive Unilateral Improvements of a Coalition, CTSUI). For $H\subseteq N$, $s^t,s^{t^{\prime }}\in S$ with $H\ne \varnothing$, $s^{t_1}$ is the CTSUI of H in state $s^t$ such that $s^{t_1}\in t{R}_H^{+}\left(s^t\right)$ and the following conditions hold:

(1) $H\subseteq U\left(t\right)$, where $U\left(t\right)\subseteq N$ denotes the set of DMs not in the state transition process at time t;

(2) For each $k\in H$, DM k initiates a TSUM to state $s^{t_k}\in t{R}_k\left(s^t\right)$, and K is the set of DMs initiating TSUMs;

(3) ${\tau }_H(s^t,s^{t_1})=ma{x}_{k\in K}{\tau }_k(s^t,s^{t_k})\le T-t$, where $s^{t_1}$ is the state when the CTSUI has completed and satisfies $s^{t_1}={\cup }_{k\in K}{O}_k\left(s^{t_k}\right){\cup }_{k^{\prime }\in N-H}{O}_{k^{\prime }}\left(s^t\right)$.

Example 4. 

Using the same scenario and notation as in Examples 1, 2 and 3, let $H=\{B,C\}$. At $t=0$, $t{R}_B\left(s_2\right)=\{s_3,s_4\}$, $t{R}_B^{+}\left(s_2\right)=\left\{s_4\right\}$, $t{R}_C\left(s_2\right)=\{s_5,s_6\}$, and $t{R}_C^{+}\left(s_2\right)=\left\{s_5\right\}$. Therefore, $s_3$, $s_4$, $s_5$, $s_6\in t{R}_H\left(s_2\right)$, $s_4$, and $s_5\in t{R}_H^{+}\left(s_2\right)$. In addition to the movement of individual DMs, there are also simultaneous moves, which are illustrated in Figure 2. As shown in Figure 2a, a CTSUM is initiated by coalition H, where B initiates a TSUM to $s_3$; C takes a TSUI to $s_5$; and the target state, ${\cup }_{k\in K}{O}_k\left(s^{t_k}\right){\cup }_{k^{\prime }\in N-H}{O}_{k^{\prime }}\left(s^t\right)=O_A\left(s_2\right)\cup O_B\left(s_3\right)\cup O_C\left(s_5\right)=\left\{YYNYN\right\}$, is $s_9$. The ToUM for the CTSUM, ${\tau }_H(s_2,s_9)$, is $\max ({\tau }_B(s_2,s_3),{\tau }_C(s_2,s_5))=2$. When B alters its option moving to $s_4$, there would be a CTSUI because $s_4\in R_B^{+}\left(s_2\right)$ and $s_5\in R_C^{+}\left(s_2\right)$, as represented in Figure 2b. In Figure 2c,d, the other two CTSUM are introduced, with calculation of the target states and ToUMs. Thus, at $t=0$, $t{R}_H\left(s_2\right)=\{s_3$, $s_4$, $s_5$, $s_6$, $s_7$, $s_8$, $s_9$, $s_{10}\}$, $t{R}_H^{+}\left(s_2\right)=\{s_4$, $s_5$, $s_{10}\}$.

In the classical graph model, DMs’ ultimate moves represent theoretical actions rather than actual sequential actions. However, because the proposed time-sensitive graph model permits multiple DMs to execute unilateral moves at the same time, after this process completes, for each DM, it is clear that the final state is not the original target state.

3.3. Stability Analysis of TSGMCR

New stability concepts for the time-sensitive graph model, including Time-Sensitive Nash (TSNash), Time-Sensitive GMR (TSGMR), Time-Sensitive SMR (TSSMR), and Time-Sensitive SEQ (TSSEQ), are defined to describe DMs’ behavioral patterns.

Definition 16 

(Time-Sensitive Nash, TSNash). For each $k\in N$, a state ($s^t\in S$) is TSNash if and only if $t{R}_k^{+}\left(s^t\right)=\varnothing$.

Definition 17 

(Time-Sensitive GMR, TSGMR). For each $k\in N$, a state ($s^t\in S$) is TSGMR if and only if, for every $s^{t_1}\in t{R}_i^{+}\left(s^t\right)$, there exists at least one $s^{t_2}\in t{R}_{N\setminus \left\{k\right\}}\left(s^{t_1}\right)$ with $s^t{⪰}_k{s}^{t_2}$.

Definition 18 

(Time-Sensitive SMR, TSSMR). For each $k\in N$, a state ($s^t\in S$) is TSSMR if and only if, for every $s^{t_1}\in t{R}_i^{+}\left(s^t\right)$, there exists at least one $s^{t_2}\in t{R}_{N\setminus \left\{k\right\}}\left(s^{t_1}\right)$ with $s^t{⪰}_k{s}^{t_2}$ and $s^t{⪰}_k{s}^{t_3}$ for all $s^{t_3}\in t{R}_k\left(s^{t_2}\right)$.

Definition 19 

(Time-Sensitive SEQ, TSSEQ). For each $k\in N$, a state ($s^t\in S$) is TSSEQ if and only if, for every $s^{t_1}\in t{R}_i^{+}\left(s^t\right)$, there exists at least one $s^{t_2}\in t{R}_{N\setminus \left\{k\right\}}^{+}\left(s^{t_1}\right)$ with $s^t{⪰}_k{s}^{t_2}$.

3.4. TSGMCR Analysis of Real-World Conflict Problem Process

The application of the TSGMCR method to real-world conflict problems follows the process illustrated in Figure 3. Initially, the core elements of the conflict—including DMs, feasible states, and their preferences—are extracted from the actual scenario, after which the UMs, UIs, and ToUMs for each DM are computed. Subsequently, the equilibrium state is determined based on the definition of time-sensitive stability. This equilibrium analysis is then integrated with the timeline to provide strategic insights, ultimately guiding the decision-making process for the involved parties.

3.5. Inter-Relationships Between Key Concepts of TSGMCR and Classic GMCR

Based on the definitions presented in Section 3.3, here, we explore the relationships between the relevant concepts in TSGMCR and classic GMCR.

3.5.1. Inter-Relationships Among UMs, UIs, TSUMs, and TSUIs

By using Definitions 1, 2, 11, and 13, one can easily determine the relationships among the sets of UMs, UIs, TSUMs, and TSUIs as depicted in Figure 4.

As shown in Figure 4, for each $k\in N$ and $s^t\in S$, there exists $t{R}_k^{+}\left(s^t\right)\subseteq t{R}_k\left(s^t\right)\subseteq R_k\left(s^t\right)$ and $t{R}_k^{+}\left(s^t\right)\subseteq R_k^{+}\left(s^t\right)$. The proof is displayed as follows:

Proof. 

If $s\in t{R}_k^{+}\left(s^t\right)$, then s satisfies all conditions of Definition 13, obviously, s also satisfies all conditions of Definition 11, i.e., $s\in t{R}_k\left(s^t\right)$. Similarly, it can also be concluded that $t{R}_k\left(s^t\right)\subseteq R_k\left(s^t\right)$. Therefore, $t{R}_k^{+}\left(s^t\right)\subseteq t{R}_k\left(s^t\right)\subseteq R_k\left(s^t\right)$ and $t{R}_k^{+}\left(s^t\right)\subseteq R_k^{+}\left(s^t\right)$ hold. □

3.5.2. Inter-Relationships Among Time-Sensitive Stabilities

The inter-relationships among time-sensitive stabilities and classical stabilities are summarized in Figure 5. For $k\in N$, if s is TSNash stable, s is also stable under TSSEQ, TSSMR, and TSGMR; if state s is stable under TSSEQ or TSSMR, s must be TSGMR stable.

As shown in Figure 5, the following relationships can be obtained:

$$\begin{array}{cc}\hfill S_k^{TSNash}\subseteq & S_k^{TSSMR}\subseteq S_k^{TSGMR}\hfill \\ \hfill S_k^{TSNash}\subseteq & S_k^{TSSEQ}\subseteq S_k^{TSGMR}\hfill \end{array}$$

The proofs are similar to the relationships among Nash, SEQ, SMR, and GMR, as given in [1,21].

Proof. 

If $s\in S^{TSNash}$, then $t{R}_k^{+}\left(s^t\right)=\varnothing$ and Definition 18 is satisfied, so $s\in S^{TSSMR}$. If $s\in S^{TSSMR}$, then Definition 18 implies that Definition 17 is satisfied, which means $s\in S^{TSGMR}$. Hence, $S_k^{TSNash}\subseteq S_k^{TSSMR}\subseteq S_k^{TSGMR}$ holds. If $s\in S^{TSNash}$, then Definition 19 is satisfied because $t{R}_k^{+}\left(s^t\right)=\varnothing$, so $s\in S^{TSSEQ}$. Based on Definition 17, if $s\in S^{TSSEQ}$, then $s\in S^{TSGMR}$. □

4. Case Studies

4.1. Background of Jakarta–Bandung High-Speed Railway Project Conflict

The Jakarta–Bandung High-Speed Railway (HSR) project, as the first high-speed rail system in Indonesia and Southeast Asia, epitomizes the strategic convergence between China’s Belt and Road Initiative and Indonesia’s national development plan [30]. This USD 6.07 billion megaproject, initiated in 2016 with an original completion target of 2019, encountered a 51-month delay, culminating in its 2023 opening to traffic [31]. The protracted implementation process exposed fundamental tensions across multiple dimensions [32], including technological conflict between Chinese rail standards (CTCS-3) and Indonesia’s volcanic terrain adaptability requirements, additional requirements imposed by international financial institutions in project financing assessments, and protest activities triggered by environmental groups’ concerns over mangrove forests and endangered species’ habitats [33].

4.2. Modeling the Jakarta-Bandung HSR Conflict

The Jakarta–Bandung HSR conflict involves four key DMs: the Indonesian government (IDN), the Chinese government (CN), the China Development Bank (CDB), and Environmental Protection Organizations (EPOs). IDN’s primary strategic objective centers on leveraging the railway project to foster regional economic integration while reinforcing its position as Southeast Asia’s transportation hub. IDN has two options: (1) adopting the full Chinese technology supply chain and (2) insisting on localization adjustments. For CN, the project represents a critical opportunity to promote its high-speed rail technical standards abroad, establishing a flagship demonstration initiative under the Belt and Road Initiative framework. Consequently, CN must choose between (1) strictly enforcing Chinese technical standards and (2) compromising on adaptive modifications. The CDB prioritizes financial risk mitigation to prevent delays caused by geological uncertainties or environmental controversies. Its available options include (1) adding environmental clauses and prolonging approval processes and (2) suspending financial engagements to exert pressure. Meanwhile, EPO is committed to preserving tropical rainforest ecosystems and minimizing construction-related disruptions to local communities. Their strategic options include (1) organizing radical protests and (2) pursuing legal litigation. All options involved in the Jakarta–Bandung HSR conflict, along with their time parameters, are summarized in

Table 5. DMs, options, and ToUMs of the Jakarta–Bandung HSR conflict.

DMs	Options	Time / Months (N to Y)	Time / Months (Y to N)	Descriptions or Sources
IDN	1: Adopt full Chinese technology supply chain	3	6	See Sources 1,2
	2: Insist on localization adjustments	8	4
CN	3: Strictly implement Chinese standards	4	5	See Sources 1–3
	4: Compromise on technical adjustments	6	3
CDB	5: Add environmental clauses and prolong approval	5	4	CDB tends to
	6: Suspend loans to pressure	1	1	control risk 4–6
EPO	7: Radical protests	2	1	See Sources 7
	8: Legal litigation	12	6

¹ https://tdgcxb.crec.cn/CN/article/downloadArticleFile.do?attachType=PDF&id=6686, accessed on 4 May 2025; ² http://rail.ally.net.cn/html/2016/emg_int_0506/3804.html, accessed on 4 May 2025; ³ https://www.nra.gov.cn/xwzx/xwxx/xwlb/202211/t20221115_338867.shtml, accessed on 1 May 2025; ⁴ http://dmia.danareksaonline.com/Upload/20211012%20WIKA.pdf, accessed on 4 May 2025; ⁵ https://www.crecg.com/eportal/fileDir/zgztywz/resource/cms/article/10212924/10248937/2022%20Annual%20Report.pdf, accessed on 21 April 2025; ⁶ https://onlinelibrary.wiley.com/doi/epdf/10.1111/aspp.12292#accessDenialLayout, accessed on 8 May 2025.; ⁷ https://www.nishimura.com/sites/default/files/images/111201_Indonesia_E.pdf, accessed on 28 April 2025.

. Given the complexity of real-world scenarios, this paper determines the time attributes of options through consultations with domain experts. As shown in Table 5, each option has two time attributes in months: from unselected to selected (N to Y) and from selected to unselected (Y to N). Certainly, real-world scenarios are far more complex and uncertain than the simplified cases presented in Table 5. To facilitate computation and analysis, this paper simplifies the actual conditions by modeling only the key conflicts among stakeholders, including disputes over technological pathways, risk control, and environmental protection. Additionally, the time attributes of these options are determined based on real-world information. It should be noted that these time attributes represent the DMs’ "estimates", which are used to analyze the evolution of conflicts when multiple decision-makers undergo state transitions simultaneously. In the conflict, 16 months are available to reach consensus among four DMs so that the project can begin on schedule [34].

There are eight options in the Jakarta–Bandung HSR conflict, which theoretically generate $2^8$ = 256 states. After accounting for real-world constraints and eliminating infeasible combinations, 48 feasible states remain, as listed in

Table 6. Feasible states for the Jakarta–Bandung HSR conflict.

States	IDN		CN		CDB		EPO		States	IDN		CN		CDB		EPO		States	IDN		CN		CDB		EPO
	1	2	3	4	5	6	7	8		1	2	3	4	5	6	7	8		1	2	3	4	5	6	7	8
$s_1$	Y	N	Y	N	N	N	Y	N	$s_{17}$	Y	N	Y	N	N	N	N	Y	$s_{33}$	Y	N	Y	N	N	N	Y	Y
$s_2$	N	Y	Y	N	N	N	Y	N	$s_{18}$	N	Y	Y	N	N	N	N	Y	$s_{34}$	N	Y	Y	N	N	N	Y	Y
$s_3$	Y	N	N	Y	N	N	Y	N	$s_{19}$	Y	N	N	Y	N	N	N	Y	$s_{35}$	Y	N	N	Y	N	N	Y	Y
$s_4$	N	Y	N	Y	N	N	Y	N	$s_{20}$	N	Y	N	Y	N	N	N	Y	$s_{36}$	N	Y	N	Y	N	N	Y	Y
$s_5$	Y	N	Y	N	Y	N	Y	N	$s_{21}$	Y	N	Y	N	Y	N	N	Y	$s_{37}$	Y	N	Y	N	Y	N	Y	Y
$s_6$	N	Y	Y	N	Y	N	Y	N	$s_{22}$	N	Y	Y	N	Y	N	N	Y	$s_{38}$	N	Y	Y	N	Y	N	Y	Y
$s_7$	Y	N	N	Y	Y	N	Y	N	$s_{23}$	Y	N	N	Y	Y	N	N	Y	$s_{39}$	Y	N	N	Y	Y	N	Y	Y
$s_8$	N	Y	N	Y	Y	N	Y	N	$s_{24}$	N	Y	N	Y	Y	N	N	Y	$s_{40}$	N	Y	N	Y	Y	N	Y	Y
$s_9$	Y	N	Y	N	N	Y	Y	N	$s_{25}$	Y	N	Y	N	N	Y	N	Y	$s_{41}$	Y	N	Y	N	N	Y	Y	Y
$s_{10}$	N	Y	Y	N	N	Y	Y	N	$s_{26}$	N	Y	Y	N	N	Y	N	Y	$s_{42}$	N	Y	Y	N	N	Y	Y	Y
$s_{11}$	Y	N	N	Y	N	Y	Y	N	$s_{27}$	Y	N	N	Y	N	Y	N	Y	$s_{43}$	Y	N	N	Y	N	Y	Y	Y
$s_{12}$	N	Y	N	Y	N	Y	Y	N	$s_{28}$	N	Y	N	Y	N	Y	N	Y	$s_{44}$	N	Y	N	Y	N	Y	Y	Y
$s_{13}$	Y	N	Y	N	Y	Y	Y	N	$s_{29}$	Y	N	Y	N	Y	Y	N	Y	$s_{45}$	Y	N	Y	N	Y	Y	Y	Y
$s_{14}$	N	Y	Y	N	Y	Y	Y	N	$s_{30}$	N	Y	Y	N	Y	Y	N	Y	$s_{46}$	N	Y	Y	N	Y	Y	Y	Y
$s_{15}$	Y	N	N	Y	Y	Y	Y	N	$s_{31}$	Y	N	N	Y	Y	Y	N	Y	$s_{47}$	Y	N	N	Y	Y	Y	Y	Y
$s_{16}$	N	Y	N	Y	Y	Y	Y	N	$s_{32}$	N	Y	N	Y	Y	Y	N	Y	$s_{48}$	N	Y	N	Y	Y	Y	Y	Y

.

Recall that each DM’s preference information is determined by their preference statements, which are listed in descending order of importance in

Table 7. Preference statements for DMs in the Jakarta–Bandung HSR conflict.

DMs	Preference Statements	Description
IDN	1∣(2 & 4)	IDN will adopt Chinese technology or implement localization adjustments with CN’s technical adjustments
	1 IF 3	If CN enforces technical standards, IDN tendsto adopt the full Chinese supply chain
	−7 & −8	IDN will absolutely avoid protests or litigation by environmental groups
	−7 & 2	IDN tends toward localization adjustments when EPO does not choose protests
	2	IDN hopes to insist on localization adjustments
	5 IF 1	If IDN adopts Chinese technology, IDN wishes CDB can add environmental clauses
	−6 IF (2 & 4)	If localizing adjustments are made and CN compromises technically, IDN wishes CDB does not suspend loans
	−5∣−6	IDN prefers avoiding CDB’s prolonged approval or loan suspension
	2 IF −4	If CN abandons technical compromise, IDN will accept localization
CN	1 & 3	CN will enforce standards and expects IDN to adopt Chinese technology
	3∣(4 & 1)	CN will enforce Chinese standards or compromise technically if IDN adopts Chinese technology
	4 IF 2	If IDN chooses localization, CN wishes to compromise technically
	−8 & (−7∣−6)	CN wants to avoid litigation and prevent either protests or CDB loan suspension
	−4 IF 5	If CDB adds environmental clauses, CN rejects technical compromise
	−3 IF −1	If IDN does not adopt Chinese technology, CN will not implement Chinese standards strictly
	−7	CN hopes EPO rejects radical protests
	−6 IF (1 & 3)	If CN enforces standards and expects IDN to adopt Chinese technology, CN wishes CDB will not suspend loans
	−5 & 3	CDB will reject adding environmental clauses when CN implements Chinese standards strictly
CDB	5∣(6 & −7)	CDB will prolong approval or suspend loans if no protests occur
	5 IF (3∣4)	If CN enforces standards or compromises, CDB wishes to add environmental clauses
	−8 & (−2∣−4)	CDB tends to avoid litigation and ensure IDN avoids localization or CN avoids compromise
	2 & 4	When CN compromises on technical adjustments, IDN may insist on localization adjustments
	−6 IF 1	If IDN adopts Chinese technology, CDB will not suspend loans
	−7∣−8	CDB prioritizes evading direct actions by environmental groups
	1 & 3	CN will enforce standards and expects IDN to adopt Chinese technology
EPO	2 & 4	When CN compromises on technical adjustments, IDN may insist on localization adjustments
	7∣(8 &−1)	EPO will launch protests or litigation if IDN rejects Chinese technology
	7 IF 3	If CN enforces standards, EPO wishes to protest
	5	EPO hopes that CDB can add environmental clauses and prolong approval
	−1 & −3 & −5	EPO trends to oppose Chinese technology, enforced standards, and CDB’s prolonged approval
	−7 IF 2	If IDN tends to localization adjustments, EPO does not choose protests
	(7 & 8) IF 1	If IDN adopts full Chinese technology, EPO tends to radical protests and legal litigation
	8 IF 6	If CDB suspends loans, EPO will switch to litigation
	−2∣−4	CDB wishes to accept either IDN’s localization or CN’s technical compromise

. Based on Equations (1) and (2), we can calculate the truth values and preference scores for each state, along with their corresponding preference rankings, all of which are presented in

Table 8. Preference scores of all feasible states for DMs.

IDN	States	$s_{20}$	$s_{24}$	$s_{28}$	$s_{32}$	$s_{36}$	$s_4$	$s_{40}$	$s_8$	$s_{12}$	$s_{44}$	$s_{16}$	$s_{48}$	$s_{23}$	$s_{39}$	$s_7$	$s_{21}$
	Scores	447	447	443	441	415	415	415	415	411	411	409	409	399	399	399	398
	States	$s_{37}$	$s_5$	$s_{15}$	$s_{31}$	$s_{47}$	$s_{13}$	$s_{29}$	$s_{45}$	$s_{11}$	$s_{19}$	$s_{27}$	$s_3$	$s_{35}$	$s_{43}$	$s_1$	$s_{17}$
	Scores	398	398	397	397	397	396	396	396	391	391	391	391	391	391	390	390
	States	$s_{25}$	$s_{33}$	$s_{41}$	$s_9$	$s_{18}$	$s_{22}$	$s_{26}$	$s_{30}$	$s_{10}$	$s_2$	$s_{34}$	$s_{38}$	$s_{42}$	$s_6$	$s_{14}$	$s_{46}$
	Scores	390	390	390	390	63	63	63	61	31	31	31	31	31	31	29	29
CN	States	$s_1$	$s_5$	$s_{17}$	$s_{21}$	$s_{25}$	$s_{29}$	$s_{33}$	$s_{37}$	$s_{41}$	$s_9$	$s_{13}$	$s_{45}$	$s_3$	$s_7$	$s_{19}$	$s_{27}$
	Scores	507	506	479	478	477	476	475	474	473	473	472	472	250	234	222	222
	States	$s_{11}$	$s_{35}$	$s_{43}$	$s_{23}$	$s_{31}$	$s_{15}$	$s_{39}$	$s_{47}$	$s_2$	$s_6$	$s_{18}$	$s_{26}$	$s_{22}$	$s_{30}$	$s_{10}$	$s_{34}$
	Scores	218	218	218	206	206	202	202	202	179	178	151	151	150	150	147	147
	States	$s_{42}$	$s_{14}$	$s_{38}$	$s_{46}$	$s_4$	$s_8$	$s_{20}$	$s_{28}$	$s_{12}$	$s_{36}$	$s_{44}$	$s_{24}$	$s_{32}$	$s_{16}$	$s_{40}$	$s_{48}$
	Scores	147	146	146	146	122	106	94	94	90	90	90	78	78	74	74	74
CDB	States	$s_5$	$s_{14}$	$s_6$	$s_7$	$s_{13}$	$s_{15}$	$s_{16}$	$s_{24}$	$s_{32}$	$s_8$	$s_{40}$	$s_{48}$	$s_{21}$	$s_{22}$	$s_{23}$	$s_{30}$
	Scores	119	118	118	118	115	114	110	110	110	110	108	108	103	102	102	102
	States	$s_{37}$	$s_{38}$	$s_{39}$	$s_{46}$	$s_{29}$	$s_{31}$	$s_{45}$	$s_{47}$	$s_{28}$	$s_{26}$	$s_{25}$	$s_{27}$	$s_1$	$s_{10}$	$s_2$	$s_3$
	Scores	101	100	100	100	99	98	97	96	78	70	67	66	23	22	22	22
	States	$s_9$	$s_{11}$	$s_{12}$	$s_{20}$	$s_4$	$s_{36}$	$s_{44}$	$s_{17}$	$s_{18}$	$s_{19}$	$s_{33}$	$s_{34}$	$s_{35}$	$s_{42}$	$s_{41}$	$s_{43}$
	Scores	19	18	14	14	14	12	12	7	6	6	5	4	4	4	1	0
EPO	States	$s_{24}$	$s_{32}$	$s_{40}$	$s_{48}$	$s_8$	$s_{16}$	$s_{20}$	$s_{28}$	$s_{36}$	$s_4$	$s_{44}$	$s_{12}$	$s_{37}$	$s_{39}$	$s_{45}$	$s_{47}$
	Scores	494	494	486	486	486	484	478	478	470	470	470	468	239	239	239	239
	States	$s_5$	$s_7$	$s_{13}$	$s_{15}$	$s_{38}$	$s_{46}$	$s_6$	$s_{14}$	$s_{33}$	$s_{35}$	$s_{41}$	$s_{43}$	$s_1$	$s_3$	$s_{11}$	$s_9$
	Scores	235	235	233	233	231	231	231	229	207	207	207	207	203	203	201	201
	States	$s_2$	$s_{34}$	$s_{42}$	$s_{10}$	$s_{22}$	$s_{30}$	$s_{18}$	$s_{26}$	$s_{23}$	$s_{31}$	$s_{19}$	$s_{27}$	$s_{21}$	$s_{29}$	$s_{17}$	$s_{25}$
	Scores	199	199	199	197	175	175	143	143	107	107	75	75	43	43	11	11

.

Using state $s_2$ as the initial state, the TSGMCR equilibrium for the Jakartai–Bandung HSR conflict is calculated, with the results shown in Figure 6. In Figure 6, the topmost arrow represents the timeline from $t=0$. Possible moves by DMs are indicated by solid arrows, where the length along the timeline denotes to the ToUM of the move. A state depicted with dashed lines indicates that the state is less preferred to the beginning state for the DM, while a state with a double solid line is an equilibrium.

In Figure 6, IDN can reach state $s_1$ through a TSUI at t = 4, which can be written as $s_1^4$. Simultaneously, CDB has three possible choices: (1) a TSUI with a 5-month ToUM to $s_5$, (2) a TSUI with a 5-month ToUM to $s_{13}$, or (3) a TSUM to $s_9$. Note that in the TSUM list and TSUI list of CDB at $s_2^0$, the target states of the three choices are $s_6$, $s_{10}$, and $s_{14}$, respectively. Due to ${\tau }_{IDN}(s_2,s_1)<{\tau }_{CDB}(s_2,s_6)={\tau }_{CDB}(s_2,s_{10})={\tau }_{CDB}(s_2,s_{14})$, IDN completes its TSUI earlier than CDB. The state reached by CDB is updated to $s_5$, $s_{13}$, and $s_9$ because IDN changes its option into $\left\{YN\right\}$. For CDB, both choices (1) and (2) qualify as TSUIs, with identical preference ratios of 19.2. In contrast, choice (3) represents a move that leads to a decline in preference and, thus, does not constitute a TSUI. Hence, CDB has two possible TSUIs, the target states of which are $s_5$ and $s_{13}$. At $t=0$, EPO has no TSUIs. EPO possesses only one TSUI choice, requiring 12 months after IDN has finished a state transition to $s_1$ at $t=4$. However, considering that its ToUM, ${\tau }_{EPO}(s_1,s_{33})$, exceeds CDB’s ${\tau }_{CDB}(s_2,s_6)$ and ${\tau }_{CDB}(s_2,s_{10})$, there are two possible scenarios: if CDB selects $s_5$, EPO’s TSUI leads to state $s_{37}$; if CDB chooses $s_{13}$, the final state will be $s_{45}$. Both $s_{37}$ and $s_{45}$ are equilibrium states where no DM has an incentive to leave unilaterally. In fact, for CDB, $s_{37}$ has a higher preference score than $s_{45}$. Note that the deadline of the Jakarta–Bandung HSR conflict (T) is 16 months. Hence, as shown in Figure 6, $(s_{45},s_{37})$ is not a TSUI for CDB because it exceeds the deadline time (T). Thus, when $s_2$ serves as the status quo, the Jakarta-=Bandung HSR conflict ultimately stabilizes at an equilibrium state of $s_{37}$ or $s_{45}$, where IDN compromises with CN by accepting CN’s technical standards, aiming to reduce construction time. However, this selection fails to account for IDN’s geological and ecological constraints, which would inevitably trigger CDB’s risk control mechanisms (options 5 and 6) and initiate strong opposition from EPO. Such developments would most likely result in the suspension of the high-speed rail project, potentially even necessitating contract renegotiation. In reality, the Jakarta–Bandung HSR project has encountered numerous obstacles and delays from the very beginning of its construction [35].

In order to visually compare the advantages of the proposed TSGMCR method, the classical GMCR method was applied to this case, as shown in Figure 7. In Figure 7, there is only one final state, i.e., $s_{37}$. Since there are no time constraints, DMs can execute as many moves as they want until an equilibrium state is reached. Moreover, as illustrated Figure 7, the moves of DMs are executed in an inflexible sequence, which offers limited guidance for reality. Compared with Figure 6, the results derived from the classical GMCR lack certain critical details, including the process of simultaneous moves by DMs and the temporal aspects of actions. Evidently, the classical graph model’s computational results fail to capture the temporal dynamics of state transitions or the evolutionary trajectory of conflicts. The principal contribution of this paper’s proposed TSGMCR framework resides in its capacity to offer a novel analytical perspective for real-world conflict scenarios—one that more accurately reflects actual conflict dynamics and better accommodates the complexities of strategic interactions in practice. If the time-sensitive graph model proposed by He [28] were applied to analyze this case, the CDB could only act after the IDN had completed its move. This outcome significantly deviates from reality, as illustrated in Figure 6, where DMs in the actual scenario typically act simultaneously. Finally, future research directions could explore how reducing implementation timelines for critical strategies might influence conflict outcomes, building upon the foundational insights established in this study.

5. Conclusions

In this study, a novel time-sensitive GMCR approach is proposed to address multistakeholder strategic conflicts incorporating a time attribute. The proposed model more accurately captures real-world conflict dynamics by allowing multiple DMs to execute simultaneous state transitions, with opponents’ strategy selections remaining unobservable until their transitions are finalized. Specifically, two core concepts, TSUM and TSUI, are defined in this paper, incorporating both the simultaneous decision-making scenarios and time attribute of state transitions. Within the TSGMCR framework, novel concepts of TSUM and TSUI of a coalition are introduced, addressing the limitations of the classic graph model’s alternating-move assumption. Furthermore, the novel time-sensitive stabilities are established and rigorously proved, along with their theoretical inter-relationships, including those among TSNash, TSSEQ, TSSMR, and TSGMR. Empirical validation through a high-speed rail conflict case demonstrates that the proposed TSGMCR method yields more realistic outcomes than the classic graph model.

Additionally, the TSGMCR approach can be further extended through multiple dimensions, such as an uncertain time attribute, time-inconsistent preferences [36,37], uncertain preference [29], or status quo analysis [38,39]. Another interesting research direction is to investigate the critical paths in conflict evolution, which can effectively assist in advanced applications in risk management and optimization decision-making.