A Stochastic Nash Equilibrium Problem for Crisis Rescue

Abstract

This paper proposes a two-stage stochastic non-cooperative game model to solve relief supplies procurement and distribution optimization of multiple rescue organizations in crisis rescue. Rescue organizations with limited budgets minimize rescue costs through relief supply procurement, storage, and transportation in an uncertain environment. Under a mild assumption, we establish the existence and uniqueness of the equilibrium point and derive the optimality conditions by using the duality theory, characterizing the saddle point in the Lagrange framework. The problem is further reformulated as a constraint system governed by Lagrange multipliers, and its optimality is characterized by the Karush–Kuhn–Tucker condition. The economic interpretation of the multipliers as shadow prices is elucidated. Numerical experiments verify the effectiveness of the model in cost optimization in crisis rescue scenarios.

1. Introduction

Crisis response refers to the emergency measures implemented in the event of disasters or crisis situations and other urgent circumstances, with the goal of minimizing losses and casualties. In the face of limited financial budgets, relief organizations need to ensure that they can provide prompt and effective rescue in emergencies. It is important to have a plan in place before a crisis occurs. Many scholars have conducted in-depth research in this field [1,2,3,4].

Coordinating the rescue operations of multiple organizations in crisis relief is a major challenge. Game theory provides a powerful framework for analyzing strategic interactions among multiple decision-makers. In the context of crisis relief operations, Nash equilibrium [5], where no organization can benefit by unilaterally changing its strategy while others’ strategies remain fixed, offers crucial insights into multi-agent coordination under competition. Deterministic Nash equilibrium models have been extensively applied to resource allocation problems, where all parameters (e.g., demand, transportation costs) are assumed to be known with certainty. These models typically formulate each organization’s optimization problem as follows: $\underset{x_i}{\min }{f}_i(x_i,x_{-i})\mathrm{s}.\mathrm{t}.g_i\left(x_i\right)\le 0$, where $x_{-i}$ denotes competitors’ decisions, $f_i$ represents cost functions, and $g_i$ captures operational constraints.

However, the realistic background of crisis rescue is full of uncertainties. When decision-makers are confronted with unexpected events (such as the scale of disasters, fluctuations in relief supply demands, and the probability of road damage), the traditional deterministic Nash equilibrium fails to capture the defects of demand fluctuations and disaster evolution. So the deterministic Nash Equilibrium needs to be extended to the Stochastic Nash Equilibrium. Let $\xi \left(\omega \right)$ denote a random vector defined on probability space $(\Omega ,\mathcal{F},P)$. The strategy set of each rescue organization is ${\mathcal{S}}_i$. A stochastic Nash equilibrium requires $f_i(x_i^{\ast },x_{-i}^{\ast },\xi \left(\omega \right))\ge f_i(x_i,x_{-i}^{\ast },\xi \left(\omega \right)),\forall x_i\in {\mathcal{S}}_i,\mathrm{a}.\mathrm{s}.\omega \in \Omega$. This extension necessitates advanced analytical tools such as stochastic variational inequalities (SVI) and scenario-based decomposition algorithms, which handle probabilistic constraints and multi-stage decision processes.

Recent decades have witnessed notable developments in the field of game theory and optimization model of humanitarian crisis relief. In 2014, Toyasaki and Wakolbinger [6] developed a novel game-theoretic framework to study donation competition among humanitarian agencies. Subsequently, in 2016, Anna Nagurney [7] et al. established a generalized Nash equilibrium network model of non-governmental organizations’ post-disaster humanitarian relief, providing a more comprehensive perspective for the decision-making of humanitarian relief. In 2018, Coles and Zhang [8] applied non-cooperative game theory to the study of partnership selection in the disaster response and recovery stage, analyzing how different organizations choose appropriate partners in the disaster response and recovery stage to achieve more efficient rescue and recovery work. More recently, in 2020, the two-stage stochastic programming model constructed by Anna Nagurney and Mojtaba Salarpour [9] revealed the optimization mechanism of resource allocation for humanitarian organizations in the scenario of multi-agency collaborative disaster relief. In 2025, Hassani, Dana and Anna Nagurney [10] constructed a multi-period and multi-commodity equilibrium model of the international agricultural product trade network, and took the Russia–Ukraine war as an example to quantify the impact mechanism of the war on global agricultural product flows, prices and food security. Furthermore, novel stochastic processes like weird Brownian motion [11] and scaled fractional Brownian motion [12] can model non-Gaussian and non-ergodic anomalous diffusion in crisis scenarios (e.g., disaster propagation, sudden changes in resource demand), capturing extreme events or complex spatiotemporal dependencies. This provides novel perspectives for advancement. The theory of random variational inequalities has also achieved remarkable development. Ashok Ganguly and Kamal Wadhwa [13] strictly demonstrated the existence and uniqueness conditions of understanding by constructing a specific type of stochastic variational inequality model. S. Nanda and S. Pani [14] studied the existence results and complementarity problems of stochastic variational inequalities in Hilbert Spaces. Gwinner and Raciti [15] studied the internal relationship between stochastic variational inequalities and some equilibrium problems. Georgia Fargetta and Antonino Maugeri [16] employed stochastic variational inequality theory to analyze inter-hospital competition for medical resources in emergency disaster scenarios. Significant progress has also been made in multi-stage stochastic variational inequalities. With ongoing research and advances in stochastic variational inequalities, Rockafellar et al. [17] pioneered the extension of their study from single-stage to multi-stage frameworks in 2017. Chen and Pong [18] innovatively reconstructed the two-stage stochastic variational inequality problem as a recourse two-stage stochastic optimization problem based on the minimum expected residual criterion. In 2019, in order to solve the monotone multi-stage stochastic variational inequality, Rockafellar and Sun [19] proposed the asymptotic hedging method. They innovatively constructed a two-parameter asymptotic hedging algorithm, effectively analyzing the Lagrange variational inequality problem [20]. In 2020, Li and Zhang [21] converted the conventional convex two-stage stochastic programming model into a two-stage stochastic variational inequality formulation. In 2021, Sun and Chen [22] explicitly established the theoretical framework of two-stage stochastic variational inequalities. This work was the first to integrate the dual-stage decision-making mechanism into variational inequality models, characterizing optimality conditions for multi-stage optimization and game-theoretic problems under stochastic environments. In 2024, Jiang and Sun [23] investigated the discrete approximation of two-stage stochastic variational inequalities when the second-stage problem had multiple solutions, and employed the progressive hedging algorithm to validate the effectiveness of the discrete approximation method.

To address the complex interaction of competition, uncertainty and multi-stage planning in crisis rescue, this paper presents a stochastic optimization framework for coordinating relief supply procurement and distribution among multiple organizations during crisis events. The proposed two-stage non-cooperative game model addresses key challenges, including pre-disaster preparedness costs, emergency procurement strategies, penalty mechanisms for unmet demand, and organizations’ operational capacities (budgets and donations). By transforming the problem into a stochastic variational inequality formulation, we establish theoretical guarantees for equilibrium solutions [24] while accounting for real-world uncertainties. The model provides a systematic approach to prevent resource waste and improve coordination efficiency among relief organizations, offering both theoretical insights and practical decision support for humanitarian operations under uncertainty. We present the first-order optimality conditions of the Lagrange multiplier characterizing this problem [25,26,27]. We established the existence of saddle points for the problem and reformulated the primal problem into its dual counterpart, thus enabling effective solution derivation. We reconstructed the two-stage random variational inequality into a Lagrange multiplier regulation system through existence demonstration [28]. Lagrange multiplier is the shadow price in aid procurement. By adjusting Lagrange multiplier, the allocation of disaster relief supplies can be better optimized. This means that rescue organizations are able to meet the needs of more rescue points.

This paper makes four key contributions: (1) It develops a novel two-stage stochastic non-cooperative game model integrating procurement, storage, and transportation for multiple relief organizations, bridging game theory and stochastic optimization to advance humanitarian decision-making; (2) under mild assumptions such as convexity and continuous differentiability of the cost functions, the existence of stochastic Nash equilibrium solutions is proven, strengthening the theoretical foundation of crisis rescue research; (3) for general probability distributions, it characterizes second-stage equilibrium through infinite Lagrange duality with economic interpretations of multipliers as shadow prices; (4) numerical experiments using the PHM algorithm validate the model’s effectiveness, demonstrating its practical utility in optimizing relief operations under uncertainty.

The organizational structure of our paper proceeds as follows. In Section 2, we propose a two-phase stochastic framework for modeling competition in rescue supplies allocation. In Section 3, we formalize the Stochastic Nash Equilibrium principle underlying our framework and demonstrate the existence and uniqueness of equilibrium solutions. In Section 4, the existence of saddle points in the Lagrange framework of the second-stage stochastic problem is strictly proved. And we further reformulate the crisis relief problem as a Lagrangian-regulated system. We also apply the progressive hedging algorithm to some numerical examples in Section 5. Our research outcomes and final deductions are compiled in Section 6.

2. The Stochastic Nash Equilibrium Model for Crisis Relief

2.1. The Stochastic Model

Crisis events feature inherent unpredictability in terms of time, location, severity and regional impact. To address these uncertainties, relief organizations implement a two-phase operational framework. Pre-Crisis Planning Phase: Organizations proactively strengthen readiness by identifying and preparing storage points, shelters, and logistical networks. These preparatory measures ensure a foundation for rapid deployment when a crisis strikes. Post-Crisis Implementation Phase: Once the disaster scenario materializes—revealing specific demands and conditions—organizations leverage their pre-established infrastructure while dynamically executing additional actions. This phase integrates prior preparedness with real-time adjustments to optimize supply delivery and aid distribution. By structuring operations into these complementary stages, relief agencies balance proactive planning with responsive adaptation, enhancing their capacity to mitigate uncertainties and deliver effective humanitarian assistance.

The basic symbols used in our model are detailed in Abbreviations Section.

In this chapter, we aim to establish a stochastic Nash equilibrium model to characterize the dynamic game mechanism among multiple rescue organizations in allocating and procuring relief supplies during crisis scenarios. We assume there are W rescue centers participating in providing relief materials to D disaster areas. The model includes W rescue centers (relief supply storage nodes) and G procurement points (markets). Large-scale regional disasters often cause supply chain disruptions and transportation route failures, necessitating a multi-market procurement mechanism. Variations in prices and supply capacities across markets drive rescue organizations to dynamically optimize their strategies. Through this multi-market model, organizations select optimal procurement points and transportation modes, prioritizing low-cost markets under budget constraints while relying on multi-market procurement to address supply shortages during crises.

There are N types of relief supplies, such as water, food, and medical materials. In the pre-disaster preparedness phase, each organization formulates strategic procurement plans based on limited budgets. A rescue center procures quantity ($x_{gw}^n$) of relief supply n from market g and transports it to rescue center w via transportation mode m for storage. The relief supply flow moves from node G to node W. This pre-decision phase balances procurement costs (${\rho }_g^{n1}$), transportation fees ($x_{gw}^n$), and warehouse operating costs (${\tau }_w$), while adhering to regional supply caps (${\zeta }_n$) and budget constraints ($B_d$). When a disaster scenario ($\omega$) occurs, rescue organizations enter the post-disaster dynamic game phase. Facing uncertain demand, they must urgently deploy stored supplies to disaster areas ($y_{wd}^n\left(\omega \right)$). If pre-stocked supplies are insufficient, emergency premium procurement (${\rho }_g^{n2}$) is triggered, where transportation costs ($c_{wd}^{m2}$, $c_{gd}^{m2}$) may surge due to route damage. Relief supply flows then shift from nodes W to D and nodes G to D. To balance cost minimization and demand fulfillment, a penalty function (${\varphi }_d^n\left(\omega ,z_d^n\left(\omega \right)\right)$) is introduced. For example, high penalties incentivize greater pre-disaster stockpiling, while low penalties may reduce emergency procurement. Additionally, social media coverage of the disaster amplifies public awareness, prompting donations from compassionate social groups (${\chi }_d^n\left(\omega ,z_d^n\left(\omega \right)\right)$) based on perceived shortages. These donations partially alleviate the burden on rescue organizations. In particular, we rewrite the relevant problems of each disaster response location d as a two-stage stochastic Nash equilibrium problem:

$$\begin{array}{cc}\hfill min\sum _{g\in G}\left(\sum _{n\in N}{\rho }_g^{n1}{x}_{gw}^n+\sum _{m\in M}{c}_{gw}^{m1}\left(x_{gw}\right)\right)& +\sum _{w\in W}{\tau }_w\sum _{g\in G}\sum _{n\in N}{x}_{gw}^n+\sum _{w\in W}\sum _{m\in M}{c}_{wd}^{m1}\left(x_{wd}\right)\hfill \\ & +{\mathbb{E}}_{\xi }\left[{{\rm Y}}_d(x,\xi \left(\omega \right))\right]\hfill \end{array}$$

(1)

subject to:

$$x_{gw}^n\ge 0,\forall n\in N,\forall g\in G.$$

(2)

$$x_{wd}^n\ge 0,\forall n\in N,\forall w\in W.$$

(3)

$$\sum _{w\in W}{\tau }_w\sum _{g\in G}\sum _{n\in N}{x}_{gw}^n+\sum _{g\in G}\left(\sum _{n\in N}{\rho }_g^{n1}{x}_{gw}^n+\sum _{m\in M}{c}_{gw}^{m1}\left(x_{gw}\right)\right)\le B_d,\forall d\in D,$$

(4)

$$\sum _{w\in W}{x}_{wd}^n\ge {\eta }_d^n,\forall n\in N,$$

(5)

$$\sum _{w\in W}{x}_{wd}^n\le {\zeta }_n,\forall n\in N,$$

(6)

$$x_{gw}^n\ge x_{wd}^n,\forall n\in N,$$

(7)

$$x_{wd}^n\ge 0,\forall n\in N,\forall w\in W.$$

(8)

The objective function (1) minimizes the total procurement expenses, transportation fees, and warehousing charges for the initial supply strategy, plus the anticipated value of the phase two outcomes under disaster scenarios, ${{\rm Y}}_d(x,\xi \left(\omega \right))$. Constraints (2), (3) and (8) are non-negative requirements for variables; Constraint (4) is the budget of each rescue organization; Constraint (5) ensures that the volume of relief supplies transported to disaster location d meets or exceeds the specified demand; Constraint (6) represents the supply capacity limit for each relief supplies category n; Constraint (7) indicates that the number of different types of relief supplies transported to the relief point cannot exceed the amount of relief supplies purchased. Ensuring the existence of a solution requires that condition ${\eta }_d^n\le {\zeta }_n,\forall n,d$ holds. Specifically, ${{\rm Y}}_d(x,\xi \left(\omega \right))$ represents the optimum solution to the subsequent stochastic Nash equilibrium problem:

$$\begin{array}{cc}\hfill {{\rm Y}}_d\left(x,\xi \left(\omega \right)\right)=& min\sum _{w\in W}\sum _{m\in M}{c}_{wd}^{m2}(\omega ,y\left(\omega \right))+\sum _{g\in G}\sum _{n\in N}{\rho }_g^{n2}{q}_{gd}^n\left(\omega \right)+\sum _{g\in G}\sum _{m\in M}{c}_{gd}^{m2}(\omega ,q\left(\omega \right))\hfill \\ \hfill & +\sum _{n\in N}{\varphi }_d^n\left(\omega ,z_d^n\left(\omega \right)\right)-\sum _{n\in N}{\chi }_d^n\left(\omega ,z_d^n\left(\omega \right)\right)\hfill \end{array}$$

(9)

subject to

$$\sum _{w\in W}{y}_{wd}^n(\omega )+\sum _{g\in G}{q}_{gd}^n\left(\omega \right)+z_d^n\left(\omega \right)\ge {\eta }_d^n\left(\omega \right),\forall n\in N,P-a.s.,$$

(10)

$$\sum _{w\in W}{y}_{wd}^n(\omega )+\sum _{w\in W}{x}_{wd}^n+\sum _{g\in G}{q}_{gd}^n\left(\omega \right)\le {\zeta }_n\left(\omega \right),\forall n\in N,P-a.s.,$$

(11)

$$z_d^n\left(\omega \right)\le \alpha {\eta }_d^n\left(\omega \right),\forall n\in N,P-a.s.,$$

(12)

$$x_{gw}^n\ge 0,\forall n\in N,\forall g\in G,P-a.s.,$$

(13)

$$x_{wd}^n\ge 0,\forall n\in N,\forall w\in W,P-a.s.,$$

(14)

$$y_{wd}^n\left(\omega \right)\ge 0,\forall w\in W,\forall n\in N,P-a.s.,$$

(15)

$$q_{wd}^n\left(\omega \right)\ge 0,\forall w\in W,\forall n\in N,P-a.s.,$$

(16)

$$z_d^n\left(\omega \right)\ge 0,\forall n\in N,P-a.s.$$

(17)

The objective function (9) reduces the aggregate expenditure and second-phase shortage penalties. Constraint (10) specifies that the phase two provision, plus the shortage quantity, must meet or exceed the second-stage requirement. Constraint (11) defines the supply capacity limit for each category n of relief supplies. Constraints (12)–(17) enforce the non-negative condition on all decision variables. We postulate that $z_d^n\left(\omega \right)\le \alpha {\eta }_d^n\left(\omega \right),\alpha \in \left[0,1\right],P-a.s.$, namely, unfulfilled demand is capped at a specified fraction of the initial stage demand. To ensure feasibility in the second-stage restrictions, we assume ${\eta }_d^n\left(\omega \right)\le min\left\{{\zeta }_n\left(\omega \right),{\displaystyle \frac{{\eta }_d^n}{\alpha }}\right\},\forall d,n,P-a.s.$ The two-stage crisis response optimization can be reformulated as the subsequent minimization formulation:

$$\begin{array}{cc}\hfill min\sum _{g\in G}\left(\sum _{n\in N}{\rho }_g^{n1}{x}_{gw}^n+\sum _{m\in M}{c}_{gw}^{m1}\left(x_{gw}\right)\right)& +\sum _{w\in W}{\tau }_w\sum _{g\in G}\sum _{n\in N}{x}_{gw}^n+\sum _{w\in W}\sum _{m\in M}{c}_{wd}^{m1}\left(x_{wd}\right)\hfill \\ \hfill & +{\mathbb{E}}_{\xi }\left[{{\rm Y}}_d(x,\xi \left(\omega \right))\right]\hfill \end{array}$$

(18)

where the constraint is (2)–(8) and (9)–(17).

2.2. Stochastic Nash Equilibrium Problem

We assume that $c_{gw}^{m1}\left(x_{gw}\right)$ a.e in the probability measure space $\Omega$ are continuously differentiable and convex for all w, g, m. And if $x_{gw}^n\ge 0$, ${\nabla }_{x_{gw}^n}{c}_{gw}^{m1}\left(x_{gw}\right)\ge 0$ and $J_{x_{gw}^n}\left({\nabla }_{x_{gw}^n}{c}_{gw}^{m1}\left(x_{gw}\right)\right)\ge 0$ hold, where $J_x\left(A\right)$ represents the Jacobian matrix of the vector-valued function A with respect to x. $c_{wd}^{m1}(·)$ a.e in the probability measure space $\Omega$, are continuously differentiable and convex for all w, d, m. And when we have $x_{wd}^n\ge 0$, we have ${\nabla }_{x_{wd}^n}{c}_{wd}^{m1}\left(x_{wd}\right)\ge 0$ and $J_{x_{wd}^n}\left({\nabla }_{x_w^n}{c}_{wd}^{m1}\left(x_{wd}\right)\right)\ge 0$; $c_{wd}^{m2}(\omega ,·)$ a.e in the probability measure space $\Omega$, are continuously differentiable and convex for all w, d, m. And for any given $\omega$, if there is $y_{wd}^n\left(\omega \right)\ge 0$, then there is ${\nabla }_{y_{wd}^n\left(\omega \right)}{c}_{wd}^{m2}(\omega ,y\left(\omega \right))\ge 0$ and $J_{y_{wd}^n\left(\omega \right)}\left({\nabla }_{y_{wd}^n\left(\omega \right)}{c}_{wd}^{m2}(\omega ,y\left(\omega \right))\right)\ge 0$; $c_{gd}^{m2}(\omega ,·)$ a.e in $\Omega$ are continuously differentiable and convex for all g, d, m. And for any given $\omega$, if $q_{gd}^n\left(\omega \right)\ge 0$, then ${\nabla }_{q_{gd}^n\left(\omega \right)}{c}_{gd}^{m2}(\omega ,q\left(\omega \right))\ge 0$ and $J_{q_{gd}^n\left(\omega \right)}\left({\nabla }_{y_{wd}^n\left(\omega \right)}{c}_{gd}^{m2}(\omega ,q\left(\omega \right))\right)\ge 0$; ${\varphi }_d^n(\omega ,·)$ a.e in $\Omega$, are continuously differentiable and convex for all d, n. And for any given $\omega$, if $z_d^n\left(\omega \right)\ge 0$, then ${\nabla }_{z_d^n\left(\omega \right)}{\varphi }_d^n\left(\omega ,z_d^n\left(\omega \right)\right)\ge 0$ and $J_{z_d^n\left(\omega \right)}\left({\nabla }_{z_d^n\left(\omega \right)}{\varphi }_d^n\left(\omega ,z_d^n\left(\omega \right)\right)\right)\ge 0$; ${\chi }_d^n(\omega ,·)$ a.e in $\Omega$ are continuously differentiable and convex for all d, n. And for any given $\omega$, if $z_d^n\left(\omega \right)\ge 0$, then ${\nabla }_{z_d^n\left(\omega \right)}{\chi }_d^n\left(\omega ,z_d^n\left(\omega \right)\right)\ge 0$, and $J_{z_d^n\left(\omega \right)}\left({\nabla }_{z_d^n\left(\omega \right)}{\chi }_d^n\left(\omega ,z_d^n\left(\omega \right)\right)\right)\ge 0$.

As the Nash equilibrium is central to our analysis, we specify the fundamental constraint sets. We define

$${\mathcal{S}}_i\equiv \{x^i=(x_{gw}^n,x_{wd}^n)|\mathrm{Constraints}(2-8)\mathrm{hold}\}$$

where ${\mathcal{S}}_i$ represents the feasible strategy set for each relief organization in the first stage. And we let ${\mathcal{S}}^1\equiv {\prod }_{i=1}^D{\mathcal{S}}_i$. ${\mathcal{S}}^1$ represents the joint feasible strategy set of all relief organizations in the first stage. The collective feasible region $\mathcal{K}$ is defined by $\mathcal{K}\equiv \{x=(x_{wd}^n,y_{wd}^n,q_{gd}^n,z_d^n)|\mathrm{Constraints}\left(10–17\right)\mathrm{hold}\}$. $\mathcal{K}$ represents the joint feasible strategy set of all relief organizations in the second stage (under disaster scenario $\omega$). The feasible set ${\mathcal{S}}^2\equiv {\mathcal{S}}^1\cap \mathcal{K}$. ${\mathcal{S}}^2$ represents the joint feasible region encompassing the strategies of all rescue organizations across both the first and second stages. Additionally, let us make $U_d=\left\{{\left(y_d,q_d,z_d\right)=(y_{wd}\left(\omega \right),q_{gd}\left(\omega \right),z_{dn}\left(\omega \right))}_{w,g,n}\in R^{W+G+N}:\left(10\right)–\left(12\right),\left(15\right)–\left(17\right)\mathrm{hold}\right\}$.

We rewrite the objective function (18) as ${\mathcal{Q}}_d(\omega ,x^1,x^2,y\left(\omega \right),q\left(\omega \right),z\left(\omega \right))$, namely

$$\begin{array}{cc}\hfill {\mathcal{Q}}_d\left(\omega ,x^1,x^2,y\left(\omega \right),q\left(\omega \right),z\left(\omega \right)\right)& =\sum _{g\in G}\left(\sum _{n\in N}{\rho }_g^{n1}{x}_{gw}^n+\sum _{m\in M}{c}_{gw}^{m1}\left(x_{gw}\right)\right)+\sum _{w\in W}{\tau }_w\sum _{g\in G}\sum _{n\in N}{x}_{gw}^n\hfill \\ \hfill & +\sum _{w\in W}\sum _{m\in M}{c}_{wd}^{m1}\left(x_{wd}\right)+{\mathbb{E}}_{\xi }\left[{{\rm Y}}_d\left(x,\xi \left(\omega \right)\right)\right].\hfill \end{array}$$

We require, for $\forall \mathrm{m},\mathrm{w},\mathrm{h},\mathrm{n}$, the gradient function of the decision variable to satisfy the local Lipschitz continuity:

For each $x_{gw}^n,{x_{gw}^n}^{\prime }\in {\mathcal{S}}_i$, there exists $L_{gw}>0$, such that

$$∥{\nabla }_{x_{gw}}{c}_{gw}^{m1}\left(x_{gw}\right)-{\nabla }_{x_{gw}}{c}_{gw}^{m1}\left(x_{gw}^{\prime }\right)∥\le L_{gw}∥x_{gw}^n-{x_{gw}^n}^{\prime }∥$$

Analogous Lipschitz constants $L_{wd}>0$ exist for $c_{wd}^{m1}$.

For each $y_{wd}^n\left(\omega \right)$, there exists $L_{wd}^{m2}\left(\omega \right)\in L^2(\Omega )$, such that

$$\parallel {\nabla }_y{c}_{wd}^{m2}(\omega ,y)-{\nabla }_y{c}_{wd}^{m2}(\omega ,y^{\prime })\parallel \le L_{wd}^m\left(\omega \right)\parallel y_{wd}^n\left(\omega \right)-y_{wd}^n{\left(\omega \right)}^{\prime }\parallel .$$

Analogous Lipschitz stochastic processes $L_{gd}^{m2}\left(\omega \right)$, $L_{\varphi }\left(\omega \right)$, $L_{\chi }\left(\omega \right)$ exist for $c_{gd}^{m2}$, ${\varphi }_d^n$, and ${\chi }_{d^{\prime }}^n$, respectively.

Additionally, the gradients $\nabla c_{wd}^{m2}(\omega ,·),\nabla c_{gd}^{m2}(\omega ,·),\nabla {\varphi }_d^n(\omega ,·),\mathrm{and}\nabla {\chi }_d^n(\omega ,·)$ are Carathéodory functions (measurable in $\omega$, continuous in decision variables).

Below, we present the definition of stochastic Nash equilibrium.

Definition 1.

A vector of rescue supplies $(x^{1\ast },x^{2\ast },y^{\ast },q^{\ast },z^{\ast })\in {\mathcal{S}}^2$ is a stochastic Nash equilibrium if, for each $d\in D$,

$$\begin{array}{cc}\hfill & {\mathcal{Q}}_d\left(\omega ,x_d^{1\ast },x_d^{2\ast },y_d^{\ast }\left(\omega \right),q_d^{\ast }\left(\omega \right),z_d^{\ast }\left(\omega \right),x_{-d}^{1\ast },x_{-d}^{2\ast },y_{-d}^{\ast }\left(\omega \right),q_{-d}^{\ast }\left(\omega \right),z_{-d}^{\ast }\left(\omega \right)\right)\hfill \\ \hfill & \le {\mathcal{Q}}_d\left(\omega ,x_d^1,x_d^2,y_d\left(\omega \right),q_d\left(\omega \right),z_d\left(\omega \right),x_{-d}^{1\ast },x_{-d}^{2\ast },y_{-d}^{\ast }\left(\omega \right),q_{-d}^{\ast }\left(\omega \right),z_{-d}^{\ast }\left(\omega \right)\right),\hfill \end{array}$$

for all $(x_d^1,x_d^2,y_d\left(\omega \right),q_d\left(\omega \right),z_d\left(\omega \right))\in {\mathcal{S}}^2$, P-a.s., where $x_{-d}^1,x_{-d}^2,y_{-d}\left(\omega \right),q_{-d}\left(\omega \right),z_{-d}\left(\omega \right)$ represents the quantity of relief supplies and the unsatisfied requirements across all relief points excluding d.

We now recast the stochastic Nash equilibrium problem as a stochastic variational inequality within the variational analysis framework.

Theorem 1.

Under the assumption that the overall cost functions are convex and continuously differentiable, a vector $(x^{1\ast },x^{2\ast },y^{\ast },q^{\ast },z^{\ast })\in {\mathcal{S}}^2$ is an equilibrium solution of the relief supply problem if and only if it satisfies the following variational inequality:

$$\begin{array}{cc}\hfill & \sum _{g\in G}\sum _{w\in W}\sum _{n\in N}\left({\rho }_g^{n1}+\sum _{m\in M}{\displaystyle \frac{\partial c_{gw}^{m1}\left(x_{gw}^{\ast n}\right)}{\partial x_{gw}^n}}+{\tau }_w\right)\times (x_{gw}^n-x_{gw}^{\ast n})+\hfill \\ \hfill & +\sum _{w\in W}\sum _{d\in D}\sum _{n\in N}\left(\sum _{m\in M}{\displaystyle \frac{\partial c_{wd}^{m1}\left(x_{wd}^{\ast n}\right)}{\partial x_{wd}^n}}\right)\times (x_{wd}^n-x_{wd}^{\ast n})\hfill \\ \hfill & +\sum _{d\in D}\sum _{n\in N}{\int }_{\Omega }[\sum _{w\in W}\left(\sum _{m\in M}{\displaystyle \frac{\partial c_{wd}^{m2}(\omega ,y^{\ast }\left(\omega \right))}{\partial y_{wd}^n}}\right)\times (y_{wd}^n\left(\omega \right)-y_{wd}^{\ast n}\left(\omega \right))\hfill \\ \hfill & +\left({\displaystyle \frac{\partial {\varphi }_d^n(\omega ,z_d^n\left(\omega \right))}{\partial z_d^n}}-{\displaystyle \frac{\partial {\chi }_d^n(\omega ,z_d^n\left(\omega \right))}{\partial z_d^n}}\right)\hfill \\ \hfill & +\sum _{g\in G}\left({\rho }_g^{n2}+\sum _{m\in M}{\displaystyle \frac{\partial c_{gd}^{m2}(\omega ,q^{\ast }\left(\omega \right))}{\partial q_{gd}^n}}\right)\times (q_{gd}^n\left(\omega \right)-q_{gd}^{\ast n}\left(\omega \right))]d\omega \ge 0\hfill \end{array}$$

(19)

Notably, our analysis of the stochastic Nash equilibrium hinges on the convexity of cost functions and constraints. Theorem 1’s variational inequality formulation is derived from the first-order optimality conditions of the convex optimization problems faced by each rescue organization.

Remark 1.

${\rho }_g^{n1}\ge 0$ forms the first-stage gradient term together with the transportation cost gradient $\nabla c_{gw}^{m1}$ and storage cost ${\tau }_w$ ensures the Jacobian matrix $U_1$. ${\rho }_g^{n1}\ge 0$ remains positive semidefinite, reinforcing the convexity of the first-stage cost functions. Excessive ${\rho }_g^{n1}$ prompts rescue organizations to reduce procurement volume in Market g. ${\rho }_g^{n2}$ interacts with $\nabla c_{gd}^{m2}$ to influence the gradient term of emergency procurement $q_{gd}^n$, enhancing the diagonal dominance of matrix $\mathfrak{N}\left(\xi \left(\omega \right),\omega \right)$ and guaranteeing monotonicity. ${\rho }_g^{n2}$ increases marginal transportation costs, reflecting the premium on relief supplies during the second stage. Storage cost ${\tau }_w$ ensures the non-negativity of the gradient term ${\nabla }_{x_{gw}^n}{c}_{gw}^{m1}\left(x_{gw}\right)+{\tau }_w$ in Jacobian matrix E, further strengthening operator monotonicity. Higher ${\tau }_w$ increases inventory holding costs at rescue centers, prompting implementation of lean inventory management strategies.

3. Analysis of the Existence and Convergence

In this chapter, we will systematically investigate the well-posedness of the stochastic Nash equilibrium problem under a two-stage stochastic variational inequality framework. Now, we present the relevant concepts required.

Suppose for any given $\omega \in E$, when $\xi \left(\omega \right)\ge 0$, the matrix

$$\begin{array}{cc}\hfill \mathfrak{N}\left(\xi \left(\omega \right),\omega \right)& =\left(\begin{array}{c}{\displaystyle \sum _{w\in W}}\left({\displaystyle \sum _{m\in M}}{\displaystyle \frac{\partial c_{w1}^{m2}\left(\omega ,y^{\ast }\left(\omega \right)\right)}{\partial y_{wd}^1}}+{\displaystyle \sum _{g\in G}}\left({\rho }_g^{12}+{\displaystyle \sum _{m\in M}}{\displaystyle \frac{\partial c_{g1}^{m2}({\omega }_j,q^{\ast }\left(\omega \right))}{\partial q_{gd}^1}}\right)\right)\\ ⋮\\ {\displaystyle \sum _{w\in W}}\left({\displaystyle \sum _{m\in M}}{\displaystyle \frac{\partial c_{wd}^{m2}\left(\omega ,y^{\ast }\left(\omega \right)\right)}{\partial y_{wd}^N}}+{\displaystyle \sum _{g\in G}}\left({\rho }_g^{N2}+{\displaystyle \sum _{m\in M}}{\displaystyle \frac{\partial c_{gd}^{m2}({\omega }_j,q^{\ast }\left(\omega \right))}{\partial q_{gd}^N}}\right)\right)\end{array}\right)\hfill \\ \hfill & +\left(\begin{array}{c}{\displaystyle \sum _{w\in W}}\left({\displaystyle \frac{\partial {\varphi }_d^1(\omega ,z_d^1\left(\omega \right))}{\partial z_d^1}}-{\displaystyle \frac{\partial {\chi }_d^1(\omega ,z_d^1\left(\omega \right))}{\partial z_d^1}}\right)\\ ⋮\\ {\displaystyle \sum _{w\in W}}\left({\displaystyle \frac{\partial {\varphi }_d^N(\omega ,z_d^N\left(\omega \right))}{\partial z_d^N}}-{\displaystyle \frac{\partial {\chi }_d^N(\omega ,z_d^N\left(\omega \right))}{\partial z_d^N}}\right)\end{array}\right)\hfill \end{array}$$

if the Jacobian matrix with respect to $\xi \left(\omega \right)$ is a semi-definite matrix.

Under the framework of existence demonstration, our objective function (1) can be reconstructed into the following functional form:

$$-\mathcal{F}(\mathrm{x},\xi \left(\omega \right))\in N_{\mathbb{k}}(x,\xi \left(\omega \right))$$

where $\xi \left(\omega \right)=\left(y_{wd}^n\left(\omega \right),q_{gd}^n\left(\omega \right),z_d^n\left(\omega \right)\right)\in {\mathbb{R}}^{NWD+NGD+ND}$, $\omega \in E$, $x=(x_{gw}^n,x_{wd}^n)\in {\mathbb{R}}^{NWD+NGD}$, $\mathbb{k}={\mathbb{R}}^{2NWD+NGD+NGW+ND}$.

Lemma 1

([29]). Let the set $\mathbb{K}\subseteq {\mathbb{R}}^n$ be a closed convex set while the function

$$\mathcal{F}:\mathbb{K}\to {\mathbb{R}}^{2NWD+NGW+ND}$$

is continuous. If $\mathcal{F}$ is pseudo-monotone on $\mathbb{K}$, then the solution set of $\mathcal{F}$ is non-empty and bounded if and only if

$${\mathbb{K}}_{\infty }\cap \left[-{\left(\mathcal{F}\left(\mathbb{K}\right)\right)}^{\ast }\right]=\left\{0\right\}.$$

Remark 2.

The criterion ${\mathbb{K}}_{\infty }\cap \left[-\left(\mathcal{F}{\left(\mathbb{K}\right)}^{\ast }\right)\right]=\left\{0\right\}$ ensures that the solution set is non-empty and bounded by excluding non-trivial directions in the recession cone ${\mathbb{K}}_{\infty }$ that align with the dual cone of $\mathcal{F}\left(\mathbb{K}\right)$. If ${\left(\mathcal{F}\left(\mathbb{K}\right)\right)}^{\ast }$ contains nonzero vectors, the condition ${\mathbb{K}}_{\infty }\cap \left[-\left(\mathcal{F}{\left(\mathbb{K}\right)}^{\ast }\right)\right]=\left\{0\right\}$ may fail, implying potential unboundedness or non-existence of solutions. To preserve solution existence, additional constraints (e.g., strict monotonicity or compactness of $\mathbb{K}$) must be imposed to ensure that no nonzero recession direction $d\in {\mathbb{K}}_{\infty }$ satisfies $-d\in {\left(\mathcal{F}\left(\mathbb{K}\right)\right)}^{\ast }$.

Lemma 2.

For any partitioned vector $v=(v_1,v_2)\in {\mathbb{R}}^n\times {\mathbb{R}}^m$, if the block matrix $\mathrm{J}$ satisfies

$$\mathrm{J}=\left[\begin{array}{cc}{U}_1& S_1\\ -S_1^{\top }& U_2\end{array}\right],$$

with $U_1,U_2\ge 0$, then:

$$\langle Jv,v\rangle =\langle U_1{v}_1,v_1\rangle +\langle U_2{v}_2,v_2\rangle \ge 0.$$

From Lemma 2, it follows that the antisymmetry of the skew-symmetric coupling terms $S_1$ and $-S_1^{\top }$ ensures that the mutual cancellation of cross-derivative terms eliminates disturbances from off-diagonal elements to the non-negativity of the quadratic form. Consequently, when the matrix blocks $U_1$ and $U_2$ are positive semi-definite, the symmetric component of matrix $\mathrm{J}$ predominates in governing its positive semi-definiteness.

Theorem 2.

If the previous hypotheses hold and there exists at least one set of $\overline{x}(·)=\left({\overline{x}}_{gd}^n,{\overline{x}}_{wd}^n,{\overline{y}}_{wd}^n\left(\omega \right),{\overline{q}}_{gd}^n\left(\omega \right),{\overline{z}}_d^n\left(\omega \right)\right)\in \mathbb{K}$ such that $\overline{x}(·)\in int{\left({\mathbb{K}}_{\infty }\right)}^{\ast }$, then the solution set of the function $\mathcal{F}$ is non-empty and bounded.

Proof of Theorem 2.

The first-order conditions of the variational inequality can be expressed as the monotonicity of the gradient mapping. To formalize this, we construct a block-structured Jacobian matrix:

$$J_{(x,\xi (\omega \left)\right)}={\left(\begin{array}{cc}s& s_2\\ s_1& 0\end{array}\right)}_{\left((NWD+NGD)(NWD+NGD+ND+1)\right)},$$

where $\mathrm{s}={\left(\begin{array}{cc}{U}_1& 0\\ 0& U_2\end{array}\right)}_{(2NWD+2NGD+ND)}$, $S_2=-S_1^T$, $S_1=\left(U_3{U}_4\right)$, $U_1={\left(\begin{array}{ccc}{\sum }_{g\in G}{\displaystyle \frac{{\partial }^2{c}_{w1}^{m2}\left(x_{gw}\right)}{\partial {\left(x_{gw}^1\right)}^2}}+{\sum }_{d\in D}{\displaystyle \frac{{\partial }^2{c}_{wd}^{m2}\left(x_{wd}\right)}{\partial {\left(x_{wd}^1\right)}^2}}& \dots & 0\\ ⋮& ⋮& ⋮\\ 0& \dots & {\sum }_{g\in G}{\displaystyle \frac{{\partial }^2{c}_{w1}^{m2}\left(x_{gw}\right)}{\partial {\left(x_{gw}^N\right)}^2}}+{\sum }_{d\in D}{\displaystyle \frac{{\partial }^2{c}_{wd}^{m2}\left(x_{wd}\right)}{\partial {\left(x_{wd}^N\right)}^2}}\end{array}\right)}_{(NWD+NGD)}$, $U_2=\left(\begin{array}{c}{\displaystyle \sum _{w\in W}}\left({\displaystyle \sum _{m\in M}}{\displaystyle \frac{\partial c_{w1}^{m2}\left(\omega ,y^{\ast }\left(\omega \right)\right)}{\partial y_{wd}^1}}+{\displaystyle \sum _{g\in G}}\left({\rho }_g^{12}+{\displaystyle \sum _{m\in M}}{\displaystyle \frac{\partial c_{g1}^{m2}({\omega }_j,q^{\ast }\left(\omega \right))}{\partial q_{gd}^1}}\right)\right)\\ ⋮\\ {\displaystyle \sum _{w\in W}}\left({\displaystyle \sum _{m\in M}}{\displaystyle \frac{\partial c_{wd}^{m2}\left(\omega ,y^{\ast }\left(\omega \right)\right)}{\partial y_{wd}^N}}+{\displaystyle \sum _{g\in G}}\left({\rho }_g^{N2}+{\displaystyle \sum _{m\in M}}{\displaystyle \frac{\partial c_{gd}^{m2}({\omega }_j,q^{\ast }\left(\omega \right))}{\partial q_{gd}^N}}\right)\right)\end{array}\right)$+

$\left(\begin{array}{c}{\displaystyle \sum _{w\in W}}\left({\displaystyle \frac{\partial {\varphi }_d^1(\omega ,z_d^1\left(\omega \right))}{\partial z_d^1}}-{\displaystyle \frac{\partial {\chi }_d^1(\omega ,z_d^1\left(\omega \right))}{\partial z_d^1}}\right)\\ ⋮\\ {\displaystyle \sum _{w\in W}}\left({\displaystyle \frac{\partial {\varphi }_d^N(\omega ,z_d^N\left(\omega \right))}{\partial z_d^N}}-{\displaystyle \frac{\partial {\chi }_d^N(\omega ,z_d^N\left(\omega \right))}{\partial z_d^N}}\right)\end{array}\right)$, $U_3={\left(\begin{array}{ccc}1& \dots & 0\\ ⋮& \ddots & ⋮\\ 0& \dots & 1\end{array}\right)}_{(NWD+NGD)}$,

$U_4={\left(\begin{array}{ccc}{D}_1& \dots & 0\\ ⋮& \ddots & ⋮\\ 0& \dots & D_1\end{array}\right)}_{(NWD+NGD)\times (NWD+NGD+ND)}$, $D_1={\left(\left(\begin{array}{c}-1\\ ⋮\\ -1\end{array}\right)\right)}_{1\times {\displaystyle \frac{NWD+NGD+ND}{NWD+NGD}}}$.

For any $(x_{gw}^n,x_{wd}^n,y_{wd}^n\left(\omega \right),q_{gd}^n\left(\omega \right),z_d^n\left(\omega \right))\in \mathbb{K}$, $s_1$ and $s_2$ are mutually symmetric. $U_1$ is a positive semidefinite matrix, $U_2=\mathfrak{N}(\xi \left(\omega \right),\omega )$ is also a positive semidefinite matrix. So ${\mathrm{J}}_{(\mathrm{x},\xi (\omega \left)\right)}$ is a semidefinite matrix, i.e.,

Since the cost function is continuously differentiable and convex, the function $\mathcal{F}(\mathrm{x},\xi (\omega \left)\right)$ is continuous and convex. First-stage constraints (linear inequalities, convex costs) define a polyhedral convex ${\mathcal{S}}^1$. Since both ${\mathcal{S}}^1$ and $\mathcal{K}$ are closed convex sets, ${\mathcal{S}}^2$ is likewise a closed convex set. Second-stage constraints are affine and convex P-a.s., making $\mathcal{K}$ closed and convex under the probability measure. Since ${\mathcal{S}}^1$ and $\mathcal{K}$ are subject to the constraints of the financial budget, supply upper bounds, and demand lower bounds, they are bounded. Consequently, ${\mathcal{S}}^2$ is bounded. Therefore, the feasible region ${\mathcal{S}}^2$ forms a compact set. So $\mathcal{F}(\mathrm{x},\xi (\omega \left)\right)$ is monotonic on $\mathbb{K}$.

There is only one condition for $-\mathcal{T}\in \mathbb{K}$ vector $\mathcal{T}:x_{gw}^n\le 0,x_{wd}^n\le 0,y_{wd}^n\left(\omega \right)\le 0,q_{gd}^n\left(\omega \right)\le 0,z_d^n\left(\omega \right)\le 0$ and at least one element ${\mathcal{T}}_1\in (x_{gw}^n,x_{wd}^n,y_{wd}^n\left(\omega \right),q_{gd}^n\left(\omega \right),z_d^n\left(\omega \right))$, such that ${\mathcal{T}}_1<0$.

From Theorem 2, we know that for any given point $\mathcal{T}$, there exists point $\overline{x}=({\overline{x}}_{gd}^n,{\overline{x}}_{wd}^n,{\overline{y}}_{wd}^n\left(\omega \right),{\overline{q}}_{gd}^n\left(\omega \right),{\overline{z}}_d^n\left(\omega \right))$, such that $({\overline{x}}_{gd}^n,{\overline{x}}_{wd}^n,{\overline{y}}_{wd}^n\left(\omega \right),{\overline{q}}_{gd}^n\left(\omega \right),{\overline{z}}_d^n\left(\omega \right))\in int{\left({\mathbb{K}}_{\infty }\right)}^{\ast }$. Then, for any non-zero $d\in {\mathbb{K}}_{\infty }$, there holds ${\overline{x}}^{\top }d>0.$

When $d=-\mathcal{T}$, we have $d=-\mathcal{T}$, hence ${\overline{x}}^{\top }(-\mathcal{T})>0$. So we get ${\overline{x}}^{\top }\left(\mathcal{T}\right)<0$. From the monotonicity of $\mathcal{F}$ and the definition of the variational inequality, we have

$$\mathcal{F}{\left(\overline{x}\right)}^{\top }(x-\overline{x})\ge 0,\forall x\in \mathbb{K}.$$

Letting $x=\overline{x}+\lambda \mathcal{T}$, when $\lambda \to 0^{+}$ holds, we have $x\in \mathbb{K}$. Then,

$$\mathcal{F}{\left(\overline{x}\right)}^{\top }(\overline{x}+\lambda \mathcal{T}-\overline{x})=\lambda \mathcal{F}{\left(\overline{x}\right)}^{\top }\mathcal{T}\ge 0$$

Therefore, $\mathcal{F}{\left(\overline{x}\right)}^{\top }\mathcal{T}\ge 0$. Because the gradient of $\mathcal{F}\left(\overline{x}\right)$ aligns with the direction of $\overline{x}$, this leads to a contradiction with ${\overline{x}}^{\top }\left(\mathcal{T}\right)>0$.

If $\overline{x}(·)\in int{\left({\mathbb{K}}_{\infty }\right)}^{\ast }$, then ${\mathbb{K}}_{\infty }$ is pointed. To address the case where $\mathcal{F}{\left(\mathbb{K}\right)}^{\ast }$ contains nonzero vectors, we note that monotonicity of $\mathcal{F}$ implies ${\left(\mathcal{F}\left(\mathbb{K}\right)\right)}^{\ast }\subseteq {\mathbb{K}}_{\infty }^{\ast }$. Hence, if $\overline{x}(·)\in int{\left({\mathbb{K}}_{\infty }\right)}^{\ast }$, then for any nonzero $d\in {\mathbb{K}}_{\infty }$, ${\overline{x}}^{\top }d\le 0$, which contradicts $-d\in {\left(\mathcal{F}\left(\mathbb{K}\right)\right)}^{\ast }$. Therefore, ${\mathbb{K}}_{\infty }\cap \left[-{\left(\mathcal{F}\left(\mathbb{K}\right)\right)}^{\ast }\right]=\left\{0\right\}$ holds, ensuring solution existence and boundedness.

From Lemma 1, it follows that the solution set of $\mathcal{F}(x,\xi \left(\omega \right))$ is non-empty and bounded. □

Above, we have proven the existence of the equilibrium solution; next, we will proceed to prove its uniqueness.

Theorem 3.

If the following strengthened Lipschitz conditions hold

$$\left\{\begin{array}{cc}\hfill {\mu }_{gw}I& \le J(\nabla c_{gw}^{m1})\le L_{gw}I\hfill \\ \hfill {\mu }_{wd}I& \le J(\nabla c_{wd}^{m1})\le L_{wd}I\hfill \\ \hfill {\mu }_{wd}{\left(\omega \right)}^{m2}I& \le J(\nabla c_{wd}^{m2})\le L_{wd}^{m2}\left(\omega \right)I\hfill \\ \hfill {\mu }_{gd}{\left(\omega \right)}^{m2}I& \le J(\nabla c_{gd}^{m2})\le L_{gd}^{m2}\left(\omega \right)I\hfill \\ \hfill {\mu }_{\varphi }\left(\omega \right)I& \le J(\nabla {\varphi }_d^n)\le L_{\varphi }\left(\omega \right)I\hfill \\ \hfill {\mu }_{\chi }\left(\omega \right)I& \le J(\nabla {\chi }_d^n)\le L_{\chi }\left(\omega \right)I\hfill \end{array}\right.$$

with ${\mu }_{gw}>0$, ${\mu }_{wd}>0$, ${inf}_{\Omega }{\mu }_{wd}\left(\omega \right)>0$, ${inf}_{\Omega }{\mu }_{gd}\left(\omega \right)>0$, ${inf}_{\Omega }{\mu }_{\varphi }\left(\omega \right)>0$ and ${inf}_{\Omega }{\mu }_{\chi }\left(\omega \right)>0$ P-a.s., then the composite operator $\mathcal{F}$ is strongly monotone:

$$\langle \mathcal{F}\left(x\right)-\mathcal{F}\left(x^{\prime }\right),x-x^{\prime }\rangle \ge \gamma {\parallel x-x^{\prime }\parallel }^2,$$

$$\gamma =min\left\{{\mu }_{gw},{\mu }_{wd},\underset{\Omega }{inf}{\mu }_{wd}\left(\omega \right),\underset{\Omega }{inf}{\mu }_{gd}\left(\omega \right),\underset{\Omega }{inf}{\mu }_{\varphi }\left(\omega \right),\underset{\Omega }{inf}{\mu }_{\chi }\left(\omega \right)\right\}.$$

Theorem 4

(Uniqueness under Strong Monotonicity). Suppose the Jacobian matrix $\mathfrak{N}\left(\xi \left(\omega \right),\omega \right)$ in variational inequality (19) is strongly monotone, i.e., there exists $\gamma >0$ such that for all feasible strategies $x=(x^1,x^2,y,q,z),x^{\prime }=(x^{1^{\prime }},x^{2^{\prime }},y^{\prime },q^{\prime },z^{\prime })\in {\mathcal{S}}^2$:

$$\langle \mathfrak{N}(x,\xi \left(\omega \right))-\mathfrak{N}(x^{\prime },\xi \left(\omega \right)),x-x^{\prime }\rangle \ge \gamma {\parallel x-x^{\prime }\parallel }^{2}P-\mathrm{a}.\mathrm{s}.$$

Then the stochastic Nash equilibrium $(x^{1\ast },x^{2\ast },y^{\ast },q^{\ast },z^{\ast })$ is unique.

Proof of Theorem 4.

The variational inequality (19) in the paper is defined as follows:

$$\sum _{g\in G}{\int }_{\Omega }\left[{\nabla }_x{\mathcal{Q}}_d(x,\xi \left(\omega \right))·(x-x^{\ast })\right]dP\left(\omega \right)\ge 0,\forall x\in {\mathcal{S}}^2,$$

And $\mathfrak{N}\left(\xi \left(\omega \right),\omega \right)$ representing the Jacobian of the gradient terms. By strengthened Lipschitz conditions, $\mathfrak{N}\left(\xi \left(\omega \right),\omega \right)$ is strongly monotone with modulus $\gamma \ge 0$. Then,

$$\langle \mathfrak{N}(x,\xi \left(\omega \right))-\mathfrak{N}(x^{\prime },\xi \left(\omega \right)),x-x^{\prime }\rangle \ge \gamma {\parallel x-x^{\prime }\parallel }^2\forall x,x^{\prime }\in {\mathcal{S}}^2,\mathrm{P}-\mathrm{a}.\mathrm{s}.$$

Suppose there exist two distinct equilibria $x^{\ast }$, $x^{\ast \ast }\in {\mathcal{S}}^2$. Since both $x^{\ast }$ and $x^{\ast \ast }$ satisfy the variational inequality (19), we have

$${\int }_{\Omega }〈\mathfrak{N}(x^{\ast },\xi \left(\omega \right)),x^{\ast \ast }-x^{\ast }〉dP\left(\omega \right)\ge 0,$$

$${\int }_{\Omega }〈\mathfrak{N}(x^{\ast \ast },\xi \left(\omega \right)),x^{\ast }-x^{\ast \ast }〉dP\left(\omega \right)\ge 0.$$

Thus, we get

$$in{t}_{\Omega }〈\mathfrak{N}(x^{\ast },\xi \left(\omega \right))-\mathfrak{N}(x^{\ast \ast },\xi \left(\omega \right)),x^{\ast }-x^{\ast \ast }〉dP\left(\omega \right)\le 0.$$

By the strong monotonicity

$$\langle \mathfrak{M}(x^{\ast },\xi \left(\omega \right))-\mathfrak{M}(x^{\ast \ast },\xi \left(\omega \right)),x^{\ast }-x^{\ast \ast }\rangle \ge \gamma {\parallel x^{\ast }-x^{\ast \ast }\parallel }^{2}P-\mathrm{a}.\mathrm{s}.$$

Then,

$${\int }_{\Omega }\gamma {\parallel x^{\ast }-x^{\ast \ast }\parallel }^{2}dP\left(\omega \right)\le {\int }_{\Omega }〈\mathfrak{N}(x^{\ast },\xi \left(\omega \right))-\mathfrak{N}(x^{\ast \ast },\xi \left(\omega \right)),x^{\ast }-x^{\ast \ast }〉dP\left(\omega \right)$$

Consequently, we get

$$\gamma {\int }_{\Omega }{\parallel x^{\ast }-x^{\ast \ast }\parallel }^{2}dP\left(\omega \right)\le 0.$$

Since $\gamma >0$, then

$${\int }_{\Omega }{\parallel x^{\ast }-x^{\ast \ast }\parallel }^{2}dP\left(\omega \right)=0\Rightarrow x^{\ast }=x^{\ast \ast }P-\mathrm{a}.\mathrm{s}.$$

□

Here, we need to pay special attention. Under the assumptions that ${\varphi }_d^n(\omega ,·)$ is convex and ${\chi }_d^n(\omega ,·)$ is concave for all d, n, $\omega$, the penalty/incentive terms ${\displaystyle \frac{\partial {\varphi }_d^n}{\partial z_d^n}}-{\displaystyle \frac{\partial {\chi }_d^n}{\partial z_d^n}}$ ensure the positive semi-definiteness of $\mathfrak{N}\left(\xi \left(\omega \right),\omega \right)$. Specifically, Convexity of ${\varphi }_d^n$: Since ${\varphi }_d^n(\omega ,z_d^n)$ is convex in $z_d^n$, ${\displaystyle \frac{{\partial }^2{\varphi }_d^n}{\partial {\left(z_d^n\right)}^2}}\ge 0$. Concavity of ${\chi }_d^n$: Since ${\chi }_d^n(\omega ,z_d^n)$ is concave in $z_d^n$, ${\displaystyle \frac{{\partial }^2{\chi }_d^n}{\partial {\left(z_d^n\right)}^2}}\le 0$.

Thus, the combined term ${\displaystyle \frac{\partial {\varphi }_d^n}{\partial z_d^n}}-{\displaystyle \frac{\partial {\chi }_d^n}{\partial z_d^n}}$ contributes a non-negative diagonal entry to $\mathfrak{N}\left(\xi \left(\omega \right),\omega \right)$, reinforcing its positive semi-definiteness. This property is critical for guaranteeing the monotonicity of the variational inequality (19) and the existence of equilibrium solutions.

4. Lagrange Multiplier Regulation System

In this section, we will construct the Lagrangian multiplier adjustment system for the stochastic Nash equilibrium problem in crisis rescue.

Theorem 5.

The vector $(y_d^{\ast },q_d^{\ast },z_d^{\ast })\in U_d$, for all $d\in D$, is an optimal solution of the second-stage problem (8)–(11) if and only if $(y_d^{\ast },q_d^{\ast },z_d^{\ast })\in U_d$ solves the variational inequality

$$\begin{array}{cc}\hfill & \sum _{n\in N}{\int }_{\Omega }[\sum _{w\in W}\left(\sum _{m\in M}{\displaystyle \frac{\partial c_{wd}^{m2}(\omega ,y^{\ast }\left(\omega \right))}{\partial y_{wd}^n}}\right)·(y_{wd}^n\left(\omega \right)-y_{wd}^{\ast n}\left(\omega \right))\hfill \\ \hfill & +\sum _{g\in G}\left({\rho }_g^{n2}+\sum _{m\in M}{\displaystyle \frac{\partial c_{gd}^{m2}(\omega ,q^{\ast }\left(\omega \right))}{\partial q_{gd}^n}}\right)·(q_{gd}^n\left(\omega \right)-q_{gd}^{\ast n}\left(\omega \right))\hfill \\ \hfill & +\left({\displaystyle \frac{\partial {\varphi }_d^n(\omega ,z_d^n\left(\omega \right))}{\partial z_d^n}}-{\displaystyle \frac{\partial {\chi }_d^n(\omega ,z_d^n\left(\omega \right))}{\partial z_d^n}}\right)·(z_d^n\left(\omega \right)-z_d^{\ast n}\left(\omega \right))]\mathrm{d}P\left(\omega \right)\ge 0,\hfill \\ \hfill & \forall (y_d,q_d,z_d)\in {\mathcal{U}}_d.\hfill \end{array}$$

(20)

See Appendix A for the formal proof of Theorem 5.

Theorem 6.

Let $(y_d^{\ast },q_d^{\ast },z_d^{\ast })\in U_d$ be a solution to (20) a.e. in Ω. Let us say two functions $f_{\mathrm{d}}^a\left(\omega \right)={\sum }_{m\in M}{\displaystyle \frac{\partial c_{wg}^{m2}(\omega ,y^{\ast }\left(\omega \right))}{\partial y_{wg}^n}}$, $\forall w\in W$, $\forall n\in N$, $f_{\mathrm{d}}^{\mathrm{c}}\left(\omega \right)={\rho }_g^{n2}+{\sum }_{m\in M}{\displaystyle \frac{\partial c_{wh}^{m2}\left(\omega ,q^{\ast }\left(\omega \right)\right)}{\partial q_{gd}^n}}$, $\forall g\in G$, $\forall n\in N$, and $f_{\mathrm{d}}^{\mathrm{c}}\left(\omega \right)={\displaystyle \frac{\partial {\varphi }_d^n(\omega ,z_d^n\left(\omega \right))}{\partial z_d^n}}-{\displaystyle \frac{\partial {\chi }_d^n(\omega ,z_d^n\left(\omega \right))}{\partial z_d^n}}$, $\forall n\in N$. We draw the following conclusions:

$$\begin{array}{c}\left(a\right)\left\{\begin{array}{cc}{y}_{wd}^n\left(\omega \right)\ge 0,\hfill & if\omega \in Y^1=\left\{\omega \in \Omega |f_d^a\left(\omega \right)=\underset{\omega \in \Omega }{min}{f}_d^a\left(\omega \right)\right\},\hfill \\ y_{wd}^n\left(\omega \right)=0,\hfill & if\omega \in Y^2=\left\{\omega \in \Omega |f_d^a\left(\omega \right)>\underset{\omega \in \Omega }{min}{f}_d^a\left(\omega \right)\right\},\hfill \end{array}\right.\forall w\in W,n\in \mathbb{N};\hfill \\ \left(b\right)\left\{\begin{array}{cc}{q}_{gd}^n\left(\omega \right)\ge 0,\hfill & if\omega \in Q^1=\left\{\omega \in \Omega |f_d^b\left(\omega \right)=\underset{\omega \in \Omega }{min}{f}_d^b\left(\omega \right)\right\},\hfill \\ q_{gd}^n\left(\omega \right)=0,\hfill & if\omega \in Q^2=\left\{\omega \in \Omega |f_d^b\left(\omega \right)>\underset{\omega \in \Omega }{min}{f}_d^b\left(\omega \right)\right\},\hfill \end{array}\right.\forall g\in G,n\in \mathbb{N};\hfill \\ \left(c\right)\left\{\begin{array}{cc}{z}_d^n\left(\omega \right)\ge 0,\hfill & if\omega \in Z^1=\left\{\omega \in \Omega |f_d^c\left(\omega \right)=\underset{\omega \in \Omega }{min}{f}_d^c\left(\omega \right)\right\},\hfill \\ z_d^n\left(\omega \right)=0,\hfill & if\omega \in Z^2=\left\{\omega \in \Omega |f_d^c\left(\omega \right)>\underset{\omega \in \Omega }{min}{f}_d^c\left(\omega \right)\right\},\hfill \end{array}\right.\forall n\in \mathbb{N}.\hfill \end{array}$$

In other words, if there are three functions $f_d^a\left(\omega \right)$, $f_d^b\left(\omega \right)$, $f_d^c\left(\omega \right)\in L^2(\Omega ,P,\mathbb{R})$, such that (a)–(c) hold, then $(y_d^{\ast },q_d^{\ast },z_d^{\ast })\in U_d$ is the solution to problem (20).

The proof of Theorem 6 is provided in the Appendix A.

Here, we need to make a special statement: the term $f_{\mathrm{d}}^{\mathrm{c}}\left(\omega \right)={\displaystyle \frac{\partial {\varphi }_d^n(\omega ,z_d^n\left(\omega \right))}{\partial z_d^n}}-{\displaystyle \frac{\partial {\chi }_d^n(\omega ,z_d^n\left(\omega \right))}{\partial z_d^n}}$ represents the net marginal penalty for unmet demand. When $f_{\mathrm{d}}^{\mathrm{c}}\left(\omega \right)$ takes the minimum value, the rescue organization prioritizes pursuing the minimum rescue cost rather than fully meeting the rescue material demands of the disaster-stricken area. The rescue organization is in a cost-optimal state, meaning that marginal cost of further reducing unmet demand (emergency procurement or transportation costs) would surpass the marginal benefits gained from such reductions (penalty mitigation or donation incentives). So there is $z_d^n\left(\omega \right)\ge 0$. When $f_{\mathrm{d}}^{\mathrm{c}}\left(\omega \right)$ is greater than the minimum value, it means that our marginal penalty cost surges or the existing relief supplies are sufficient. The rescue organization must redistribute the relief supplies to meet the relief supply demands in the disaster-stricken area as much as possible. Rescue organizations must prioritize fully meeting demands because the marginal penalty of unmet needs or the cost of reduced donations now exceeds the marginal cost of stockpiling or procurement. This forces organizations to adjust strategies (such as activating emergency procurement or reallocating resources) to avoid higher costs. So we have $z_d^n\left(\omega \right)=0$.

We recalled the definition of the tangent cone. Suppose A represents a real normed space and, given an element $a\in A$ the set:

$$T_C\left(a\right):=\{a\in A:b=\underset{n\to \infty }{lim}{\lambda }_n\left(a_n-a\right),{\lambda }_n\in \mathbb{R},{\lambda }_n>0,\forall n\in \mathbb{N},a_n\in A,\forall n\in \mathbb{N},\underset{n\to \infty }{lim}{a}_n=a\}$$

is formally defined as the tangent cone to C at a.

Theorem 7.

If $(y_d^{\ast },q_d^{\ast },z_d^{\ast })\in U_d$ is a stochastic Nash equilibrium to problem (20), then $(y_d^{\ast },q_d^{\ast },z_d^{\ast })$ satisfies $T_C\left((y_d^{\ast },q_d^{\ast },z_d^{\ast })\right)\cap (-\infty ,0)=\varnothing$.

The proof of Theorem 7 can be found in the Appendix A.

Theorem 8.

$(y_d^{\ast },q_d^{\ast },z_d^{\ast })\in U_d$ is a stochastic Nash equilibrium to problem (20) if and only if there exists ${\lambda }_d^{\ast 1n}\left(\omega \right)$, ${\lambda }_d^{\ast 2n}\left(\omega \right)$, ${\nu }_d^{\ast 1n}\left(\omega \right)$, ${\nu }_g^{\ast 1n}\left(\omega \right)$, ${\pi }_d^{\ast 1n}\left(\omega \right)$, ${\pi }_d^{\ast 2n}\left(\omega \right)\in L^2(\Omega ,P,{\mathbb{R}}^{+})$ such that ${{(y}_d^{\ast },q_d^{\ast },z_d^{\ast },\lambda }_d^{\ast 1n}\left(\omega \right)$, ${\lambda }_d^{\ast 2n}\left(\omega \right)$, ${\nu }_d^{\ast 1n}\left(\omega \right)$, ${\nu }_g^{\ast 1n}\left(\omega \right)$, ${\pi }_d^{\ast 1n}\left(\omega \right)$, ${\pi }_d^{\ast 2n}\left(\omega \right))$ is a saddle point of the Lagrange functional.

Proof of Theorem 8.

Assuming that $(y_d^{\ast },q_d^{\ast },z_d^{\ast })\in U_d$ is a solution to (20). We set

$$\begin{array}{cc}\hfill & {\phi }_d(\gamma ,q,z)=\sum _{n\in N}{\int }_{\Omega }[\sum _{n\in N}\left(\sum _{m\in M}{\displaystyle \frac{\partial c_{wd}^{m2}(\omega ,y^{\ast }\left(\omega \right))}{\partial y_{wd}^n}}\times \left(y_{wd}^n\left(\omega \right)-y_{wd}^{\ast n}\left(\omega \right)\right)\right)\hfill \\ \hfill & +\sum _{g\in G}\left(p_g^{n2}+\sum _{m\in M}{\displaystyle \frac{\partial c_{gd}^{m2}(\omega ,q^{\ast }\left(\omega \right))}{\partial q_{gd}^n}}\right)\times \left(q_{gd}^n\left(\omega \right)-q_{gd}^{\ast n}\left(\omega \right)\right)\hfill \\ \hfill & +\left({\displaystyle \frac{\partial {\varphi }_d^n(\omega ,z_d^{\ast n}\left(\omega \right))}{\partial z_d^n}}-{\displaystyle \frac{\partial {\chi }_d^n(\omega ,z_d^{\ast n}\left(\omega \right))}{\partial z_d^n}}\right)\times \left(z_d^n\left(\omega \right)-z_d^{\ast n}\left(\omega \right)\right)]\mathrm{d}P\left(\omega \right)\ge 0,\hfill \\ \hfill & \forall (y_d,q_d,z_d)\in U_d.\hfill \end{array}$$

We consider the Lagrange function:

$$\begin{array}{cc}\hfill & {\mathcal{L}}_{\mathrm{d}}(y,q,z,\lambda ,\nu ,\pi )\hfill \\ \hfill & ={\phi }_{\mathrm{d}}(y,q,z)+{\int }_{\Omega }\sum _{n\in N}{\lambda }_{\mathrm{d}}^{1n}\left(\omega \right)\left(-\sum _{w\in W}{y}_{wd}^n\left(\omega \right)-\sum _{g\in G}{q}_{gd}^n\left(\omega \right)-z_d^n\left(\omega \right)+{\eta }_d^n\left(\omega \right)\right)\mathrm{d}P\left(\omega \right)\hfill \\ \hfill & +{\int }_{\Omega }\sum _{n\in N}{\lambda }_{\mathrm{d}}^{2n}\left(\omega \right)\left(\sum _{w\in W}{y}_{wd}^n\left(\omega \right)+\sum _{w\in W}{x}_{wd}^n+\sum _{g\in G}{q}_{gd}^n\left(\omega \right)-{\zeta }_n\left(\omega \right)\right)\mathrm{d}P\left(\omega \right)\hfill \\ \hfill & +{\int }_{\Omega }\sum _{n\in N}\sum _{w\in W}{\nu }_{\mathrm{d}}^{1n}\left(\omega \right)\left(-y_{wd}^n\left(\omega \right)\right)\mathrm{d}P\left(\omega \right)+{\int }_{\Omega }\sum _{n\in N}\sum _{g\in G}{\nu }_{\mathrm{g}}^{1n}\left(\omega \right)\left(-q_{gd}^n\left(\omega \right)\right)\mathrm{d}P\left(\omega \right)\hfill \\ \hfill & +{\int }_{\Omega }\sum _{n\in N}{\pi }_{\mathrm{d}}^{1n}\left(\omega \right)\left(-z_d^n\left(\omega \right)\right)\mathrm{d}P\left(\omega \right)+{\int }_{\Omega }\sum _{n\in N}{\pi }_{\mathrm{d}}^{2n}\left(\omega \right)\left(z_d^n\left(\omega \right)-\alpha {\eta }_d^n\left(\omega \right)\right)\mathrm{d}P\left(\omega \right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \forall y\in L^2(\Omega ,P,{\mathbb{R}}^{WD}),q\in L^2(\Omega ,P,{\mathbb{R}}^{GD}),z\in L^2(\Omega ,P,{\mathbb{R}}^N),\hfill \\ \hfill & {\lambda }_d^{1n}\left(\omega \right),{\lambda }_d^{2n}\left(\omega \right),{\nu }_d^{1n}\left(\omega \right),{\nu }_g^{1n}\left(\omega \right){\pi }_d^{1n}\left(\omega \right),{\pi }_d^{2n}\left(\omega \right)\in L^2(\Omega ,P,{\mathbb{R}}^{+})\hfill \end{array}$$

Since $S_{feasible}\cap S_{descent}=\varnothing$ is already proven, it exists ${\lambda }_d^{\ast 1n}\left(\omega \right)$, ${\lambda }_d^{\ast 2n}\left(\omega \right)$, ${\nu }_d^{\ast 1n}\left(\omega \right)$, ${\nu }_g^{\ast 1n}\left(\omega \right)$, ${\pi }_d^{\ast 1n}\left(\omega \right)$, ${\pi }_d^{\ast 2n}\left(\omega \right)\ge 0,P-a.s.$ such that ${{(y}_d^{\ast },q_d^{\ast },z_d^{\ast },\lambda }_d^{\ast 1n}\left(\omega \right)$, ${\lambda }_d^{\ast 2n}\left(\omega \right)$, ${\nu }_d^{\ast 1n}\left(\omega \right)$, ${\nu }_g^{\ast 1n}\left(\omega \right)$, ${\pi }_d^{\ast 1n}\left(\omega \right)$, ${\pi }_d^{\ast 2n}\left(\omega \right))$ is a saddle point of Lagrange functional

$$\begin{array}{cc}\hfill & {\mathcal{L}}_{\mathrm{d}}(y^{\ast },q^{\ast },z^{\ast },\lambda ,\nu ,\pi )\le {\mathcal{L}}_{\mathrm{d}}(y^{\ast },q^{\ast },z^{\ast },{\lambda }^{\ast },{\nu }^{\ast },{\pi }^{\ast })\le {\mathcal{L}}_{\mathrm{d}}(y,q,z,{\lambda }^{\ast },{\nu }^{\ast },{\pi }^{\ast }),\hfill \\ \hfill & \forall (y_{\mathrm{d}},q_{\mathrm{d}},z_{\mathrm{d}})\in U_{\mathrm{d}},{\lambda }_{\mathrm{d}}^{1n}\left(\omega \right),{\lambda }_{\mathrm{d}}^{2n}\left(\omega \right),{\nu }_{\mathrm{d}}^{1n}\left(\omega \right),{\nu }_{\mathrm{g}}^{1n}\left(\omega \right),{\pi }_{\mathrm{d}}^{1n}\left(\omega \right),{\pi }_{\mathrm{d}}^{2n}\left(\omega \right)\ge 0,P-a.s.\hfill \\ \hfill & {\lambda }_{\mathrm{d}}^{\ast 1n}\left(\omega \right)(-\sum _{w\in W}{y}_{wd}^{\ast n}\left(\omega \right)-\sum _{g\in G}{q}_{gd}^{\ast n}\left(\omega \right)-z_d^{\ast n}\left(\omega \right)+{\eta }_d^n\left(\omega \right))=0,P-a.s.\hfill \\ \hfill & {\lambda }_{\mathrm{d}}^{\ast 2n}\left(\omega \right)(\sum _{w\in W}{y}_{wd}^{\ast n}\left(\omega \right)+\sum _{w\in W}{x}_{wd}^n+\sum _{g\in G}{q}_{gd}^{\ast n}\left(\omega \right)-{\zeta }_n\left(\omega \right))=0,P-a.s.\hfill \\ \hfill & {\nu }_{\mathrm{d}}^{\ast 1n}\left(\omega \right)(-y_{wd}^{\ast n}\left(\omega \right))=0,{\pi }_{\mathrm{d}}^{\ast 1n}\left(\omega \right)(-z_d^{\ast n}\left(\omega \right))=0,P-a.s.\hfill \\ & {\pi }_{\mathrm{d}}^{\ast 2n}\left(\omega \right)(z_d^{\ast n}\left(\omega \right)-a{\eta }_d^n\left(\omega \right))=0,v_{\mathrm{g}}^{\ast 1n}\left(\omega \right)(-q_{gd}^{\ast n}\left(\omega \right))=0,P-a.s.\hfill \end{array}$$

$$\begin{array}{cc}\hfill Thus,0& ={\mathcal{L}}_d\left(y^{\ast },q^{\ast },z^{\ast },{\lambda }^{\ast },{\nu }^{\ast },{\pi }^{\ast }\right)\le {\mathcal{L}}_d\left(y,q,z,{\lambda }^{\ast },{\nu }^{\ast },{\pi }^{\ast }\right)\hfill \\ \hfill & =\sum _{n\in N}{\int }_{\Omega }[\sum _{w\in W}\left(\sum _{m\in M}{\displaystyle \frac{\partial c_{wg}^{m2}(\omega ,y^{\ast })}{\partial y_{wg}^n}}-{\lambda }_{\mathrm{d}}^{\ast 1n}+{\lambda }_{\mathrm{d}}^{\ast 2n}-{\nu }_{\mathrm{d}}^{\ast 1n}\right)·\left(y_{wd}^{\mathrm{n}}-y_{wd}^{\ast n}\right)\hfill \\ \hfill & +\sum _{g\in G}\left({\rho }_g^{n2}+\sum _{m\in M}{\displaystyle \frac{\partial c_{gd}^{m2}(\omega ,q^{\ast })}{\partial q_{gd}^n}}-{\lambda }_{\mathrm{d}}^{\ast 1n}+{\lambda }_{\mathrm{d}}^{\ast 2n}-{\nu }_{\mathrm{g}}^{\ast 1n}\right)·\left(q_{gd}^n-q_{gd}^{\ast n}\right)\hfill \\ \hfill & +\left({\displaystyle \frac{\partial {\varphi }_d^n(\omega ,z_d^{\ast n})}{\partial z_d^n}}-{\displaystyle \frac{\partial {\chi }_d^n(\omega ,z_d^{\ast n})}{\partial z_d^n}}-{\pi }_{\mathrm{d}}^{\ast 1n}+{\pi }_{\mathrm{d}}^{\ast 2n}-{\lambda }_{\mathrm{d}}^{\ast 1n}\right)·\left(z_d^n-z_d^{\ast n}\right)]\mathrm{d}P\left(\omega \right),\hfill \\ \hfill & \forall (y_d,q_d,z_d)\in U_d.\hfill \end{array}$$

Setting $q_{gd}^n\left(\omega \right)=q_{gd}^{\ast n}\left(\omega \right)$, $z_d^n\left(\omega \right)=z_d^{\ast n}\left(\omega \right)$ and $y_{wd}^n\left(\omega \right)=y_{wd}^{\ast n}\left(\omega \right){+\delta }_{wd}^n\left(\omega \right)$ where ${\delta }_{wd}^n\left(\omega \right)\in L^2\left(\Omega ,P,{\mathbb{R}}^{WD}\right)$, then

$$\begin{array}{c}\hfill \sum _{n\in N}{\int }_{\Omega }\left[\sum _{w\in W}(\sum _{m\in M}{\displaystyle \frac{\partial c_{Wg}^{mz}(\omega ,y^{\ast }\left(\omega \right))}{\partial y_{Wg}^n}}-{\lambda }_{\mathrm{d}}^{\ast 1n}\left(\omega \right)+{\lambda }_{\mathrm{d}}^{\ast 2n}\left(\omega \right)-{\nu }_{\mathrm{d}}^{\ast 1n}\left(\omega \right))\right]{\delta }_{wd}^n\left(\omega \right)dP\left(\omega \right)=0,P-a.s.\end{array}$$

that we get

$$\begin{array}{c}\hfill \sum _{m\in M}{\displaystyle \frac{\partial c_{wg}^{m2}(\omega ,y^{\ast }\left(\omega \right))}{\partial y_{wg}^n}}-{\lambda }_{\mathrm{d}}^{\ast 1n}\left(\omega \right)+{\lambda }_{\mathrm{d}}^{\ast 2n}\left(\omega \right)-{\nu }_{\mathrm{d}}^{\ast 1n}\left(\omega \right)=0,P-a.s..\end{array}$$

Analogously, we can obtain

$$\begin{array}{c}\hfill {\rho }_g^{n2}+\sum _{m\in M}{\displaystyle \frac{\partial c_{gd}^{m2}(\omega ,q^{\ast }\left(\omega \right))}{\partial q_{gd}^n}}-{\lambda }_{\mathrm{d}}^{\ast 1n}\left(\omega \right)+{\lambda }_{\mathrm{d}}^{\ast 2n}\left(\omega \right)-{\nu }_{\mathrm{g}}^{\ast 1n}\left(\omega \right)=0,P-a.s.,\end{array}$$

and

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial {\varphi }_d^n(\omega ,z_d^{\ast n}\left(\omega \right))}{\partial z_d^n}}-{\displaystyle \frac{\partial {\chi }_d^n(\omega ,z_d^{\ast n}\left(\omega \right))}{\partial z_d^n}}-{\pi }_{\mathrm{d}}^{\ast 1n}\left(\omega \right)+{\pi }_{\mathrm{d}}^{\ast 2n}\left(\omega \right)-{\lambda }_{\mathrm{d}}^{\ast 1n}\left(\omega \right)=0,P-a.s..\end{array}$$

□

We transform the variational inequality (20) into a Lagrange multiplier regulated system:

$$\begin{array}{cc}\hfill & min\sum _{g\in G}(\sum _{n\in N}{\rho }_g^{n1}{x}_{gw}^n+\sum _{m\in M}{c}_{gw}^{m1}\left(x_{gw}\right))+\sum _{w\in W}{\tau }_w\sum _{g\in G}\sum _{m\in N}{x}_{gw}^n+\sum _{w\in W}\sum _{m\in M}{c}_{wd}^{m1}\left(x_{wd}\right)\hfill \\ \hfill & +{\int }_{\Omega }{\Psi }_d\left(x,\xi \left(\omega \right)\right)dP\left(\omega \right)\hfill \\ \hfill & \sum _{g\in G}\sum _{n\in N}{\rho }_g^{n1}{x}_{gw}^n+\sum _{g\in G}\sum _{m\in M}{c}_{gw}^{m1}\left(x_{gw}\right)+\sum _{w\in W}{\tau }_w\sum _{g\in G}\sum _{n\in N}{x}_{gw}^n\le B_d,\forall d\in D,\hfill \\ \hfill & \sum _{w\in W}{x}_{wd}^n\ge {\eta }_d^n,\sum _{w\in W}{x}_{wd}^n\le {\zeta }_n,x_{gw}^n\ge x_{wd}^n,\forall n\in N,\hfill \\ \hfill & x_{gw}^n\ge 0,\forall n\in N,\forall g\in \mathrm{G},\hfill \\ \hfill & x_{wd}^n\ge 0,\forall n\in N,\forall w\in W,\hfill \\ \hfill & \sum _{m\in M}{\displaystyle \frac{\partial c_{Wg}^{m2}(\omega ,y^{\ast }\left(\omega \right))}{\partial y_{wg}^n}}-{\lambda }_{\mathrm{d}}^{\ast 1n}\left(\omega \right)+{\lambda }_{\mathrm{d}}^{\ast 2n}\left(\omega \right)-{\nu }_{\mathrm{d}}^{\ast 1n}\left(\omega \right),P-a.s.,\hfill \\ \hfill & {\rho }_g^{n2}+\sum _{m\in M}{\displaystyle \frac{\partial c_{gd}^{mz}(\omega ,q^{\ast }\left(\omega \right))}{\partial q_{gd}^n}}-{\lambda }_{\mathrm{d}}^{\ast 1n}\left(\omega \right)+{\lambda }_{\mathrm{d}}^{\ast 2n}\left(\omega \right)-{\nu }_{\mathrm{g}}^{\ast 1n}\left(\omega \right),P-a.s.,\hfill \\ \hfill & {\displaystyle \frac{\partial {\varphi }_d^n(\omega ,z_d^{\ast n}\left(\omega \right))}{\partial z_d^n}}-{\displaystyle \frac{\partial {\chi }_d^n(\omega ,z_d^{\ast n}\left(\omega \right))}{\partial z_d^n}}-{\pi }_{\mathrm{d}}^{\ast 1n}\left(\omega \right)+{\pi }_{\mathrm{d}}^{\ast 2n}\left(\omega \right)-{\lambda }_{\mathrm{d}}^{\ast 1n}\left(\omega \right),P-a.s.,\hfill \\ \hfill & {\lambda }_{\mathrm{d}}^{\ast 1n}\left(\omega \right)(-\sum _{w\in W}{y}_{wd}^{\ast n}\left(\omega \right)-\sum _{g\in G}{q}_{gd}^{\ast n}\left(\omega \right)-z_d^{\ast n}\left(\omega \right)+{\eta }_d^{\ast n}\left(\omega \right)),P-a.s.,\hfill \\ \hfill & {\lambda }_{\mathrm{d}}^{\ast 2n}\left(\omega \right)(\sum _{w\in W}{y}_{wd}^{\ast n}\left(\omega \right)+\sum _{w\in W}{x}_{wd}^n+\sum _{g\in G}{q}_{gd}^{\ast n}\left(\omega \right)-{\zeta }_n\left(\omega \right)),P-a.s.,\hfill \\ \hfill & v_{\mathrm{d}}^{\ast 1n}\left(\omega \right)(-y_{wd}^{\ast n}\left(\omega \right)),{\pi }_{\mathrm{d}}^{\ast 1n}\left(\omega \right)(-z_d^{\ast n}\left(\omega \right)),P-a.s.\hfill \\ \hfill & {\pi }_{\mathrm{d}}^{\ast 2n}\left(\omega \right)(z_d^{\ast n}\left(\omega \right)-\alpha {\eta }_d^n\left(\omega \right)),v_{\mathrm{g}}^{\ast 1n}\left(\omega \right)(-q_{gd}^{\ast n}\left(\omega \right)),P-a.s.\hfill \end{array}$$

This analysis examines the key control variables governing relief operations: ${\lambda }_d^{\ast 1n}\left(\omega \right)$, ${\lambda }_d^{\ast 2n}\left(\omega \right)$, ${\nu }_d^{\ast 1n}\left(\omega \right)$, ${\nu }_g^{\ast 1n}\left(\omega \right)$, ${\pi }_d^{\ast 1n}\left(\omega \right)$, ${\pi }_d^{\ast 2n}\left(\omega \right)$ regulates procurement quantities, while ${\lambda }_d^{\ast 1n}\left(\omega \right)$ and ${\lambda }_d^{\ast 2n}\left(\omega \right)$ control demand and supply availability across stages. Variables ${\nu }_d^{\ast 1n}\left(\omega \right)$ and ${\nu }_g^{\ast 1n}\left(\omega \right)$ manage distribution flows from warehouses and direct purchases to disaster areas, respectively, with ${\pi }_d^{\ast 1n}\left(\omega \right)$ and ${\pi }_d^{\ast 2n}\left(\omega \right)$ tracking unmet demand. We have

$$\begin{array}{c}\hfill \sum _{m\in M}{\displaystyle \frac{\partial c_{wg}^{m2}(\omega ,y^{\ast }\left(\omega \right))}{\partial y_{wa}^n}}-{\lambda }_{\mathrm{d}}^{\ast 1n}\left(\omega \right)+{\lambda }_{\mathrm{d}}^{\ast 2n}\left(\omega \right)-{\nu }_{\mathrm{d}}^{\ast 1n}\left(\omega \right)=0,P-a.s.\end{array}$$

If $y_{wg}^n>0$, then ${\nu }_d^{\ast 1n}\left(\omega \right)=0,P-a.s.$, and ${\sum }_{m\in M}{\displaystyle \frac{\partial c_{wg}^{m2}\left(\omega ,y^{\ast }\left(\omega \right)\right)}{\partial y_{wg}^n}}={\lambda }_d^{\ast 1n}\left(\omega \right)-{\lambda }_d^{\ast 2n}\left(\omega \right),P-a.s.$. The marginal cost of transporting relief items from the warehouse to the disaster area equals the algebraic difference between the fluctuation coefficient of disaster demand and the elasticity coefficient of material supply. Moreover, if ${\lambda }_d^{\ast 1n}\left(\omega \right)=0$, ${\lambda }_d^{\ast 2n}\left(\omega \right)>0,P-a.s.$, we get ${\sum }_{m\in M}{\displaystyle \frac{\partial c_{wg}^{m2}\left(\omega ,y^{\ast }\left(\omega \right)\right)}{\partial y_{wg}^n}}=-{\lambda }_d^{\ast 2n}\left(\omega \right),P-a.s.$, and the marginal transportation cost exhibits a decreasing trend. If ${\lambda }_d^{\ast 1n}\left(\omega \right)>0,{\lambda }_d^{\ast 2n}\left(\omega \right)=0,P-a.s.$, we get ${\sum }_{m\in M}{\displaystyle \frac{\partial c_{wg}^{m2}\left(\omega ,y^{\ast }\left(\omega \right)\right)}{\partial y_{wg}^n}}={\lambda }_d^{\ast 1n}\left(\omega \right)P-a.s.$ and the system’s marginal cost exhibits an upward trajectory.

We have ${\rho }_g^{n2}+{\sum }_{m\in M}{\displaystyle \frac{\partial c_{gd}^{m2}(\omega ,q^{\ast }\left(\omega \right))}{\partial q_{gd}^n}}-{\lambda }_d^{\ast 1n}\left(\omega \right)+{\lambda }_d^{\ast 2n}\left(\omega \right)-{\nu }_g^{\ast 1n}\left(\omega \right),P-a.s.$. If $q_{gd}^{\ast n}\left(\omega \right)>0$, then ${\nu }_d^{\ast 1n}\left(\omega \right)=0,P-a.s.$, and ${\rho }_g^{n2}+{\sum }_{m\in M}{\displaystyle \frac{\partial c_{gd}^{m2}(\omega ,q^{\ast }\left(\omega \right))}{\partial q_{gd}^n}}={\lambda }_d^{\ast 1n}\left(\omega \right)-{\lambda }_d^{\ast 2n}\left(\omega \right),P-a.s.$ namely, the marginal transportation cost for relief commodities between procurement centers and disaster zones equilibrates with the differential between demand regulation parameters and supply availability indicators in the marketplace. Moreover, if ${\lambda }_d^{\ast 1n}\left(\omega \right)=0$, ${\lambda }_d^{\ast 2n}\left(\omega \right)>0,P-a.s.$, we get ${\rho }_g^{n2}+{\sum }_{m\in M}{\displaystyle \frac{\partial c_{gd}^{m2}(\omega ,q^{\ast }\left(\omega \right))}{\partial q_{gd}^n}}=-{\lambda }_d^{\ast 2n}\left(\omega \right),P-a.s.$, and the marginal cost decreases. If ${\lambda }_d^{\ast 1n}\left(\omega \right)>0$, ${\lambda }_d^{\ast 2n}\left(\omega \right)=0,P-a.s.$, we get ${\rho }_g^{n2}+{\sum }_{m\in M}{\displaystyle \frac{\partial c_{gd}^{m2}(\omega ,q^{\ast }\left(\omega \right))}{\partial q_{gd}^n}}={\lambda }_d^{\ast 1n}\left(\omega \right),P-a.s.$ and the marginal cost increases.

We have ${\displaystyle \frac{\partial {\varphi }_d^n(\omega ,z_d^{\ast n}\left(\omega \right))}{\partial z_d^n}}-{\displaystyle \frac{\partial {\chi }_d^n(\omega ,z_d^{\ast n}\left(\omega \right))}{\partial z_d^n}}-{\pi }_d^{\ast 1n}\left(\omega \right)+{\pi }_d^{\ast 2n}\left(\omega \right)-{\lambda }_d^{\ast 1n}\left(\omega \right),P-a.s.$, If ${\lambda }_d^{\ast 1n}\left(\omega \right)=0,P-a.s.$, and ${\displaystyle \frac{\partial {\varphi }_d^n(\omega ,z_d^{\ast n}\left(\omega \right))}{\partial z_d^n}}-{\displaystyle \frac{\partial {\chi }_d^n(\omega ,z_d^{\ast n}\left(\omega \right))}{\partial z_d^n}}={\pi }_d^{\ast 1n}\left(\omega \right)-{\pi }_d^{\ast 2n}\left(\omega \right),P-a.s.$, we find that the marginal penalty is determined by the dynamic difference of the supply and demand regulation parameters. If ${0<z}_d^{\ast n}\left(\omega \right)<\alpha {\eta }_d^n\left(\omega \right),P-a.s.$, then ${\pi }_d^{\ast 1n}\left(\omega \right)={\pi }_d^{\ast 2n}\left(\omega \right)=0$, and ${\displaystyle \frac{\partial {\varphi }_d^n(\omega ,z_d^{\ast n}\left(\omega \right))}{\partial z_d^n}}-{\displaystyle \frac{\partial {\chi }_d^n(\omega ,z_d^{\ast n}\left(\omega \right))}{\partial z_d^n}}=0,P-a.s.$, namely, the equilibrium state falls within the cost-minimizing tolerance range, resulting in null marginal penalties. If ${\pi }_d^{\ast 1n}\left(\omega \right)>0$, then $z_d^{\ast n}\left(\omega \right)=0,P-a.s.$ This corresponds to the ideal scenario of perfect demand fulfillment. If ${\pi }_d^{\ast 2n}\left(\omega \right)>0$, then ${\pi }_d^{\ast 1n}\left(\omega \right)=0,P-a.s.$, and ${\displaystyle \frac{\partial {\varphi }_d^n(\omega ,z_d^{\ast n}\left(\omega \right))}{\partial z_d^n}}-{\displaystyle \frac{\partial {\chi }_d^n(\omega ,z_d^{\ast n}\left(\omega \right))}{\partial z_d^n}}=-{\pi }_d^{\ast 2n}\left(\omega \right),P-a.s.$, namely, a diminishing-returns-to-penalty property emerges in the system.

When $f_d^c\left(\omega \right)=min{f}_d^c\left(\omega \right)$, the optimality conditions require ${\displaystyle \frac{\partial {\varphi }_d^n}{\partial z_d^n}}-{\displaystyle \frac{\partial {\chi }_d^n}{\partial z_d^n}}={\pi }_d^{\ast 1n}\left(\omega \right)-{\pi }_d^{\ast 2n}\left(\omega \right)$, at which point the system is in an equilibrium state where marginal penalty and supply cost are balanced. The system allows for some unmet requirements. When $f_d^c\left(\omega \right)>min{f}_d^c\left(\omega \right)$, the system needs to weigh the local and global costs to force the rescue team to fully meet the rescue requirements.

Remark 3.

The monotonicity condition $J_{(x,\xi (\omega \left)\right)}\ge 0$ remains preserved under coefficient scaling because $\mathrm{If}{\rho }_g^{n1}\le {\rho }_g^{n2},\mathrm{then}{\lambda }_d^{\ast 1n}\left(\omega \right)-{\lambda }_d^{\ast 2n}\left(\omega \right)\ge 0(byKKTconditions)$, ensuring the positive semi-definiteness of $\mathfrak{N}\left(\xi \left(\omega \right),\omega \right)$ matrix.

5. Stochastic Equilibrium Numerical Example

In this section, we provide a numerical example for demonstrative purposes. We consider purchase locations (g = 2), two relief centers (w = 2), two relief points (d = 2), two different relief supplies (n = 2), one mode of transportation (m = 1), and four scenarios. Penalty costs, which involve unmet demand, depend on the number of supplies not delivered in a day. Computational experiments were conducted via the progressive hedging algorithm (PHM) [30,31], a scenario-based decomposition technique for stochastic optimization. A seminal algorithm has adapted for multistage stochastic variational inequalities (SVI) and their stochastic Lagrange formulations. PHM decomposes complex problems into independent subproblems across different scenarios, coordinating decisions at each stage step by step, making it particularly suitable for multi-stage dynamic decision-making in crisis rescue operations. Uncertainties such as fluctuating demand and supply disruptions in crisis rescue are modeled using scenario trees. PHM allows for independent optimization under different scenarios, then coordinates a global solution through multipliers. These multipliers can be interpreted as “shadow prices”, reflecting the marginal cost of resource shortages. This provides decision-makers with real-time guidance for adjustments (e.g., prioritizing resource allocation to high-shortage areas), which is closely aligned with the Lagrangian regulation system proposed in this paper. The cost functions is given by

$$\begin{array}{rrrrrrrrrrrr}10& 12& 0.0026& 0.017& 0.1& 0.0009& 0.007& 12& 14& 0.01& 0.005& \mathrm{18,000}\\ \hfill 9& 7& 0.0003& 0.0005& 0.001& 0.0006& 0.0007& 10& 15& 8& 0.6& \mathrm{10,000}\end{array}$$

Initially, we concentrates on the flows $x_{gw}^n$ of the primary phase. The analysis reveals

$$\begin{array}{cc}\hfill & x_{11}^1=2091.7;x_{12}^1=0;x_{21}^1=286.62;x_{22}^1=6959.9;\hfill \\ \hfill & x_{11}^2=1507.7;x_{12}^2=16.233;x_{21}^2=0;x_{22}^2=6720;\hfill \end{array}$$

From observing the flows $x_{wd}^n$ in the second stage, we find

$$\begin{array}{cc}\hfill & x_{11}^1=623.53;x_{12}^1=565.53;x_{21}^1=3423;x_{22}^1=3380.1;\hfill \\ \hfill & x_{11}^2=399.96;x_{12}^2=384.62;x_{21}^2=3373.9;x_{22}^2=3353.7;\hfill \end{array}$$

During the subsequent phase, under crisis scenarios, demand fulfillment becomes non-negotiable through resource allocation optimization. Where the needs of the rescue site cannot be met, the rescue center will do its best to meet the needs to minimize penalties. We find that a high penalty coefficient forces rescue organizations to meet the needs of the disaster-stricken areas as much as possible.

Table 1. Numerical results of the two-stage stochastic model with a high penalty coefficient.

Supplies	Flows	${\mathit{\omega }}_1$	${\mathit{\omega }}_2$	${\mathit{\omega }}_3$	${\mathit{\omega }}_4$
$n=1$	$y_{11}^1\left(\omega \right)$	724.96	129.72	258.34	1576.3
	$y_{12}^1\left(\omega \right)$	741.49	64.034	285.84	1335.8
	$y_{21}^1\left(\omega \right)$	0	0	0	0.76996
	$y_{22}^1\left(\omega \right)$	0	0	0	783.5
n = 2	$y_{11}^2\left(\omega \right)$	0	1012.5	58.339	208.91
	$y_{12}^2\left(\omega \right)$	0	1037.3	85.834	187.97
	$y_{21}^2\left(\omega \right)$	0	0	0	0
	$y_{22}^2\left(\omega \right)$	0	0	0	42.806
n = 1	$q_{11}^1\left(\omega \right)$	0	0	0	0
	$q_{12}^1\left(\omega \right)$	0	0	0	8.5888
	$q_{21}^1\left(\omega \right)$	0	0	0	0
	$q_{22}^1\left(\omega \right)$	0	0	0	0
n = 2	$q_{11}^2\left(\omega \right)$	0	0	0	10.629
	$q_{12}^2\left(\omega \right)$	0	0	0	169.59
	$q_{21}^2\left(\omega \right)$	0	0	0	0
	$q_{22}^2\left(\omega \right)$	0	0	0	0
n = 1	$z_1^1\left(\omega \right)$	0	0	0	0
	$z_2^1\left(\omega \right)$	0	0	60,934	0
n = 2	$z_1^2\left(\omega \right)$	0	0	0	0
	$z_2^2\left(\omega \right)$	0	0	0	60,866

enumerates the computational findings.

Next, we consider a low penalty function or sufficient social donations (failure to meet the demand for a certain type of relief supplies would not lead to severe consequences). The cost function is as follows:

$$\begin{array}{rrrrrrrrrrrr}10& 12& 0.0026& 0.017& 0.1& 0.0009& 0.007& 12& 14& 0.01& 0.005& 900\\ 9& 7& 0.0003& 0.0005& 0.001& 0.0006& 0.0007& 10& 15& 8& 0.6& 600\end{array}$$

We obtain the flows $x_{gw}^n$ for the first stage as follows:

$$\begin{array}{cc}\hfill & x_{11}^1=1640;x_{12}^1=0;x_{21}^1=0;x_{22}^1=7700;\hfill \\ \hfill & x_{11}^2=1560;x_{12}^2=0;x_{21}^2=0;x_{22}^2=6720;\hfill \end{array}$$

We get the flows $x_{wd}^n$ for the second stage:

$$\begin{array}{cc}\hfill & x_{11}^1=668.43;x_{12}^1=705.14;x_{21}^1=3899.8;x_{22}^1=3800.2;\hfill \\ \hfill & x_{11}^2=437.8;x_{12}^2=427.56;x_{21}^2=3362.7;x_{22}^2=3346.4;\hfill \end{array}$$

We find that when the penalty coefficient is set too low or the social donation of materials is sufficient, rescue organizations do not tend to purchase additional rescue materials in the market to meet the needs of the disaster-stricken areas. We know that a low penalty function or adequate donations can reduce the emergency procurement motivation of rescue organizations. The results are shown in

Table 2. Numerical results of the two-stage stochastic model with a low penalty coefficient.

Supplies	Flows	${\mathit{\omega }}_1$	${\mathit{\omega }}_2$	${\mathit{\omega }}_3$	${\mathit{\omega }}_4$
$n=1$	$y_{11}^1\left(\omega \right)$	0	0	0	508.72
	$y_{12}^1\left(\omega \right)$	0	67.18	186.33	536.32
	$y_{21}^1\left(\omega \right)$	0	0	0	0
	$y_{22}^1\left(\omega \right)$	0	0	0	0
$n=2$	$y_{11}^2\left(\omega \right)$	0	964.12	0	239.15
	$y_{12}^2\left(\omega \right)$	0	1020.8	0	256.67
	$y_{21}^2\left(\omega \right)$	0	0	0	0
	$y_{22}^2\left(\omega \right)$	0	0	0	54.342
$n=1$	$q_{11}^1\left(\omega \right)$	0	0	0	0
	$q_{12}^1\left(\omega \right)$	0	0	0	0
	$q_{21}^1\left(\omega \right)$	0	0	0	0
	$q_{22}^1\left(\omega \right)$	0	0	0	0
$n=2$	$q_{11}^2\left(\omega \right)$	0	0	0	0
	$q_{12}^2\left(\omega \right)$	0	0	0	0
	$q_{21}^2\left(\omega \right)$	0	0	0	0
	$q_{22}^2\left(\omega \right)$	0	0	0	0
$n=1$	$z_1^1\left(\omega \right)$	0	0	0	0
	$z_2^1\left(\omega \right)$	0	0	0	0
$n=2$	$z_1^2\left(\omega \right)$	0	0	0	60,912
	$z_2^2\left(\omega \right)$	0	0	0	60,888

.

For the above results, a high penalty coefficient (${\varphi }_d^n\left(\omega \right)$) will force rescue organizations to prioritize the supply of urgently needed relief materials, while a low penalty coefficient will allow rescue organizations to pursue the minimization of rescue costs; that is, to tolerate strategic shortages within a certain range. Therefore, ${\varphi }_d^n\left(\omega \right)$ can be used as an effective tool for top decision-makers to regulate the efficiency of the rescue area. Sufficient social donations (${\chi }_d^n\left(\omega \right)$) will reduce emergency procurement behavior in the second stage, but excessive social donations will lead to the accumulation of relief materials. The top decision-makers can classify the penalty coefficients based on the severity of the disaster-stricken areas to accelerate the rescue response. The Lagrange multiplier (${\lambda }_d^{\ast 1n}\left(\omega \right)$) can be used as a real-time shortage indicator. When the rescue organization detects B, it should immediately adjust the distribution strategy of relief supplies. The top decision-makers can monitor the priority of relief supplies in real time through Lagrange multipliers, and social organizations can adjust their donation strategies based on “shadow prices” to reduce the waste of rescue resources caused by game conflicts.

6. Conclusions

This study constructs a two-stage stochastic Nash equilibrium framework to coordinate joint disaster relief operations among multiple rescue organizations under uncertainty. The model fundamentally advances crisis relief optimization by integrating three critical dimensions that are frequently overlooked in existing research: (1) competitive-cooperative dynamics among budget-constrained organizations; (2) multi-stage decision-making under incomplete information; and (3) an endogenous penalty mechanism reflecting operational costs and social responsibilities. First, rescue agencies achieve cost minimization under financial constraints, reflecting real-world trade-offs between prepositioned reserves and emergency procurement. Second, cost functions are modeled as differentiable convex functions that align with empirical logistics data patterns through gradient monotonicity, supporting stochastic variational inequality modeling. Third, unmet demand compliance ($z_d^n\left(\omega \right)\le \alpha {\eta }_d^n\left(\omega \right)$) is enforced to eliminate the distortion of “zero-rescue” equilibria. Our model resolves service conflicts among multiple rescue organizations in overlapping disaster zones through non-cooperative game theory, preventing severe resource waste. A social donation function ${\chi }_d^n\left(\omega ,z_d^n\left(\omega \right)\right)$ captures empathy-driven fund flows based on shortage visibility, enabling government agencies to better support affected areas through subsidies and penalty mechanisms. Additionally, interactions between first-stage inventory ($x_{gw}$) and second-stage emergency procurement ($q_{gd}^n\left(\omega \right)$) via Lagrangian multipliers facilitate adaptive fiscal strategies. Theoretically, we establish a solution that addresses the existence and uniqueness for the stochastic variational inequality formulation under mild regularity conditions (convexity and continuous differentiability of cost functions), significantly expanding game-theoretic models’ applicability in resource allocation. Through infinite-dimensional Lagrangian duality theory, equilibrium states are characterized as a Lagrange multiplier-regulated system. Consequently, resource prioritization can be adjusted via shadow prices (Lagrangian multipliers) to optimize demand fulfillment rates across relief points.

The results demonstrate that relief centers can dynamically reallocate supplies to meet evolving needs. Uncertainty fundamentally impacts crisis relief success, necessitating constant preparedness for resource redistribution. In future research, we will extend the two-stage framework to multi-stage rolling decisions by introducing Markov decision processes to capture temporal dependencies in disaster evolution, requiring solutions for policy update mechanisms in high-dimensional state spaces. Our model does not address nonconvexities but provides a tractable equilibrium solution under convexity. So we plan to incorporate loss-aversion characteristics (e.g., effects of irrational behavior on allocation fairness) and non-convex functions. This will involve employing hemivariational inequalities and stochastic evolutionary processes to model irrational behaviors in crisis relief scenarios.