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A Generalized Nash Equilibrium-Seeking Scheme for Trauma Resuscitation
Promise Ekpo, Angelique Taylor, Lekan Molu
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 92%
Last extracted: 7/8/2026, 6:58:46 PM
Summary
This paper formulates trauma resuscitation as a distributed generalized Nash equilibrium (GNE) seeking game to optimize healthcare workers' decisions in high-stakes clinical environments. By modeling team dynamics over a time-varying communication graph, the framework algorithmically balances skill proficiency, alacrity, fairness, and communication efficiency. The proposed distributed optimization method aims to improve patient outcomes by coordinating workload distribution, resource allocation, and inter-team communication under coupled inequality constraints.
Entities (11)
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Promise Ekpo → authored → A Generalized Nash Equilibrium-Seeking Scheme for Trauma Resuscitation
confidence 99% · Promise Ekpo Angelique Taylor Lekan Molu Cornell University, NY.
Lekan Molu → authored → A Generalized Nash Equilibrium-Seeking Scheme for Trauma Resuscitation
confidence 99% · Promise Ekpo Angelique Taylor Lekan Molu Cornell University, NY. Bala Cynwyd, PA. lekanmolu@scriptedonachip.com
Angelique Taylor → authored → A Generalized Nash Equilibrium-Seeking Scheme for Trauma Resuscitation
confidence 99% · Promise Ekpo Angelique Taylor Lekan Molu Cornell University, NY.
Trauma Resuscitation → modeledas → Generalized Nash Equilibrium
confidence 95% · This motivates our socio-technical formulation of trauma resuscitation as a distributed generalized Nash equilibrium (GNE)-seeking game with coupled inequality constraints.
Healthcare Workers → participatein → Generalized Nash Equilibrium
confidence 93% · Each healthcare worker is a player i∈𝒱 with objective Ji... in a game Γ(𝒱,Ω,{Ji},K)
Generalized Nash Equilibrium → operatesover → Time-varying Communication Graph
confidence 92% · This method is optimized over a time-varying communication graph.
Advanced Life Support Protocol → guides → Trauma Resuscitation
confidence 90% · We validate the distributed GNEP algorithm on a trauma resuscitation scenario with 7 healthcare workers executing the Advanced Life Support (ALS) cardiac arrest protocol
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Abstract
Abstract:Trauma resuscitation is a clinical process for treating life-threatening physiological disorders in safety-critical environments, driven by the experience of healthcare workers (HCWs). Designing and optimizing quantifiable metrics that accurately capture HCW decisions may augment current resuscitation procedures with the potential to improve patient outcomes. This motivates our socio-technical formulation of trauma resuscitation as a distributed generalized Nash equilibrium (GNE)-seeking game with coupled inequality constraints. This method is optimized over a time-varying communication graph. We introduce novel insights from clinical experience to model HCWs behavior. This work facilitates the best possible resuscitation outcome given HCWs workloads, schedules, competencies, and limited resources.
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- Source: https://arxiv.org/abs/2605.22661v2
- Canonical: https://arxiv.org/abs/2605.22661v2
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A Generalized Nash Equilibrium-Seeking Scheme for Trauma Resuscitation Promise Ekpo Angelique Taylor Lekan Molu Cornell University, NY. poe6, amt298@cornell.edu. Bala Cynwyd, PA. lekanmolu@scriptedonachip.com Abstract Trauma resuscitation is a clinical process for treating life-threatening physiological disorders in safety-critical environments, driven by the experience of healthcare workers (HCWs). Designing and optimizing quantifiable metrics that accurately capture HCW decisions may augment current resuscitation procedures with the potential to improve patient outcomes. This motivates our socio-technical formulation of trauma resuscitation as a distributed generalized Nash equilibrium (GNE)-seeking game with coupled inequality constraints. This method is optimized over a time-varying communication graph. We introduce novel insights from clinical experience to model HCWs behavior. This work facilitates the best possible resuscitation outcome given HCWs’ workloads, schedules, competencies, and limited resources. keywords: Cyber-physical and human systems (CPHS); Social computing; Game theory. 1 Introduction Trauma resuscitation (TR) is a collaborative, often conflicting, coordination effort among a team of healthcare workers (HCWs). It is often characterized by significant differential in (i) skillset; (i) mental and physical alacrity; and (i) intra-team communication skills. These factors are shaped by HCWs’ workplace experience and can degrade over time. The degradation may spur burnout and fatigue — undermining productivity; miscommunication informs duplicated execution of tasks — engendering redundancy. Across medical organizational hierarchies, through verbal and aural cues (Sarcevic et al., 2012; Ogundare et al., 2025) HCWs can improve decision quality if a systematic and quantifiable algorithmic framework is integrated into workflows to track performance. Previous efforts have integrated autonomous systems to support team dynamics (Taylor et al., 2024; Tanjim et al., 2025), deliver supplies (Salinas-Avila et al., 2022), or sanitize surgical rooms (Sun and Yeung, 2007). Figure 1: Communication graph at k=0k=0. Nodes are n=7n=7 healthcare workers; edges link pairs within radius r=200r=200 ft. Player positions yield a time-varying neighbor structure. A well-executed TR procedure involves a deliberative process where skill-informed task delegation by clinic leaders and responsibilities hand-off among peers are well-executed in spite of time constraints. We set forth to design socio-technical process that (i) supports clinical decision-making through automated task allocation among HCWs; (i) maximizes HCW skill levels while counterbalancing other HCWs’ skill levels; and (i) enhances communication efficacy whilst reducing duplicate information exchange. This work studies quantifiable algorithms in high-stakes socio-technical medical settings to enhance patient outcomes. We take the view of a game-theoretic, collaborative TR setting, informed by our observations in a representative medical simulation training environment for HCWs at Weill Cornell Emergency Medicine in New York City111The BASE Camp event for interprofessional pediatric emergency medicine team training in the emergency care of critically ill and injured children was employed (Weill Cornell Medicine, 2025).. We provide a mathematical characterization between observed skill levels and task models in the algorithmic framework we present. The role of shared mental models and decentralized communication workflows has been described in high-duress trauma resuscitation contexts (Sarcevic et al., 2012; Batra et al., 2026). Co-design studies have proposed cognitive aids and augmented-reality systems to improve information flow and situational awareness for distributed clinical teams (Taylor et al., 2024). MATEC (Cho et al., 2025) and TraumaFlow (Neumann et al., 2024) demonstrate that software agents can align team actions with clinical protocols with centralized controllers. Distributed optimization in games over networks may be characterized with generalized Nash equilibrium (GNE)-seeking algorithms that better enhance multi-agent socio-technical systems with coupling constraints (Dave et al., 2022). Models incorporating agent dynamics, constraints, and communication topology align well with real-world healthcare teams. In these environments, agents exhibit rationality under partial observability. The rest of this paper is structured as follows: §2 describes the notations and games background. Our hypothesis is presented in §3 and it is evaluated on numerical test environments in §4. We conclude the paper in §5. 2 Background and Preliminaries 2.1 General Notations The set of real (non-negative) numbers is denoted, ℝ(ℝ+)R\,(R_+), the m-dimensional vector space is ℝmR^m, and the n×n× n dimensional real matrix is denoted ℝn×nR^n× n. A time-varying variable, x is denoted x(t)x(t). For a function f that depends on x(t)x(t), we write f(x;t)f(x;t). The absolute value of scalar x is |x||x|. The Euclidean norm of the vector x∈ℝnx ^n is ‖x‖:=x⊤x\|x\|:= x x. The n-dimensional vector of ones (zeroes) is 1n(0n)1_n(0_n). For a differentiable function J(x)J(x), the first-order derivative of J(x)J(x) is denoted ∇xJ(x) _xJ(x). For a set Ω and variable x, the Euclidean projection of x onto Ω is Ω(x)=argminx∈Ω‖x−x′‖2P_ (x)=argmin_x∈ \|x-x \|_2. The set ⊂ℝmS ^m is a convex set if for α>0α>0 and for every x,y∈x,y , αx+(1−α)y∈α x+(1-α)y . The cardinality of a S is denoted ⌊⌋ . Suppose that S is a closed convex set, and (x)P_S(x) is a projection of x onto the set S, then there is a unique element (x)∈P_S(x) such that ‖x−(x)‖=infy∈‖x−y‖\|x-P_S(x)\|= _y \|x-y\|. Similarly, the Euclidean projection of x onto the set Ω is denoted PΩ(x)∥=argminy∈Ω∥x−y∥P_ (x)\|=argmin_y∈ \|x-y\|, 2.2 Games Taxonomy Let a game be denoted as Γ(,Ω,J) (V, ,J), where V are the players 1,…,n\1,…,n\, each with action xix_i belonging in a constraint set Ωi∈ℝm _i ^m and a local cost profile Ji:Ωi→ℝJ_i: _i so that J=J1,⋯,JnJ=\J_1,·s,J_n\. The action profile of the game is Ω:=Ω1×⋯×Ωn := _1×·s× _n. The vector formed by all players’ actions is x:=(xi,x−i)≜xii=1nx:=(x_i,x_-i) \x_i\_i=1^n, where the vector formed by all the players’ actions except those of player i is x−i=xi′i′=1,i′≠in∈ℝn−ix_-i=\x_i \_i =1,\,i ≠ i^n ^n_-i where n−i:=n−nin_-i:=n-n_i. Define Kip:=xi∈Ωi|hi(xi)≤0K_i^p:=\x_i∈ _i\,|\,h_i(x_i)≤ 0\ as the set of each individual player’s constraint set, and Ks:=x∈Ω|g(x)≥0mK^s:=\x∈ \,|\,g(x)≥ 0_m\ as the set of shared constraints, where g(x)=c⊤x−d,for c=[c1⊤,⋯,cn⊤]⊤,ci∈ℝmg(x)=c x-d,\,for c=[c_1 ,·s,c_n ] ,\ c_i ^m and d∈ℝd . It follows that player i’s action is constrained along two directions: constraints that depend on the action of other players i.e.g(x)≥0\ g(x)≥ 0, and individual constraints that depend on player i’s action i.e.i.e. xi∈Kix_i∈ K_i. Contrary to popular formulations in GNE-seeking literature, we have chosen g(x)≥0g(x)≥ 0 since the skills, time indices, and other parameters we are optimizing for cannot be negative. 2.3 Games Machinery Let us set a few definitions and preliminary results in motion. Denote the feasible set of player i’ action by Ki(x−i):=xi|(xi,x−i)∈KK_i(x_-i):=\x_i\ |\ (x_i,x_-i)∈ K\, where K:=Ks∩(K1×⋯Kn)K:=K^s∩(K_1×·s K_n). Suppose we define the generalized Nash equilibrium problem (GNEP) as Γ(,Ω,J,K) (V, ,J,K), then the individual players in the game “freeze” other players’ actions x−ix_-i as exogenous variables, to solve minxi∈Ωi _x_i∈ _i Ji(xi,x−i),∀i∈ J_i(x_i,x_-i), ∀\,\,i subject to xi∈Ki(x−i). x_i∈ K_i(x_-i). (GNEP) Definition 1 (Neighbors of a Player) We define the neighbors i(t)N_i(t) of player i at time t as the set of all agents that lie within a predefined radius, rir_i. i(t)=j∣(j,i)∈ℰN_i(t)=\j (j,i) \. Definition 2 (Generalized Nash Equilibrium) In problem (GNEP) , the action profile x⋆:=(xi⋆,x−i⋆)x :=(x_i ,x_-i ) is a GNE if Ji(xi⋆,x−i⋆)≤Ji(xi,x−i⋆),∀xi∈Ωi,∀i∈J_i(x_i ,x_-i )≤ J_i(x_i,x_-i ),\,∀\,x_i∈ _i,\,∀ i . Assumptions are made for feasibility of the solution (GNEP). Assumption 1 (Interconnectivity) The game occurs over a time-varying connected222That is, any two nodes in V are connected by a path. and undirected333That is, (i,j)∈ℰ⇔(j,i)∈ℰ(i,j) (j,i) . weighted communication graph ()(t):,ℰG(A)(t):\V,E\ where t>0t>0 with edges ℰ⊂×E ×V and adjacency matrix :=[aij]∈ℝn×nA:=[a_ij] ^n× n. Player i is connected to (can receive information from) player j if (i,j)∈ℰ(i,j) and vice versa. Let aij>0a_ij>0 if (i,j)∈ℰ(i,j) and aij=0a_ij=0 otherwise. For a D=diag(d11,⋯,dnn)∈ℝn×nD=diag(d_11,·s,d_n) ^n× n, where dii=∑j=1naijd_i= _j=1^na_ij for all i∈i , we have ()G(A)’s Laplacian as ℒ:=D−∈ℝn×nL:=D-A ^n× n Assumption 2 (Cost Function) For i∈i , Ωi∈ℝm _i ^m is a nonempty, convex, and closed set; and the cost Ji(xi,x−i)J_i(x_i,x_-i) is continuously differentiable on Ω and convex in xix_i for every fixed x−ix_-i. Define ∇J(x)=[∇x1J1⊤(x),⋯,∇xnJn⊤(x)]⊤∇ J(x)=[ _x_1J_1 (x),·s, _x_nJ_n (x)] as the single-value game mapping. Assumption 3 (Monotonicity and Game-mapping) The single-valued mapping ℳ:ℝm→ℝmM:R^m ^m is strongly monotone on the action constraint set Ω∈ℝm ^m if there exists a constant μ>0μ>0 for any x,x′∈Ω, where x≠x′x,x ∈ , where x≠ x such that (∇J−∇J′)⊤(x−x′)≥μ‖x−x′‖2(∇ J-∇ J ) (x-x )≥μ\|x-x \|^2 for any ∇J∈ℳ(x)∇ J (x), ∇J′∈ℳ(x′)∇ J (x ). Assumption 4 (Constraint Function) The inequality constraint function g(x)≥0g(x)≥ 0 is continuously differentiable and convex in x. Also, the feasible action set K is nonempty, convex, and closed — satisfying the Slater’s condition i.e.i.e. , x⋆∈int(Ω)x ( )444The relative interior of the set Ω is denoted int(Ω)int( ). such that g(x⋆)≤0mg(x )≤ 0_m. Lemma 1 (Existence of a GNEP) Let the GNEP of Def. 2 be given and suppose that (i) Ωi∈ℝm _i ^m be a nonempty, convex, and compact such that for i∈i , Ki(x−i)K_i(x_-i) is nonempty, convex, closed, and KiK_i is both upper and lower semi-continuous; and (i) the local cost Ji(xi,x−i)J_i(x_i,x_-i) is quasi-convex on Ki(x−i)K_i(x_-i) for every i∈i . Then, a GNE exists. pf (Existence of GNEP) Follows directly from Thm. 4.1 in (Facchinei and Kanzow, 2010). Remark 1 When assumptions 2, 3, and 4 are fulfilled, then the GNEP Γ(,Ω,J,K) (V, ,J,K) satisfies the existence condition x⋆x in Lemma 1. Lemma 2 (Optimality of the GNE) Suppose that there exists a multiplier λi⋆∈ℝm _i ^m, ∀i∈∀\,i . The optimality condition for each player can be found via the KKT conditions ∇xiJi(xi⋆,x−i⋆) _x_iJ_i(x_i ,x_-i ) +⟨λi⋆,∇xg(x⋆)⟩=0n,∀i∈,λi⋆∈, + _i , _xg(x ) =0_n,\,\,∀\,\,i , _i , 0m≤ 0_m≤ λi⋆⟂g(xi⋆,x−i⋆)≥0m \, _i \,\,g(x_i ,x_-i )≥ 0_m (1) under the assumption of continuous differentiability. This follows from Thm. 4.6 in Facchinei and Kanzow (2010). As a system of equations, (1) can be written as J J (x⋆,λ⋆)=0, (x ,λ )=0, 0≤ 0≤\,\, λ⋆⟂g(x⋆)≥ 0, λ g(x )\,\,≥\,\,0, (2) for g(x⋆)=[g⊤(x1⋆),⋯,g⊤(xn⋆)]⊤,λ=[λ1⋆⊤,⋯,λn⋆⊤]⊤, g(x )=[g (x_1 ),·s,g (x_n )] ,\,λ=[ _1 ,·s, _n ] , and J(x⋆,λ⋆)=[∇x1J1⊤(x1⋆),⋯,∇xnJn⊤(xn⋆)]⊤. J(x ,λ )=[ _x_1J_1 (x_1 ),·s, _x_nJ_n (x_n )] . With the constraint qualification, the x portion of (2) is a first-order necessary condition for the GNEP; and under appropriate convexity assumptions, the x part solves the GNEP so that (2) is sufficient condition for the GNEP. Remark 2 Lemma 2 is the so-called variational inequality problem VI(X,∇J(x))VI(X,∇ J(x)) whose solution set is a special class of GNEs with all Lagrange multipliers equal. That is, the solution x⋆x to the VI(X,∇J(x))VI(X,∇ J(x)) under the KKT conditions (1) satisfies the GNE x⋆x if λ1⋆=⋯=λn⋆=λ⋆ _1 =·s= _n =λ . 3 A GNEP Trauma Resuscitation Scheme In this section, we present our formalism of clinical TR workflows into an games and optimization problem. 3.1 Clinical model Table 1: Representative HCW skill attributes (ALS protocol). Taxonomy Meaning CPR Cardiopulmonary resuscitation SHOCK Deliver defibrillation shock RHYTHM_CHECK Evaluate rhythm every 2 min AIRWAY_MGMT Airway interventions EPI Epinephrine administration TEAM_COMM Team communication Each healthcare worker is a player i∈i with objective JiJ_i (e.g. , start CPR), individual constraints KipK_i^p (e.g. , hemorrhage control), and shared constraints KisK_i^s (e.g. , breathing or pulse checks) in a game Γ(,Ω,Ji,K) (V, ,\J_i\,K) where Ki=Kis∩KipK_i=K_i^s∩ K_i^p. Table 1 itemizes typical HCW skill attributes inspired by the ALS code card for adult cardiac arrest (American Red Cross, 2025). The inequality constraint hi(xi)≤0h_i(x_i)≤ 0 represents each player’s clinical role assignment. The state xix_i of each player is xi(t):=[si,ai(t),fi(t),υi(t)]⊤ x_i(t):=[\,s_i,\ a_i(t),\,f_i(t),\, _i(t)\,] (3) with skill proficiency index sis_i and time-dependent decision variables: alacrity index ai(t)a_i(t), fairness index fi(t)f_i(t), and communication efficiency υi(t) _i(t). Here sis_i is a fixed parameter characterizing player i’s competence, while [ai(t),fi(t),υi(t)][\,a_i(t),f_i(t), _i(t)\,] constitutes player i’s action and the state of the dynamical system that converges to the Nash equilibrium. The action profile lies in Ω=K1×⋯×Kn⊂ℝ3n =K_1×·s× K_n ^3n. The skill proficiency index si∈[0,1]s_i∈[0,1] is a fixed scalar averaging player i’s competence over past assignments and training (nursing, clinical certifications, physician training), calibrated by the institution. Performance degradation due to fatigue is captured indirectly through the time-varying alacrity index ai(t)a_i(t). Players’ transient step response is described as a second-order system (Nise, 2019) σi(t) _i(t) =1−11−ζ2exp(−ζωnt)cos(ωn =1- 1 1-ζ^2 (-ζ _nt) ( _n 1−ζ2t−ϕ) 1-ζ^2\,t-φ) (4) where ϕ=arctan(ζ/(1−ζ2))φ= (ζ/( 1-ζ^2)) for damping ratio ζ and natural frequency ωn _n. We use an underdamped model (ζ<1ζ<1) since human physiological responses to stress overshoot before settling (Rosenblum et al., 2025). We adopt ωni:=2πbi/60 _n^i:=2π b_i/60 where bib_i is player i’s resting heart rate (bpm), a proxy for the engagement transient time.555In practice, sensor noise in bib_i would be handled by standard filtering (e.g. , multisensor fusion, low-pass smoothing). Alacrity measure: The alacrity index aia_i encodes a player’s readiness to execute tasks within their skillset. We model aia_i via the rise time trt_r of (4), which is the time for σ(t)σ(t) to go from 10% to 90% of its final value. Fixing ζ=k<1ζ=k<1 and using ωnit _n^it as the normalized time variable, ai(t)a_i(t) is found as ai(t)=Δσ(t)/ωni,t≥0. a_i(t)= σ(t)/ _n^i, t≥ 0. (5) Fairness in workload distribution should be encouraged when players share resources (Ekpo et al., 2025b, a). If one worker bears workload beyond their capacity, the uneven throughput becomes untenable for efficient resuscitation. We adopt the Jain fairness index (Jain et al., 1984), a convex, scale-invariant metric, bounded in [1/n,1][1/n,1]: fi(w;t)=(∑i=1nw(t))2n∑i=1nw(t)2,w(t),t≥0 f_i(w;t)= ( _i=1^nw(t) )^2n _i=1^nw(t)^2, w(t),t≥ 0 (6) which measures the equity of work w allocated to player i. Each player maintains a local team-level estimate f^i(t) f_i(t) obtained by running dynamic averaging consensus on ∑jwj _jw_j and ∑jwj2 _jw_j^2. The per-task weights δτ∈[0,1] _τ∈[0,1] that define the workload wi(t)=∑τ∈i(t)δτw_i(t)= _τ _i(t) _τ are specified in Section 4. Communication efficiency. We define the communication efficacy as a function of (i) settling time; (i) engagement level; and (i) response time. The settling time tsit_s^i is the time for σi(t) _i(t)’s damped oscillations to reach and stay within ±2%± 2\% of steady state, approximated as tsi=−ln(ϵ1−ζ2)/ζωn t_s^i=- (ε 1-ζ^2)/ζ _n (7) with ϵ=0.02ε=0.02. The response time is the difference between reaching steady state and being within 10% of σi(t) _i(t)’s final value upon a task handover. The timely response is t¯i=kΔtij t_i=k t_ij where Δtij t_ij is the inter-call time between players i and j for all j∈x−ij∈ x_-i. The engagement level for player i is a running mean of visit counts from neighbors iN_i up to discretized time K: ei(t)=1Tn−i∫t0T(∑k=1νij(t))t,∀j∈x−i.e_i(t)= 1Tn_-i _t_0^T ( _k=1^K _ij(t) )dt, ∀\,j∈ x_-i. A “visit” is a discrete communication event (verbal handoff, task delegation, closed-loop confirmation) recorded at each timestep k. The communication efficiency index is υi(t)=α(tsi+t¯i)+(1−α)ei(t),α∈(0,1). _i(t)=α(t_s^i+ t_i)+(1-α)e_i(t),\,\,α∈(0,1). (8) 3.2 Local costs and dynamics Problem 1 We want a distributed algorithm for a GNEP Γ(,Ω,∇J(x),K) (V, ,∇ J(x),K) for the socio-technical game with coupled constraints, K. The local cost for player i is the quadratic objective between player i and its neighbors iN_i: Ji(xi,x−i;t):=Ji(xi;t)+⟨Ji(x−i;t)⟩ri J_i(x_i,x_-i;t):=J_i(x_i;t)+ J_i(x_-i;t) _r_i (9) where Ji(xi;t) J_i(x_i;t) =‖si‖2+‖ai(t)‖2+‖fi(w;t)‖2+‖υi(t)‖2 =\|s_i\|^2+\|a_i(t)\|^2+\|f_i(w;t)\|^2+\| _i(t)\|^2 and ⟨Ji(x−i;t)⟩ri J_i(x_-i;t) _r_i =11+ni(t)(Ji(xi;t)+∑j∈i(t)Ji(x−i;t)) = 11+n_i(t) (J_i(x_i;t)+ _j _i(t)J_i(x_-i;t) ) (10) denotes the cost of neighboring players within radius ri>0r_i>0 of player i. The resuscitation coordination is modeled as a dynamic game whose actions evolve under the distributed dynamics in (11), converging to a variational generalized Nash equilibrium (v-GNE) x∗=(x1∗,…,xn∗)x^*=(x_1^*,…,x_n^*): x˙i x_i =PΩi(xi−α∇xiJi(x)+αγn∇xigi(xi)⊤λi)−xi =P_ _i\! (x_i-α _x_iJ_i(x)+ αγn\, _x_ig_i(x_i) _i )-x_i (11a) λ˙i λ_i =Pℝ+p[κ∑j∈isgn(λj−λi)−gi(xi)] =P_R_+^p\! [κ _j _isgn( _j- _i)-g_i(x_i) ] (11b) ζ˙i ζ_i =ρ∑j∈isgn(ηj−ηi),η˙i=ζi+Ji(x−i) =ρ _j _isgn( _j- _i), η_i= _i+J_i(x_-i) (11c) where α>0α>0 is the primal step size; γ>0γ>0 scales the constraint coupling, normalized by n; and κ,ρ>0κ,ρ>0 are the consensus gains for the dual and aggregation channels. The Lagrange multiplier λi∈ℝ+p _i _+^p carries one component per inequality constraint in gi(xi)g_i(x_i); ζi _i and ηi _i are auxiliary scalars implementing finite-time dynamic averaging consensus on Ji(x−i)J_i(x_-i). Initial conditions are xi(0)∈Ωix_i(0)∈ _i, λi(0)∈ℝ+p _i(0) _+^p, and ζi(0)=0 _i(0)=0. Solutions are interpreted similarly to Filippov (Cortés, 2008). Algorithm (11) extends Liang et al. (2017) to inequality constraints g(x)≥0g(x)≥ 0. The constants are chosen as κ>(n−1)h1κ>(n-1)h_1 and ρ>γ(n−1)h2ρ>γ(n-1)h_2, where h1=supi∈(supxi∈Ωi‖∇xiJi(⋅,x−i)‖supy,y′∈Ω‖y−y′‖),h_1= _i ( _x_i∈ _i\| _x_iJ_i(·,x_-i)\| _y,y ∈ \|y-y \| ), h2=supi∈(supxi∈Ωi‖gi(xi)‖),h_2= _i ( _x_i∈ _i\|g_i(x_i)\| ), computed distributively as in (Liang et al., 2017). 4 Numerical Results Figure 2: Distributed GNEP simulation for 7 players. (a) Fixed-point residual. (b) Dual variable norms. (c) Individual objectives. Setup. We validate the distributed GNEP algorithm on a trauma resuscitation scenario with 77 healthcare workers executing the Advanced Life Support (ALS) cardiac arrest protocol (American Red Cross, 2025). The synthetic dataset models a 2020-minute episode with clinically accurate CPR cycles, rhythm checks, shock delivery, drug administration, airway management, and Return of Spontaneous Circulation (ROSC) assessment. Each timestamp assigns one atomic action per agent (e.g. , Start_CPR, Deliver_Shock, Administer_Epinephrine), following the ideal ALS flowchart. Initial conditions and constraints. Workers are initialized with heterogeneous state components: alacrity in [0.5,1.0][0.5,1.0], fairness in [0.8,0.95][0.8,0.95], communication efficiency υi _i in [0.9,1.0][0.9,1.0], and fixed skill proficiencies in [0.6,0.9][0.6,0.9]. A spatial communication radius r=200r=200 ft induces time-varying neighbor sets i(t)N_i(t). The shared constraint g(x)≥0g(x)≥ 0 enforces minimum operational thresholds ai,fi,υi≥0.2a_i,f_i, _i≥ 0.2, preventing excessive fatigue or coordination degradation. All states are normalized to [0,1][0,1]. Task weighting. Each task τ is assigned a weight δτ∈[0,1] _τ∈[0,1], with negative penalties applied when a player’s assignment falls below the protocol-prescribed threshold: −1.0-1.0 for suboptimal role assignment, −0.5-0.5 for low alacrity, and −0.5-0.5 for low energy. The negative values discount the workload contribution of substandard assignments so that the resulting workload wi(t)=∑τ∈i(t)δτw_i(t)= _τ _i(t) _τ enters the Jain fairness index in (6) with the right sign. Convergence. Figure 2 shows the simulation results. Fixed-point residuals decay below the 10−310^-3 tolerance within roughly six iterations (panel a), confirming that all players reach a stable joint decision under the time-varying graph. Dual norms remain bounded throughout (panel b), indicating that the constraints remain appropriately enforced without divergence. Individual objectives reach heterogeneous steady values consistent with the KKT conditions of a v-GNE (panel c); the spread across players reflects each player’s fixed skill sis_i, since each one optimizes its own cost rather than a shared team cost. Communication topology. Figure 1 shows the communication graph at k=0k=0, with seven healthcare workers within a 200200-ft radius. Edges connect player pairs whose Euclidean distance falls below the radius. Since player positions drift after each iteration, the neighbor sets i(k)N_i(k) change over time, and the graph in subsequent iterations differs from the initial snapshot. The dynamics in (11) operate on the graph ()(k)G(A)(k) at each iteration. Scalability. Figure 3 shows how the algorithm scales with team size n. The number of iterations to reach the 10−310^-3 convergence threshold depends on how well-connected the communication graph is. As long as the graph stays well-connected as n grows, the iteration count grows only mildly. Empirically, this count grows mildly with n, from ∼124 124 at n=5n=5 to ∼147 147 at n=50n=50. Wall-clock time per iteration is dominated by the neighbor sums and scales as O(n+|ℰ|)O(n+|E|), well below the O(n2)O(n^2) worst-case reference. Figure 3: Scalability of the distributed GNEP algorithm. (a) Iteration count to reach the convergence threshold 10−310^-3 versus n. (b) Wall-clock time versus n, with O(n2)O(n^2) worst case reference (dashed). Chattering analysis. The signum function in (11) is discontinuous, which can cause small oscillations in numerical simulation. To mitigate these oscillations, we smooth the signum with a “smooth-abs” function sgnα(x):=x2+α2−α,α>0, _α(x):= x^2+α^2-α, α>0, (12) following Tassa et al. (2012). The original discontinuous dynamics are interpreted in the standard Filippov framework of Cortés (2008). Figure 4 compares the two. Smoothing preserves convergence to the v-GNE and eliminates the oscillations. 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