Nash Equilibria and Bargaining Solutions of Differential Bilinear Games

Campana, Francesca Calà; Ciaramella, Gabriele; Borzì, Alfio

doi:10.1007/s13235-020-00351-2

Cited by 5 publications

(6 citation statements)

References 20 publications

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“…Similarly, |δỹ 1 (t)| ≤ C 12 δu 2 L 2 (0,T ) , |δỹ 2 (t)| ≤ C 21 δu 1 L 2 (0,T ) , t ∈ (0, T ); (21) see [9] for a proof. By using these estimates in Eq.…”

Section: Numerical Experimentsmentioning

confidence: 95%

“…We denote with U ad = U (1) ad × U (2) ad and U = L 2 (0, T ; R m ) × L 2 (0, T ; R m ). Notice that we have a uniform bound on |y(u 1 , u 2 )(t)|, t ∈ [0, T ], that holds for any u ∈ U ad ; see [9]. By using the map (u 1 , u 2 ) → y = y(u 1 , u 2 ), we can introduce the reduced objec-…”

Section: Pmp Characterization Of Nash Gamesmentioning

confidence: 99%

“…We remark that existence of a NE point can be proved subject to appropriate conditions on the structure of the differential game, including the choice of T . For our purpose, we assume existence of a Nash equilibrium (u * 1 , u * 2 ) ∈ U ad , and refer to [9] for a review and recent results in this field.…”

Section: Definition 1 the Functions (U *mentioning

confidence: 99%

See 2 more Smart Citations

On the SQH Method for Solving Differential Nash Games

Campana

Borzì

2021

J Dyn Control Syst

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A sequentialquadratic Hamiltonian schemefor solving open-loop differential Nash games is proposed and investigated. This method is formulated in the framework of the Pontryagin maximum principle and represents an efficient and robust extension of the successive approximations strategy for solving optimal control problems. Theoretical results are presented that prove the well-posedness of the proposed scheme, and results of numerical experiments are reported that successfully validate its computational performance.

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“…Similarly, |δỹ 1 (t)| ≤ C 12 δu 2 L 2 (0,T ) , |δỹ 2 (t)| ≤ C 21 δu 1 L 2 (0,T ) , t ∈ (0, T ); (21) see [9] for a proof. By using these estimates in Eq.…”

Section: Numerical Experimentsmentioning

confidence: 95%

Section: Pmp Characterization Of Nash Gamesmentioning

confidence: 99%

See 1 more Smart Citation

On the SQH Method for Solving Differential Nash Games

Campana

Borzì

2021

J Dyn Control Syst

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“…In order to address the nonlinear coupling among the strategies adopted by the players, a relaxation scheme (e.g. [ 41 ]) is employed for the NE. The relaxation scheme is implemented in algorithm (5.1).…”

Section: Numerical Schemes For Solving Nash and Stackelberg Equilibriamentioning

confidence: 99%

An evolutionary differential game for regulating the role of monoclonal antibodies in treating signalling pathways in oesophageal cancer

Alajmi,

Roy

2024

R. Soc. Open Sci.

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This work presents a new framework for a competitive evolutionary game between monoclonal antibodies and signalling pathways in oesophageal cancer. The framework is based on a novel dynamical model that takes into account the dynamic progression of signalling pathways, resistance mechanisms and monoclonal antibody therapies. This game involves a scenario in which signalling pathways and monoclonal antibodies are the players competing against each other, where monoclonal antibodies use Brentuximab and Pembrolizumab dosages as strategies to counter the evolutionary resistance strategy implemented by the signalling pathways. Their interactions are described by the dynamical model, which serves as the game’s playground. The analysis and computation of two game-theoretic strategies, Stackelberg and Nash equilibria, are conducted within this framework to ascertain the most favourable outcome for the patient. By comparing Stackelberg equilibria with Nash equilibria, numerical experiments show that the Stackelberg equilibria are superior for treating signalling pathways and are critical for the success of monoclonal antibodies in improving oesophageal cancer patient outcomes.

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“…This changes the features of the optimisation problem drastically. In [13] on the other hand, the question of existence and computation of Nash equilibria in bilinear problems, but for ODE models. Our paper is, to the best of our knowledge, a first contribution to the qualitative analysis of L ∞ −L 1 constrained bilinear optimal control problems with a cost function that is not of tracking-type.…”

Section: Bibliographical Referencesmentioning

confidence: 99%

Spatial ecology, optimal control and game theoretical fishing problems

Mazari¹,

Ruiz-Balet²

2022

Preprint

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Of paramount importance in both ecological systems and economic policies are the problems of harvesting of natural resources. A paradigmatic situation where this question is raised is that of fishing strategies. Indeed, overfishing is a well-known problem in the management of live-stocks, as being too greedy may lead to an overall dramatic depletion of the population we are harvesting. A closely related topic is that of Nash equilibria in the context of fishing policies. Namely, two players being in competition for the same pool of resources, is it possible for them to find an equilibrium situation?The goal of this paper is to provide a detailed analysis of these two queries (i.e optimal fishing strategies for single-player models and study of Nash equilibria for multiple players games) by using a basic yet instructive mathematical model, the logistic-diffusive equation. In this framework, the underlying model simply reads −µ∆θ = θ(K(x) − α(x) − θ) where K accounts for natural resources, θ for the density of the population that is being harvested and α = α(x) encodes either the single player fishing strategy or, when dealing with Nash equilibria, a combination of the fishing strategies of both players. This article consists of two main parts. The first one gives a very fine characterisation of the optimisers for the singleplayer game where one aims at solving sup α ´Ω αθ, under L ∞ and L 1 constraints on the fishing strategies α. In particular, we show that, depending on the value of these constraints, this optimal control problem may behave like a convex or, conversely, concave problem. We also provide a detailed analysis of the large diffusivity limit of this problem. In the case where two players are involved, we rather write α as α1 + α2 where αi, the fishing strategy of the i-th player, also satisfies L ∞ and L 1 constraints. Defining I1 := ´Ω αiθ we aim at finding a Nash equilibrium. We prove the existence of Nash equilibria in several different regimes and investigate several related qualitative queries, for instance providing examples of the wellknown tragedy of commons.Our study is completed by a variety of numerical simulations that illustrate our results and allow us to formulate open questions and conjectures.

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Nash Equilibria and Bargaining Solutions of Differential Bilinear Games

Cited by 5 publications

References 20 publications

On the SQH Method for Solving Differential Nash Games

On the SQH Method for Solving Differential Nash Games

An evolutionary differential game for regulating the role of monoclonal antibodies in treating signalling pathways in oesophageal cancer

Spatial ecology, optimal control and game theoretical fishing problems

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