# Strategic Stability Analysis of a Nonlinear Security Game

Jed Boufaied, Jean Le Hénaff, Hélène Le Cadre

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## To cite this version:

Jed Boufaied, Jean Le Hénaff, Hélène Le Cadre. Strategic Stability Analysis of a Nonlinear Security Game. ROADEF 2024, Mar 2024, Amiens, France. hal-04490283

HAL Id: hal-04490283

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Distributed under a Creative Commons Attribution 4.0 International License
# Strategic Stability Analysis of a Nonlinear Security Game

Jed Boufaied, Jean Le Hénaff, Hélène Le Cadre

<figure><img src="image_1.png" /></figure>

## To cite this version:

Jed Boufaied, Jean Le Hénaff, Hélène Le Cadre. Strategic Stability Analysis of a Nonlinear Security Game. ROADEF 2024, Mar 2024, Amiens, France. hal-04490283

HAL Id: hal-04490283

https://inria.hal.science/hal-04490283v1

Submitted on 5 Mar 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_2.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Strategic Stability Analysis of a Nonlinear Security Game

Jed Boufaied¹, Jean Le Hénaff¹,², Hélène Le Cadre¹

¹ Inria, CNRS, Univ. de Lille, Centrale Lille, UMR 9189 - CRISTAL, F-59000 Lille, France.
{jed.boufaied, helene.le-cadre}@inria.fr

² CEA/DAM/Direction de l'Analyse Stratégique, F-91297 Arpajon, France.
jean.le-henaff@ponts.org

**Keywords** : Security Game; Nash Equilibrium; Coercivity; Equilibrium Learning; Partial Equilibrium Model

## 1 Introduction

The primary goal of Nations is to maximize their perceived security, which requires each Nation to develop an effective deterrence strategy [1]. In classical game theoretic models, deterrence as a military strategy is based upon a standard, iterated two-player prisoner's dilemma model by which cooperation can be assured if a rational player understands the punishment or cost of defection [2].

However, many of the actions taken by Nations in pursuit of the maximization of their perceived security might, in turn, lead to a decrease rather than an increase in their perceived security. This dilemma is explained by the fact that the increase in military capability of one Nation might be interpreted as a risk that the arming Nation will use it to perform an attack in the future.

In this context, in order to re-establish the balance of power, the other Nations could either increase their own military capabilities or initiate a preemptive strike. Choosing the first option may result in a security spiral, in which several Nations are tied in an arms race, and may lead to war in the long run. Such a situation is known as the security dilemma [3], and is one possible outcome of arms race models.

If the goal of game-theoretic models is to quantify the outcomes of situations of conflict between armed Nations, non-proliferation policies and containment treaties are often designed with the goal to stabilize the rivalry between Nations and reduce strategic risks.

## 2 Problem Statement

We model the interactions among Nations possessing both conventional and nuclear weapons, i.e., mixed armament, using noncooperative game theory relying on [5]. Contrary to most papers which focus on the dynamics of arms procurement, we focus on the steady-state analysis of the game, which captures the long-term solution of the dynamical model, and is formulated as a one-shot (static) game involving nonlinear and non-differentiable utility functions.

On the quantitative side, we extend classical results from noncooperative game theory, to characterize the game equilibria. On the qualitative side, we assess the effects of long-term investments in nuclear weapons on the efficiency and strategic stability of the Nations' international system considering a duopoly. The latter result is then extended to an arbitrary large number of Nations.

Finally, it is worth noticing that world weapons production is concentrated in the hands of a small number of producers who are self-sufficient for their own government procurement and have incentives to trade with recipients, which are engaged in regional antagonisms and are reliant on weapons imports. In this context, we assume a fixed division of Nations into
# Strategic Stability Analysis of a Nonlinear Security Game

Jed Boufaied¹, Jean Le Hénaff¹,², Hélène Le Cadre¹

¹ Inria, CNRS, Univ. de Lille, Centrale Lille, UMR 9189 - CRISTAL, F-59000 Lille, France.

`{jed.boufaied, helene.le-cadre}@inria.fr`

² CEA/DAM/Direction de l'Analyse Stratégique, F-91297 Arpajon, France.

`jean.le-henaff@ponts.org`

**Keywords** : Security Game; Nash Equilibrium; Coercivity; Equilibrium Learning; Partial Equilibrium Model

## 1 Introduction

The primary goal of Nations is to maximize their perceived security, which requires each Nation to develop an effective deterrence strategy [1]. In classical game theoretic models, deterrence as a military strategy is based upon a standard, iterated two-player prisoner's dilemma model by which cooperation can be assured if a rational player understands the punishment or cost of defection [2].

However, many of the actions taken by Nations in pursuit of the maximization of their perceived security might, in turn, lead to a decrease rather than an increase in their perceived security. This dilemma is explained by the fact that the increase in military capability of one Nation might be interpreted as a risk that the arming Nation will use it to perform an attack in the future.

In this context, in order to re-establish the balance of power, the other Nations could either increase their own military capabilities or initiate a preemptive strike. Choosing the first option may result in a security spiral, in which several Nations are tied in an arms race, and may lead to war in the long run. Such a situation is known as the security dilemma [3], and is one possible outcome of arms race models.

If the goal of game-theoretic models is to quantify the outcomes of situations of conflict between armed Nations, non-proliferation policies and containment treaties are often designed with the goal to stabilize the rivalry between Nations and reduce strategic risks.

## 2 Problem Statement

We model the interactions among Nations possessing both conventional and nuclear weapons, i.e., mixed armament, using noncooperative game theory relying on [5]. Contrary to most papers which focus on the dynamics of arms procurement, we focus on the steady-state analysis of the game, which captures the long-term solution of the dynamical model, and is formulated as a one-shot (static) game involving nonlinear and non-differentiable utility functions.

On the quantitative side, we extend classical results from noncooperative game theory, to characterize the game equilibria. On the qualitative side, we assess the effects of long-term investments in nuclear weapons on the efficiency and strategic stability of the Nations' international system considering a duopoly. The latter result is then extended to an arbitrary large number of Nations.

Finally, it is worth noticing that world weapons production is concentrated in the hands of a small number of producers who are self-sufficient for their own government procurement and have incentives to trade with recipients, which are engaged in regional antagonisms and are reliant on weapons imports. In this context, we assume a fixed division of Nations into
arms suppliers and recipients. Assuming that suppliers are split into a coalition of allies, and
 neutrals, we aim to understand whether a common defense policy can be beneficial in ensuring
 the stability of the allies' coalition and a security increase with respect to the no coalition
 benchmark setting.

3 Contributions

We propose a multipolar security model which addresses the specific logic of nuclear strategies
in a context involving mixed armament, i.e., both conventional and nuclear weapons. In [5],
considering a new way to model the security perceived at the Nation-wide level, we extended
the classical results from Rosen [4] on compact-convex games to unbounded convex games
relying on the uniform coercivity property<fnref n="1" /> of the Nations’ utility functions. This allowed us,
in a second step, to prove the existence and uniqueness (under mild assumptions) of an interior
point Nash Equilibrium solution of the one-shot game.

In this talk, we extend the work from [5] according to the following three directions :

(i) We propose and implement a regularization algorithm relying on the proximal point method, to compute in a distributed fashion the interior point Nash Equilibrium;

(ii) We provide an analytical proof that nuclear weapons have a stabilizing effect for an arbitrary large number of Nations; therefore generalizing the numerical results obtained for the duopoly case;

(iii) We provide a partial equilibrium model of the international arms market and compare
the outcomes of different market designs, assessing the stability of the allies' coalition
and increase in security by comparison with the no coalition benchmark as a function of
the size of the allies' coalition [6].

Références

[1] Koepsell, D., Stankova, K. : Non-Proliferation Regimes, Immoral and Risky : A Game-Theoretic Approach. International Journal of World Peace 27(2), 63–83 (2012)

[2] Axelrod, R. : The Evolution of Cooperation. ISBN (1984)

[3] Herz, J.H. : Idealist Internationalism and the Security Dilemma. World Politics, 2(2), 171–201 (1950)

[4] Rosen, J.B. : Existence and Uniqueness of Equilibrium Points for Concave N-Person Games. Econometrica 33(3), 520 (1965)

[5] Le Hénaff, Le Cadre, H. : Modeling and Analysis of a Nonlinear Security Game with Mixed Armament, Proceedings of the Conference on Decision and Game Theory for Security (Gamesec 2023)

[6] Le Hénaff, J., Boufaied, J., Le Cadre, H. : On Stability of Nonlinear Armsbuilding Security Game. Preprint HAL (2024) https://inria.hal.science/hal-04411599

<footnote n="1">1. Let $J_n$ be the utility function of Nation $n$ and $x_n \in \mathcal{U}_n$ be Nation $n$'s strategy. then $-J_n$ is coercive in $x_n$ if $\lim_{\|x_n\| \to +\infty} J_n(x_n, x_{-n}) = +\infty, \forall x_{-n} \in \prod_{m \neq n} \mathcal{U}_m$.</footnote>
arms suppliers and recipients. Assuming that suppliers are split into a coalition of allies, and
neutrals, we aim to understand whether a common defense policy can be beneficial in ensuring
the stability of the allies' coalition and a security increase with respect to the no coalition
benchmark setting.

# 3 Contributions

We propose a multipolar security model which addresses the specific logic of nuclear strategies
in a context involving mixed armament, i.e., both conventional and nuclear weapons. In [5],
considering a new way to model the security perceived at the Nation-wide level, we extended
the classical results from Rosen [4] on compact-convex games to unbounded convex games
relying on the uniform coercivity property<fnref n="1" /> of the Nations' utility functions. This allowed us,
in a second step, to prove the existence and uniqueness (under mild assumptions) of an interior
point Nash Equilibrium solution of the one-shot game.

In this talk, we extend the work from [5] according to the following three directions :

(i) We propose and implement a regularization algorithm relying on the proximal point method, to compute in a distributed fashion the interior point Nash Equilibrium;

(ii) We provide an analytical proof that nuclear weapons have a stabilizing effect for an arbitrary large number of Nations; therefore generalizing the numerical results obtained for the duopoly case;

(iii) We provide a partial equilibrium model of the international arms market and compare the outcomes of different market designs, assessing the stability of the allies' coalition and increase in security by comparison with the no coalition benchmark as a function of the size of the allies' coalition [6].

Références

[1] Koepsell, D., Stankova, K. : Non-Proliferation Regimes, Immoral and Risky : A Game-Theoretic Approach. International Journal of World Peace 27(2), 63–83 (2012)

[2] Axelrod, R. : The Evolution of Cooperation. ISBN (1984)

[3] Herz, J.H. : Idealist Internationalism and the Security Dilemma. World Politics, 2(2), 171–201 (1950)

[4] Rosen, J.B. : Existence and Uniqueness of Equilibrium Points for Concave N-Person Games. Econometrica 33(3), 520 (1965)

[5] Le Hénaff, Le Cadre, H. : Modeling and Analysis of a Nonlinear Security Game with Mixed Armament, Proceedings of the Conference on Decision and Game Theory for Security (Gamesec 2023)

[6] Le Hénaff, J., Boufaied, J., Le Cadre, H. : On Stability of Nonlinear Armsbuilding Security Game. Preprint HAL (2024) https://inria.hal.science/hal-04411599

<footnote n="1">1. Let $J_n$ be the utility function of Nation $n$ and $x_n \in \mathcal{U}_n$ be Nation $n$'s strategy. then $-J_n$ is coercive in $x_n$ if $\lim_{\|x_n\| \to +\infty} J_n(x_n, x_{-n}) = +\infty$, $\forall x_{-n} \in \prod_{m \neq n} \mathcal{U}_m$.</footnote>