Jean Le Hénaff
email: jean.le-henaff@ponts.org

Hélène Le Cadre
email: helene.le-cadre@inria.fr

Modeling and Analysis of a Nonlinear Security Game with Mixed Armament

Keywords: Security Game, Nash Equilibrium, Uniform Coercivity, Strategic Stability

We formulate a security game in a context of mixed armament acquisition, involving a finite set of Nations in strategic relationship, with utility functions which are nonlinear and non-differentiable on the boundary of their sets of definition. Since we want to study the long-term effect of the Nations' investment in nuclear weapons, we focus on the steady-state analysis of the game. This requires us to extend the classical results from Rosen on compact-convex games, to unbounded convex games, relying on the coercivity property of the Nations' utility functions. In addition, we prove the existence and uniqueness, under mild assumptions, of an interior point Nash Equilibrium solution of the game. Simulations are performed in case of a duopoly, highlighting the efficiency loss reduction and stabilizing effect of nuclear armaments by comparison with the conventional-only setting.

Introduction

The primary goal of Nations is to maximize their perceived security, which requires each Nation to develop an effective deterrence strategy [START_REF] Koepsell | Non-Proliferation Regimes, Immoral and Risky: A Game-Theoretic Approach[END_REF]. 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 [START_REF] Kennan | Containment, Then and Now[END_REF][START_REF] Axelrod | The Evolution of Cooperation[END_REF][START_REF] Schlosser | Iterated games and arms race models[END_REF]. Models in this area are often inspired by the USA and USSR-allied, or at least Western vs Eastern blocs, nuclear arms race, and formulate the security dilemma in arms procurement as a two-player noncooperative game in normal form [START_REF] Snyder | Prisoner's Dilemma and Chicken Models in International Politics[END_REF][START_REF] Plous | The Nuclear Arms Race: Prisoner's Dilemma or Perceptual Dilemma[END_REF]. However, many of the actions taken by Nations in pursuit of the maximization of their perceived security -such as arms procurement and the development of new military technologies -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, responding to increases in arms procurement and defense expenditure by others by arming themselves more and more heavily, and may lead to war in the long run. Such a situation is known as the security dilemma [START_REF] Herz | Idealist Internationalism and the Security Dilemma[END_REF][START_REF] Jervis | Cooperation under the Security Dilemma[END_REF], and is one possible outcome of arms race models.

Standard literature dealing with arms race models [START_REF] Wallace | Non-Linear Arms race Models[END_REF][START_REF] Richardson | Arms and Insecurity[END_REF] focuses on the dynamic process of the arms race over the time or on fitting real data collected over the years using this model. In this setting, game theory is a useful tool to model interactions between Nations with conflicting interests. The classical literature dealing with nuclear strategy using game theory focuses mainly on discrete decision space repeated games [START_REF] O'neill | Game Theory Models of Peace and War[END_REF] and differential games. Within repeated games, iterated prisoners' dilemmas between two Nations [START_REF] Majeski | Arms races as iterated prisoner's dilemma games[END_REF][START_REF] Lichbach | When Is an Arms Rivalry a Prisoner's Dilemma?: Richardson's Models and 2 × 2 Games[END_REF] have been extensively studied. At each iteration, each Nation has a choice between a high or low level of arms. In the static version, each Nation's dominant strategy is to choose a high level of arms. The Nash Equilibrium outcome of the static prisoner dilemma model for arms race is therefore that both Nations choose high. As a result, the static game outcome is worse for both Nations than if both had chosen a low level of arms. However, in reality, the game is not played once and for all, but is an ongoing series of decisions, i.e., the prisoner dilemma is played repeatedly by the two Nations. This opens the possibility for cooperation to emerge through rewards and punishment strategies, e.g., tit-for-tat [START_REF] Fudenberg | The Theory of Learning in Games[END_REF].

A somewhat equivalent framework stems from Richardson-type models of arms race [START_REF] Lichbach | Stability in Richardson's Arms Races and Cooperation in Prisoner's Dilemma Arms Rivalries[END_REF][START_REF] Simaan | Formulation of Richardson's Model of Arms Race from a Differential Game Viewpoint[END_REF], which provide parametrized systems of differential equations to represent the evolution of the weapons stockpile. These equations can be derived as closed-loop Nash Equilibria of linear-quadratic differential games that implement an interpretable reasoning of players [START_REF] Simaan | Formulation of Richardson's Model of Arms Race from a Differential Game Viewpoint[END_REF][START_REF] Melese | Reversing the Arms Race: A Differential Game Model[END_REF][START_REF] Moriarty | Differential Game Theory Applied to a Model of the Arms Race[END_REF][START_REF] Simaan | Nash Equilibrium Strategies for the Problem of Armament Race and Control[END_REF]. However, military strategies that hinge on nuclear weapons imply that Nations' evaluation of the effect of each other's actions should be nonlinear, and Richardson-type's models fail to represent accurately the specific nature of nuclear armaments [START_REF] Deger | Optimal control and differential game models of military expenditure in less developed countries[END_REF][START_REF] Dunne | The Econometrics of Military Arms Races[END_REF]. Indeed, contrary to conventional war, nuclear warfare allows for significant preemptive and retaliatory strikes whose magnitude depend nonlinearly on the initial stockpiles. A two-stage game model may show that a second-strike capability decreases as the exponential of the ratio of the initial stockpiles [START_REF] Bracken | Multipolar Nuclear Stability: Incentives to Strike and Incentives to Preempt[END_REF].

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. For example, during the Cold War, several agreements between the USA and the USSR, such as the intermediate-range nuclear forces (INF) treaty, helped balance nuclear-able weapons so that each side could survive a preemptive nuclear attack with a sufficiently large stockpile of ballistic missiles to launch a retaliatory strike, in the context of nuclear strategies heavily influenced by the concept of MAD (mutually assured destruction). Nowadays, 9 countries are thought to possess nuclear weapons. Within them, only 5 -the USA, Russia, China, France and the United Kingdom -have signed the treaty on the non-proliferation of nuclear weapons (NPT), the second most widely ratified treaty after the United Nations charter. Under the NPT, each of the 191 parties 'undertakes to pursue negotiations in good faith on effective measures relating to the cessation of the nuclear arms race at an early date and to nuclear disarmament, and on a treaty on general and complete disarmament under strict and effective international control' [START_REF]Treaty on the Non-Proliferation of Nuclear Weapons[END_REF]. The security architecture is completed by the comprehensive nuclear-test-ban treaty (CTBT) with 177 parties, which prohibits nuclear tests and as such limits the possibility to develop nuclear weapons1 .

Problem Statement

We model the interactions among Nations possessing both conventional and nuclear weapons, i.e., mixed armament, using noncooperative game theory. 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. On the quantitative side, a first difficulty appears in modeling the Nations' utility functions such that they reflect the security perceived at a Nation-wide level -thus, depending on the Nation's armament strategy and on that of its rivals. This leads us to define utility functions which are nonlinear and non-differentiable on the boundary of their sets of definition. A second difficulty arises from this modeling choice and requires that we extend classical results from noncooperative game theory, to characterize the game equilibria. On the qualitative side, we aim to assess the effects of long-term investments in nuclear weapons on the efficiency and strategic stability of the Nations' international system.

Main 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. The model is formulated in a dynamic setting, but since we want to study the stabilizing and security dilemma reduction potential of mixed armament, we focus on the one-shot game steady-state formulation, which captures the long-term effects of the competition among Nations. This paper provides three important contributions to the state of the art. First, considering a new way to model the security perceived at the Nation-wide level, that takes inspiration from [START_REF] Bracken | Multipolar Nuclear Stability: Incentives to Strike and Incentives to Preempt[END_REF], we extend the classical results from Rosen [START_REF] Rosen | Existence and Uniqueness of Equilibrium Points for Concave N-Person Games[END_REF] on compactconvex games to unbounded convex games relying on the coercivity property of the Nations' utility functions. This allows us, in a second step, to prove the existence of an interior point Nash Equilibrium solution of the one-shot game. The proof of uniqueness of the interior point Nash Equilibrium follows, under mild assumption on the game parameters. In a last step, our model is used to determine numerically that nuclear weapons have a stabilizing effect and can potentially reduce the efficiency loss by comparison with a conventional-only setting, therefore providing possible guidelines for current treaties refinements.

Paper Organisation

This paper is organized as follows. In Section 1, we provide an overview of the game theoretic approaches for arms procurement competition modeling and how their outcomes can be used to design containment treaties with stabilizing effects. We also introduce the problem statement, and main contributions. In Section 2, the arms building model is formulated as a one-shot noncooperative game. Conditions for the existence and uniqueness of an interior point Nash Equilibrium solution of this game are studied in Section 3. Analytical results and numerical illustrations in case of a duopoly are discussed in Section 4. We conclude in Section 5.

The Mixed Armament Competition Model

The international system is first and foremost made up of Nations, that we will indifferently call players. In international relations [START_REF] Achen | A State with Bureaucratic Politics is Representable as a Unitary Rational Actor[END_REF], an organization is said to be unitary if and only if its decisions contain all the information useful for its interaction with the international system. Moreover, an organization is said to be rational if its decision-making process consists in maximizing its (wellordered) preferences through its actions, given the information available to it and its anticipation of what other organizations might do. Hence, in our work, Nations can be represented as rational unitary players. Note that no hypothesis is made a priori on the order of the preferences.

We aim to study the strategies of Nations to invest in nuclear and conventional weapons in order to maximize their security while minimizing their storage and R&D costs. We start by placing ourselves in a framework of discrete time evolution. Nations seek to ensure their own security within a certain time horizon (finite or infinite), and in anticipation of a crisis. They can acquire conventional and nuclear weapons to maximize their security, while bearing storage and R&D costs. We choose a realistic framework [START_REF] Waltz | Theory of international politics[END_REF], and assume that Nations are primarily defensive and secondarily aggressive. This means that they seek through conventional and nuclear armaments to ensure their own security, and that their marginal gain from arming themselves increases with the levels of adversary armaments.

Dynamics of Arms Building

Let N be a finite set of N Nations, and T be the horizon of the game (possibly infinite). We propose a model with continuous decision spaces, i.e., the Nations (as players) do not decide whether to attack or not, but they decide on the power drawn from their investment in nuclear and conventional weapons. Models with continuous decision spaces prove themselves more realistic [START_REF] Basar | Dynamic Noncooperative Game Theory[END_REF][START_REF] Koepsell | Non-Proliferation Regimes, Immoral and Risky: A Game-Theoretic Approach[END_REF][START_REF] Simaan | Formulation of Richardson's Model of Arms Race from a Differential Game Viewpoint[END_REF][START_REF] Simaan | Nash Equilibrium Strategies for the Problem of Armament Race and Control[END_REF]. We might assume that the power drawn for each category of armament depends linearly on the Nation's arms production and that the coefficients relating the arms production to the power drawn out of it differ between the categories of weapons. In absence of numerical values for these coefficients, we will not consider this level of details in the present work. In what follows, τ = {nuc} or {conv} and denotes nuclear or conventional.

Let u τ n (t) be the power drawn from the quantity of weapons of class τ produced by Nation n ∈ N and x τ n (t) be the power drawn from its stockpile of weapons of class τ at time period t. We define u n (t) def = u τ n (t) τ , the column vector which contains the power drawn from the production of Nation n for each class of weapons, u(t) def = col (u n (t)) n be the stack of the power drawn from the weapons production by the N Nations. Similarly, x n (t) def = x τ n (t) τ is the column vector which contains the power drawn from stockpile of Nation n for each class of weapons, and x(t) def = col (x n (t)) n be the stack of the power drawn from the stockpiles of the N Nations. Furthermore, we define the following sequences:

x def = (x(t)) t and u n def = (u n (t)) t .
The dynamics of Nation n's power drawn from stockpile of class τ weapons is defined as follows:

∀n ∈ N , x τ n (t + 1) = x τ n (t) + u τ n (t) -νx τ n (t), (1)
with an obsolescence rate ν ≥ 0. The players' utility at time period t writes:

J n u n (t), x(t) = f n x(t) -c S n x n (t) -c D n u n (t) , (2)
where f n (•) is the security function of player n, c S n (•) its storage (operational) cost and c D n (•) its development (R&D) cost. The players may have local constraints, e.g., finite budget, maximum capacity of production, etc., which may evolve dynamically. We denote U n (t) ⊂ R 2 + the set of u n (t) that preserve these constraints, i.e., the feasibility set of player n. Each player n ∈ N maximizes the sum J n (•) over time of its future utility, with discounting rate ρ n ∈]0; 1[, n ∈ N :

V n (x) = max un(t)∈Un(t),∀t Jn u n , x ,
where Jn (u n , x)

def = T t=0 ρ t n J n u n (t), x(t) . (3)
We aim to study this model in the long run, with ν = 0. In what follows, we denote x n def = lim t→+∞ x n (t) player n's steady state, with lim t→+∞ u(t) = 0. Using a slight abuse of notation, the static model is given by:

V n (x) = max xn∈R 2 + Jn (x) where Jn (x) def = J n 0, x . (4)
with the decision variable now being Nation n's power drawn from its stockpiles,

x n ∈ U n . Let x -n def = col (x m
) m̸ =n , ∀n be the vector that contains the stack of the power drawn from the stockpiles of weapons of all Nations in N except n.

Strategic Relationship

For each Nation, the international system is divided into opponents, neutrals and allies (commercial or military). A commercial alliance is a binding agreement allowing transfer of military capabilities between parties. A military alliance is a binding agreement establishing a coalition of military interests and capabilities between parties. To simplify the setting, we will not consider alliances in the following. In what follows, we make the assumption that the decisions of Nations are not affected by those of neutrals.

We now want to structure the set of Nations, by taking into account how their decisions impact those of the others. For each Nation n ∈ N , the set of opponents σ(n) ⊂ N is defined by a binary relation of hostility, represented by the hostility function σ(•). It is reasonable to assume that a Nation is never its own opponent and that any opponent of a Nation must treat the considered Nation as an opponent in return. Therefore, we assume that the relation of hostility is irreflexive and symmetric, i.e.:

∀n ∈ N , n / ∈ σ(n), ∀n, m ∈ N , m ∈ σ(n) ⇐⇒ n ∈ σ(m).
We introduce the strategic relation as the reflexive-transitive closure of the hostility relation:

, m, l ∈ N such that n ̸ = m, m ̸ = l, n ̸ = l, n ∼ m and m ∼ l mean that there exist k, k ′ ∈ N such that m = σ k (n) and l = σ k ′ (m).
Then, by composition of the strategic relation, we get that l = σ k+k ′ (n). Therefore, the strategic relation is transitive. Now, ∀n ∈ N , σ 0 (n) = {n}, thus the strategic relation is an equivalence relation.

⊓ ⊔

Lemma 1 implies that the decisions of Nations belonging to the same equivalence class are independent of the decisions of the Nations out of it. Hence, we can split the set of players into a finite number of equivalence classes. This allows to decompose the main noncooperative game into the same number of noncooperative games, that can be solved independently. In what follows, we assume this procedure has already been done, and focus on a game in which the N Nations share strategic relations:

Assumption 1 All Nations in N are in strategic relationship.
In particular, if the game is nontrivial and consists of at least two players, then every Nation has an opponent.

Target Value

For Nation n ∈ N , the target value of adverse armament of class τ is denoted by x τ σ(n) , which is a function of x τ -n . Later, we will call it Nation n target value. This value represents the level of threat caused by the set of opponents. Several modeling choices are possible. For instance, Nations preparing for a one-off crisis will evaluate the target value as the maximum level of armaments of class τ of the set of opponents. In any case, it must satisfy the following assumption:

Assumption 2 There exist κ -, κ + > 0 input coefficients that do not depend on the power drawn from the weapon stockpiles, such that:

κ -max m∈σ(n) x τ m ≤ x τ σ(n) ≤ κ + max m̸ =n x τ m .
Remark 1. Target values defined as the maximum, the sum or the mean over σ(n) of the power drawn from the weapon stockpiles satisfy Assumption 2:

max m∈σ(n) x τ m ≤ max m∈σ(n) x τ m ≤ max m̸ =n x τ m , max m∈σ(n) x τ m ≤ m∈σ(n) x τ m ≤ N max m̸ =n x τ m , 1 N max m∈σ(n) x τ m ≤ m∈σ(n) x τ m |σ(n)| ≤ max m̸ =n x τ m .
Let us discuss briefly the interpretation of Assumption 2. The right hand side of the inequality ensures that Nations use armaments levels of others to compute the level of threat they face, while the left hand side ensures that Nations do not ignore the greatest opponent in the computation of the threat they face.

Security Functions

The perceived security of a Nation n ∈ N is a function of the power drawn from its own weapon stockpile x n and of its target value x σ(n) , where x σ(n) is the column vector which contains the target value of Nation n in each weapon class. As Nations seek to maximize their perceived security, it increases with their effective weapon stockpile and decreases with the opponents' effective weapon stockpiles. Using deterrence terminology, the effective stockpile value can be defined as the maximum available quantity of armaments, a Nation is both credible, and capable to use in a potential conflict. We shall assume that Nations have a credible threat in using the military capabilities we consider. However, the effectiveness of the weapon stockpiles may depend on the quality and type of armaments as well as on the military doctrine of each Nation. We distinguish between two classes of armaments: nuclear (strategic) weapons allowing for preemptive strikes, and conventional (tactical) weapons which can be used on the battlefield. Therefore, Nation n's security function can be decomposed as follows:

f n (x n , x -n ) = τ α τ f τ n (x n , x -n ), where τ α τ = 1, α τ ≥ 0, ∀τ. (5)
Extrinsic and intrinsic perceived security. For each class τ of armaments, we decompose f τ n into two parts: the extrinsic part, which represents the loss of perceived security due to the environment, and the intrinsic part, which represents the gain of perceived security due to the possession of armaments of class τ . The extrinsic perceived security of Nation n brought by class τ armaments coincides with its opponents' effective stockpile of class τ . Now, as Nations are primarily defensive and secondarily aggressive, the intrinsic perceived security of Nation n brought by class τ armaments is modeled as a linear function of Nation n's effective stockpile value of armaments of class τ . Its slope is a concave function ϕ(•) of the weapons they are supposed to deter. Nuclear strategies are generally dual-purpose, contrary to conventional strategies. Thus, the slope is ϕ

x conv σ(n) +x nuc σ(n) ξn
for nuclear, and ϕ

x conv σ(n) ξn
for conventional, where ξ n > 0 is a threat threshold which captures Nation n's peacefulness. To go further we need to evaluate the effective stockpile value for nuclear and conventional armaments.

Effective stockpiles. Consider first conventional weapons. It is reasonable to assume that Nations are able to use at some point in a potential conflict all of the conventional weapons they possess. Therefore, Nation n's power drawn from effective stockpile of conventional weapons is equal to the power drawn from its actual stockpile of conventional weapons x conv n . Now, consider nuclear weapons. Depending on multiple factors, Nations can be cautious and prepare for an adversarial first strike by seeking a second-strike capability, or they can hinge on their preemptive first-strike capability to deter their counterparts. Therefore, stochastic multi-stage games can be used to model the succession of strikes [START_REF] Bracken | Multipolar Nuclear Stability: Incentives to Strike and Incentives to Preempt[END_REF]. We use this setting to derive the closed form expression of the expected number of remaining weapons in case of a conflict between two armed Nations, after a single preemptive strike.

Following [START_REF] Bracken | Multipolar Nuclear Stability: Incentives to Strike and Incentives to Preempt[END_REF], we consider two Nations called n and m ∈ N , n ̸ = m. Nation n holds W n ∈ N weapons. It is attacking Nation m, which holds W m ∈ N * weapons. Nation n aims to destroy with a preemptive strike as much as possible of Nation m's weapons. Assume that Nation n's weapons have an accuracy λ n > 0, i.e., each weapon has a probability p n = 1 -e -λn of destroying its target. Assume that both weapons and targets are indistinguishable. Let W r m be the random variable that gives the number of remaining weapons for Nation m after a single preemptive strike of Nation n. Let K def

= Wn

Wm and

w def = W n -KW m .
Proposition 1. After an optimal single preemptive strike by Nation n, the expected number of remaining weapons for Nation m is:

E[W r m ] = (W m -w) exp(-Kλ n ) + w exp (-(K + 1)λ n ) , = W m exp -λ n W n W m , if K ∈ N. Proof. Let W n = 1, W n and W m = 1, W m . For all k ∈ W m , let g(k) ∈ W n
be the number of weapons assigned by Nation n to target k. We order the targets such that g(•) is decreasing. Notice that W r m is a function of g(•), but to simplify the notation we omit this dependence. Nation n has to solve the following optimization problem:

min g(•) E [W r m ] , s.t. k∈Wm g(k) = W n , (6)
which consists in finding a minimizer of a function over a non-empty finite set, whence the minimizer exists. We argue that the (unique) minimizer is given by

g(k) = K + 1 if 1 ≤ k ≤ W n -KW m , K else. (7)
First, consider a strike on a single target k ∈ W m . Up to a reordering, let (X l ) l∈ 1,g(k) be a collection of Bernoulli independent and identically distributed random variables with parameter p n . X l = 1 means that weapon l destroys its target k, with probability

p n = 1 -e -λn . Let Y k = g(k) l=1 X l . Target k remains intact if and only if Y k = 0, with probability P (Y k = 0) = e -g(k)λn . Nation m has W r m = k∈Wm 1 {Y k =0}
remaining weapons. Therefore:

E [W r m ] = k∈Wm E 1 {Y k =0} = k∈Wm P (Y k = 0) = k∈Wm e -g(k)λn .
Let g(•) be a minimizer of problem [START_REF] Deger | Optimal control and differential game models of military expenditure in less developed countries[END_REF]. Let us show that g(0) ≤ g(W m ) + 1 using on a proof by contradiction. Assume the converse:

g(0) ≥ g(W m ) + 2. As g(•) is decreasing, there exists l, l ′ such that g(0) = g(l) > g(l + 1) ≥ g(l ′ ) = g(W m ) and l ′ is minimal. Define g ♯ (•) by g ♯ (l) = g(l) -1, g ♯ (l ′ ) = g(l + 1), g ♯ (k) = g(k), ∀k ∈ W m \ {l, l ′ },
g ♯ is decreasing and sums to W n . Note that g(l ′ ) + 1 = g(W m ) + 1 < g(0) = g(l).

Whence, one can compute:

E [W r m ]| g ♯ -E [W r m ]| g = (e λn -1
) e -λng(l) -e -λn(g(l ′ )+1) < 0, which contradicts that g be minimal. Whence, the minimizer g is decreasing and satisfies g(0) ≤ g(W m ) + 1, i.e., it is given by (7). In turn, the minimum is:

E [W r m ] = (W m -w) exp (-Kλ n ) + w exp (-(K + 1)λ n ) . Now, if W n = KW m , i.e., w = 0, then E [W r m ] = W m exp (-Kλ n ). ⊓ ⊔
Nation n's opponents have x τ σ(n) weapons of class τ . Thus Proposition 1 shows that the effective power drawn from stockpile value of class τ weapons, after a single preemptive strike by Nation n with nuclear weapons, is given by

x τ σ(n) exp -λ n x conv n x τ σ(n)
.

Generic security function. Combining all of the above, we can define a generic security function for both conventional and nuclear armaments. Let a, b ∈ {0, 1}. Define Nation n's security function for conventional armaments as:

f conv n (x n , x -n ) = x conv n exp -λ σ(n) x nuc σ(n) x conv n a ϕ x conv σ(n) ξ n -x conv σ(n) exp -λ n x nuc n x conv σ(n) b , (8)
and Nation n's security function for nuclear armaments as

f nuc n (x n , x -n ) = x nuc n exp -λ σ(n) x nuc σ(n) x nuc n a ϕ x conv σ(n) + x nuc σ(n) ξ n -x nuc σ(n) exp -λ n x nuc n x nuc σ(n) b , (9)
where λ σ(n) is a measure of the accuracy of the nuclear weapons of Nation n's opponents. In general, Nations might choose among four doctrine choices, shaping their security functions. Indeed, each Nation will choose whether it relies on its vulnerable second-strike capability (a = 1) or not (a = 0); and whether it relies on its preemptive first strike capability (b = 1) or not (b = 0). We focus on the case (a, b) = (0, 1), more representative of modern nuclear military capabilities. Therefore, applying (5), the security function of Nation n ∈ N will be given by:

f n (x n , x -n ) = α nuc x nuc n ϕ (x conv σ(n) + x nuc σ(n) )/ξ n -x nuc σ(n) exp -λ n x nuc n /x nuc σ(n) + α conv x conv n ϕ x conv σ(n) /ξ n -x conv σ(n) exp -λ n x nuc n /x conv σ(n) , (10)
with α nuc + α conv = 1, α nuc , α conv ≥ 0, λ n is Nation n's nuclear armaments accuracy, and ξ n > 0 is a Nation-specific constant which models Nation n's peacefulness. Remind that ϕ(•) is a concave, strictly increasing function. We also assume ϕ(0) = 0, ϕ ′ (0) = 1, ϕ (3) (0) > 0 and that ϕ(x) = o(x) at infinity.

Remark 2.

In what follows, we will choose ϕ : s → ln(1 + s) in numerical computations. Furthermore, for technical purposes, let k > 0, ϕ k : s → min{ϕ(s), ϕ(k)} and denote Jk n (•) the utility function defined as Jn (•) by replacing ϕ(•) by ϕ k (•).

Let ∇ n • Jn (x) def =
⟨∇x n Jn(x), xn⟩ ∥xn∥ be the directional derivative of Jn (•) along x n . Also, observe that under Assumption 2, the following inequality holds:

∇ n • Jn (x n , x -n ) ≤ λ + ϕ √ 2 κ + ξ ∥ x∥ -Č∥x n ∥, ∀n ∈ N (11)
where x = ( x τ ) τ with x τ = max We observe that f n (.,

x -n ) is a C ∞ function on R * + 2
. It is also well-defined on ∂(R * + ) 2 \ {0}. Moreover, around this set, it is continuous and bounded thus one can extend it by continuity at 0 by the value 0. However, it is not differentiable on ∂(R * + ) 2 . Therefore, we can not apply directly standard results from game theory which require that the players' utility functions are defined on a closed-convex domain and differentiable. The analysis of the one-shot game will therefore require to deal, first, with the degenerate case of the boundary and, second, to develop theoretical tools to study objective functions defined on open unbounded domains.

Analysis of the One-Shot Game

Let U def = n U n . We define G def = N , U, ( Jn ) n as the one-shot (static) game involving a set N of N armed Nations. For all n ∈ N , Nation n's utility function is derived from (4), with target value satisfying Assumption 2. The security functions are defined in ( 5), and we assume quadratic cost functions, c S n (•), ∀n:

c S n (x n ) = 1 2 C conv n (x conv n ) 2 + 1 2 C nuc n (x nuc n ) 2 ,
where C conv n , C nuc n > 0 are storage marginal costs for conventional and nuclear weapons respectively, assuming c D n (0) = 0. Nation n's utility function takes the form:

Jn (x n , x -n ) = f n (x) -c S n (x n ).
We will analyze the outcome of the noncooperative game G relying on the classical Nash Equilibrium, as solution concept.

Definition 2 (Nash Equilibrium). A Nash Equilibrium x * = (x * n ) n of G is a vector of power drawn from the stockpiles of weapons, such that Jn (x * ) ≥ Jn (x n , x * -n ), ∀x n ∈ U n , ∀n ∈ N .

We recall below the definition of a concave game.

Definition 3 (Concave N -player game). Let E def = N n=1
E n be a product of Euclidean spaces (E n ) n . Let U ⊂ E be a convex subset of E and for all n ∈ N assume the utility functions Jn : U → R are continuous. The game G, where each player n ∈ N solves the parametrized optimization problem:

max xn∈En Jn (x n , x -n ) s.t. x ∈ U, ∀n ∈ N , is called a concave game if for all n ∈ N , x n → Jn (x n , x -n ) is a concave function for each fixed value of x -n such that x ∈ U. If U is compact, then it is called a compact-concave game.
A classical result [START_REF] Rosen | Existence and Uniqueness of Equilibrium Points for Concave N-Person Games[END_REF]Theorem 1] ensures the existence of Nash Equilibria for concave N -player games whose joint strategy set is a compact convex. In the follow up, we extend this result to the case where: (i) the joint strategy set is a closed convex; and, (ii) the players' utility functions satisfy a coercivity property. Finally, we prove the existence of an interior point Nash Equilibrium for the N -player static game G.

Remark 3. Note that contrary to the finite-horizon game involving a finite number of time steps in the dynamic game, decision variables cannot be normalized in the one-shot game. Indeed, as we study the equilibria in the long-run, i.e., steady states, without requiring a priori budget constraints, there is no reason that Nash Equilibria take bounded values when time goes to infinity, e.g., in the case of a security spiral.

The following proposition deals with the degenerate cases where, at a Nash Equilibrium, a Nation would disarm and give up a weapons class. We show that this kind of solution requires all Nations to disarm for the same weapons class.

Proposition 2 (Disarmament). Let x be a Nash Equilibrium. Let τ ∈ {nuc, conv} and n ∈ N . The following statements hold true:

1. If x τ σ(n) = 0, then for all m ∈ σ(n), x τ m = 0; 2. If x τ n = 0, then for all m ∈ N , x τ m = 0.
Proof. To prove the first statement, let

x τ σ(n) = 0. Then, for all m ∈ σ(n), 0 ≤ x τ m ≤ max ℓ∈σ(n) x τ ℓ ≤ x τ σ(n) κ -= 0, by Assumption 2.
Therefore, ∀m ∈ σ(n), x τ m = 0. Now, we want to prove the second statement. First, let us show that if x τ n = 0, then ∀m ∈ σ(n), x τ m = 0 by contraposition. Assume that there exists m ∈ σ(n) such that x τ m > 0, which implies x τ σ(n) > 0 using the first statement. Now, as x τ σ(n) > 0, a direct computation yields a lower bound on the gradient of the security function at

(x nuc n , x conv n ) ∈ (R + ) 2 : ∂fn ∂x τ n (x n , x -n ) ≥ α τ ϕ x τ σ(n)
ξn > 0, because ϕ(•) is strictly increasing and ϕ(0) = 0.

Note that, at x τ n = 0,

∂c S n ∂x τ n (x n ) = C τ n x τ n = 0. Therefore, ∂ Jn ∂x τ n (x n , x -n ) = ∂f n ∂x τ n (x n , x -n ) - ∂c S n ∂x τ n (x n ) = ∂f n ∂x τ n (x n , x -n ) > 0.
Hence, all strategies with x τ n = 0 are dominated, i.e., if x is a Nash Equilibrium, we have that

x τ n > 0. Therefore, if x τ n = 0, then ∀m ∈ σ(n), x τ m = 0. Now, assume that x τ n = 0. Then, by induction, ∀k ∈ N, ∀m ∈ σ k (n), x τ m = 0. As N is a strategic relationship class, N = σ |N | (n). Therefore, x τ m = 0. ⊓ ⊔ Let U † def = U ∩ (R * + ) 2N and U † n def = U n ∩ (R * + ) 2 .
As a consequence of Proposition 2, there exists a trivial Nash Equilibrium which coincides with a "general and complete disarmament" strategy (x = 0). In the following sections, we prove the existence and uniqueness of interior points Nash Equilibria x * ∈ U † . An interior point Nash Equilibrium does not lie on the boundary of (R * + ) 2 .

Existence of a Nash Equilibrium

It is well-known that every compact-concave game admits a Nash Equilibrium.

As the utility functions of the game G are concave in their own strategy space, we can build a compact concave game G ′ by imposing constraints to G, ensuring the existence of a Nash Equilibrium for G ′ . Let U ′ ⊂ U be a compact convex.

Lemma 2. Assume that a Nash Equilibrium is reached at an interior point of U ′ . Then, since the players' utility functions are concave in their own strategy space, it is also a Nash Equilibrium for the game with strategy space U.

Proof. The proof relies directly on the concavity of the objective functions.

Using this property, we can show under further assumptions on the asymptotic behavior of the utility functions that the boundedness of U is not required. Definition 4. Let Ψ : U ⊂ E → R be a continuous function. We say that Ψ (•) is uniformly coercive in x n over E n if and only if there exists r n > 0 and g :

E n → R such that: ∀x ∈ U, ∇ n • Ψ (x) ≥ g(x n ), ∀y ∈ E n , ∥y∥ ≥ r n =⇒ g(y) > 0.
where

∇ n • Ψ (x) = ⟨∇x n Ψ (x), xn⟩ ∥xn∥ is the directional derivative of Ψ (•) along x n .
Using [START_REF] Herz | Idealist Internationalism and the Security Dilemma[END_REF] and that ϕ k (•) are bounded, we check that the opposite of objective functions -Jk n (•), ∀n are uniformly coercive over their own strategy space. Proposition 3. Let G = (N , U, (Ψ n ) n ) be a concave game. Assume U ⊂ E is closed and -Ψ n (•) is uniformly coercive over its own strategy space U n , ∀n. Then, G admits a Nash Equilibrium.

Proof. Let B n (r) be the closed ball of radius r > 0 in E n . Let B(r

) def = N n=1 B n (r).
Hence, Γ (r) def = B(0, r) ∩ U ⊂ U is the intersection of two closed convex sets and is bounded, thus it is a compact convex. Then, G is a compact concave game over Γ (r) and admits a Nash Equilibrium, denoted by x * . Now, for all n ∈ N , -Ψ n (•) is uniformly coercive over E n . Hence, there exists r n > 0 and

g n (•) such that ∀x ∈ U, -∇ n • Ψ n (x) ≥ g(x n ), ∀y ∈ E n , ∥y∥ ≥ r n =⇒ g(y) > 0.
Set r > max n {r n }. If there existed n ∈ N such that x n ∈ ∂B n (r), then the optimality conditions would write:

∇ n • Ψ n (x) ≥ 0 with ∥x n ∥ = r, Now, ∥x n ∥ = r ≥ r n thus ∂Ψ ∂n (x) ≤ -g n (x n ) < 0,
which is a contradiction. Therefore, for r > 0 large enough, Nash Equilibria of G are either interior points of Γ (r) ⊂ U or boundary points of U. Notice that in the former case, by Lemma 2, they are also interior points Nash Equilibria for the game with strategy space U. Therefore, the game G admits a Nash Equilibrium.

⊓ ⊔

Proposition 4. There exists a Nash Equilibrium solution of G Proof. From Proposition 3 that for all k > 0, the game G k def = N , U, ( Jk n ) n admits a Nash Equilibrium x * k . Thus, the directional derivative of Jk n (•) at x * k must be non-negative. For all ζ > 0, let

F (ζ) = λ + ϕ √ 2 κ+ Č ξ
ζ . From [START_REF] Herz | Idealist Internationalism and the Security Dilemma[END_REF], we infer that z ≤ F (z) where z = Č x * k . As ϕ(•) is strictly increasing, for all p ∈ N, z ≤ F p (z). Now, the assumptions on ϕ imply that F has a unique fixed point z 0 over R * + that depends only on ϕ, λ, κ+ Č ξ and that F p (z) converges to z 0 . Therefore, z ≤ z 0 . Let r > z 0 / Č and k > √ 2 κ+ ξ r. Hence, one can check that the games Gk = N , U ∩ B(r), ( Jk n ) n and G = N , U ∩ B(r), ( Jn ) n are identical. Now, let x * a Nash Equilibrium of G. As ∥x * ∥ ≤ z/ Č ≤ z 0 / Č < r, x * does not lie on ∂B(r) thus it is a Nash Equilibrium of G.

⊓ ⊔ Theorem 1. The game G admits an interior point Nash Equilibrium x * ∈ U † under the sufficient conditions:

C nuc n ξ n < 2κ -α nuc n , C conv n ξ n < κ -α conv n . Proof. Let ε > 0 and Γ ε = n,τ {x τ n ≥ ε} ⊂ U † . From Proposition 3, as Γ ε is a closed convex set, the game G ♯ def = N , Γ ε , ( Jn )
n admits a Nash equilibrium x * . Reasoning by contradiction, suppose that x * ∈ ∂Γ ε . This means that there exists n ∈ N and τ ∈ {nuc, conv} such that x * τ n = ε. Hence, using first order Taylor-Lagrange expansion, at least one of the following inequalities holds true: In the rest of the paper, we assume that Assumption 3 holds.

∂ Jn ∂x conv n (x * ) ≥ κ -α conv -C conv n ξ n ε ξ n + o(ε), ∂ Jn ∂x nuc n (x * ) ≥ 2κ -α nuc -C nuc n ξ n ε ξ n + o(ε). If G parameters are chosen such that κ -α conv > C conv n ξ n , 2κ -α nuc > C nuc n ξ n , for ε small enough, ∂ Jn
Proposition 5. The game G † is strongly monotone.

Proof.

Let F n def = ∂ Jn(x) ∂x nuc n ∂ Jn(x) ∂x conv n T
be the gradient of player n's utility with respect to its own actions. We get: ∀x n ∈ U, ∂ Jn(x)

∂x conv n = α conv ϕ(x conv σ(n) /ξ n ) - C conv n x conv n
, which implies that Jn (•) reaches its maximum in the variable

x conv n at (x conv n ) * = α conv C conv n ϕ x conv σ(n) ξn , if (x conv
∂x nuc n = α nuc ϕ x conv σ(n) +x nuc σ(n) ξn + λ n τ α τ exp -λ n x nuc n x τ σ(n) -C nuc n x nuc n .
Consider the pseudo-Hessian matrix H nuc of the players' utilities considering only nuclear weapons where each n th row and m th column component is given as

H nuc n,m = ∂F nuc n ∂x nuc m
, n, m ∈ N , i.e., in details:

H nuc n,m =    -λ 2 n τ α τ x τ σ(n) exp -λ n x nuc n x τ σ(n) -C nuc n if m = n, α nuc ξn ϕ ′ x conv σ(n) +x nuc σ(n) ξn + λ 2 n α nuc x nuc n (x nuc σ(n) ) 2 exp -λ n x nuc n x nuc σ(n) ∂x nuc σ(n) ∂x nuc m if m ̸ = n.
Thus -H nuc is a Z-matrix as its-off diagonal entries are negative. Under Assumption 3, we observe that -H nuc is strictly diagonally dominant thus -H nuc is an M -matrix, i.e., a Z-matrix with eigenvalues whose real parts are nonnegative [START_REF] Young | Iterative solution of large linear systems[END_REF]. Thus, H nuc is negative definite. Furthermore, letting

F def = col (F n ) n , as ∂Fn ∂x conv n
is constant for all n ∈ N , the inequality (y-x) T F (y)-F (x) < 0, ∀x ̸ = y follows from [START_REF] Rosen | Existence and Uniqueness of Equilibrium Points for Concave N-Person Games[END_REF]Theorem 6] and the game G † is strongly monotone.

⊓ ⊔ Theorem 2. The game G has a unique interior point Nash Equilibrium x * ∈ U † .

Proof. Let us consider that there exists two Nash Equilibria x * and x * * solutions of the game G † . As both x * and x * * are Nash Equilibria they must satisfy the stationarity condition. Mutliplying the first order condition with y -x * at point x * and y -x * * at point x * * , we get:

(y -x * ) T F (x * ) = 0, ∀y ∈ U † , (12a)
(y -x * * ) T F (x * * ) = 0, ∀y ∈ U † . (12b)
Taking (12a) in y = x * and (12b) in y = x * * and summing the two above equations, we get: (x * * -x * ) T (F (x * * ) -F (x * )) = 0, which contradicts the fact that the game G † is strongly monotone from Proposition 5. Therefore, the game G † admits a unique Nash Equilibrium. ⊓ ⊔

Distributed Nash Equilibrium Seeking

When more than two Nations are involved, algorithmic methods need to be developed to compute Nash Equilibria.

Proposition 6. The pseudo-gradient F (x) of the game G is Lipschitz continuous in x ∈ U † .

Proof. The proof relies on the Triangle Inequality and Mean Value Theorem. Due to space limit, it is omitted.

⊓ ⊔

We assume that the Nations are in a full information feedback setting.

Assumption 4 Each player n ∈ N can get the decisions of all its opponents x -n ∈ m̸ =n U m .

Under Propositions 5 and 6, and Assumption 4, various gradient-based algorithms can be proposed to compute the interior point Nash Equilibrium solution of G and are proved to converge [START_REF] Facchinei | Finite-Dimensional Variational Inequalities and Complementarity Problems[END_REF]. We will implement a regularization algorithm for monotone game, called proximal point method (PPM). PPM is an alternative method to distributed gradient descent schemes, whose interest lies in the fact that it avoids having to coordinate the players in their steplength choice. At iteration l, x l is the solution of a Variational Inequality (VI) of the type: (y -x) T F l (x) ≤ 0, ∀y ∈ U, with F l (x) = F (x) -θ(x -x l ), and θ > 0 is a regularization parameter which implicitly determines iteration bounds to reach a prescribed error level [START_REF] Nesterov | Dual extrapolation and its applications to solving variational inequalities and related problems[END_REF]. In this setting, the regularization parameters are required to be the same for all the players. Extensions of iterative proximal point method for monotone games where each player can independently select and adapt its algorithm parameter after each iteration [START_REF] Kannan | Distributed Computation of Equilibria in Monotone Nash Games via Iterative Regularization Techniques[END_REF] exist, but will not be considered in the current version of the work. In practice, the PPM algorithm can be implemented by solving at each iteration l:

x l+1 n = arg max u∈Un Jn (u, x l -n ) -θ∥u -x l n ∥ 2 , ∀n ∈ N .
4 Analysis and Simulations in Case of a Duopoly

Analytical Analysis

In the case of a duopoly (N = 2), we provide an analytical characterization of the interior point Nash Equilibrium of G. We define

N def = {(n, m) ∈ N 2 | n ̸ = m}.
For all (n, m) ∈ N , we let x m denote x -n and σ(n) def = {m}. Then, the interior point Nash Equilibrium of G is obtained by solving the following system:

max x∈(R * + ) 2 Jn (x n , x m ), ∀(n, m) ∈ N ,
leading to the following first order (necessary) optimality conditions:

∂ Jn ∂x conv n (x n , x m ) = 0, ∂ Jn ∂x nuc n (x n , x m ) = 0, ∀(n, m) ∈ N ,
which can be written explicitly as the following system of equations:

∀(n, m) ∈ N , C conv n x conv n -α conv ϕ x conv m ξ n = 0, (13a)
C nuc n x nuc n -λ n τ exp - λ n x nuc n x τ m -α nuc ϕ x nuc m +x conv m ξ n = 0. (13b) Let (n, m) ∈ N . Define R conv n : x → α conv C conv n ϕ x ξn . Then one can rewrite (13a) as x conv n = R conv n (x conv m ).
Hence, when dealing with two players, the Nash Equilibrium components (x conv n ) n are found at the intersection (χ 2 , χ 1 ) of curves (u, R conv

Y -λ n exp - λ n ρ n Y X + exp - λ n ρ n Y µ n -α nuc ϕ(X +µ n ) = 0, ∀n ∈ N . ( 15
)
Proposition 7. Equation (15) has a unique solution.

Proof. Let ρ, µ, λ > 0 and α ∈ (0, 1). For all X, Y ≥ 0, define

h(X, Y ) =    Y -λ exp -λ ρ Y X -λ exp -λ ρ Y µ -αϕ(X + µ) if X ̸ = 0, Y -λ exp -λ ρ Y µ -αϕ(µ) else.
For all X ≥ 0, Y → h(X, Y ) is continuous and strictly increasing over R + . As h(X, 0) ≤ -λ < 0 and lim is negligible, i.e., we assume that preemptive nuclear strikes onto conventional targets achieve little security gain when compared to nuclear targets. Setting it to 0 yields the following equation:

Y -λ n exp - λ n ρ n Y X -α nuc ϕ(X + µ n ) = 0, ∀n ∈ N . (16)
In the following, we provide a closed-form expression of the solution of ( 16), which relies on the special function W(•), defined as the principal branch of the Lambert W-function [7] , solution to we w = r, r ≥ 0. W(•) cannot be expressed in terms of elementary functions, although some approximations, bounds, and integral representations, e.g., W(x) = 1 π π 0 ln 1 + x sin t t e t cot t dt are well-known. For all X, ρ, λ, µ > 0, let G(X; ρ, λ, µ) def = α nuc ρ ϕ(X + µ) + 1 λ XW λ exp -α nuc λ ρ ϕ(X + µ) X .

Fig. 1. In the special case of the duopoly, the interior point Nash Equilibrium is found graphically at the intersection point of the best-reply maps (R nuc n )n=N1,N2 of each Nation as defined in [START_REF] Majeski | Arms races as iterated prisoner's dilemma games[END_REF]. The model's parameters are given in Table 1.

Proposition 8. The functional equation [START_REF] Lichbach | Stability in Richardson's Arms Races and Cooperation in Prisoner's Dilemma Arms Rivalries[END_REF] with unknown X → Y (X) admits a unique solution on C ∞ (R * + ) which satisfies :

Y = ρ n G(X; ρ n , λ n , µ n ). (17)
Proof. Let n ∈ N . For all X, Y > 0, define w = λn ρnX (Y -α nuc ϕ(X + µ n )) and r = λ n exp -α nuc λn ρn ϕ(X+µn) X

. One has the expression

w -r exp(-w) = λ n ρ n X Y -λ n exp - λ n ρ n Y X -α nuc ϕ(X + µ n ) .
Consequently, (X, Y ) ∈ (R * + ) 2 satisfies (16) if and only if we w = r, i.e., if and only if w = W(r), which writes as [START_REF] Lichbach | When Is an Arms Rivalry a Prisoner's Dilemma?: Richardson's Models and 2 × 2 Games[END_REF].

⊓ ⊔

From Proposition 8, we infer the closed-form expressions of the best-replies:

x nuc n = R nuc n (x nuc m ) = ξ n G x nuc m ξ n ; C nuc n ξ n , λ n C nuc n ξ n , ∀(n, m) ∈ N . (18)

Numerical Results and Discussions

As shown in Figure 1, Nash Equilibria can be numerically obtained when computing the intersection points of the curves drawn by the best-reply maps R nuc n computed as with [START_REF] Majeski | Arms races as iterated prisoner's dilemma games[END_REF] and α conv = α nuc = 1 2 . Note that the security and the utility of both Nations with mixed armaments in a mutual deterrence relationship are greater than with only conventional weapons. Indeed, from Table 1 we can compute the efficiency gap between the social optimum of disarmament and the interior point Nash Equilibrium. With only conventional armaments the efficiency gap is of 108 whereas with mixed armaments it drops to 74, representing an inefficiency reduction of 31%. This can be interpreted as a stabilizing effect of nuclear armaments.

Depending on the observed efficiency loss, it might be interesting, in an extension of our work, to provide a method for specifying a Pareto dominating solution that depends on the Nations' threat thresholds.

Conclusion

We formulate a security game in a context of mixed armament acquisition, involving a finite set of Nations in strategic relationship. Our model incorporates key elements such as resource allocation to R&D and storage, defense capabilities, and geopolitical considerations, to provide a realistic representation of the arms procurement process. To analyze the long-term solution of the dynamical model, we study the steady states of the dynamical game by solving the associated one-shot game. On the theoretical side, we provide two important contributions to the state of the art. First, we extend classical results about compact-convex games to unbounded convex games relying on the coercivity property of the utility functions. Second, our results showcase the stabilizing effect of nuclear armaments and provide possible guidelines for current treaties refinements or for the understanding of various arms competition scenarios, including multilateral arms competitions and asymmetric weaponry.

Future research could focus on studying other identified factors such as alliances and the role of emerging technologies. Additionally, empirical studies and use case analyses would be valuable to validate the model outcomes and assess its real-world applicability. Finally, we anticipate that our contributions will serve as a valuable tool for policymakers, analysts, and researchers, assisting in the development of effective arms control measures and promoting stability in international relations.

Definition 1 (Lemma 1 .

 11 Strategic relation). We say that two Nations n and m are in strategic relationship and write n ∼ m if a (possibly degenerate) sequence of hostility relations connects n to m, i.e., if there exists k ∈ N such that m ∈ σ k (n). The strategic relation is an equivalence relation. Proof. By definition, the strategic relation is symmetric. It is transitive because ∀n

  Čτ m , λ = max m∈N λ m , and ξ = max m∈N ξ m . An analogous inequality holds for Jk n (•), ϕ k (•) .

3 . 2

 32 ) > 0. It contradicts the necessary optimality condition∂ ∂x τ n Jn (x * ) ≤ 0 thus x * ∈ U † . Hence, G † def= N , U † , ( Jn ) n admits the same Nash Equilibrium x * , which is also an interior point Nash Equilibrium of G. ⊓ ⊔ Uniqueness of the Interior Point Nash Equilibrium Assumption 3 ∀n ∈ N , 1 -

n)

  * is admissible, or, else, at the border of the interval of definition of x conv n . Similarly, for Nation n's other decision variable, we get: ∂ Jn(x)

1 (

 1 u)) u>0 and (R conv 2 (v), v) v>0 . Consider the following functional equations with unknowns R nuc n : x → y: C nuc n y -λ n exp -∀n ∈ N . (14) Then (13b) is equivalent to R nuc n (x nuc m ) = x nuc n by setting x = x nuc m and y = x nuc n . Let µ n = χn ξn , ρ n = C nuc n ξ n and the change of variables (X, Y ) = ( x nuc ξn , C nuc n y nuc ), then (14) recasts as:

Y

  →+∞ h(X, Y ) = +∞, it is one-to-one from R + to a set containing 0. Thus it has a unique zero on R + and (15) has a unique solution. ⊓ ⊔ Due to limited space and for the sake of simplicity, we assume in what follows that λ n exp -λn ρn Y µn

= 5 ×

 5 ) and security (italic-black) functions at Nash Equilibrium. Parameters: ξN1 = 200, ξN2 = 150, C conv n = C nuc n 10 -3 , λn = 0.7, ∀n ∈ {N1, N2}.

Table 1 .

 1 Utility (bold-red

		Nash Equilibrium Strategy Conventional-only Mixed armaments
		(x conv n	, x nuc n )	Utility Security Utility Security
	Nation 1	(52, 149)	-53	6	-45	14
	Nation 2	(60, 141)	-55	7	-29	33

Currently, while it is already enforced by parties, the CTBT has not yet entered into force because 8 out of 44 Annex

Nations have not ratified it.


Thomas Houet
email: thomas.houet@univ-tlse2.fr

Peter H Verburg

Thomas R Loveland

Monitoring and modelling landscape dynamics

Introduction

Changes in land cover and land use are among the most pervasive and important sources of recent alterations of the Earth's land surface. Land changes significantly affect key aspects of Earth System functioning, for example in contributing to local and regional climate change as well as to global climate warming, impacting biodiversity and water quality, or increasing soil degradation [START_REF] Vitousek | Human domination of Earth's ecosystems[END_REF][START_REF] Stohlgren | Evidence that local land use practices influence regional climate and vegetation patterns in adjacent natural areas[END_REF][START_REF] Houghton | The US carbon budget: Contributions from land-use change[END_REF]. Landscape dynamics studies integrating human-environment interactions and related to environmental issues have become increasingly important. Over the years these studies moved away from a focus on detecting and identifying land use and land cover changes [START_REF] Lambin | The causes of land-use and land-cover change: moving beyond the myths[END_REF][START_REF] Loveland | Introduction: Special Issue on Global Land Cover Mapping and Validation[END_REF][START_REF] Loveland | A strategy for estimating the rates of recent United States land-cover changes[END_REF] and understanding driving forces of landscape changes [START_REF] Bürgi | Driving forces of landscape changecurrent and new directions[END_REF][START_REF] Antrop | Why landscapes of the past are important for the future? Landscape and Urban Planning[END_REF] to modelling present land systems for predicting land cover changes (Veldkamp et Lambin 2001;[START_REF] Corgne | Land cover change prediction with a new theory of plausible and a paradoxical reasoning[END_REF][START_REF] Hepinstall | Predicting land cover change and avian community responses in rapidly urbanizing environments[END_REF] and exploring possible futures of landscapes [START_REF] Verburg | Land use change modelling: current practice and research priorities[END_REF][START_REF] Kok | Integrated Assessment of the land system: The future of land use[END_REF]. Such studies were and are still largely supported by national and international global environmental change programs such as the Land Use and Cover Change program [START_REF] Lambin | Land-use and land-cover change (LUCC): Implementation strategy[END_REF], the Global Land Project [START_REF] Glp | Science Plan and Implementation Strategy[END_REF], and the US Climate Change Research Program [START_REF] Loveland | Chapter 6: Land Use/Land Cover Change[END_REF]. They contribute to improving the understanding of natural-human interactions and they advance monitoring and modelling of landscape dynamics needed to meet the challenges of land change science [START_REF] Turner | The emergence of land change science for global environmental change and sustainability[END_REF].

Monitoring landscape and land use/cover changes is considered as an essential first step to assist the identification of driving forces [START_REF] Bürgi | Driving forces of landscape changecurrent and new directions[END_REF]) and provide the data needed for modelling. Projection of futures landscape changes requires and understanding and integration of past landscape trends, current land change processes and feedbacks, and the incorporation of plausible assumptions or scenarios. Obviously, monitoring and modelling of landscape dynamics strongly depends on the scale and objectives of the planned applications. This, in turn, determines the different techniques needed. Thus, landscape change models should be appropriate for simulating identified social, economic, and ecological processes, and their dynamics and interactions that shape landscapes. Land change models that are founded in land use theory and that consider land use history is also important and offer new opportunities for interdisciplinary research.

Landscape dynamics studies were and are often driven by disciplines other than landscape ecology although techniques and concepts from landscape ecology have consistently contributed to landscape change studies [START_REF] Naveh | Some remarks on recent developments in landscape ecology as a transdisciplinary ecological and geographical science[END_REF][START_REF] Bürgi | Integrative methods to study landscape changes[END_REF]Antrop 2002). For example, the remote sensing community is regularly involved in monitoring landscape dynamics. Various disciplines, such as geography, offer diversified points of view and approaches that enable understanding the dynamics of landscape structure and function at different spatial and temporal scales.. The need for the integration of landscape history, including past/current dynamics and feedbacks has been increasingly recognized [START_REF] Caspersen | Contribution of land-use history to carbon accumulation in U.S. forest[END_REF][START_REF] Nabuurs | Temporal evolution of the European forest sector carbon sink from 1950 to 1999[END_REF][START_REF] Antrop | Why landscapes of the past are important for the future? Landscape and Urban Planning[END_REF][START_REF] Rhemtulla | Why history matters in landscape ecology[END_REF][START_REF] Claessens | Modelling interactions and feedback mechanisms between land use change and landscape processes[END_REF][START_REF] Gillson | Landscapes in time and space[END_REF]. This special issue focuses on multidisciplinary research in land change science that illustrate how innovative data integration, analytical methods and techniques, and perspectives from geography, ecology, agronomy and computer sciences can help foster cross-disciplinary research with landscape ecologists [START_REF] Hobbs | Future landscapes and the future of landscape ecology[END_REF].

This special issue adds to the existing literature on the state of the art on landscape dynamics studies [START_REF] Turner | The emergence of land change science for global environmental change and sustainability[END_REF][START_REF] Milne | Integrated modelling of natural and social systems in land change science[END_REF][START_REF] Kok | Integrated Assessment of the land system: The future of land use[END_REF]Lambin and Geist 2006). It also illustrates both current advances and challenges for monitoring and modelling landscape dynamics. Selected papers in this volume illustrate key dimensions of land change science, highlight possible new directions of research such as the reinforcement of multi-scale [START_REF] Verburg | Simulating feedback in land use and land cover change models[END_REF] or human/nature [START_REF] Milne | Integrated modelling of natural and social systems in land change science[END_REF] modelling approaches, and present integrated approaches to project future landscape changes [START_REF] Kok | Integrated Assessment of the land system: The future of land use[END_REF]. Most of the papers in this issue were selected from presentations at the international symposium "Spatial landscape modelling: from dynamic approaches to functional evaluations" that took place in Toulouse (France) from 3rd to 5th of June 2008 (http://w3.geode.univ-tlse2.fr/rtpmodelisation/).

Monitoring and modelling landscape dynamics: current practices, limits and new directions

Monitoring landscape dynamics is essential for understanding the complex interactions between social, environmental and geophysical processes [START_REF] Munroe | Issues in spatially explicit statistical land use/cover change (LUCC) models: Examples from western Honduras and the Central Highlands of Vietnam[END_REF]. Land use and land cover change studies are often based on information on landscape structure and composition at different spatial and temporal resolution that is derived from remotely sensed data [START_REF] Loveland | Development of a Global Land Cover Characteristics Database and IGBP DISCover from 1-km AVHRR Data[END_REF][START_REF] Goetz | Crisis in Earth observation[END_REF][START_REF] Coops | Demonstration of a satellite-based index to monitor habitat at continental-scales[END_REF]. However, there are several challenges associated with developing the remote sensing inputs needed to understand landscape dynamics. The first challenge is to maintain continuous time series of low cost imagery [START_REF] Turner | The emergence of land change science for global environmental change and sustainability[END_REF]. The planned 2012 launch of the Landsat Data Continuity Mission, for example, extends the Landsat record an additional 5-10 years. New data acquisition systems with high spatial and temporal resolutions (i.e. Formosat, VENµS missions) are also extremely promising sources of remotely sensed data that will further enable monitoring landscape dynamics at fine spatial and temporal scales. Improvement of techniques combining multi-scale / multi-sensors / multi-source data studies show potential to rescale land cover data [START_REF] Gardner | A new approach for rescaling land cover data[END_REF] in order to improve common landscape monitoring methods. New direction to better understand landscape dynamics include moving from land cover to land use systems such as the detection of cropping systems. The paper of Lazrak et al (this issue) in this volume lays out an innovative use of data mining techniques to detect landscape regularities over time.

This special issue also presents new directions in modelling landscape dynamics. Agent-based models have primarily been used to simulate local land use and land cover changes processes with a focus on decision making [START_REF] Le | Land-use dynamic simulator (LUDAS): a multi-agent system model for simulating spatio-temporal dynamics of coupled humanlandscape system. I. Structure and theoretical specification[END_REF][START_REF] Matthews | Agent-based land-use models: a review of applications[END_REF][START_REF] Parker | Multi-agent systems for the simulation of land use and land use change: a review[END_REF]Bousquet and Le Page 2001). Valbuena et al (this issue) present an agent-based modelling approach that is also applicable at the regional scale and links individual decision making to changes in landscape structure. Innovation is not only found in new or improved models, but also in the combination of existing models. Gaucherel et al (2009) reports on research in which multiple process-based models are coupled to evaluate ecological and aesthetic impacts of landscape changes. Verburg et al (this issue) illustrate the application of multiple models at different scales to explore possible landscape trajectories in Europe. Along with with Valbuena et al (this issue), Sohl et al (this issue), and Houet et al (this issue), Verburg et al (this issue) point out the need for improved integrated land change models that connect local-to-global scales and land use pattern with land use change processes. If models are becoming more and more efficient to simulate processes at multiple scales, there is an obvious strong convergence between local scales models attempting to take into account global driving forces and regional/global models looking for finer spatial rendering at elementary landscape units [START_REF] Sohl | The FORE-SCE model: a practical approach for projecting land use change using scenario-based modeling[END_REF][START_REF] Castella | Combining top-down and bottom-up modelling approaches of land use/cover change to support public policies: Application to sustainable management of natural resources in northern Vietnam[END_REF][START_REF] Verburg | A multi-scale, multi-model approach for analyzing the future dynamics of European land use[END_REF].

The papers in this issue also highlight land change science challenges associated with improved prediction of land use change (Lambin et Veldkamp 2001), better integration of the land system [START_REF] Kok | Integrated Assessment of the land system: The future of land use[END_REF], and the need for advanced scenario-based studies [START_REF] Verburg | Simulating feedback in land use and land cover change models[END_REF]. Sohl et al (this issue) and Houet et al (this issue) each propose a framework, at regional and local scales respectively, to include landscape trends and histories, to choose, parameterize, and validate models, and to explore multiple land change scenarios. Whatever the spatial scale of such approaches, data availability remains an ongoing challenge thus reinforcing the importance of landscape monitoring using multisource data (historical maps, remotely sensed data, etc). Gibon et al (this issue) illustrates the need for an integrated and participatory approach that considers socio-ecological processes in the modelling and elaboration of scenarios. Regarding exploration of alternative land change futures, Verburg et al (this issue) assess possible future landscape changes based on contrasted scenarios. This provides a good indicator of likely future land configurations. This future dimension of landscape dynamic studies provides a helpful tool to delimit the envelope of possible landscape futures and to define the plausibility of the occurrence of futures landscape changes.

Finally, all of the papers in this special issue indicate that 'standard' monitoring and modelling techniques are not always valid or the most appropriate. Land cover monitoring does not always detect subtle changes within the landscape, e.g., changes in landscape elements and change in land management practices. Models should better represent such subtle landscape changes given the importance of such changes for the functioning of the landscape (Houet et al this issue). At the same time land use (cropping) systems that show specific landscape temporal patterns and cycles that are not easily observed from traditional land cover data, but have importance for the dynamics in the landscape (Lazrak et al this issue). Moreover, landscape dynamics studies need to go further in combining modelling approaches and techniques (Gaucherel et al this issue; Gibon et al this issue; Sohl et al this issue; Valbuena et al this issue) and projecting landscape dynamics to reduce uncertainties of the futures of landscape (Verburg et al this issue; Houet et al this issue). The temporal dimension of landscapes has to be considered as important as the spatial dimension in order to monitor, model, and assess human/nature interactions. Collectively, these considerations open new directions for research in land change science.

Acknowledgements

This special issue is one of the results of the symposium "Spatial landscape modelling: from dynamic approaches to functional evaluations" that occurred in Toulouse in June 2008, supported by the French Thematic Interdisciplinary Network on «Landscape and Environment» of the department Human and Social Sciences of CNRS (National Centre for Scientific Research), the Region Midi-Pyrénées, the Toulouse Le Mirail University and the GEODE UMR 5602. We would like to warmly thank all the reviewers that have contributed to this issue by their constructive comments.


Sergio Hernández

Mikael Feracci

Carolina Trajano De Jesus

Priscila El Kazzi

Rafik Kaci

Laura Garlatti

Clemence Mondielli

Fabrice Bailly

Philippe Cotelle

Franck Touret

Carolina Trajano De Jesus

Xavier De Lamballerie

Bruno Coutard

Etienne Decroly

Bruno Canard

François Ferron
email: francois.ferron@univ-amu.fr.

Karine Alvarez
email: karine.alvarez@univ-amu.fr

Identification of potent inhibitors of arenavirus and SARS-CoV-2 exoribonucleases by fluorescence polarization assay

Keywords: Phosphohydrolase activity, Arenaviridae, Coronaviridae, exoribonuclease activity, fluorescence polarization, screening, inhibitors, IC50, cellular assays. Highlights

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.

Introduction

Arenaviruses are emerging RNA viruses associated with fatal neurological and hemorrhagic diseases in humans. Currently there is only ribavirin as FDA-approved antiviral. For coronaviruses, in the context of the global SARS-CoV-2 pandemic, it is essential to continue to develop potent antivirals, in support of the vaccine approach.

These RNA viruses encode for proteins, forming a replication/transcription complex (RTC) that orchestrates viral replication [START_REF] Ferron | Transcription and replication mechanisms of Bunyaviridae and Arenaviridae L proteins[END_REF]Knipe and Howley, 2013;Ogando et al., 2019). The peculiarity of Arenaviridae and Coronaviridae families, is the unique viral 3'-5'-exoribonuclease (ExoN) activity described so far [START_REF] Hastie | Structure of the Lassa virus nucleoprotein reveals a dsRNA-specific 3′ to 5′ exonuclease activity essential for immune suppression[END_REF][START_REF] Snijder | Unique and Conserved Features of Genome and Proteome of SARS-coronavirus, an Early Split-off From the Coronavirus Group 2 Lineage[END_REF], The ExoN activity carried by arenavirus nucleoprotein (N) or the nsp14 protein of coronavirus participates to the suppression of the host innate immune response [START_REF] Martínez-Sobrido | Identification of Amino Acid Residues Critical for the Anti-Interferon Activity of the Nucleoprotein of the Prototypic Arenavirus Lymphocytic Choriomeningitis Virus[END_REF][START_REF] Martínez-Sobrido | Differential Inhibition of Type I Interferon Induction by Arenavirus Nucleoproteins[END_REF][START_REF] Martínez-Sobrido | Inhibition of the Type I Interferon Response by the Nucleoprotein of the Prototypic Arenavirus Lymphocytic Choriomeningitis Virus[END_REF][START_REF] Becares | Mutagenesis of Coronavirus nsp14 Reveals Its Potential Role in Modulation of the Innate Immune Response[END_REF][START_REF] Lei | Activation and evasion of type I interferon responses by SARS-CoV-2[END_REF] , moreover the nsp14 displays an additional role in maintening the genome integrity [START_REF] Bouvet | RNA 3'-end mismatch excision by the severe acute respiratory syndrome coronavirus nonstructural protein nsp10/nsp14 exoribonuclease complex[END_REF][START_REF] Ferron | Structural and molecular basis of mismatch correction and ribavirin excision from coronavirus RNA[END_REF]. The arenavirus and coronavirus ExoN belonging to the same DEDD superfamily (DED/EDh subfamily), are structurally homolog [START_REF] Zuo | Exoribonuclease superfamilies: structural analysis and phylogenetic distribution[END_REF], and characterized by the presence of four conserved acidic residues (Asp and Glu) [START_REF] Bouvet | RNA 3'-end mismatch excision by the severe acute respiratory syndrome coronavirus nonstructural protein nsp10/nsp14 exoribonuclease complex[END_REF][START_REF] Yekwa | Arenaviridae exoribonuclease presents genomic RNA edition capacity[END_REF], essential for the binding of two metal cations (Mg 2+ and/or Mn 2+ ) involved in the RNA hydrolysis mechanism [START_REF] Steitz | A general two-metal-ion mechanism for catalytic RNA[END_REF], always proceeding from the 3'-to the 5'-direction.

Because of its biological significance to the viral life cycle, ExoN is an attractive target for drug-design [START_REF] Papageorgiou | Brothers in Arms: Structure, Assembly and Function of Arenaviridae Nucleoprotein[END_REF][START_REF] Subissi | SARS-CoV ORF1bencoded nonstructural proteins 12-16: replicative enzymes as antiviral targets[END_REF].

In drug development efforts using ExoN of mammarenavirus and coronavirus as targets, assays making use of radiolabeled substrate [START_REF] Baddock | Characterization of the SARS-CoV-2 ExoN (nsp14ExoN-nsp10) complex: implications for its role in viral genome stability and inhibitor identification[END_REF][START_REF] Bouvet | RNA 3'-end mismatch excision by the severe acute respiratory syndrome coronavirus nonstructural protein nsp10/nsp14 exoribonuclease complex[END_REF][START_REF] Ferron | Structural and molecular basis of mismatch correction and ribavirin excision from coronavirus RNA[END_REF][START_REF] Saramago | New targets for drug design: Importance of nsp14/nsp10 complex formation for the 3'-5' exoribonucleolytic activity on SARS-CoV-2[END_REF][START_REF] Yekwa | Arenaviridae exoribonuclease presents genomic RNA edition capacity[END_REF][START_REF] Yekwa | Activity inhibition and crystal polymorphism induced by active-site metal swapping[END_REF] or mass spectrometry [START_REF] Scholle | Label-Free Screening of SARS-CoV-2 NSP14 Exonuclease Activity Using SAMDI Mass Spectrometry[END_REF] are available but barely adapted to HT screening. Recently, HTS assays using FRET were applied to screen SARS-CoV-2 ExoN [START_REF] Canal | Identifying SARS-CoV-2 antiviral compounds by screening for small molecule inhibitors of nsp14/nsp10 exoribonuclease[END_REF][START_REF] Chen | Synergistic Inhibition of SARS-CoV-2 Replication Using Disulfiram/Ebselen and Remdesivir[END_REF][START_REF] Rona | The NSP14/NSP10 RNA repair complex as a Pan-coronavirus therapeutic target[END_REF].

Here we present a convenient method using fluorescence polarization (FP) to assess the ExoN activity and its inhibition. The recorded FP signal is altered proportionally to the size of the fluorescent RNA probe which is hydrolyzed into smaller fragments. We validate the method on SARS-CoV-2, Lymphocytic Choriomeningitis (LCMV) and Machupo viruses (MACV) ExoN. The method is sensitive, robust, amenable to miniaturization (384 well plates) and allowed us to screen a focused library of 113 metal chelators over the three-targeted viral ExoN, validating the proof-of-concept of the assay. We selected several hit inhibitors and determined by FP their IC50. The hit compounds were tested in cellular assays.

Materials and methods

Products and reagents.

Aurintricarboxyclic acid (40) was purchased from Acros Organics. Ribavirin (82) and Favipiravir (83) were purchased from Carbosynth. The library (113 compounds) containing phenyl-DKA, piperidinyl-DKA, indolyl-DKA, benzofuranyl-DKA, triazolyl-DKA (DKA families), polyphenols (POP family), Nhydroxyisoquinoline-1,3-diones (HID family) and miscellaneous compounds have been described previously [START_REF] Saez-Ayala | Metal chelators for the inhibition of the lymphocytic choriomeningitis virus endonuclease domain[END_REF], 2018). The hit compounds 3, 4, 18, 55 and 57 have been synthesized and their synthesis will be published in due course. All compounds were resuspended in 100 % DMSO at 20 mM and stored at -20 •C. The 22-mer RNA 5'-UGACGGCCCGGAAAACCGGGCC-3' containing FAM dye at the 5'end (5'-FAM-RNA) was purchased from Microsynth AG (Switzerland).

Protein expression and purification.

All proteins were expressed in E. Coli. The LCMV and MACV ExoN plasmids used for this study were described in [START_REF] Yekwa | Arenaviridae exoribonuclease presents genomic RNA edition capacity[END_REF]. For the expression of the SARS-CoV-2 nsp10 and nsp14, the synthetic genes were purchased from Twist Bioscience (USA). The protein production methods are detailed in the supplementary data file (see also figure S1 for elution profiles).

Set-up of exonuclease activity conditions based on fluorescent polarization assay

Optimization of 5'-FAM-RNA concentration, ExoN/RNA ratio, temperature, metal ion nature and concentration

Reactions were performed in 20 µl total volume in a buffer containing 40 mM Tris (pH 8), 5 mM DTT and 2 or 5 mM MnCl2. For each 5'-FAM-RNA concentration (0.05, 0.1, 0.25, 0.5 and 1 µM), a range of concentration of ExoN was tested from a ratio of 5-fold less enzyme up to 10-fold more enzyme (0.01 to 10 µM) than 5'-FAM-RNA. For SARS-CoV-2 ExoN, the molar ratio of nsp14:nsp10 complex in the reactions was always kept at 1:4, as optimized previously [START_REF] Bouvet | RNA 3'-end mismatch excision by the severe acute respiratory syndrome coronavirus nonstructural protein nsp10/nsp14 exoribonuclease complex[END_REF] to enhance nsp14 activity.

To find the optimal temperature, metal ion nature and concentration, the 5'-FAM-RNA concentration was kept at 0.1 µM and ExoN concentration ranged from 0.1 to 1.6 µM. The activity was tested in the presence of MnCl2 or MgCl2 at 1, 2 or 5 mM. The FP signal was recorded either at 25°C or at 37°C. Negative controls tested used consisted in the replacement of MnCl2 by CaCl2 or, depletion of the metal ion or the enzyme. The reactions started by the addition of 5'-FAM-RNA and the fluorescence polarization (FP) was read in the Pherastar FSX (BMG Labtech) using the 480 nm excitation and 520 nm emission filter, during 30 minutes at 25°C or 37°C, every 30 seconds. The gain was set up using the negative control which contained the 5'-FAM-RNA in the presence of heat-denatured nuclease. After 30 minutes, the reactions were stopped by the addition of an equal volume of loading buffer (8M urea containing 10 mM EDTA) and the digestion products were then loaded in 7 M urea containing 20% (wt/vol) polyacrylamide gels (acrylamide/bisacrylamide ratio 19:1) buffered with TBE and visualized using a Fluorescent Image Analyzer Typhoon (GE Healthcare).

Time course of ExoN assay

Reactions of 20 µl total volume were performed in the buffer mentioned above, with 2 or 5 mM MnCl2 for SARS-CoV-2 or arenaviruses ExoN, respectively. The 5'-FAM-RNA concentration was fixed at 0.1 µM. The SARS-CoV-2 nsp14/nsp10 ExoN complex concentration was ranged from 0.2 to 1 µM and pre-incubation at room temperature for 5 minutes was performed to allow the complex formation. The LCMV and MACV ExoN concentration was ranged from 0.1 to 1.6 µM. The reaction started with the addition of 5'-FAM-RNA and The FP signal was recorded as mentioned previously. At end point, an aliquot of the sample was examined in urea PAGE as mentioned above. The assays were done in triplicate.

Screenings and IC50 determination.

Screenings

The screenings at 5 µM and 20 µM inhibitor concentration were performed in 384-wells with flat bottom Nunc plates, in 20 µL total volume reaction. Reactions were performed in a buffer containing 40 mM Tris (pH 8), 5 mM DTT, with 2 or 5 mM MnCl2 for SARS-CoV-2 or arenaviruses ExoN, respectively. The 5'-FAM-RNA and ExoN concentrations were fixed at 0.1 µM. and 0.4 µM, respectively. Compounds (1-113) were added to the reaction with a final concentration of 5 µM and 20 µM in 5% DMSO. 8 reactions mix with ExoN were used as positive controls and 8 reactions mix with heat-denatured ExoN were used as negative controls.

Reactions were initiated by adding the 5'-FAM-RNA and the FP signal was recorded as mentioned previously.

To calculate the percentage of inhibition, a time correction was applied for the delayed initiation of the reaction due to the use of a multichannel pipette. The percentage of inhibition at a given time was calculated as follows:

𝑖𝑛ℎ𝑖𝑏𝑖𝑡𝑖𝑜𝑛% = 100 × (𝐹𝑃-𝑎𝑣𝑒𝑟𝑎𝑔𝑒𝑜𝑓𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒𝑐𝑜𝑛𝑡𝑟𝑜𝑙𝑠) (𝑎𝑣𝑒𝑟𝑎𝑔𝑒𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒𝑐𝑜𝑛𝑡𝑟𝑜𝑙𝑠-𝑎𝑣𝑒𝑟𝑎𝑔𝑒𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒𝑐𝑜𝑛𝑡𝑟𝑜𝑙𝑠)
where FP correspond to the fluorescent polarization signal of a compound.

The time selected for doing this calculation was the time when the signal of the positive control reached the plateau (30 minutes). To assess the robustness of the assay we calculate the z' value.(Formula and representative scatter plot in supplementary figure S3).

IC50 determination

Reactions were performed in the same experimental conditions as used for the screenings. The compound concentrations varied from 0.2 to 12.5 µM for LCMV ExoN, from 0.4 to 25 µM for MACV ExoN and from 0.5 to 16 µM for SARS-CoV-2 ExoN nsp14/nsp10 complex. The reaction started with the addition of 5'-FAM-RNA and the FP signal was recorded as mentioned previously. The percentage of inhibition was calculated as indicated in the previous section. The curves of the percentage of inhibition with respect to the inhibitor concentration in a logarithmic scale were fitted in Graphpad Prism software using a four parameters equation.

The assays were done in triplicate.

Cellular assays

LCMV minigenome assays (MG) were adapted from a previously described procedure [START_REF] Saez-Ayala | Metal chelators for the inhibition of the lymphocytic choriomeningitis virus endonuclease domain[END_REF], 2018). SARS-CoV-2 cellular assays were performed as previously described [START_REF] Touret | Preclinical evaluation of Imatinib does not support its use as an antiviral drug against SARS-CoV-2[END_REF][START_REF] Touret | In vitro screening of a FDA approved chemical library reveals potential inhibitors of SARS-CoV-2 replication[END_REF]. Protocols are detailed in the supplementary data file.

Results and discussion

Fluorescence polarization (FP) is a reliable and sensitive tool for monitoring enzymatic reaction, by determining the difference of polarization signals [START_REF] Zhang | A versatile molecular beacon-like probe for multiplexed detection based on fluorescence polarization and its application for a resettable logic gate[END_REF]. FP -based assay is a robust method commonly used in drug discovery [START_REF] Lea | Fluorescence polarization assays in small molecule screening[END_REF][START_REF] Uri | What is the current value of fluorescence polarization assays in small molecule screening?[END_REF]. The FP signal recorded is proportional to the molecular weight (MW) of a fluorescent molecule [START_REF] Kwok | SNP genotyping with fluorescence polarization detection[END_REF][START_REF] Latif | Fluorescence polarization in homogeneous nucleic acid analysis II: 5'-nuclease assay[END_REF]. We decided to apply this method to viral ExoN activity, by monitoring the size of a fluorescent-labeled RNA probe which is altered in the course of the nuclease, reflecting the enzymatic activity [START_REF] Liu | A real-time fluorescence polarization activity assay to screen for inhibitors of bacterial ribonuclease P[END_REF][START_REF] Zhang | A versatile molecular beacon-like probe for multiplexed detection based on fluorescence polarization and its application for a resettable logic gate[END_REF]. The assay was first optimized.

Optimization of experimental conditions of ExoN activity based on FP assay

To set-up a screening assay, we explored 5'-FAM-RNA substrate and ExoN concentration and ratio, metal ion co-factors, temperature and reaction duration. Some RNA substrate and optimal conditions were already described [START_REF] Bouvet | RNA 3'-end mismatch excision by the severe acute respiratory syndrome coronavirus nonstructural protein nsp10/nsp14 exoribonuclease complex[END_REF][START_REF] Saramago | New targets for drug design: Importance of nsp14/nsp10 complex formation for the 3'-5' exoribonucleolytic activity on SARS-CoV-2[END_REF][START_REF] Yekwa | Arenaviridae exoribonuclease presents genomic RNA edition capacity[END_REF][START_REF] Yekwa | Activity inhibition and crystal polymorphism induced by active-site metal swapping[END_REF]. A 22mer RNA that forms a stable hairpin in its 3' end has been reported to be a valuable substrate for both ExoN. Moreover, for SARS-CoV-2 ExoN, nsp10 able to stimulate nsp14 ExoN activity [START_REF] Bouvet | RNA 3'-end mismatch excision by the severe acute respiratory syndrome coronavirus nonstructural protein nsp10/nsp14 exoribonuclease complex[END_REF][START_REF] Saramago | New targets for drug design: Importance of nsp14/nsp10 complex formation for the 3'-5' exoribonucleolytic activity on SARS-CoV-2[END_REF] was added in the reaction (ratio 1:4 of nsp14:nsp10).

We first determined the optimal 5'-FAM-RNA concentration that provides sufficient and stable FP signal, and the ratio of ExoN and 5'-FAM-RNA. For each 5'-FAM-RNA concentration (0.05 to 1µM), single turnover (STO) conditions (excess of enzyme versus substrate) and also multiple turnover (MTO) conditions (excess of substrate versus enzyme) were tested. A significant change in the FP signal was observed only under STO conditions. 0.1µM concentration of 5'-FAM-RNA was selected to provide a reproducible and stable FP signal.

To optimize the controls, several combinations were tested (Supplementary figure S2). For all negative controls, there was no variation between the initial and final FP values. However, we observe that the initial FP values are lower without MnCl2. This result is consistent with the study of Liu et al [START_REF] Liu | A real-time fluorescence polarization activity assay to screen for inhibitors of bacterial ribonuclease P[END_REF], who reported that cations can affect the FP signal. The negative controls were then prepared by mixing 5'-FAM-RNA with MnCl2 and without ExoN.

Regarding the nature of the metal ion, MnCl2 and MgCl2 were tested. An increased activity is observed with MnCl2 (data not shown). The concentration of MnCl2 that shows the highest reduction in FP signal is 5 mM for arenaviruses [START_REF] Yekwa | Arenaviridae exoribonuclease presents genomic RNA edition capacity[END_REF][START_REF] Yekwa | Activity inhibition and crystal polymorphism induced by active-site metal swapping[END_REF] and 2 mM for SARS-CoV-2 [START_REF] Baddock | Characterisation of the SARS-CoV-2 ExoN (nsp14ExoN-nsp10) complex: implications for its role in viral genome stability and inhibitor identification[END_REF][START_REF] Saramago | New targets for drug design: Importance of nsp14/nsp10 complex formation for the 3'-5' exoribonucleolytic activity on SARS-CoV-2[END_REF]. Finally, the temperature of the assay is 25°C since the incubation at 37°C leads to a significant evaporation of the sample.

Assessing the SARS-CoV-2, LCMV and MACV ExoN activity by FP

While the 5'-FAM-RNA substrate was fixed at 0.1 µM, 0.1 to 1.6 µM of ExoN were tested. The FP signal was measured for 30 min. The FP curves and end point RNA degradation products examined in urea PAGE, are gathered in figure 1.

The negative controls (figure 1, brown curves, in panels A, B and C) present flattened curves, as expected.

Strikingly, a small increase in the initial FP value is observed, corresponding to the time required for the formation of the catalytic complex. This is particularly visible with SARS-CoV-2 nsp14, which requires its activation by the cofactor nsp10. Indeed, nsp14 ExoN N-terminus is flexible, when nsp10 is not bound the catalytic site of nsp14 is distorted not allowing the catalytic ions to activate the water molecule for the nucleophilic attack [START_REF] Ferron | Structural and molecular basis of mismatch correction and ribavirin excision from coronavirus RNA[END_REF].

For all tested ExoN, except for the lowest concentration, we observe a reduction of the FP signal with time.

The decrease of the FP value is correlated with the hydrolysis of the 5'-FAM-RNA, as confirmed on the urea PAGE analysis (figure 1, panels D, E andF). The difference between initial and final FP value after 30 min is proportional to the number of nucleotides removed from the substrate. By increasing the ExoN concentration, we observe an increase in the FP curves slope but the FP signals reach a plateau, which might be related to the inability to remove any extra nucleotide when the RNA fragments are to shorter.

The method efficiency was validated by the difference of FP values between the negative control (figure 1, brown curves, in panels A, B and C) and one concentration after 30 min reaction. This difference was selected as significant enough, with DFP +/-30 to 60 units of mP. A molar ratio of 1:4 between the 5'-FAM-RNA (0.1 µM) and ExoN (0.4 µM) was used for the screening. The robustness of the assay was assessed by calculating the z' factor (supplementary methods and figure S3). For LCMV, MACV and SARS-CoV-2 we obtained z' value of 0.72, 0.74 and 0.69, respectively.

Screening of a focused library against the SARS-CoV-2, LCMV, and MACV ExoN using FP

Because ExoN activity is metal-dependent, we screened a focused library of 113 metal chelators that we have previously developed [START_REF] Saez-Ayala | Metal chelators for the inhibition of the lymphocytic choriomeningitis virus endonuclease domain[END_REF], 2018), in order to demonstrate the robustness of our FP method.

All compounds of the library were selected on the basis of competitive inhibition by metal chelation approach.

Several patterns were chosen on the chelating scaffolds, in order to explore electronic, structural, and steric features critical for the binding process and active site accommodation. Two screenings were performed using 20 µM and 5 µM of compounds and the percentages of inhibition, deduced from FP curves, are gathered in figure 2 (panels A, B, and C) and in supplementary figure S3, respectively. Aurintricarboxyclic Acid (ATA, 40), a polyphenol (POP), was included as a positive control as it has been previously described as a nuclease inhibitor [START_REF] Baddock | Characterization of the SARS-CoV-2 ExoN (nsp14ExoN-nsp10) complex: implications for its role in viral genome stability and inhibitor identification[END_REF][START_REF] Canal | Identifying SARS-CoV-2 antiviral compounds by screening for small molecule inhibitors of nsp14/nsp10 exoribonuclease[END_REF][START_REF] Huang | Identification of Inhibitors for the DEDDh Family of Exonucleases and a Unique Inhibition Mechanism by Crystal Structure Analysis of CRN-4 Bound with 2-Morpholin-4-ylethanesulfonate (MES)[END_REF], acting as non-specific metal chelator [START_REF] Liu | A real-time fluorescence polarization activity assay to screen for inhibitors of bacterial ribonuclease P[END_REF]. Compound 40 inhibits 100% of the different ExoN activity at 20 µM (Figure 2, Table 1) and 52%, 39% and 23% at 5 µM (Supplementary figure S3), respectively against LCMV, MACV and SARS-CoV-2 ExoN.

We identified 5 inhibitors (65, 67, 77, 78 and 91) of LCMV ExoN including 3 common with MACV ExoN (Figure 2 and Table 1) with efficiency between 70 and 100% inhibition at 20 µM. Compounds 65, 67, 77, 78 belong to N-hydroxyisoquinoline-1,3-diones (HID) family and compound 91 belongs to β-diketo acid (DKA) family were previously described by us as potent specific metal chelators [START_REF] Saez-Ayala | Metal chelators for the inhibition of the lymphocytic choriomeningitis virus endonuclease domain[END_REF], 2018). 9 hit compounds (18,40,46,55,68,77,78,91 and 92) inhibiting SARS-CoV-2 ExoN with efficiency between 70 and 100% inhibition at 20 µM were also identified (Figure 2, Table1, Supplementary figure S3).

Compounds 18 and 55 have never been described before. DKA 91 is the more efficient compound against all ExoN, showing inhibition at both 5 µM and 20 µM, with 87 to 100% inhibition at 20 µM and 37 to 100% inhibition at 5 µM.

IC50 measurement of hit compounds by FP

IC50 determination was performed for all hit compounds displaying more than 70% inhibition at 20 µM concentration. The dose-response curves and IC50 values determined by hill plot curve fitting all hit compounds and reference compound 40, are gathered in figure 3 and table 1. The FP curves and the end point digestion products examined in urea PAGE are gathered in Supplementary Figures S5 to S8. IC50 determination was also performed for compounds 3, 4, 44 and 57 (table 1), although they did not display high inhibition during the primary screening.

As expected, the reference compound ATA 40 was a potent inhibitor with IC50 values of 7.05 ± 0.78, 5.57 ± 0.41 and 5.50 ± 0.19 µM, for LCMV, MACV and SARS-CoV-2 ExoN, respectively. The highest inhibition observed with the DKA 91 during the two screenings was confirmed in the IC50 values. DKA 91 displayed equivalent efficiency to inhibit LCMV and MACV ExoN with respectively 5.63 ± 0.35 and 5.42 ± 0.46 µM IC50 values. The inhibition is slightly more pronounced for SARS-CoV-2 ExoN with IC50 of 1.86 ± 0.16 µM.

For other hit compounds, IC50 values ranged between 6 and 24 µM. Notably, these values correlated perfectly with the inhibition % obtained by both screenings at 20 and 5 µM, confirming the robustness and reproducibility of the assay. Moreover, The FP signal correlated with the degree of digestion observed upon PAGE analysis (Supplementary Figures S5 to S8). While the HID family (compounds 65, 67, 68, 77 and 78) showed a rather broad spectrum of efficacy for both arenavirus and coronavirus ExoNs, the DKA family (phenyl-DKA, piperidinyl-DKA and triazolyl-DKA, compounds 3, 4, 18, 44, 55, 57, 91 and 92) showed a clear specificity for the SARS-CoV-2 ExoN.

LCMV minigenome activity

Arenavirus replication can be modeled using a recombinant plasmid minigenome system comprising the viral RdRp (L), the nucleoprotein (N), and a LCMV genomic RNA [START_REF] Ortiz-Riaño | Arenavirus reverse genetics for vaccine development[END_REF]. Previous studies have shown the efficiency of this method [START_REF] Saez-Ayala | Metal chelators for the inhibition of the lymphocytic choriomeningitis virus endonuclease domain[END_REF]. We tested the hit compounds 40, 65, 67, 77, 78

and 91, at 100 and 50 µM. Because ribavirin (82) is the only licensed antiviral agent with activity against arenaviruses [START_REF] Mccormick | Lassa fever. Effective therapy with ribavirin[END_REF][START_REF] Moreno | Ribavirin Can Be Mutagenic for Arenaviruses ▿[END_REF], it was chosen as a reference compound. Favipiravir (83), having demonstrated broad-spectrum activity against a number of RNA viruses, was also added in the screening. Since the compounds were supposed to target the polymerase function, the assay was particularly suited by measuring the GLuc reporter gene (Gluc activity %, Supplementary Figure S9). We also tested the cytotoxic effect of the compounds at 50 and 100 µM (Cell viability %, Supplementary Figure S9).

The ribavirin (82) resulted in a 76% and 47% inhibition of GLuc activity, at 100 and 50 µM respectively, but was associated with important cytotoxic effect with 35 and 50 % cell viability, as described previously [START_REF] Mendenhall | T-705 (Favipiravir) Inhibition of Arenavirus Replication in Cell Culture[END_REF]. Unexpectedly, favipiravir (83) displayed weak inhibition (30% at 100 μM) without cell viability reduction [START_REF] Mendenhall | T-705 (Favipiravir) Inhibition of Arenavirus Replication in Cell Culture[END_REF]. ATA (40) displayed no antiviral effect, but, cytotoxicity as described previously [START_REF] Andrew | Aurintricarboxylic acid inhibits apoptosis and supports proliferation in a haemopoietic growth-factor dependent myeloid cell line[END_REF][START_REF] Haimsohn | Aurintricarboxylic Acid Induces a Distinct Activation of the IGF-I Receptor Signaling within MDA-231 Cells[END_REF]. Compounds 65, 67 and 77 displayed slight effect (25 to 50% inhibition) on GLuc expression but associated with cytotoxic effect, as observed for ribavirin (82). Compound 78 displayed weak GLuc expression inhibition (30 % at 50 μM) without cytotoxicity, compared to ribavirin (82).

SARS-Cov-2 in vitro evaluation

The full library of 113 metal chelators was screened in a SARS-CoV-2 infected cell-based assay. We performed a primary screen for SARS-Cov-2 in VeroE6 cells [START_REF] Touret | In vitro screening of a FDA approved chemical library reveals potential inhibitors of SARS-CoV-2 replication[END_REF] and we used remdesivir as a reference compound [START_REF] Pruijssers | Remdesivir Inhibits SARS-CoV-2 in Human Lung Cells and Chimeric SARS-CoV Expressing the SARS-CoV-2 RNA Polymerase in Mice[END_REF][START_REF] Wang | Remdesivir and chloroquine effectively inhibit the recently emerged novel coronavirus (2019-nCoV) in vitro[END_REF]. This fast and reliable assay, based on the cell viability, measures the ability of a compound to prevent the cytopathic effect induced by the viral replication at 50 µM concentration.

Our primary screen identified 31 hit compounds that prevent cytopathic effect by at least 40 % (Supplemental figure S10). To go further, we next performed dose-response analysis in VeroE6 TMPRSS2 [START_REF] Touret | Preclinical evaluation of Imatinib does not support its use as an antiviral drug against SARS-CoV-2[END_REF][START_REF] Weiss | Niclosamide shows strong antiviral activity in a human airway model of SARS-CoV-2 infection and a conserved potency against the Alpha (B.1.1.7), Beta (B.1.351) and Delta variant (B.1.617.2)[END_REF]. We used VeroE6/TMPRSS2 cells to avoid off target false-positive compounds as it has been shown for Chloroquine [START_REF] Hoffmann | Chloroquine does not inhibit infection of human lung cells with SARS-CoV-2[END_REF]. Remdesivir, as reference compound, displayed EC50 of 1.5±1.1 μM with CC50 > 10 μM.

Of the 31 primary hits, 19 compounds displaying between 40 and 100% inhibition were analyzed in doseresponse assays. 7 compounds from the same family and 5 compounds displaying between 40 and 70% inhibition were set aside. Additionally, the 8 hit compounds from the biochemical screening displaying between 0 and 40% inhibition were also analyzed. Here, we confirmed the antiviral activity of ATA ( 40) against SARS-Cov-2 in cell culture [START_REF] Canal | Identifying SARS-CoV-2 antiviral compounds by screening for small molecule inhibitors of nsp14/nsp10 exoribonuclease[END_REF] with EC50 of 27.23±20.68 μM. We were unable to confirm primary efficiency of 6 potential hit compounds and we identified 5 compounds that displayed significant cytotoxicity.

Finally, we identified 4 new compounds 3, 4, 44, and 57, that displayed a robust antiviral activity with EC50 between 8 and 38 μM without cytotoxicity and 2 compounds 46 and 55 that exert a potent antiviral activity in the same range but also showing signs of cytotoxicity, without reaching the CC50 value (table 1 and supplementary figure S11) . Compounds 46 and 55 were identified during the biochemical screening on the SARS-CoV-2 ExoN, confirming the ExoN targeting.

Conclusion

We proved in this study that fluorescence polarization (FP) method is reliable, sensitive to monitor nuclease activity and feasible in 384-well plates. The method is robust, rapid and non-destructive allowing the visualization of the RNA substrate degradation using gels as orthogonal method of confirmation. We identified several hit inhibitors of viral arenavirus (LCMV and MACV) and coronavirus (SARS-CoV-2) ExoN activity and determined by fluorescence polarization their IC50 at a micromolar level. Then we validated their activities in cell culture assays and we found compounds displaying efficiency at micromolar level against SARS-CoV-2. These results highlight the effectiveness of fluorescence polarization assay in the screening of compounds libraries for the discovery of antivirals.

Legend to figures

Figure 1 .

 1 Figure 1. ExoN activity measured by FP for LCMV (panel A), MACV (panel B) and SARS-CoV-2 (panel C).

Figure 2 .

 2 Figure 2. Screening of focused library of metal chelators (113) against LCMV (panel A), MACV (panel B)

We thank Pr C Drosten for providing the SARS-CoV-2 BavPat strain through EVA GLOBAL. A part of the work was done on the Aix Marseille University antivirals drug design platform "AD2P".We thank Thi Hong Van Nguyen and Adrien Delpal for technical assistance.

Supplementary data associated with this article can be found, in the online version, at

Funding and Acknowledgments

This work was supported by grants from the Ministry of the Armed Forces (DGA) -Defense Innovation Agency (AID) and the French National Research Agency (ANR-18-ASTR-0010-01, PaNuVi), Fondation pour la Recherche Medicale (Chemistry for Medecine, DCM20181039531), SCORE project H2020 SC1-PHE Coronavirus-2020 (grant#101003627). Priscila El-Kazzi and Rafik Kaci were funded by Fondation Mediterranée Infection (Infectiopole Sud), Laura Garlatti was funded by DGA and Aix-Marseille University (fellowship N° DGA01D19024292 AID). The Carolina Trajano De Jesus master's internship was supported by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior -Brasil (CAPES) -Finance Code 001.