## 1 Introduction  
After the 11 September 2001 terrorist attacks on the World Trade Center and the Pentagon in the United States, interest in counter-terrorism methods with limited resources has rapidly increased. However, allocating scarce defensive resources to optimize an objective function is one of the classic problems in Operations Research [1], being an important and difficult task [2]. For the defense agencies in many countries, weapon system of systems (WSOS) development planning is a critical process used to develop countermeasures (CMs) to confront the threats posed by other hostile countries [3]. The planning solution focuses on how to allocate resources to different weapons to effectively eliminate the threat from hostile countries. In prior studies on weapons development planning, a decision maker (DM) seeks to maximize a single objective function, such as capability, profit, and effectiveness. For example, [PERSON] et al. [4] established a WSOS planning model by maximizing the expected effectiveness as the optimization target. [PERSON] [5] compared the long-term defense planning systems of seven Nordic countries and found that most countries aimed to improve their own profit. [PERSON] et al. [6] proposed an optimization model to minimize the capability gaps to find the best development scheme by comparing different combinations of weapons, based on the given capability requirements. Throughout the literature research, however, did not mention the change of the opponent. In contrast, this paper considered minimizing the threat caused by the enemy as the objective function. Herein, we addressed an arms race between two parties, \(R\) and \(B\). Each party considers the potential threat from the other party. \(R\) develops one or more weapons to gain operational advantages over \(B\), whereas \(B\) attempts to develop one or more CMs that would neutralize, or at least mitigate, the threat from \(R\)[7]. Resource allocation in adversarial settings, also called the research of arms races using mathematical models, has been studied since the 1930s. However, most of these studies focused only on the strategic aspect but not the operational aspect. In 1935, [PERSON] [8] first proposed a simple coupled differential equation model to solve the effects of an arms race between rival states, thus successfully predicting World War II. Later, [PERSON] [9] analyzed a three-nation arms race by solving linear difference equations based on [PERSON]'s study. Similar to most previous research, [PERSON]'s model focused only on the strategic aspect and did not consider the relationship between research and development (R&D) investments and time, nor the cooperation relationships among different weapons. Afterwards, [PERSON] [3] established and solved equations to optimize the function of each side rather than the constrained nonlinear optimization model. Recently, [PERSON] et al. [10] proposed a model for determining optimal resource allocation by combining game theory with a simple multi-attribute utility model. The defender first allocated her resources amongst all combinations of countermeasures and targets, and then the attacker striked with his best response to this allocation. [PERSON] [11] investigated efficient computation schemes for allocating two defensive resources to multiple sites to protect against possible attacks by an adversary. The availability of the two resources was constrained and the effectiveness of each may vary over the site. [PERSON] [12] modelled the attacker's behavior using multi-attribute prospect theory to account for the attacker's multiple objectives and deviations from rationality. These articles mainly modelled adversary's behavior from macroscopic and strategic view, and considered neither the time required to develop new countermeasures, nor the relationship between expenditures and development gap. In addition, budgets were not modelled explicitly.