## 4 Modeling WSN LPP as a Matrix Game
### Deterministic Survival Times  
Let \(\mathbf{H}\in\mathbb{R}^{D\times A}\) be a matrix, where the \(d\)th row and \(a\)th column element \(h_{d,a}\) represents the survival time when an adversary picks an attack mechanism \(a\in A\) and a defender picks a defense mechanism \(d\in D\). Let us assume that \(h_{d,a}\) is crisp and deterministic. The expected payoff \(E\) of the game, gain for the defender (loss for the adversary), is given as follows:  
\[E=\mathbf{x}^{T}\mathbf{H}\mathbf{y} \tag{1}\]  
where \(\mathbf{x}=[x_{1},\ldots,x_{|D|}]^{T}\), \(x_{d}\) is the probability of using the strategy \(d\) by the defender, \(\mathbf{y}=[y_{1},\ldots,y_{|A|}]^{T}\), and \(y_{a}\) is the probability of using the strategy \(a\) by the adversary. In the case of mixed strategies, the objective of the defender is to maximize the minimum expected payoff, whereas the objective of the adversary is to minimize the maximum expected payoff. **Lemma 1**. : _The Operations Research (OR) model to identify the best mixed strategy for the defender (also known as max-minimizer) can be written as:_  
\[\max_{\mathbf{x}\geq\mathbf{0}}:\min_{a\in A}\left\{\mathbf{h}_{a}^{T} \mathbf{x}|\mathbf{e}^{T}\mathbf{x}=1\right\} \tag{2}\]  
_where \(\mathbf{h}_{a}\) is the \(a\)th column of matrix \(\mathbf{H}\), \(\mathbf{e}\) represents a vector containing all ones, and its size can be identified from the context._  
Proof. : Let \(E_{d}(\mathbf{x})\) be the worst expected payoff value of the defender for any arbitrary choice of \(\mathbf{x}\), defined as:  
\[E_{d}(\mathbf{x})=\min_{ \begin{subarray}{c}\mathbf{e}^{T}\mathbf{y}=1,\\ \mathbf{y}\geq\mathbf{0}\end{subarray} }\left\{\mathbf{x}^{T}\mathbf{H} \mathbf{y}\right\} \tag{3}\]  
The worse expected value can be re-written as:  
\[E_{d}(\mathbf{x})= \min_{ \begin{subarray}{c}\mathbf{e}^{T}\mathbf{y}=1,\\ \mathbf{y}\geq\mathbf{0}\end{subarray} }\left\{\mathbf{x}^{T}\mathbf{H} \mathbf{y}\right\} \tag{4a}\] \[= \min_{ \begin{subarray}{c}\mathbf{e}^{T}\mathbf{y}=1,\\ \mathbf{y}\geq\mathbf{0}\end{subarray} }\left\{\sum_{a\in A}(\mathbf{H}^{T} \mathbf{x})_{a}y_{a}\right\}\] (4b) \[= \min_{a\in A}\{(\mathbf{H}^{T}\mathbf{x})_{a}\}\] (4c) \[= \min_{a\in A}\{\mathbf{h}_{a}^{T}\mathbf{x}\} \tag{4d}\]  
where \((\mathbf{H}^{T}\mathbf{x})_{a}\) is the \(a\)th element of vector \(\mathbf{H}^{T}\mathbf{x}\). Equation (4c) follows from the theory of linear programming, which states that the optimal solution of a linear function \(\sum_{a\in A}(\mathbf{H}^{T}\mathbf{x})_{a}y_{a}\) on a non-empty simplex \(\{\mathbf{e}^{T}\mathbf{y}=1,\mathbf{y}\geq 0\}\) lies at one of the \(|A|\) extreme points. Thus, the OR model that provides the best mixed strategy for the defender (also known as max-minimizer) can be written as:  
\[\max_{ \begin{subarray}{c}\mathbf{e}^{T}\mathbf{x}=1,\\ \mathbf{x}\geq\mathbf{0}\end{subarray} }:\min_{a\in A}\left\{\mathbf{h}_{a}^{T} \mathbf{x}\right\} \tag{5}\]  
The above formulation is the same as Formulation (2). **Corollary 1**. : _The OR model to identify the best mixed strategy for the attacker can be written as:_  
\[\min_{\mathbf{y}\geq\mathbf{0}}:\max_{d\in D}\left\{\mathbf{h}_{d}^{T} \mathbf{y}|\mathbf{e}^{T}\mathbf{y}=1\right\} \tag{6}\]_where \(\mathbf{h}_{d}\) is the \(d\)th row of matrix \(\mathbf{H}\)._  
**Lemma 2**: _The proposed game has a mixed strategy Nash equilibrium._  
Proof: Based on the linear programming's weak duality theory [34], the following holds:  
\[\min_{a\in A}\{\mathbf{h}_{a}^{T}\mathbf{x}|\mathbf{e}^{T}\mathbf{x}=1,\ \mathbf{x}\geq \mathbf{0}\}\ \leq\ \max_{d\in D}\{\mathbf{h}_{d}^{T}\mathbf{y}|\mathbf{e}^{T}\mathbf{y}=1, \mathbf{y}\geq\mathbf{0}\} \tag{7}\]  
Notice that both Formulations (2) and (6) are feasible. From the linear programming's strong duality theory, it can be concluded that the optimal objective values for both formulations are equal and finite, say \(v\).