## 4 Computation of \(\epsilon\) Proper Equilibria
### Mixed 0-1 Linear Formulation  
It is possible to resolve all numerical issues due to the non-linear formulation if \(\epsilon\) is set to a fixed value. The following Corollary 2 can be formulated. It establishes a 0-1 mixed linear formulation from Proposition 4. Corollary 2: _For a given \(\epsilon>0\), consider the following 0-1 mixed linear program_  
\[ \begin{array}{ll}(\mathfrak{P}_{\bullet})&\max\sigma\\ &\text{subject to}\\ &\hat{x}_{ik}-\epsilon\leq x_{ik}\leq\hat{x}_{ik}+\epsilon,\ \ k\in\{1,2,\ldots,m_{i}\},\ i\in N,\\ &(X_{1},\ldots,X_{n})\in\Omega_{\epsilon}^{\sigma}.\end{array} \]  
_If the optimal objective value of \((\mathfrak{P}_{\bullet})\) is equal to 0, or \((\mathfrak{P}_{\bullet})\) is unfeasible, the Nash equilibrium \(\hat{X}=(\hat{X}_{1},\ldots,\hat{X}_{n})\) is not proper._The proof of this corollary is trivial (see Proof of Proposition 3). Hence, we can rely on the 0-1 mixed linear optimization results to certify that an equilibrium is non-proper. Computationally, this can be done using exact arithmetics. Exact arithmetics perform operations on real numbers defined as rationals instead of floats or doubles. A rational number consists of two very large integers, namely a numerator and a denominator. Overloading the elementary operators to handle algebraic operations on these rationals is fundamental. A Greatest Common denominator function is useful to reduce the size of the stored Big Integers. Exact arithmetics was successfully used by [PERSON] et al. [25] and [PERSON] et al. [26] in the context of two-person games equilibrium enumeration and refinement. The automatic detection of non-proper Nash equilibria can then safely be carried out for polymatrix games. **Case 2: \(\epsilon(\sigma)\) converges to zero** If \(\epsilon(\sigma)\) converges towards zero, we can conclude on the existence of at least one sequence of \(\epsilon\)-proper equilibria necessarily close to the given Nash equilibrium \(\hat{X}\) we are testing for properness. Computationally speaking, unless exact arithmetics is used here, this may not be enough to conclude on the properness of the given Nash equilibrium. Therefore, it is imperative to use exact arithmetics to get consistent results. As an extension of Corollary 2, the following Corollary 3 can be formulated, when \(\epsilon\) is set to a fixed value. **Corollary 3**. : _For some \(\epsilon>0\), consider the mixed 0-1 linear program \((\mathfrak{P}_{\bullet})\) of corollary 2:_  
\[ \begin{array}{ll}(\mathfrak{P}_{\bullet})&\sigma(\epsilon)=\max\sigma\\ &\text{subject to}\\ &\hat{x}_{ik}-\epsilon\leq x_{ik}\leq\hat{x}_{ik}+\epsilon,\ \ k\in\{1,2,\ldots,m_{i}\},\ i\in N,\\ &(X_{1},\ldots,X_{n})\in\Omega_{\epsilon}^{\sigma}.\end{array} \]  
_If the optimal objective value \(\sigma(\epsilon)\) of \((\mathfrak{P}_{\bullet})\) converges to 0 and remains strictly positive, when \(\epsilon\) converges to 0, then \((\hat{X}_{1},\ldots,\hat{X}_{n})\) is a proper equilibrium._  
Proof. : Let \((X_{1}(\epsilon),\ldots,X_{n}(\epsilon),\sigma(\epsilon))\) be the optimal solution to (4), for some given equilibrium profile \((\hat{X}_{1},\ldots,\hat{X}_{n})\). If \(\sigma(\epsilon)\) is strictly positive, Proposition (1) ensures that \((X_{1}(\epsilon),\ldots,X_{n}(\epsilon))\) is an \(\epsilon\)-proper equilibrium. Hence, if the optimal objective value \(\sigma(\epsilon)\) converges to 0 and remains strictly positive when \(\epsilon\) converges to 0, \((X_{1}(\epsilon),\ldots,X_{n}(\epsilon))\) remains an \(\epsilon\)-proper equilibrium and converges to the equilibrium \((\hat{X}_{1},\ldots,\hat{X}_{n})\). Therefore, \((\hat{X}_{1},\ldots,\hat{X}_{n})\) is a proper equilibrium. Since it is based on a linear formulation, this corollary resolves all numerical issues due to the nonlinear formulation of Proposition 3. First, we illustrate our results with a polymatrix game where one of the extreme Nash equilibria is not perfect. With the following example, we show how the Corollary 3 can be applied.