## 3 Set of \(\epsilon\)-Proper Equilibria  
The _proper_ refinement originates from the assumption that reasonable players should try much harder to avoid more costly mistakes than they should try to avoid less costly ones. While it is well known that any proper equilibrium is always perfect, a perfect Nash equilibrium may be non-proper. Given a finite game in strategic form, [PERSON] [21] defines a _proper_ equilibrium as the limit of an infinite sequence of \(\epsilon\)-proper equilibria, when \(\epsilon\) converges to zero. We here apply [PERSON]'s definition of the \(\epsilon\)-proper equilibrium to polymatrix games. Let us first note \(A_{i. }^{k}\) and \(A_{i. }^{q}\) as the \(k^{th}\) and \(q^{th}\) rows of the payoff matrix \(A_{i. }\) of player \(i\), respectively. Definition 1: A polymatrix game profile \(\hat{X}=(\hat{X}_{1},\ldots,\hat{X}_{n})\) is \(\epsilon\)-proper, for \(\epsilon>0\), if for all \(i\in N\):  
\[\text{if }\sum_{j\
eq i}\ A_{ij}^{k}\hat{X}_{j}<\sum_{j\
eq i}\ A_{ij}^{q} \hat{X}_{j},\text{then}\]  
\[\hat{x}_{ik}\leq\epsilon\hat{x}_{iq},\,k,q\in\{1,2,\ldots,m_{i}\}, \tag{1}\]  
\[\text{and }\hat{x}_{ik}>0,\,k\in\{1,2,\ldots,m_{i}\}. \tag{2}\]  
For a given \(\epsilon\geq 0\) and \(\sigma\geq 0\), we now introduce the following set \(\Omega_{\epsilon}^{\sigma}\) such that:  
\[\Omega_{\epsilon}^{\sigma} = \{X=(X_{1},\ldots,X_{n}):\forall i\in N,\quad\exists\ u_{i}\text{ and }L\ \text{ such that}\] \[\sigma \leq x_{ik},\ \forall k\in\{1,2,\ldots,m_{i}\},\] \[\text{and }\forall\ k,q\in\{1,2,\ldots,m_{i}\},\ k\
eq q:\] \[\sum_{j\
eq i}\ A_{ij}^{k}X_{j}\leq\sum_{j\
eq i}\ A_{ij}^{q}X_{j} +Lu_{qk}^{i},\] \[x_{ik}+u_{kq}^{i}\leq\epsilon x_{iq}+1,\] \[u_{kq}^{i}+u_{jk}^{i}\leq 1,\text{ with }k<q,\] \[u_{kq}^{i}\in\{0,1\}.\}\]  
In the above, \(u_{kq}^{i}\) is a binary variable, and the parameter \(L\in\Re^{+}\) is set to a sufficiently large value. The following Proposition 1 gives a novel formulation of the \(\epsilon\)-properness conditions and shows that each element of \(\Omega_{\epsilon}^{\sigma}\) is an \(\epsilon\)-proper equilibrium. Proposition 1: _If \(X=(X_{1},\ldots,X_{n})\in\Omega_{\epsilon}^{\sigma}\), for \(\epsilon>0\) and \(\sigma>0\), then \(X\) is an \(\epsilon\)-proper equilibrium._  
Proof: Assume that \(X=(X_{1},\ldots,X_{n})\in\Omega_{\epsilon}^{\sigma}\), for \(\epsilon>0\) and \(\sigma>0\). Let \(k\) and \(q\) be indices in \(\{1,2,\ldots,m_{i}\}\), such that \(k\
eq q\). The inequalities \(u_{kq}^{i}+u_{qk}^{i}\leq 1\) ensure that the binary combination \(u_{kq}^{i}=1\) and \(u_{qk}^{i}=1\) is impossible. Moreover,  
* if \(u_{kq}^{i}=0\) and \(u_{qk}^{i}=0\), then \(\sum_{j\
eq i}\ A_{ij}^{k}X_{j}=\sum_{j\
eq i}\ A_{ij}^{q}X_{j}\);if \(u^{i}_{kq}=1\), then \(ex_{ik}\leq x_{ik}\leq ex_{iq}\leq x_{iq}\) implies that \(x_{iq}\geq ex_{ik}\) and \(u^{i}_{qk}=0\), thus \(\sum\limits_{j\
eq i}A^{k}_{ij}X_{j}\leq\sum\limits_{j\
eq i}A^{q}_{ij}X_{j}\). It follows that conditions (1) are satisfied. Finally, with \(0<\sigma\leq x_{ik}\), for all \(k\in\{1,2,\ldots,m_{i}\}\), conditions (2) are satisfied. Proposition 2 states that every \(\epsilon\)-proper equilibrium belongs to \(\Omega^{\sigma}_{\epsilon}\), for every sufficiently small value of \(\sigma\). Proposition 2: _If \(X=(X_{1},\ldots,X_{n})\) is an \(\epsilon\)-proper equilibrium for a given \(\epsilon>0\), there exists a real number \(\sigma>0\), such that \((X_{1},\ldots,X_{n})\in\Omega^{\sigma}_{\epsilon}\), for \(0\leq\sigma\leq\sigma\)._  
Proof: If a profile \(X=(X_{1},\ldots,X_{n})\) is an \(\epsilon\)-proper equilibrium for some \(\epsilon>0\), for all \(i\in N\), conditions (1) can be reformulated using the binary variables \(u^{i}_{kq}\), for all \(k\
eq q\in\{1,2,\ldots,m_{i}\}\):  
\[\text{If }\left\{ \begin{array}{l}\sum\limits_{j\
eq i}A^{k}_{ij}X_{j} <\sum\limits_{j\
eq i}A^{q}_{ij}X_{j},\\ x_{ik}\leq\epsilon x_{iq},\end{array} \right.\text{ then }\left\{