Construction and application of hyper-inverse conflict models based on the sequential stability

Abstract

Based on the graph model for conflict resolution (GMCR), an analytical framework is proposed to assist the focal decision maker (DM) with an informational advantage in resolving real-world hyper-inverse conflict situations considering the sequential (SEQ) stability. The hyper-inverse conflict resolution aims to obtain the opponent’s misunderstanding preferences of focal DM, which can assist the focal DM in taking the initiative in a conflict. Among all stabilities in GMCR, the SEQ stability is selected in the present study, because this stability concept provides a logical basis for a DM to sanction the opponent, which reflects the case for many conflicts. Moreover, a nonlinear binary optimization model rooted in the matrix representation of SEQ stability is constructed to capture the cause and process of the abnormal individual stability (or stabilities) in the hypergame. Then, a solution procedure is designed for solving the optimization model to obtain the preferences that are misunderstood by the opponent. Finally, an illustrative example of two DMs in an environmental management conflict is studied to demonstrate how the proposed framework can be conveniently employed in practice.

Appendix A: Proof of Lemma 1

Proof

$\widetilde{f}_j^{\text {SEQ}}(s)$ presents the misunderstood stability function of DMj, it is in line with Definition 3. Then we recall Eq. (5) here:

$$\begin{aligned} \widetilde{f}_j^{\text {SEQ}}(s)=[e_s^T(J_j\circ P_j^+)]\cdot [(E-{\text {sign}}(J_i\circ \widetilde{P}_{ij}^+\cdot (E-P_{j}^+-I)^T))e_s] \end{aligned}$$

(18)

where $e_s$ expresses an m-dimensional column vector with $s^{th}$ element 1 and all other elements 0, I denotes an $m\times m$ matrix whose diagonal elements being 1 while others 0, $c(s,q)$ presents the (s,q) entry of I.

The linearization procedure of it is given as follows. According to Xu et al. (2007, 2008, 2018a), formula 18 is equivalent to

$$\begin{aligned} \sum \limits _{p=1}^m J_{j}(s,p)\cdot P_{j}^+(s,p)\cdot [1-{\text {sign}}(\sum \limits _{q=1}^m (J_i(p,q)\cdot \widetilde{P}_{ij}^+(p,q))\cdot (1-P_{j}^+(s,q)-c(s,q)))] \end{aligned}$$

(19)

Introducing a new decision variable $t_{j}(s,p)$, let

$$\begin{aligned} t_{j}(s,p)=\sum \limits _{q=1}^m (J_i(p,q)\cdot \widetilde{P}_{ij}^+(p,q))\cdot (1-P_{j}^+(s,q)-c(s,q)) \end{aligned}$$

(20)

Then, the formula 18 can be written as

$$\begin{aligned} \widetilde{f}_{j}^{\text {SEQ}}(s)=\sum \limits _{p=1}^m J_{j}(s,p)\cdot P_{j}^+(s,p)\cdot (1-{\text {sign}}(t_{j}(s,p))) \end{aligned}$$

(21)

To resolve the“sign” function, let $z_{j}(s,p)={\text {sign}}(t_{j}(s,p))$ be an additional variable. Since $t_{j}(s,p)\in \{0,1,2,\ldots ,m\}$, and $z_{j}(s,p)$ is binary, $z_{j}(s,p)$ can be equivalently defined by

$$\begin{aligned} z_{j}(s,p)\le t_{j}(s,p)\le m z_{j}(s,p) \end{aligned}$$

(22)

From formula 22, it is clear that

• if $t_{j}(s,p)=0$, then $z_{j}(s,p)\le 0\le m\cdot z_{j}(s,p)$, and thus $z_{j}(s,p)=0$; while

• If $t_{j}(s,p)>0$, then $m\cdot z_{j}(s,p)\ge t_{j}(s,p)>0$. We must have $z_{j}(s,p)=1$.

Therefore, formula 18 is equivalent to

$$\begin{aligned} \widetilde{f}_{j}^{\text {SEQ}}(s)=\sum \limits _{p=1}^m J_{j}(s,p)\cdot (P_{j}^+(s,p)- P_{j}^+(s,p)\cdot z_{j}(s,p)) \end{aligned}$$

(23)

Then, one can review the program of Model (I) as

$$\begin{aligned}&\sum \limits _{p=1}^m J_{j}(s,p)\cdot (P_{j}^+(s,p)- P_{j}^+(s,p)\cdot z_{j}(s,p))>0 \end{aligned}$$

(24)

$$\begin{aligned}&t_{j}(s,p)=\sum \limits _{q=1}^m (J_i(p,q)\cdot \widetilde{P}_{ij}^+(p,q))\cdot (1-P_{ij}^+(s,q)-c(s,q)) \end{aligned}$$

(25)

$$\begin{aligned}&z_{j}(s,p)\le t_{j}(s,p)\le mz_{j}(s,p) \end{aligned}$$

(26)

The variable $t_{j}(s,p)$ is replaced by its definition, so that all remaining variables are binary.