基于图模型的不确定偏好下冲突决策共识模型

doi:10.12305/j.issn.1001-506X.2025.01.20

Abstract

Abstract:

Aiming at the possible uncertainty preference information of decision makers (DMs), the consensus theory of graph model for conflict resolution (GMCR) under the simple preference is extended to uncertain scenarios considering DMs do not have consensus or self-interested behavior, but also have aggressive or conservative risk attitudes. After that, a consensus model for conflict decision-making under the uncertain preference is constructed. The uncertain preference structure is introduced into conflict consensus theory, and the logic representations of conflict consensus and dissent preferences and stabilities under four uncertain preference types are respectively defined. The corresponding matrix representation is given to facilitate the stable solution calculation and analysis and promote the stability algorithm realization. The established conflict decision consensus model under the uncertain preference is applied to the aviation industry conflict to obtain the optimal solution. The results show that the model provides effective strategy suggestions for resolving complex conflicts in uncertain environments.

Key words: uncertain preference, graph model for conflict resolution (GMCR), decision consensus, matrix representation, aviation industy conflict

CLC Number:

C934

Jinmuzi ZHANG, Haiyan XU, Lu CHEN. Consensus model under uncertain preference based on graph model for conflict resolution[J]. Systems Engineering and Electronics, 2025, 47(1): 191-201.

Figures/Tables 15

Table 1

Table 2

Table 3

Table 4

Definition of stability logic under uncertain preferences"

偏好类型	稳定性	一步转移(对任一状态)	对手反击(至少存在一个)	三步反应(对任一状态)	稳定性判定条件
a型(激进型)	$\mathrm{Nash}_{\mathrm{a}}$	$R_{i}^{+, U}(s)=\varnothing$	－	－	$R_{i}^{+, U}(s)=\varnothing$
	$\mathrm{GMR}_{\mathrm{a}}$	$s_{1} \in R_{i}^{+}, U(s)$	$s_{2} \in R_{j}\left(s_{1}\right)$	－	$s \succeq_{i} s_{2}$
	$\mathrm{SEQ}_{\mathrm{a}}$	$s_{1} \in R_{i}^{+}, U(s)$	$s_{2} \in R_{j}^{+}{ }^{U}\left(s_{1}\right)$	－	$s \succeq_{i} s_{2}$
	SMR$_{\text {a }}$	$s_{1} \in R_{i}^{+}, U(s)$	$s_{2} \in R_{j}\left(s_{1}\right)$	$s_{3} \in R_{i}\left(s_{2}\right)$	$s_{\check{i}} s_{2}, s \succeq_{i} s_{3}$
b型(混合型)	Nash$_{\text {b }}$	$R_{i}^{+}(s)=\varnothing$	－	－	$R_{i}^{+}(s)=\varnothing$
	$\mathrm{GMR}_{\mathrm{b}}$	$s_{1} \in R_{i}^{+}(s)$	$s_{2} \in R_{j}\left(s_{1}\right)$	－	$s \succeq_{i} s_{2}$
	${\rm{SEQ}}_{\text {b }}$	$s_{1} \in R_{i}^{+}(s)$	$s_{2} \in R_{j}^{+, U}\left(s_{1}\right)$	－	$s \succeq_{i} s_{2}$
	$\mathrm{SMR}_{\mathrm{b}}$	$s_{1} \in R_{i}^{+}(s)$	$s_{2} \in R_{j}\left(s_{1}\right)$	$s_{3} \in R_{i}\left(s_{2}\right)$	$s \succeq_{i} s_{2}, s \succeq_{i} s_{3}$
c型(混合型)	$\mathrm{Nash}_{\mathrm{c}}$	$R_{i}^{+, U}(s)=\varnothing$	－	－	$R_{i}^{+, U}(s)=\varnothing$
	$\mathrm{GMR}_{\mathrm{c}}$	$s_{1} \in R_{i}^{+}, U(s)$	$s_{2} \in R_{j}\left(s_{1}\right)$	－	$s \succeq_{i} s_{2}$或$s U_{i} s_{2}$
	$\mathrm{SEQ}_{\mathrm{c}}$	$s_{1} \in R_{i}^{+}, U(s)$	$s_{2} \in R_{j}^{+, U}\left(s_{1}\right)$	－	$s \succeq_{i} s_{2}$或$s U_{i} s_{2}$
	$\mathrm{SMR}_{\mathrm{c}}$	$s_{1} \in R_{i}^{+}, U(s)$	$s_{2} \in R_{j}\left(s_{1}\right)$	$s_{3} \in R_{i}\left(s_{2}\right)$	$s \succeq_i s_2$或$s U_i s_2, s \succeq_i s_3$或$s U_i s_3$
d型(保守型)	$\mathrm{Nash}_{\text {d }}$	$R_{i}^{+}(s)=\varnothing$	－	－	$R_{i}^{+}(s)=\varnothing$
	$\mathrm{GMR}_{\mathrm{d}}$	$s_{1} \in R_{i}^{+}(s)$	$s_{2} \in R_{j}\left(s_{1}\right)$	－	$s \succeq_{i} s_{2}$或$s U_{i} s_{2}$
	$\mathrm{SEQ}_{\mathrm{d}}$	$s_{1} \in R_{i}^{+}(s)$	$s_{2} \in R_{j}^{+, U}\left(s_{1}\right)$	－	$s \succeq_{i} s_{2}$或$s U_{i} s_{2}$
	$\mathrm{SMR}_{\mathrm{d}}$	$s_{1} \in R_{i}^{+}(s)$	$s_{2} \in R_{j}\left(s_{1}\right)$	$s_{3} \in R_{i}\left(s_{2}\right)$	$s \succeq_{i} s_{2}$或$s U_{i} s_{2}, s \succeq_{i} s_{3}$或$s U_{i} s_{3}$

Table 4

Table 5

Table 6

Logical definitions of UC and UD preferences"

类型	UC改良集	UD改良集	UC改良可达集	UD改良可达集
a型	$\begin{gathered}\bar{\varPhi}_{i \mathrm{a}}^{c+}(s)= \varPhi_{i}^{+, U}(s) \bigcap \varPhi_{j}^{+, U}(s)\end{gathered}$	$\begin{gathered}\bar{\varPhi}_{i \mathrm{a}}^{d+}(s)= \varPhi_{i}^{+, U}(s) \bigcap \varPhi_{j}^{-}(s)\end{gathered}$	$\begin{gathered}\bar{R}_{i \mathrm{a}}^{c+}(s)= R_{i}(s) \bigcap \bar{\varPhi}_{i \mathrm{a}}^{c+}(s)\end{gathered}$	$\begin{gathered}\bar{R}_{i \mathrm{a}}^{d+}(s)= R_{i}(s) \bigcap \bar{\varPhi}_{i \mathrm{a}}^{d+}(s)\end{gathered}$
b型	$\begin{aligned} \bar{\varPhi}_{i \mathrm{b}}^{c+}(s)= \varPhi_{i}^{+}(s) \bigcap \varPhi_{j}^{+, U}(s)\end{aligned}$	$\begin{gathered}\bar{\varPhi}_{i \mathrm{b}}^{d+}(s)= \varPhi_{i}^{+}(s) \bigcap \varPhi_{j}^{-}(s)\end{gathered}$	$\begin{gathered}\bar{R}_{i \mathrm{b}}^{c+}(s)= R_{i}(s) \bigcap \bar{\varPhi}_{i \mathrm{b}}^{c+}(s)\end{gathered}$	$\begin{gathered}\bar{R}_{i \mathrm{b}}^{d+}(s)= R_{i}(s) \bigcap \bar{\varPhi}_{i \mathrm{b}}^{d+}(s)\end{gathered}$
c型	$\begin{gathered}\bar{\varPhi}_{i{\rm{c}}}^{c+}(s)= \varPhi_{i}^{+, U}(s) \bigcap \varPhi_{j}^{+, U}(s)\end{gathered}$	$\begin{gathered}\bar{\varPhi}_{i{\rm{c}}}^{d+}(s)= \varPhi_{i}^{+}, U(s) \bigcap \varPhi_{j}^{-}(s)\end{gathered}$	$\begin{gathered}\bar{R}_{i{\rm{c}}}^{c+}(s)= R_{i}(s) \bigcap \bar{\varPhi}_{i{\rm{c}}}^{c+}(s)\end{gathered}$	$\begin{gathered}\bar{R}_{i{\rm{c}}}^{d+}(s)= R_{i}(s) \cap \bar{\varPhi}_{i{\rm{c}}}^{d+}(s)\end{gathered}$
d型	$\begin{aligned} \bar{\varPhi}_{i \mathrm{d}}^{c+}(s)= \varPhi_{i}^{+}(s) \bigcap \varPhi_{j}^{+, U}(s)\end{aligned}$	$\begin{gathered}\bar{\varPhi}_{i \mathrm{d}}^{d+}(s)= \varPhi_{i}^{+}(s) \bigcap \varPhi_{j}^{-}(s)\end{gathered}$	$\begin{gathered}\bar{R}_{i \mathrm{d}}^{c+}(s)= R_{i}(s) \bigcap \bar{\varPhi}_{i \mathrm{d}}^{c+}(s)\end{gathered}$	$\begin{gathered}\bar{R}_{i \mathrm{d}}^{d+}(s)= R_{i}(s) \bigcap \bar{\varPhi}_{i \mathrm{d}}^{d+}(s)\end{gathered}$

Table 6

Table 7

Logic expression of conflict dissent stability under uncertain preference"

偏好类型	稳定性	一步转移(对任一状态)	对手反击(至少存在一个)	三步反应(对任一状态)	稳定性判定条件
a型(激进型)	UDNash$_{\text {a }}$	$\bar{R}_{i \mathrm{a}}^{d+}(s)=\varnothing$	－	－	$\bar{R}_{i \mathrm{a}}^{d+}(s)=\varnothing$
	UDGMRa	$s_{1} \in \bar{R}_{i \mathrm{a}}^{d+}(s)$	$s_{2} \in R_{j}\left(s_{1}\right)$	－	$s \succeq_{i} s_{2}$
	$\mathrm{UDSEQ}_{\mathrm{a}}$	$s_{1} \in \bar{R}_{i \mathrm{a}}^{d+}(s)$	$s_{2} \in \bar{R}_{j \mathrm{a}}^{d+}\left(s_{1}\right)$	－	$s \succeq_{i} s_{2}$
	UDSMR$_{\text {a }}$	$s_{1} \in \bar{R}_{i \mathrm{a}}^{d+}(s)$	$s_{2} \in R_{j}\left(s_{1}\right)$	$s_{3} \in R_{i}\left(s_{2}\right)$	$s \succeq_{i} s_{2}, s \succeq_{i} s_{3}$
	UDSSEQ$_{\text {a }}$	$s_{1} \in \bar{R}_{i \mathrm{a}}^{d+}(s)$	$s_{2} \in \bar{R}_{j \mathrm{a}}^{d+}\left(s_{1}\right)$	$s_{3} \in R_{i}\left(s_{2}\right)$	$s \succeq_{i} s_{2}, s \succeq_{i} s_{3}$
b型(混合型)	UDNash$_{\text {b }}$	$\bar{R}_{i \mathrm{b}}^{d+}(s)=\varnothing$	－	－	$\bar{R}_{i \mathrm{b}}^{d+}(s)=\varnothing$
	UDGMR$_{\text {b }}$	$s_{1} \in \bar{R}_{i \mathrm{b}}^{d+}(s)$	$s_{2} \in R_{j}\left(s_{1}\right)$	－	$s \succeq_{i} s_{2}$
	$\mathrm{UDSEQ}_{\mathrm{b}}$	$s_{1} \in \bar{R}_{i \mathrm{b}}^{d+}(s)$	$s_{2} \in \bar{R}_{j \mathrm{b}}^{d+}\left(s_{1}\right)$	－	$s \succeq_{i} s_{2}$
	UDSMR$_{\text {b }}$	$s_{1} \in \bar{R}_{i \mathrm{b}}^{d+}(s)$	$s_{2} \in R_{j}\left(s_{1}\right)$	$s_{3} \in R_{i}\left(s_{2}\right)$	$s \succeq_{i} s_{2}, s \succeq_{i} s_{3}$
	UDSSEQ$_{\text {b }}$	$s_{1} \in \bar{R}_{i \mathrm{b}}^{d+}(s)$	$s_{2} \in \bar{R}_{j \mathrm{b}}^{d+}\left(s_{1}\right)$	$s_{3} \in R_{i}\left(s_{2}\right)$	$s \succeq_{i} s_{2}, s \succeq_{i} s_{3}$
c型(混合型)	UDNash$_{\text {c }}$	$\bar{R}_{i{\rm{c}}}^{d+}(s)=\varnothing$	－	－	－
	UDGMR$_{\text {c }}$	$s_{1} \in \bar{R}_{i{\rm{c}}}^{d+}(s)$	$s_{2} \in R_{j}\left(s_{1}\right)$	－	$s \succeq_{i} s_{2}$或$s U_{i} s_{2}$
	$\mathrm{UDSEQ}_{\mathrm{c}}$	$s_{1} \in \bar{R}_{i{\rm{c}}}^{d+}(s)$	${s_2} \in \bar R_{j{\rm{c}}}^{d + }\left( {{s_1}} \right)$	－	$s \succeq_{i} s_{2}$或$s U_{i} s_{2}$
	$\mathrm{UDSMR}_{\mathrm{c}}$	$s_{1} \in \bar{R}_{i \mathrm{c}}^{d+}(s)$	$s_{2} \in R_{j}\left(s_{1}\right)$	$s_{3} \in R_{i}\left(s_{2}\right)$	$s \succeq_{i} s_{2}$或$s U_{i} s_{2}, s \succeq_{i} s_{3}$或$s U_{i} s_{3}$
	${\rm{UDSSE}}{{\rm{Q}}_{{\rm{c }}}}$	$s_{1} \in \bar{R}_{i{\rm{c}}}^{d+}(s)$	$s_{2} \in \bar{R}_{j \mathrm{c}}^{d+}\left(s_{1}\right)$	$s_{3} \in R_{i}\left(s_{2}\right)$	$s \succeq_{i} s_{2}$或$s U_{i} s_{2}, s \succeq_{i} s_{3}$或$s U_{i} s_{3}$
d型(保守型)	UDNash$_{\text {d }}$	$\bar{R}_{i \mathrm{d}}^{d+}(s)=\varnothing$	－	－	$\bar{R}_{i \mathrm{d}}^{d+}(s)=\varnothing$
	$\mathrm{UDGMR}_{\mathrm{d}}$	$s_{1} \in \bar{R}_{i \mathrm{d}}^{d+}(s)$	$s_{2} \in R_{j}\left(s_{1}\right)$	－	$s \succeq_{i} s_{2}$或$s U_{i} s_{2}$
	$\mathrm{UDSEQ}_{\mathrm{d}}$	$s_{1} \in \bar{R}_{i \mathrm{d}}^{d+}(s)$	$s_{2} \in \bar{R}_{j \mathrm{d}}^{d+}\left(s_{1}\right)$	－	$s \succeq_i s_2$或$s U_i s_2$
	$\mathrm{UDSMR}_{\mathrm{d}}$	$s_{1} \in \bar{R}_{i \mathrm{d}}^{d+}(s)$	$s_{2} \in R_{j}\left(s_{1}\right)$	$s_{3} \in R_{i}\left(s_{2}\right)$	$s \succeq_{i} s_{2}$或$s U_{i} s_{2}, s \succeq_{i} s_{3}$或$s U_{i} s_{3}$
	$\mathrm{UDSSEQ}_{\mathrm{d}}$	$s_{1} \in \bar{R}_{i \mathrm{d}}^{d+}(s)$	$s_{2} \in \bar{R}_{j \mathrm{d}}^{d+}\left(s_{1}\right)$	$s_{3} \in R_{i}\left(s_{2}\right)$	$s \succeq_{i} s_{2}$或$s U_{i} s_{2}, s \succeq_{i} s_{3}$或$s U_{i} s_{3}$

Table 7

Table 8

Table 9

Table 10

Table 11

Table 12

Fig.1

Table 13

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References 32

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集合类型	集合含义
$\varPhi_{i}^{+}(s)=\left\{q \in S: s \succ_{i} q\right\}$	在状态$s$处的改良偏好集
$\varPhi_{i}^{U}(s)=\left\{q \in S: s U_{i} q\right\}$	在状态$s$处的不确定偏好集
$\varPhi_{i}^{+}, U(s)=\varPhi_{i}^{+}(s) \cup \varPhi_{i}^{U}(s)$	在状态$s$处改良或不确定偏好集

移动类型	含义
$R_{i}^{+}(s)=R_{i}(s) \cap \varPhi_{i}^{+}(s)$	决策者$i$从初始状态$s$出发的改良可达集
$R_{i}^{U}(s)=R_{i}(s) \cap \varPhi_{i}^{U}(s)$	决策者$i$从初始状态$s$出发的不确定偏好可达集
$R_{i}^{+, U}(s)=R_{i}^{+}(s) \cup R_{i}^{U}(s)$	决策者$i$从初始状态$s$出发的改良或不确定偏好可达集

偏好类型	转移动机	对手制裁	稳定状态及均衡解数量
a型(激进型)	接受不确定偏好	不接受不确定偏好	最少
b型(混合型)	不接受不确定偏好	不接受不确定偏好	中等
c型(混合型)	接受不确定偏好	接受不确定偏好	中等
d型(保守型)	不接受不确定偏好	接受不确定偏好	最多

符号	解释说明	元素解释
E	元素均为1的m×m阶矩阵	-
$\circ $	哈达玛积	-
sign	适性函数(对于矩阵A第s行q列元素为sign(A(s, q)))	$\operatorname{sign}(A(s, q))=\left\{\begin{array}{l}1, A(s, q)>0 \\ 0, A(s, q)=0 \\ -1, A(s, q) <0\end{array}\right.$
J_i	可达矩阵(第s行q列元素为J_i(s, q))	$J_i(s, q)=\left\{\begin{array}{l}1, (s, q) \in A_i \\ 0, \text { 其他 }\end{array}\right.$

偏好类型	$\bar{u}_{x k h}$	$\bar{v}_{x k h}$
a型(激进型)	$\bar{u}_{a k h}=\boldsymbol{P}_{i(k, h)}^{+, U}$	$\bar{v}_{a k h}=\boldsymbol{P}_{j(k, h)}^{+, U}$
b型(混合型)	$\bar{u}_{b k h}=\boldsymbol{P}_{i(k, h)}^{+}$	$\bar{v}_{b k h}=\boldsymbol{P}_{j(k, h)}^{+, U}$
c型(混合型)	$\bar{u}_{c k h}=\boldsymbol{P}_{i(k, h)}^{+, U}$	$\bar{v}_{c k h}=\boldsymbol{P}_{j(k, h)}^{+, U}$
d型(保守型)	$\bar{u}_{d k h}=\boldsymbol{P}_{i(k, h)}^{+}$	$\bar{v}_{d k h}=\boldsymbol{P}_{j(k, h)}^{+U}$

Consensus model under uncertain preference based on graph model for conflict resolution

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矩阵类型	定义表达	元素含义
UC改良矩阵	$\overline{\boldsymbol{P}}_{i x}^{c+}=\left[\begin{array}{ccccc}\bar{p}_{x 11}^{c+} & \cdots & \bar{p}_{x 1 h}^{c+} & \cdots & \bar{p}_{x 1 m}^{c+} \\ \vdots & \ddots & \vdots & \ddots & \vdots \\ \bar{p}_{x k 1}^{c+} & \cdots & \bar{p}_{x k h}^{c+} & \cdots & \bar{p}_{x k m}^{c+} \\ \vdots & \ddots & \vdots & \ddots & \vdots \\ \bar{p}_{x m 1}^{c+} & \cdots & \bar{p}_{x m h}^{c+} & \cdots & \bar{p}_{x m m}^{c+}\end{array}\right]$	$\begin{gathered}\bar{p}_{x k h}^{c+}= \\ \left\{\begin{array}{c}1, \bar{u}_{x k h}= \\ \bar{p}_{x k h}^{c \mathrm{~d}}=1 \\ 0, \text { 其他 }\end{array}\right.\end{gathered}$
UD改良矩阵	$\overline{\boldsymbol{P}}_{i x}^{d+}=\left[\begin{array}{ccccc}\bar{p}_{x 11}^{d+} & \cdots & \bar{p}_{x 1 h}^{d+} & \cdots & \bar{p}_{x 1 m}^{d+} \\ \vdots & \ddots & \vdots & \ddots & \vdots \\ \bar{p}_{x k 1}^{d+} & \cdots & \bar{p}_{x k h}^{d+} & \cdots & \bar{p}_{x k m}^{d+} \\ \vdots & \ddots & \vdots & \ddots & \vdots \\ \bar{p}_{x m 1}^{d+} & \cdots & \bar{p}_{x m h}^{d+} & \cdots & \bar{p}_{x m m}^{d+}\end{array}\right]$	$\begin{gathered}\bar{p}_{x k h}^{d+}= \\ \left\{\begin{array}{c}1, \bar{u}_{x k h}=1 ; ~ \\ \bar{p}_{x k h}^{c d}=-1 \\ 0, \text { 其他 }\end{array}\right.\end{gathered}$
UC改良可达矩阵	$\overline{\boldsymbol{J}}_{i x}^{c+}=\boldsymbol{J}_i \circ \overline{\boldsymbol{P}}_{i x}^{c+}$	-
UD改良可达矩阵	$\overline{\boldsymbol{J}}_{i x}^{d+}=\boldsymbol{J}_i \circ \overline{\boldsymbol{P}}_{i x}^{d+}$	-

偏好类型	稳定性	UC稳定性	UD稳定性
a型(激进型)	Nash_a	$\boldsymbol{M}_i^{{\rm{UCNas}}{{\rm{h}}_{\rm{a}}}}=\overline{\boldsymbol{J}}_{i \mathrm{a}}^{c+} \cdot \boldsymbol{E}$	$\boldsymbol{M}_i^{{\rm{UDNas}}{{\rm{h}}_{\rm{a}}}}=\overline{\boldsymbol{J}}_{i \mathrm{a}}^{d+} \cdot \boldsymbol{E}$
	GMR_a	-	$\boldsymbol{M}_i^{\mathrm{UDGMR}_{\mathrm{a}}}=\overline{\boldsymbol{J}}_{i \mathrm{a}}^{d+} \cdot\left[\boldsymbol{E}-\operatorname{sign}\left(\boldsymbol{J}_j \cdot\left(\boldsymbol{P}_i^{-} \cdot=\right)^{\mathrm{T}}\right)\right]$
	SEQ_a	$\boldsymbol{M}_i^{{\rm{UCSE}}{{\rm{Q}}_{\rm{a}}}}=\overline{\boldsymbol{J}}_{i \mathrm{a}}^{c+} \cdot\left[\boldsymbol{E}-\operatorname{sign}\left(\overline{\boldsymbol{J}}_{j \mathrm{a}}^{d+} \cdot\left(\boldsymbol{P}_i^{-\cdot}=\right)^{\mathrm{T}}\right)\right]$	$\boldsymbol{M}_i^{\mathrm{UDSEQ}_{\mathrm{a}}}=\overline{\boldsymbol{J}}_{i \mathrm{a}}^{d+} \cdot\left[\boldsymbol{E}-\operatorname{sign}\left(\overline{\boldsymbol{J}}_{j \mathrm{a}}^{d+} \cdot\left(\boldsymbol{P}_i^{-.}=\right)^{\mathrm{T}}\right)\right]$
	SMR_a	-	$\boldsymbol{M}_i^{{\rm{UDSM}}{{\rm{R}}_{\rm{a}}}}=\overline{\boldsymbol{J}}_{i \mathrm{a}}^{d+} \cdot\left[\boldsymbol{E}-\operatorname{sign}\left(\boldsymbol{J}_j \cdot \boldsymbol{W}\right)\right]$
	SSEQ_a	$\boldsymbol{M}_i^{{\rm{UCSSE}}{{\rm{Q}}_{\rm{a}}}}=\overline{\boldsymbol{J}}_{i \mathrm{a}}^{c+} \cdot\left[\boldsymbol{E}-\operatorname{sign}\left(\overline{\boldsymbol{J}}_{j \mathrm{a}}^{d+} \cdot \boldsymbol{W}\right)\right]\\\boldsymbol{W}=\left(\boldsymbol{P}_i^{-\cdot}=\right)^{\mathrm{T}} \circ\left[\boldsymbol{E}-\operatorname{sign}\left(\boldsymbol{J}_i \cdot\left(\boldsymbol{P}_i^{+, U}\right)^{\mathrm{T}}\right)\right]$	$\begin{gathered}\boldsymbol{M}_i^{{\rm{UDSSE}}{{\rm{Q}}_{\rm{a}}}}=\overline{\boldsymbol{J}}_{i \mathrm{a}}^{d+} \cdot\left[\boldsymbol{E}-\operatorname{sign}\left(\overline{\boldsymbol{J}}_{j \mathrm{a}}^{d+} \cdot \boldsymbol{W}\right)\right] \\ \boldsymbol{W}=\left(\boldsymbol{P}_i^{-}=\right)^{\mathrm{T}} \circ\left[\boldsymbol{E}-\operatorname{sign}\left(\boldsymbol{J}_i \cdot\left(\boldsymbol{P}_i^{+, U}\right)^{\mathrm{T}}\right)\right]\end{gathered}$
b型(混合型)	Nash_b	$\boldsymbol{M}_i^{{\rm{UCNas}}{{\rm{h}}_{\rm{b}}}}=\overline{\boldsymbol{J}}_{i \mathrm{b}}^{c+} \cdot \boldsymbol{E}$	$\boldsymbol{M}_i^{\mathrm{UDNash}_{\mathrm{b}}}=\overline{\boldsymbol{J}}_{i \mathrm{b}}^{d+} \cdot \boldsymbol{E}$
	GMR_b	-	$\boldsymbol{M}_i^{{\rm{UDGM}}{{\rm{R}}_{\rm{b}}}}=\overline{\boldsymbol{J}}_{i \mathrm{b}}^{d+} \cdot\left[\boldsymbol{E}-\operatorname{sign}\left(\boldsymbol{J}_j \cdot\left(\boldsymbol{P}_i^{-\cdot}=\right)^{\mathrm{T}}\right)\right]$
	SEQ_b	$\boldsymbol{M}_i^{\mathrm{UCSEQ}_{\mathrm{b}}}=\overline{\boldsymbol{J}}_{i \mathrm{b}}^{c+} \cdot\left[\boldsymbol{E}-\operatorname{sign}\left(\overline{\boldsymbol{J}}_{j \mathrm{b}}^{d+} \cdot\left(\boldsymbol{P}_i^{-} \cdot=\right)^{\mathrm{T}}\right)\right]$	$\boldsymbol{M}_i^{{\rm{UDSE}}{{\rm{Q}}_{\rm{b}}}}=\overline{\boldsymbol{J}}_{i \mathrm{b}}^{d+} \cdot\left[\boldsymbol{E}-\operatorname{sign}\left(\overline{\boldsymbol{J}}_{j \mathrm{b}}^{d+} \cdot\left(\boldsymbol{P}_i^{-.}=\right)^{\mathrm{T}}\right)\right]$
	SMR_b	-	$\begin{gathered}\boldsymbol{M}_i^{{\rm{UDSM}}{{\rm{R}}_{\rm{b}}}}=\overline{\boldsymbol{J}}_{i \mathrm{b}}^{d+} \cdot\left[\boldsymbol{E}-\operatorname{sign}\left(\boldsymbol{J}_j \cdot \boldsymbol{W}\right)\right] \\ \boldsymbol{W}=\left(\boldsymbol{P}_i^{-.}=\right)^{\mathrm{T}} \circ\left[\boldsymbol{E}-\operatorname{sign}\left(\boldsymbol{J}_i \cdot\left(\boldsymbol{P}_i^{+, U}\right)^{\mathrm{T}}\right)\right]\end{gathered}$
	SSEQ_b	$\boldsymbol{M}_i^{\mathrm{UCSSEQ}}{ }_{\mathrm{b}}=\overline{\boldsymbol{J}}_{i \mathrm{b}}^{c+} \cdot\left[\boldsymbol{E}-\operatorname{sign}\left(\overline{\boldsymbol{J}}_{j \mathrm{b}}^{d+} \cdot \boldsymbol{W}\right)\right]\\\boldsymbol{W}=\left(\boldsymbol{P}_i^{-}=\right)^{\mathrm{T}} \circ\left[\boldsymbol{E}-\operatorname{sign}\left(\boldsymbol{J}_i \cdot\left(\boldsymbol{P}_i^{+}, U\right)^{\mathrm{T}}\right)\right]$	$\boldsymbol{M}_i^{{\rm{UDSSE}}{{\rm{Q}}_{\rm{b}}}}=\overline{\boldsymbol{J}}_{i \mathrm{b}}^{d+} \cdot\left[\boldsymbol{E}-\operatorname{sign}\left(\overline{\boldsymbol{J}}_{j \mathrm{b}}^{d+} \cdot \boldsymbol{W}\right)\right]\\\boldsymbol{W}=\left(\boldsymbol{P}_i^{-} \cdot=\right)^{\mathrm{T}} \circ\left[\boldsymbol{E}-\operatorname{sign}\left(\boldsymbol{J}_i \cdot\left(\boldsymbol{P}_i^{+}, U\right)^{\mathrm{T}}\right)\right]$
c型(混合型)	Nash$_{\text {c }}$	$\boldsymbol{M}_i^{{\rm{UCNas}}{{\rm{h}}_{\rm{c}}}}=\overline{\boldsymbol{J}}_{\text {ic }}^{c+} \cdot \boldsymbol{E}$	$\boldsymbol{M}_i^{{\rm{UDNas}}{{\rm{h}}_{\rm{c}}}}=\overline{\boldsymbol{J}}_{\text {ic }}^{d+} \cdot \boldsymbol{E}$
	$\mathrm{GMR}_{\mathrm{c}}$	－	$\boldsymbol{M}_{i}^{\mathrm{UDGMR}_{\mathrm{c}}}=\overline{\boldsymbol{J}}_{i{\rm{c}}}^{d+} \cdot\left[\boldsymbol{E}-\operatorname{sign}\left(\boldsymbol{J}_{j} \cdot\left(\boldsymbol{P}_{i}^{-}, =, U\right)^{\mathrm{T}}\right)\right]$
	$\mathrm{SEQ}_{\mathrm{c}}$	$\boldsymbol{M}_i^{{\rm{UCSE}}{{\rm{Q}}_{\rm{c}}}}=\overline{\boldsymbol{J}}_{i{\rm{c}}}^{c+} \cdot\left[\boldsymbol{E}-\operatorname{sign}\left(\overline{\boldsymbol{J}}_{j \mathrm{c}}^{d+} \cdot\left(\boldsymbol{P}_{i}^{-} \cdot=, U\right)^{\mathrm{T}}\right)\right]$	$\boldsymbol{M}_i^{{\rm{UDSE}}{{\rm{Q}}_{\rm{c}}}}=\overline{\boldsymbol{J}}_{i{\rm{c}}}^{d+} \cdot\left[\boldsymbol{E}-\operatorname{sign}\left(\overline{\boldsymbol{J}}_{j \mathrm{c}}^{d+} \cdot\left(\boldsymbol{P}_i^{-, =}, \mathrm{U}\right)^{\mathrm{T}}\right)\right]$
	$\mathrm{SMR}_{\mathrm{c}}$	－	$\begin{gathered}\boldsymbol{M}_{i}^{\mathrm{UDSMR}_{\mathrm{c}}}=\overline{\boldsymbol{J}}_{i{\rm{c}}}^{d+} \cdot\left[\boldsymbol{E}-\operatorname{sign}\left(\boldsymbol{J}_{j} \cdot \boldsymbol{W}\right)\right]\\ \boldsymbol{W}=\left(\boldsymbol{P}_{i}^{-}, =, U\right)^{\mathrm{T}} \circ\left[\boldsymbol{E}-\operatorname{sign}\left(\boldsymbol{J}_{i} \cdot\left(\boldsymbol{P}_{i}^{+}\right)^{\mathrm{T}}\right)\right]\end{gathered}$
	$\mathrm{SSEQ}_{\mathrm{c}}$	$\begin{gathered}\boldsymbol{M}_{i}^{\mathrm{UCSSEQ}} =\overline{\boldsymbol{J}}_{i{\rm{c}}}^{c+} \cdot\left[\boldsymbol{E}-\operatorname{sign}\left(\overline{\boldsymbol{J}}_{j \mathrm{c}}^{d+} \cdot \boldsymbol{W}\right)\right]\\ \boldsymbol{W}=\left(\boldsymbol{P}_{i}^{--}=, \cdot\right)^{\mathrm{T}} \circ\left[\boldsymbol{E}-\operatorname{sign}\left(\boldsymbol{J}_{i} \cdot\left(\boldsymbol{P}_{i}^{+}\right)^{\mathrm{T}}\right)\right]\end{gathered}$	$\begin{gathered}\boldsymbol{M}_i^{{\rm{UDSSE}}{{\rm{Q}}_{\rm{c}}}}=\overline{\boldsymbol{J}}_{i{\rm{c}}}^{d+} \cdot\left[\boldsymbol{E}-\operatorname{sign}\left(\overline{\boldsymbol{J}}_{j c}^{d+} \cdot \boldsymbol{W}\right)\right]\\ \boldsymbol{W}=\left(\boldsymbol{P}_{i}^{--, }, =, \mathrm{U}\right)^{\mathrm{T}} \circ\left[\boldsymbol{E}-\operatorname{sign}\left(\boldsymbol{J}_{i} \cdot\left(\boldsymbol{P}_{i}^{+}\right)^{\mathrm{T}}\right)\right]\end{gathered}$
d型(保守型)	Nash$_{\text {d }}$	$\boldsymbol{M}_i^{{\rm{UCNas}}{{\rm{h}}_{\rm{d}}}}=\overline{\boldsymbol{J}}_{i {\rm{d}}}^{c+} \cdot \boldsymbol{E}$	$\boldsymbol{M}_i^{{\rm{UDNas}}{{\rm{h}}_{\rm{d}}}}=\overline{\boldsymbol{J}}_{i {\rm{d}}}^{d+} \cdot \boldsymbol{E}$
	$\mathrm{GMR}_{\mathrm{d}}$	－	$\boldsymbol{M}_i^{{\rm{UDGM}}{{\rm{R}}_{\rm{d}}}}=\overline{\boldsymbol{J}}_{i {\rm{d}}}^{d+} \cdot\left[\boldsymbol{E}-\operatorname{sign}\left(\boldsymbol{J}_{j} \cdot\left(\boldsymbol{P}_{i}^{-}, =, U\right)^{\mathrm{T}}\right)\right]$
	$\mathrm{SEQ}_{\mathrm{d}}$	$\boldsymbol{M}_i^{{\rm{UCSE}}{{\rm{Q}}_{\rm{d}}}}=\overline{\boldsymbol{J}}_{i {\rm{d}}}^{c+} \cdot\left[\boldsymbol{E}-\operatorname{sign}\left(\overline{\boldsymbol{J}}_{j \mathrm{d}}^{d+} \cdot\left(\boldsymbol{P}_{i}^{-\cdot}=, { }^{\text {d }}\right)^{\mathrm{T}}\right)\right]$	$\boldsymbol{M}_i^{{\rm{UDSE}}{{\rm{Q}}_{\rm{d}}}}=\overline{\boldsymbol{J}}_{i \mathrm{d}}^{d+} \cdot\left[\boldsymbol{E}-\operatorname{sign}\left(\overline{\boldsymbol{J}}_{j \mathrm{d}}^{d+} \cdot\left(\boldsymbol{P}_{i}^{-}, =, U\right)^{\mathrm{T}}\right)\right]$
	$\mathrm{SMR}_{\mathrm{d}}$	－	$\begin{gathered}\boldsymbol{M}_{i}^{\mathrm{UDSMR}_{\mathrm{d}}}=\overline{\boldsymbol{J}}_{i \mathrm{d}}^{d+} \cdot\left[\boldsymbol{E}-\operatorname{sign}\left(\boldsymbol{J}_{j} \cdot \boldsymbol{W}\right)\right]\\ \boldsymbol{W}=\left(\boldsymbol{P}_{i}^{-}, =, U\right)^{\mathrm{T}} \circ\left[\boldsymbol{E}-\operatorname{sign}\left(\boldsymbol{J}_{i} \cdot\left(\boldsymbol{P}_{i}^{+}\right)^{\mathrm{T}}\right)\right]\end{gathered}$
	${\rm{SSEQ}}_{\text {d }}$	$\begin{gathered}\boldsymbol{M}_i^{\mathrm{UCSSEQ}_{\mathrm{d}}}=\overline{\boldsymbol{J}}_{i \mathrm{d}}^{c+} \cdot\left[\boldsymbol{E}-\operatorname{sign}\left(\overline{\boldsymbol{J}}_{j \mathrm{d}}^{d+} \cdot \boldsymbol{W}\right)\right] \\ \boldsymbol{W}=\left(\boldsymbol{P}_i^{-}=, =, U\right)^{\mathrm{T}} \circ\left[\boldsymbol{E}-\operatorname{sign}\left(\boldsymbol{J}_i \cdot\left(\boldsymbol{P}_i^{+}\right)^{\mathrm{T}}\right)\right]\end{gathered}$	$\begin{aligned}& \boldsymbol{M}_i^{{\rm{UDSSE}}{{\rm{Q}}_{\rm{d}}}}=\overline{\boldsymbol{J}}_{i \mathrm{d}}^{d+} \cdot\left[\boldsymbol{E}-\operatorname{sign}\left(\overline{\boldsymbol{J}}_{j \mathrm{d}}^{d+} \cdot \boldsymbol{W}\right)\right] \\& \boldsymbol{W}=\left(\boldsymbol{P}_i^{-}, =, U\right)^{\mathrm{T}} \circ\left[\boldsymbol{E}-\operatorname{sign}\left(\boldsymbol{J}_i \cdot\left(\boldsymbol{P}_i^{+}\right)^{\mathrm{T}}\right)\right]\end{aligned}$

决策者	策略	含义
DM1	O₁₁: 履行合同	DM1积极履行合同，解决DM2提出的质量和性能问题
	O₁₂: 保持现状	DM1保持现状，既不解决问题，也不抵抗DM2的要求
	O₁₃: 提供补偿与改进	DM1提供经济补偿，并承诺改进飞机质量和性能问题
	O₁₄: 推迟交付并改进	DM1推迟交付新机，并在交付前解决所有质量和性能问题
DM2	O₂₁: 履行合同	DM2积极履行合同，继续接收波音787飞机，但要求DM1解决质量和性能问题
	O₂₂: 拖延合同	DM2拖延履行合同，暂时停止接收波音787飞机，直到DM1解决所有问题
	O₂₃: 取消合同并要求赔偿	DM2提议取消合同，并要求DM1赔偿已交付飞机的质量和性能问题

决策者	策略	s₁	s₂	s₃	s₄	s₅	s₆	s₇	s₈	s₉	s₁₀
DM1	O₁₁	Y	Y	Y	Y	N	Y	N	Y	N	-
	O₁₂	N	N	N	N	Y	N	Y	N	Y	-
	O₁₃	Y	N	Y	Y	Y	N	N	Y	Y	-
	O₁₄	N	Y	Y	N	N	Y	Y	Y	Y	-
DM2	O₂₁	Y	Y	Y	N	N	N	N	N	N	-
	O₂₂	N	N	N	Y	Y	Y	Y	Y	Y	-
	O₂₃	N	N	N	N	N	N	N	N	N	Y

决策者	策略声明	解释说明
DM2	－O₂₃	最不希望取消合同
	O₁₁	希望DM1积极履行合同内容，解决DM2提出的质量和性能问题
	O₂₁	希望履行合同
	－O₁₂	考虑冲突局势的稳定，DM2不希望DM1保持现状不作为
	O₂₂ if O₁₂	如果DM1保持现状不作为，DM2选择拖延履行合同
	O₁₃	希望波音提供经济补偿，并承诺改进飞机质量和性能问
	O₁₄	希望DM1推迟交付新的飞机，并在交付前解决所有质量和性能问题

决策者	部分偏好信息
DM1	$\left(s_1 U_1 s_2 U_1 s_3\right) \succ_1\left(s_4 U_1 s_6 U_1 s_8\right) \succ_1\left(s_5 U_1 s_7 U_1 s_9\right) \succ_1 s_{10}$
DM2	$s_3 \succ_2 s_1 \succ_2 s_2 \succ_2 s_8 \succ_2 s_4 \succ_2 s_6 \succ_2 s_9 \succ_2 s_5 \succ_2 s_7 \succ_2 s_{10}$