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

摘要/Abstract

摘要：

针对决策者偏好信息可能存在的不确定性, 考虑决策者不仅具有共识或利己行为, 还可能具有激进或保守的风险态度, 将简单偏好下冲突分析图模型共识理论推广到不确定场景, 构建不确定偏好下的冲突决策共识模型。具体而言, 将不确定偏好结构引入冲突共识理论, 分别定义4种不确定偏好下冲突共识与非共识偏好和稳定性的逻辑及矩阵表达, 后者不仅方便冲突的计算和分析, 更推动相关算法的实现。所建立的模型被应用于航空业冲突, 从而得到可行解决方案。结果表明, 该模型能够对不确定环境下复杂冲突的消解提供有效策略建议。

关键词: 不确定偏好, 冲突分析图模型, 决策共识, 矩阵表达, 航空业冲突

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

中图分类号:

C934

张瑾木子, 徐海燕, 陈璐. 基于图模型的不确定偏好下冲突决策共识模型[J]. 系统工程与电子技术, 2025, 47(1): 191-201.

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.

图/表 15

表1

表2

表3

表4

不确定偏好下稳定性逻辑定义"

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$s U_{i} s_{3}$

表4

表5

表6

UC与UD偏好逻辑定义"

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	$\begin{aligned} \bar{\varPhi}_{i \mathrm{d}}^{c+}(s)= \varPhi_{i}^{+}(s) \bigcap \varPhi_{j}^{+, U}(s)\end{aligned}$	$\begin{gathered}\bar{\varPhi}_{i \mathrm{d}}^{d+}(s)= \varPhi_{i}^{+}(s) \bigcap \varPhi_{j}^{-}(s)\end{gathered}$	$\begin{gathered}\bar{R}_{i \mathrm{d}}^{c+}(s)= R_{i}(s) \bigcap \bar{\varPhi}_{i \mathrm{d}}^{c+}(s)\end{gathered}$	$\begin{gathered}\bar{R}_{i \mathrm{d}}^{d+}(s)= R_{i}(s) \bigcap \bar{\varPhi}_{i \mathrm{d}}^{d+}(s)\end{gathered}$

表6

表7

不确定偏好下的冲突非共识稳定性逻辑表达"

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$s U_{i} s_{3}$

表7

表8

表9

表10

表11

表12

图1

表13

表14

参考文献 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)	$\bar{u}_{d k h}=\boldsymbol{P}_{i(k, h)}^{+}$	$\bar{v}_{d k h}=\boldsymbol{P}_{j(k, h)}^{+U}$