# State Definition for Conflict Analysis with Four-valued Logic

1st Yukiko Kato  Lynx Research Llc
Tokyo, Japan
ORCiD:0000-0003-0872-3847

###### Abstract

We examined a four-valued logic method for state settings in conflict
resolution models. Decision-making models of conflict resolution, such as game
theory and graph model for conflict resolution (GMCR), assume the description
of a state to be the outcome of a combination of strategies or the consequence
of option selection by the decision-makers. However, for a framework to
function as a decision-making system, unless a clear definition of the task of
placing information out of an infinite world exists, logical consistency
cannot be ensured, and thus, the function may be incomputable. The
introduction of paraconsistent four-valued logic can prevent incorrect state
setting and analysis with insufficient information and provide logical
validity to analytical methods that vary the analysis resolution depending on
the degree of coarseness of the available information. This study proposes a
GMCR stability analysis with state configuration based on Belnap’s four-valued
logic.

###### Index Terms:

state, four-valued logic, frame problem, conflict resolution, GMCR

*This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible.

##  I Introduction

In the decision analysis framework, a common method involves setting the
status quo and obtaining a ranking of the preferred states for a decision-
maker (DM) by repeating the pairwise comparison of states that differ from the
status quo and are more preferable for the DM. When the preferences of
multiple DMs’ conflict, some suggestions for solutions to the conflict can be
obtained by synthesizing these preferences and estimating the states in which
the entire situation is likely to settle. That is, the methodology we use to
recognize and describe the initial state is the starting point for all
subsequent classifications, analyses, and evaluations. Meanwhile, most of the
discussions in decision-making systems focus on how to derive and expand the
concepts related to preferences and stability; however, an in-depth discussion
regarding the state perception and definition does not exist. This study aims
to scrutinize the mechanism for the definition of states in conflict
resolution models, especially the graph model for conflict resolution (GMCR),
discuss the problems, introduce findings based on the Kripke structure, and
propose a new method of setting the state.

The following section clarifies the basic concepts and background of the
methods used in this study for decision and conflict resolution analysis by
overviewing the literature; essential definitions and frameworks are provided.
Section III explains the concepts and frameworks that are proposed in this
paper. Next, in Section IV, the new framework presented in Section III is
applied to the conflict cases for further observation and analysis. In the
Conclusion section V, the future research aspects based on the results of this
study are described.

##  II Underlying Concepts and Methods

GMCR, the basic framework of this paper, was derived from non-cooperative game
theory , and was developed by D.M. Kilgour, K.W. Hipel, and L. Fang[1],[2].
Extended concepts of GMCR include coalition analysis [4],[5],[28], attitude-
based analysis [6],[7], and preference strength[8]; furthermore, research has
been actively conducted using fuzzy[9], [19], [20] and grey methods[10] for
cases in which the DM preferences are uncertain. Other analysis methods with
coarser information have been proposed, such as “permissible ranges” for
preferences and analyzing them with binary information [11], [12].

GMCR adopts the concept that DMs can transition to a state that is preferable
to them within a choice of options that they can control. In game theory, the
focus of the discussion is on the DM’s “strategy,” but in GMCR, the focus is
on the “state” and “state transition” as a consequence of option selection.
The framework has more flexibility and broader descriptive capability owing to
the unique concept of state transitions, which is not addressed in game
theory.

In decision-making, we generally deal with the following essential elements: a
set of “options,” “consequences,” “maps from the option selections to the
consequences,” and a “preference structure for consequences.” An option
describes a situation in which decision-makers can make their own choice. In a
conflict, the elements of a situation to be decided by other decision-makers
are not options; similarly, elements that the decision-maker has not yet
identified but could affect his situation, such as climate or global economic
trends, are also not options. In GMCR, a state is a description of the
consequence of the DMs’ selection of options. The following subsections
provide an overview of the basic framework of the GMCR.

###  II-A Graph Model for Conflict Resolution

####  II-A1 Framework

GMCR is a framework that can be described with four tuples[1],[2]:

$(N,S,(A_{i})_{i\in N},(\succsim_{i})_{i\in N}).$ (1)

$N$ is the set of all DMs involved in the conflict, where $|N|\geq 2$. $S$ is
the set of all possible states in the conflict, where $|S|\geq 2$. If $i\in
N$, then the pairs $(S,A_{i})$ constitute a graph of DM $i$, where $S$ is the
set of all vertices and $A_{i}\subset S\times S$ is the set of all arcs. We
also assume that $(S,A_{i})$ has no loops, i.e., $(s,s)\in A$ for each $s\in
S$. $\succsim_{i}$ is the DM $i$’s preference for possible states, satisfying
reflexivity (for any $s\in S,s\succsim_{i}s$), completeness (for any
$s,s^{\prime}\in S,s\succsim_{i}s^{\prime}$ or $s^{\prime}\succsim_{i}s$), and
transitivity (for any $s,s^{\prime},s^{\prime\prime}\in S$, if
$s\succsim_{i}s^{\prime}$ and $s^{\prime}\succsim_{i}s^{\prime\prime}$, then
$s\succsim_{i}s^{\prime\prime}$). A rational DM desires the situation to
change to a more favorable state and attempts to move to the preferred states
by repeating unilateral moves: transitions over which the DM has control.

As a consequence of the DMs’ moves in GMCR, standard stability concepts,
including Nash stability (Nash)[13],[14], general metarationality (GMR) [15],
symmetric metarationality (SMR)[15], and sequential stability (SEQ) [16],[17],
exist; some underlying notions must be provided. For $i\in N$ and $s\in S$, we
define DM $i$’s reachable list from the state $s$ as the set $\\{s^{\prime}\in
S\mid(s,s^{\prime})\in A_{i}\\}$, denoted by $R_{i}(s)$. $R_{i}(s)$ is the set
of all states in the conflict that DM $i$ can move from $s$ to $s^{\prime}$ in
one step. A unilateral improvement of DM $i$ from state $s$ is defined as an
element of the reachable list of DM $i$ from $s$ (i.e., $s^{\prime}\in
R_{i}(s)$), where DM $i$ strictly prefers state $s^{\prime}$
($s^{\prime}\succ_{i}s$). Therefore, the set of unilateral improvement lists
of DM $i$ from state $s$ is the set $\\{s^{\prime}\in
R_{i}(s)|s^{\prime}\succ_{i}s\\}$, which is known as DM $i$’s unilateral
improvement list from state $s$ and denote it by $R^{+}_{i}(s)$.
$\phi^{\simeq}_{i}(s)$ denotes the set of all states that DM $i$ equally or
less prefers to state $s$; $\\{s^{\prime}\in S\mid
s\succsim_{i}s^{\prime}\\}$.

####  II-A2 Stability Concepts

The standard stability concepts in GMCR: Nash, GMR, SMR, and SEQ can be
defined based on the four tuples and the reachable lists noted in II-A1.
NASH: For $i\in N$ and state $s\in S$, $s$ is a Nash state for DM $i$ if and
only if $R^{+}_{i}(s)=\emptyset$. The set of all Nash states for DM $i$ is
denoted $S_{i}^{Nash}$. A Nash state of DM $i$ is considered stable for DM $i$
as DM $i$ cannot unilaterally improve from that state.
GMR: $s$ is the GMR state for DM $i$ if and only if $s^{\prime}\in
R_{i}^{+}(s)$, $R_{N\backslash
i}(s^{\prime})\cap\phi^{\backsimeq}_{i}(s)\neq\emptyset$. DM $i$’s GMR state
is stable for DM $i$; each DM $i$ is deterred from unilaterally improving from
that state as the countermove of other DMs leaves DM $i$ in a state equal to
or less favorable than its original state. The set of all GMR states for DM
$i$ is denoted as $S_{i}^{GMR}$.
SMR: $s$ is an SMR state for DM $i$, if and only if for all $s^{\prime}\in
R^{+}_{i}(s)$, there exists $s^{\prime\prime}\in
R_{N\backslash{i}}(s^{\prime})\cap\phi_{i}^{\backsimeq}(s)\neq\emptyset$ such
that $s^{\prime\prime\prime}\in\phi^{\backsimeq}_{i}(s)$ for all
$s^{\prime\prime\prime}\in R_{i}(s^{\prime\prime})$, i.e.,
$R_{i}(s^{\prime\prime})\subseteq\phi^{\backsimeq}_{i}(s)$. The set of all SMR
states for DM $i$ is denoted $S^{SMR}_{i}$. In DM $i$’s SMR states, the
countermove of the other DMs on the unilateral improvement of DM $i$ will
result in DM $i$ achieving a state that is no better than the original state
even if DM $i$ unilaterally moves from that state. Consequently, the original
state can be considered stable for DM $i$ because the first unilateral
improvement is not beneficial for DM $i$.
SEQ: $s$ is an SEQ state for DM $i$ if and only if for all $s^{\prime}\in
R^{+}_{i}(s),R^{+}_{N\backslash{i}}(s^{\prime})\cap\phi^{\backsimeq}_{i}(s)=\emptyset$.
The set of all SEQ states for DM $i$ is denoted as $S^{SEQ}_{i}$. In the SEQ
state of DM $i$, the credible countermove of other DMs on DM $i$’s unilateral
improvement achieves a state that is equally favorable or less favorable than
the original state to DM $i$. Therefore, the original state can be regarded as
stable for DM $i$ as DM $i$ never becomes better than the original state.
For each of these stability concepts, $s\in S$ is considered to be in
equilibrium if and only if $s$ is stable for all DMs.

Furthermore, CNash, CGMR, CSMR, and CSEQ have also been proposed as coalition
stability concepts when the DM believes that a transition to a better state
can be achieved by repeating moves in cooperation with other DMs.

####  II-A3 State

In GMCR, the alternatives of actions that the DM can control are called
options. The set of all options in a conflict is $O=\cup_{i\in N}O_{i},$ where
$i$ denotes which DM controls the options. We let $O_{i}$ represent the set of
options of DM $i$ for $i\in N$ with $o_{ij}\in O_{i}$. An option selection for
DM $i$ is a mapping $\lambda:O_{i}\rightarrow\\{0,1\\}$, such that for
$j=1,2,\dots,h_{i}$, where $o_{ij}$ is DM $i$’s $j$th option.

$\lambda(o_{ij})=\begin{cases}{1\>{{{\rm{if}\>\>DM}\,i\,{\rm{selects\,option}}}}\,o_{ij}},\\\
{0\>\rm{otherwise}}.\end{cases}$ (2)

Let $O=\bigcup_{i\in N}O_{i}$ be the set of all options for $o_{ij}\in
O_{i},i=1,2,\dots,n.$ A state is a mapping $\kappa:O\rightarrow\\{0,1\\}$,
such that for $i=1,2,\dots,n$.

$\kappa(o_{ij})=\begin{cases}{1\>{{{\rm{if}\>\>DM}\,i\,{\rm{selects\,option}}}}\,o_{ij}},\\\
{0\>\rm{otherwise}}.\end{cases}$ (3)

Let $|O|$ represent the total number of options available to the DMs. A state
can be treated as a $|O|$-dimensional column vector with 0 or 1 as the
element. $\lambda^{s}(O_{i})$ denotes DM $i$’s option selection corresponding
to state $s$ for $i=1,2,\dots,n$ and is a $|O_{i}|$-dimensional column vector
whose elements are as follows:

$\lambda^{s}(o_{ij})=\begin{cases}{1\>{{{\rm{if}\>\>DM}\,i\,{\rm{selects\,option}}}}\,o_{ij}\>\>{\rm{in\>\>state}}\>\>s},\\\
{0\>\rm{otherwise}}.\end{cases}$ (4)

Prisoners’ dilemma can be illustrated as in Table I , where “Y” indicates that
an option is selected by the DM, whereas “N” implies that the option is not
selected.

TABLE I: Prisoners’ Dilemma | DM1 | DM2
---|---|---
Option | Not Confess | Not Confess
$s_{1}$ | Y | Y
$s_{2}$ | Y | N
$s_{3}$ | Y | Y
$s_{4}$ | Y | N

The option form or binary form as shown in Table I, which expresses the
conflicts based on the selection of options, was initially proposed by Howard
[15]. Furthermore, the option form is considered one of the methods of
describing a state in the framework of GMCR, which is the underlying
conceptual foundation [1].

###  II-B Configuration of States and Information Content

In GMCR, the preference order for states is derived either by direct
evaluation of each state or by a method that satisfies the first-order
predicate logic that assumes unconditional (negation, conjunction, separation)
and conditional (iff) combinators to specify the option-prioritization method
order. In addition to fuzzy and grey, unknown [21], [22], and probabilistic
methods [23] have been studied when the preference information is ambiguous or
unknown; none of the above methods provide conceptual clarifications regarding
the basis for the states, but they focus on the viewpoint of the uncertainty
of preferences. Nevertheless, when analyzing real-world problems, information
may be insufficient at the outset of the analysis to define states with
sufficient logical consistency to withstand analysis, all at the same level of
granularity. Therefore, we consider a framework that allows for ambiguity and
uncertainty in setting the starting point of the analysis. With such
framework, we need not rewrite the original setting, but we can refine the
information granularity when the situation develops and more information is
available.

##  III State Definition by Four-valued Logic

A state in GMCR is represented as a combination of option selection by DMs
expressed as a binary value of whether DM $i$ has selected a particular
option. To define states in a manner more similar to real-world state
recognition, we consider states as information granules and an analysis system
that can simultaneously handle the states expressed in different information
granularities. Regarding the state definition, i.e., “framing” of real-world
problems, methods based on four-valued logic have been studied by Nakayama et
al.[30] mainly from the perspective of computability. The four-valued logic
uses the concept of granularity reasoning by introducing variable precision
rough set models (VPRS) by Ziarko [24] to the possible world concept in Kripke
and describes it in terms of four values, was developed by Belnap [25].

###  III-A Belnap’s Four-valued Model

Belnap’s four-valued set, represented by B4, comprises the elements of truth
value: {T, F, B(oth), N(one)}, where N indicates incompleteness, and B
indicates contradiction. The current state of an atomic proposition in the
computer can be described by four values: true, false, true and false, and
neither true nor false. The values for complex sentences were obtained by
Balnap by considering the monotonicity, which yields Table II.

TABLE II: 4-Truth Table

| N | F | T | B
---|---|---|---|---
$\neg$ | N | T | F | B

$\wedge$ | N | F | T | B
---|---|---|---|---
N | N | F | N | F
F | F | F | F | F
T | N | F | T | B
B | F | F | B | B

$\vee$ | N | F | T | B
---|---|---|---|---
N | N | N | T | T
F | N | F | T | B
T | T | T | T | T
B | T | B | T | B

###  III-B State Definition by Four-Valued Logic

Next, we define the states in Prisoner’s dilemma using B4 by viewing the
option proposition “DM $i$ does not confess” as an atomic statement. Table III
represents sixteen combinations of propositions generated by B4 .

TABLE III: Prisoner’s Dilemma -States by B4 | DM1 | DM2
---|---|---
$p$ | Not Confess | Not Confess
$1$ | T | T
$2$ | T | F
$3$ | T | N
$4$ | T | B
$5$ | F | T
$6$ | F | F
$7$ | F | N
$8$ | F | B
$9$ | N | T
${10}$ | N | F
${11}$ | N | N
${12}$ | N | B
${13}$ | B | T
${14}$ | B | F
${15}$ | B | N
${16}$ | B | B

For example, suppose there is a rumor that a pardon may be issued soon, and
the DM knows that only cases that have been closed after interrogation are
eligible for the amnesty. In such a case, we can understand that the DM may
choose B, which allows making an option depending on the status of the
execution of the pardon. N could be viewed as a DM’s situation in which the DM
is planning to escape from prison without cooperating with the interrogation.
In the general GMCR method of analysis, states are set on the assumption that
they can only be established by options that are definitely controllable by
the DM; hence, no other possibilities are considered. As a way to describe
other situations. Regarding this, the concept of “new reachability by external
factors” as a method to describe states involving non-optional elements that
cannot be controlled by DMs was proposed by Kato [12], [26]. Meanwhile, it can
be said that the state description by B4 solves the problem of the potential
state generation not by extending state transitions but by extending the state
definition itself. Unlike state generation with binary information, to set the
preference order for each state in B4 , DMs’ policy for B and N at the state
setting or framing [27] would need to be established. The preference of the DM
may be based on the objective that he prioritizes terminating the
interrogation in a shorter time, avoiding B or N regardless of the outcome of
obtaining a pardon. Alternatively, some DMs may not find the rumors of amnesty
enforcement credible and thus would prefer option B or N.

###  III-C GMCR Stability Analysis by B4

In order to perform GMCR stability analysis, reachability and preferences for
each DM must be determined.

####  III-C1 Reachability

Based on Anderson and Belnap’s entailment [29]: $\textbf{F}\preceq\textbf{B}$,
$\textbf{F}\preceq\textbf{N}$, $\textbf{B}\preceq\textbf{T}$,
$\textbf{N}\preceq\textbf{T}$, the reachability in GMCR can be defined. Thus,
we consider that DM1 can make a unilateral move when DM2’s proposition of the
initial state and the destination state is any of the following pairs: (B,B),
(B,F), (N,N), (N,F), (T,T), (T,B), (T,N).

The Figure 1 illustrates DM1’s reachability from $s_{1}$.

Figure 1: DM1’s Graph from $s_{1}$

####  III-C2 Preference

In addition to the usual preference information, each DM’s attitude toward
situations involving B and N must be assessed.

The results of the preference and stability analysis for the two DMs are
listed in Table IV, assuming that DM1 has a preference to finish the
interrogation as soon as possible in anticipation of pardon enforcement. In
contrast, DM2 has no information on the pardon and thus has a preference not
to make as clear a statement as much as possible and to be uncooperative
during the interrogation. The orders of preference for the two DMs were set as
follows:
$s_{2}\succ_{1}s_{1}\succ_{1}s_{4}\succ_{1}s_{3}\succ_{1}s_{8}\succ_{1}s_{6}\succ_{1}s_{7}\succ_{1}s_{5}\succ_{1}s_{9}\succ_{1}$,
$s_{7}\succ_{2}s_{8}\succ_{2}s_{9}\succ_{2}s_{5}\succ_{2}s_{6}\succ_{2}s_{3}\succ_{2}s_{1}\succ_{2}s_{4}\succ_{2}s_{2}\succ_{2}$.

####  III-C3 Stability Analysis

For simplicity of the stability analysis, we reduced the number of states is
reduced from 16 to 9.

TABLE IV: Prisoner’s Dilemma in B4 \- Stability Analysis  | $s_{1}$ | $s_{2}$ | $s_{3}$ | $s_{4}$ | $s_{5}$ | $s_{6}$ | $s_{7}$ | $s_{8}$ | $s_{9}$
---|---|---|---|---|---|---|---|---|---
DM1 | T | F | T | F | T | F | T | F | N
$R_{1}$ | 2,5,6,7,8,9 | 1,5,6,7,8,9 |  |  | 3,4,6 | 3,4,5 | 3,4,8,9 | 3,4,7,9 | 3,4,7,8
DM2 | T | T | F | F | N | N | B | B | B
$R_{2}$ | 3,5,7 |  | 1,5,7 |  | 1,3,7 |  | 1,3,5 |  |
Nash |  | 4* |  | ✓ |  |  |  |  |
GMR | 2* | 4* | 4* | ✓ |  |  |  |  |
SMR | 2* | 4* | 4* | ✓ |  |  |  |  |
SEQ | 2* | 4* | 4* | ✓ |  |  |  |  |
CNash |  | 4* |  | 4* |  |  |  |  |
CGMR | 2* | 4* | 4* | 4* |  |  |  |  |
CSMR | 2* | 4* | 4* | 4* |  |  |  |  |
CSEQ | 2* | 4* | 4* | 4* |  |  |  |  |
Pareto | ✓ | ✓ | ✓ |  |  |  | ✓ | ✓ |

In the Table IV, a checkmark indicates that equilibrium is established both in
the standard form and in B4, 4* represents that equilibrium is shown only in
B4. Finally, 2* indicates that equilibrium is found only in the two-valued
standard form. $R_{1}$ represents the destination states to which DM1 can
transition from the state, and $R_{2}$ represents the destinations where DM2
can transition.

The analysis shows that the original dilemma in $s_{4}$ still holds in the new
conflict setting with states represented in B4, whereas other equilibria are
newly established in $s_{2}$ and $s_{3}$. The results of the GMCR stability
analysis using states according to B4 are as follows:
$S^{Nash}=\\{s_{2},s_{4}\\}$, $S^{GMR}=\\{s_{2},s_{3},s_{4}\\}$,
$S^{SMR}=\\{s_{2},s_{3},s_{4}\\}$, $S^{CNash}=\\{s_{2},s_{4}\\}$
$S^{CGMR}=\\{s_{2},s_{3},s_{4}\\}$, $S^{CSEQ}=\\{s_{2},s_{3},s_{4}\\}$. A
strong equilibrium established in $s_{2}$, including a Nash equilibrium,
should be noted.

##  IV Reinterpretation of Conflicts in Four-Valued Logic

In the previous sections, we have presented the basic concepts of GMCR, the
new concept of states definition by introducing the four-valued logic, and
examples of its implementation. This section provides a new perspective on
Elmira conflict, a representative application analysis of GMCR.

###  IV-A State 9 in Elmira Conflict

The Elmira conflict is an environmental contamination dispute in Ontario,
Canada; numerous studies have been already conducted on this conflict using
the GMCR. Three DMs are involved in the conflict: the Ministry of Environment
(M), Uniroyal (U), and the local government (L). M discovered contamination
and issued a control order to U that included decontamination operation by U.
They want to exercise their authority efficiently. U owns questionable
chemical plants and intends to exercise its right to object, aiming to lift or
relax the control order. L represents the diverse interest groups and intends
to protect the residents and the local industrial base.

The options for each DM are as follows. M: can irreversibly modify the control
order; U: may continue to delay the objection process, irreversibly accept the
control order, or abandon the chemical plants irreversibly; or irreversibly
modify the control order; U: may continue to delay the objection process,
irreversibly accept the control order, or abandon the chemical plants
irreversibly; L: may argue for the application of the original control order.
Table V summarizes all the feasible states based on the DM’s options, and
Figure 2 represents the Graph of the conflict. The numbers in the graph
indicate the state numbers.

TABLE V: Elmira Conflict - Options and States |  | $s_{1}$ | $s_{2}$ | $s_{3}$ | $s_{4}$ | $s_{5}$ | $s_{6}$ | $s_{7}$ | $s_{8}$ | $s_{9}$
---|---|---|---|---|---|---|---|---|---|---
M | Modify | N | Y | N | Y | N | Y | N | Y | -
U | Delay | Y | Y | N | N | Y | Y | N | N | -
| Accept | N | N | Y | Y | N | N | Y | Y | -
| Abandon | N | N | N | N | N | N | N | N | Y
L | Insist | N | N | N | N | Y | Y | Y | Y | -
Figure 2: Graph Model of Elmira conflict

U’s second option where U abandons the plant, which is the cause of the
pollution, has the extreme significance in this case. $s_{9}$ in the graph
reveals that U uses this option as a powerful bargaining card. In $s_{9}$, no
other DM option selections are provided. In other words, $s_{9}$ can be
understood as the state in which U abandons the plant regardless of the other
DM’s option choices. In other words, the structure of state 9 in Elmira
conflict can be interpreted as a state in which only U’s “abandon” option
value is T, while all other options are B. Table VI shows the 9 states in B4.

TABLE VI: Elmira Conflict - States in B4 |  | $s_{1}$ | $s_{2}$ | $s_{3}$ | $s_{4}$ | $s_{5}$ | $s_{6}$ | $s_{7}$ | $s_{8}$ | $s_{9}$
---|---|---|---|---|---|---|---|---|---|---
M | Modify | F | T | F | T | F | T | F | T | B
U | Delay | T | T | F | F | T | T | F | F | B
| Accept | F | F | T | T | F | F | T | T | B
| Abandon | F | F | F | F | F | F | F | F | T
L | Insist | F | F | F | F | T | T | T | T | B

###  IV-B Interstate Conflicts-Russia Ukraine

Interstate conflicts are generally analyzed in the $2\times 2$ cooperate-
defect game structure, in which equilibrium solutions obtained by the attitude
of DMs are interpreted to have more escalatory or de-escalatory preferences.
In the case of a conflict with two DMs and one option for each (Y/N), even if
the structure of the conflict originally has no equilibrium, some equilibrium
solutions can be obtained by introducing a framework, for example, setting the
permissible ranges [11] for the DMs’ preference order. It is possible to
consider a path toward consensus building using the resolution result as a
clue. General security discussions, such as the deterrence theory, are also
based on these concepts. However, we observed that the premise of such
discussions might have been overturned by the recently erupted Russia–Ukraine
conflict. The states in B4 and the actual status of the graph of the conflict
are provided in Table VII and Figure 3, respectively. Assume that both DMs
have a positive attitude toward self.

Figure 3: Graph Model of Russia Ukraine Conflict  TABLE VII: States and Options in Russia Ukraine Conflict | Russia | Ukraine
---|---|---
Option | Attack | Attack
$s_{1}$ | T | B

Sixteen states generated by the four-value combination of $2\times 2$ games,
as shown in Table III; Only four states generated from the Y/N (T/F) selection
are subject to analysis in the general analysis method, whereas Russia: T and
Ukraine: B, which may better reflect the current situation, are not
considered. If we had analyzed the conflict using a four-valued system that
included B and N, we could predict a stalemate in T and Ukraine: B. Within a
system in which the nuclear weaponry fulfills a deterrent function, it could
be possible to respond to some extent through a binary rational analysis; it
can be said that the Russia–Ukraine conflict has already shown that such an
approach is no longer viable. Considering the current situation in which
artificial intelligence is being introduced in the security field and for
weapon control, this case suggests the danger of building a program with only
assumptions on reasonable conditions.

##  V Conclusion

In GMCR stability analysis, it is not a prerequisite that the option selection
structure underlying the state formulation be precise because it is sufficient
to know the DMs’ preference orders concerning the states. Nonetheless,
introducing a paraconsistent logic system to the state definition may improve
the computability and extensibility of GMCR analysis. In particular, as
demonstrated in Section IV, it is likely to be extremely useful in decision-
making systems in international relations and security. Four-valued logic,
which encompasses contradictions, can be a more conservative system; there are
many areas where its introduction would be worthwhile for analyzing real-world
problems.

Future work may include reexamining the $2\times 2$ international dispute
structure by introducing the four-valued logic concept and generalizing it in
the system of GMCR analysis based on studies of its relativeness to other
extended concepts such as coalition [4],[5],[28] permissible range [11], [12],
and new reachability by external factors [26].

## References

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