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An Analytical Model for Conflict DynamicsAuthor(s): N. GassSource: The Journal of the Operational Research Society, Vol. 48, No. 10 (Oct., 1997), pp. 978-987

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journal

of the

Operational Research Society (1997) 48, 978-987

?

1997 Operational Research

Society

Ltd

All rights

reserved 0160-5682/97

$12.00

An analytical

model

for conflict dynamics

N Gass

Decision Matrix, Ottawa, Canada

A coherent dynamic conflict

model is developed from basic principles.

The

governing equations

have a striking

resemblance

o the continuity equation

in

fluid

dynamics

with an additional ern for the response

to

pressureby

the

opponent.

The salient feature

of the model

is

a moving confrontation

ine which is an excellent

indicatorfor the

evolution of conflict.

Thedevelopedmodel also permits nvestigation

of the

necessary

minimum nvolvementof a third

partyactor such as an international

rganization o establisha status

quobetween the actors.The

model

is

demonstrated

on the Russian-Chechen

conflict and

the

Bosnian war.

Keywords: conflict analysis;

methodology; modelling

Introduction

With

the new political

world order, a new distribution of

power has risen

in the form

of

a multipolar system

where

the manifold

of interactions of political,

social, economics,

and military

environments tends to raise

the ambient level

of regional

conflicts potentials.

This will inevitably have an impact

on the international

crisis management, policy planning

and the structures

of

peacekeeping forces due

to the widening theatre of opera-

tions and the

new modes in which they

are conducted as

discussed by Bailey

and Ferguson.' But most important,

early recognition of potential conflicts will open additional

avenues for

conflict resolutions

as analyzed by Kaufmann2

and Bennett3 and thus will

have a higher chance

of success

to stabilize

volatile geopolitical regions.

Richardson4 started

the trend of mathematical modelling

of

conflicts

well before the Second World War and since,

numerous models based

on a wide variety of mathematical

approaches have been developed

as, for example discussed

by

Nicholson,5

Gillespie

and

Zinnes,6

Fraser and Hipel,7

and Gass.8'9

In

the

development of conflict models, two

major

problems have

to

be

overcome.

The

first is the

choice of

the governing equations which are often selected from

other

disciplines

and

adapted

to suit the

present application

without

regard to

the

mathematical

structure and

whether

it

reflects the basic

laws of the processes to be modelled.

The

other

problem

is the

choice

of

conflict parameters

and

their

numerical

values. Clearly,

the

relationships

between

actors are very complex

and many rational and subjective

considerations

influence their behaviour as described

by

Nicholson.

10

Correspondence: Dr N Gass, Decision

Matrix, 77 Havelock Street,

Ottawa, Canada,

KIS OA4.

The present

paper is an attempt

to eliminate

the first

problem by

developing a

coherent set

of differential

equations

based on

basic principles

in conflict

theory.

The

second problem

is also addressed

through the

choice

of some global parameters

which

are easier to estimate.

Nevertheless,

the underlying

numerous

subjective

factors, such

as ethnic particularities,

world opinion,

inherent

animosity,

etc., which

are often the

impetus

for irrational

actions,

are difficult to describe

in a rational

way.

The final results

of this approach

s a moving confronta-

tion line which indicates an imbalance in the status quo

between the actors.

This imbalance

can

be used,

for

example, by international

rganization

as an early

warning

signal

of

possible

conflict

escalation.

It

must be

pointed out

that the proposed

model does

not

predict future

conflicts at

a precise time but

rather

investigates the conditions

which may lead

to such

events, in a similar

way to the approaches

by

Nicholson,5

Gillespie and

Zinnes6

where conditions of stability

and

equilibrium

are studied.

Also,

the

hypothetical

question

can

be studied

of how

much

intervention by an

interna-

tional organization

s necessary

to balance

the pressure

at

the confrontation

ine and

thus stop

its movements.

The

governing

equations

are solved by using

the

commercially

available

software

tool ithink, pro-

duced

by High

PerformanceSystem Inc.,

Hanover, NH,

USA.

ithink is a

dynamic systems

tool

ideally

suited

for

tracking time-dependent

events.

Another

advantage

s that

the results are automatically

displayed

in

graphical

form

at each time

step.

With

this

feature,

an

analyst may

interfere in the computation

process

at

any given

time

to

update

or alter

various

conflict

parameters

n

order

to

study

some 'what

if

..'

questions.

A

complimentary

copy

of the

programmed

conflict

model is

available

from

the

author.


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N

Gass-Annalyticalodel

or

onflict

ynamics

79

Finally, it is worth

mentioning that the governing equa-

tions are also applicable to analyze conflicts arising in

labour relations and contractnegotiations.

Basic assumptions

Conflict situations can be

caused by many factors such as

differences in political ideologies, legal and economic

systems, ethnic and social

particularities, human rights

issues, state-sponsored errorism, ross-border nvironmen-

tal problems, territorialand

resource claims, etc. Many of

these issues may be subduedfor some time until propelled

forward

to

surface at the confrontation ine

between

the

actors giving rise to pressure.

A

detailed list of factors is

given by Gass.8

Consider the

different conflict

elements

between actors

A

and

B

depicted

n

Figure

1.

Let the individual ssues have

virtual

distances

from the

confrontation

ine

and different

speeds at which they move towards t, namely towardsthe

negotiation able, UN or WTO

forums,

or towards

military

action,etc.

Atthe

confrontation

ine,

these ssueswill be

metby

moreor ess

resistance rom he otheractorandwill causepres-

sure. Let an international

rganization

nfluence the actors

to decrease heirconfrontation

y using, for example,political

pressure,

conomic

force,

or

peacekeepingoperations.

Let the

perceived important

f the issues

S4 and

SB

of the

actors

A

and

B

be multiplied

by

the

capabilitiesCAand

CB

(political power,

economic

strength,military force)

to lend

weight

to their issues. For

example,

if an issue raises a

large

confrontationbetween a

superpower

and

a weak

opponent,

the formerdoes not need to

worry much

while the latterhas

to fear possible military actions. Conversely,

f

there is no

confrontation,hen a military mbalance s of no importance

as

is

the case between Franceand

Luxembourg,

or

example.

With these

assumptions,

wo

generalized

force

density

or

pressure

functions

pA(r'A,

t') and pB(r/B, t')

for actors A

and

B, respectively,

can be defined as

pA iA

E1

eA

C)(

r

A

N A/

EAN~~~~~~~~~~~~~~A

for all

0

eC

Lagrange

movement

f

Coordinate

ystem

confrontationine

Euler

v

Coordinate

ystem HA

L

external

AL

~~~~~~~~~~interference

withdrawal

. 0

02s>

X ee c

of

issues

8/9/2019 3010117

4/11

980 Journalf heOperationalesearchocietyol.8,No. 0

using

some

properties

of

the Dirac function yields

the

averaged pressure density of

actor

A

as

0?

r??

pA(r.A t)

=

TBHB

J J

pA(rA, t')

-00

-00

x F(rA

-

rA,

t

-

t')drdt'

,6B8 ,1 | C(t)(t )('-E

(t ))

i

00

x

F(rA

-

rA

(t'),

t

-

t)dt'

(3)

where

EA

i

EA(tI

i

(t)

=e )

CiA(/),

for all O

<

1

and likewise for

actor

B.

More details

on

the development

of

equation

3

and

the

following

derivations can

be found

in

Gass.

11

To

proceed

in

deriving

the

movement

of

the

confrontation line

F, equation

3 is

differentiated

with

respect

to global time t, yielding

N

A

'.0

A

~BBE

CiA

-t1S

C

;H

,j

CA('S6B

(1-

(t ))F

t

dt,

(4)

i

-00

and

likewise

for

actor B, where a comma

denotes partial

differentiation. The

partial

derivative

F

t

can be

expressed

as

aF

a

=

-F,lt

-

UiF,r

where

ui

=

drJ/dt'

denotes the

speed

at which an issue

moves

towards

the confrontation

line. With

this,

and

using

integration by parts, equation

4

becomes:

Aj

? B

div

c(A1

(I

4

)F

dt'

i

-00

-B

By

I

-CA Si (I -i 0)}IF

dtl

= O (S)

i

00

at

Finally, equation

5

can

be simplified by averaging

the

speeds

of the

pressure-causing

issues

in

the usual

manner as

U

(r, t)

-

L

J

Ci

sUA

(I

-

A)F

dt'

i 00

CjAS4A(l

1Ai)F

dtl)

(6)

and likewise for

actor

B.

With

this,

the

final

form

of the

conflict

equations emerge

as

A

+?pAdivUA

-T

B,By a{CA

((1

-

e)}F

dt

=

0

P't

it

i

-oo ~~at

NB

OC

a

pB

+pBdiv

UB

_

rA

lA

I

-{CiBSiB(l

B)Fdt=O

j__

at

(7)

These equationscan be

called the continuityequations of

conflict dynamics and resemble the continuity equation in

fluid dynamics. The

first

term

in

equation 7 denotes the

change

in

pressure

of the

weighted issues at the confronta-

tion line. The second term represents the

withdrawal of

issues from the confrontation ine and may be producedby

the external ntervention f an international

rganization, or

example,

or

simply by

an

actor's decision

to

drop some

demands.

A feature of this formulationis the appearance

of the

third erm

in

equations7 which governs the response of one

actor in reaction to the other

actor's issues. Depending

on

many factors,

the

response may

increase or

decrease

the

pressure at the confrontation line as will be

discussed

below.

Response to pressure

Forfurtheranalytical reatment, ome simplifications o the

response terms in equations7 are proposed.Actors

respond

differently

to encountered

response

and can either

respond

in

an aggressive, provocative, passive or submissive

way,

depending strongly

on

ethnic particularities

but

also

on

rational actors

such

as

resource imits, political strength,

or

military force.

A weak

submissive actor,

for

example,

could withdraw from contested issues when met

by

resis-

tance from a

strong

actor.

Others,

as

for example

Chech-

nya, with a tiny but very provocative force, mounts

resistance againsta comparativesuperpower,defying

mili-

tary rationality

n

pursuit

of national

goals.

Let the

weighted

issues

be

averaged,

hus

eliminating

he

summation

of

the third term. Let

the

imbalance

of

capabil-

ities between

the actors be the

driving

force behind the

response to the pressurefrom the opponent.Consider the

expression

CB

-

C)

1

A

and

CA

CB)

9

pB

for

actor

A

andB, respectively,where the symbol

0

representsa user-

defined rule to

combine the two

quantities

and

may

involve

many

rationaland

subjective

factors such as economic and

military power,

ethnic

pride,

inherent

distrust,

historical

animosity,

nternational

pinion,

etc.

One

reason for

0

is

that,

in

general,

it is

very

difficult

to

develop

an

analytical

function

for

the

response

term

to

describe

all

influences over

the entire

period

of

analysis.

Also,

it

can

be

observed that the

response

of one actor does

not

change continuously

o

infinitesimalactions of the other

actors but rathershifts in leaps and bounds dependingon

some

thresholds.

An

example

for

this

is the confrontation

between Russia and

the

Ukrainewhere threats

and

counter-

threats are

made but

the

overall relation does

not

change

with

every

turn.

It

rather emains

on

one confrontation evel

until

enough

additional

pressure

is accumulated

o

force a

jump

to

another

evel.

An

example

of

a

response

matrix

C

Op

is

given

in

Table

1

where

the

rating

A

denotes low pressure,

ittle

capability,

and liffle response, while the rating F indicates the

opposite.

Table

1 depicts the response profile of an actor who is

cautious if his capabilities are weak compared to his



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N

Gass-Annalyticalodelor

onflictynamics 81

Table

1 Response profile C

0

p

Pressure

from counterpart

0 A

B

C D E F

Capability

of actor

0

0

0

0

0 0

-A -B

A

0 0

0

0 0

-A

-A

B

0 0

0

00 0

0

C

oo

o

G

A A B

DO

0 0

A

A

B

C

E

0 0 0

A

B C

D

F 0 0

A

B

C D

E

counterpart (thus,

he

decreases

the

response

pressure by

-A)

but becomes increasingly

aggressive

in

response

when

he is strong.

Clearly,

Table

1

will

be different for each

individual case

since actors respond differently to encountered pressure.

The

response

profile for

Chechnya given

in Table 2, for

example, does

not contain any

negative quantities which

indicates

that other factors,

than strict military

considera-

tions, play

a

large

role.

Let

zA

and

CB

be the response

rates (also termed

aggres-

sion factors

if C

0p is positive)

to the pressure

from the

counterpart,

thus

targeting

the other

actor's issues

in

response.

Let the effectiveness

in

targeting

the other actor's

issues

be

denoted by

the parameter p.

The further away

an

issue

is from the confrontation

line, the less

it has surfaced

to

the

present

and the

lesser it becomes a target.

As

the

simplest choice,

let

p

be a linear

function of the

distance,

such that

A

=

PA(F

-

rB) and

9B =

B(prA

-

F)

with

the

boundary

conditions

(p(O)

=

1

and

qp(Ir

Fl

>

H)

=

0

where

H denotes the

width of a more

or less

narrow virtual area

along

F

(see Figure

1).

H can be

viewed

as

the

issue horizon

where within

this virtual area,

all issues are of concern

at

present,

while

outside,

they are

more

or less subdued until moving

forward

into the confron-

tation zone

F

?

H. Clearly, an actor

must choose

the span

of

H

such

as to include all

issues of concern.

Finally,

let the concentration

of issues

,u

be

given by

the

integral

over all

issues

inside

the

confrontation

zone

F

?

H

Table

2 Chechen

response profile

Russianpressure

pB

0

A B

C D E F

Capability

of imbalance CA-CB

-F

0 0

0 B C D

E

-E 0

0 A B C D

E

-D 0

0 B

C

D E

F

-C 0 A

B C D E

F

-B

0

A

B C D E

F

-A

0

B C D E F F

0

G

B

C D

E

F

F

normalized

over

the width

H.

Collecting

all

terms

leads

to

the

response

of

actor

B

to

the issues

of actor

A

as

RB

=

zBB{(CB

-CA)

XAp}(1

-_

)B)I

J

B

(rB,

t)drB

(8)

Again, an external actor may interact to decrease a

pressure-building esponse

since

this could

be a self-feed-

ing cycle and

any

control

would be

of

mutual

benefit to

both actors

A

and

B.

Likewise,

the

response

of actor

A

to B

is

r+H

A

RA

=

,uAy{(CA

-

CB) pB}(l

- )HA

p

(rA,

t)dr-A

(9)

With

this, equations

7 becomes

pA

+?pAdiv

UA+

RB

=

?

pB

+?pBdiv UB +RA = (

B o

These

equations

are

integrated

in

Appendix

A

Dynamic

confrontation

line

Further

analytical

treatment of

the conflict

equations

A7

and

A8,

developed

in

Appendix A,

is

possible

by choosing

the distributions

9A

=

(A(F

-

rB)

and

9B =

9B(rA

-

F)

of

the response

effectiveness

over

the

depth

of

the

issue

horizon H.

Let

these

functions

have the

simplest

form

possible with the conditions at the confrontation line

9(A(O)

=

9B(O)

=

1

and at

the horizon

sA(H)

=

YB(H)

=

0

which

is satisfied

by

a

straight

line.

Thus,

A

(1/HB)(F

-

rB)

+

1

and

9B

=

(1/HA)(rA

-

F)

+

1

with

9IA

= 1/HB and

9pB

=

1/HA.

Let

zA

=

-0,

vB

=

i-B,

and v7

=

-F

be

the velocities of

the issues and

the

confrontation

line, respectively

and

letpA(t)

=

CASA(t),

etc., by

definition,

where

CA

and

Ce

are assumed to be

constant over

the issue horizon.

With

this,

and

setting

SA(t)

SB(t)

S

A(H)

SB

(H)



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982 Journal

f

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Operational

esearchociety

ol.

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No. 0

equations A7 and A8 become

oA(vF

-

A)

+

WA

+

RB

=

(

UB(VB

+

?F)_

WB-RA =

0

where

WA

and

We

denote

the issue withdrawals

of

actors

A and B

WA =

H

A{w

+

(1 -w )A

}(

C

1)

12

WB =

HB{WB

+

(1

-

WB)?EBJ(UB

+

1)

(1)

and

RA

and

RB are the

responses

to

the

other

actors issues or

demands

RA =

HB{(CA

-

CB)

0

CBSB(t)}

X

1

- (-)S(){+

(13)

RB

=-1

BHA{(CB

_

CA)

0

CASA(t)}(I

-?

B)

X

B

S)

B(H)){2?cB}

(A

S(H)

Substituting

equations

11

into

each other

and

resolving

for

vF

leads

to the

equation

of

a

moving

confrontation

line

according to

the

pressure

differential between

the

actors.

Thus,

A

+

B)v

=AvA

-

BvB

-

WA

+

WB +

RA

-

R

(14)

The Cold War

scenario

can

serve as a

trivial

check of

equation

14. Let there

be

no

withdrawal

and no interference

by

an

external

actor.

Let

the

perceived

capabilities

and

importance

of all issues including their

speeds and issue

horizons be

equal.

With equations

12 and

13, equation

14

becomes

2cvF

=

TAH{0

09

CS(t)}B

-_,BH{O

0

CS(t)}A

The

confrontation

line

becomes

stationary

if

the

aggress-

tion rates

r

and

the pressure response

terms

0

0

CS

are

equal. During

the

Cold

War

period,

small

perturbations

about the

equilibrium

were present

reflecting

the

different

attitudes and viewpoints

vis

a

vis the balance

or

military

power

and external

political

influences.

For

this,

let

IA

=,

TB

=

oX,

R

=

{O

0

CS(t)}A

and

{O

0

CS(t)}B

=

fR.

Then,

vF

=

(1

/2)TRH(oc

-

/)

where

a

and

3

are small

perturbations about 1 which cause slow oscillations of the

confrontation line.

The

relationships

between

the

speeds

of

the

issues

towards

the

confrontation

line can be given

as

v

+

vBI, 1 (WA

+RB)

1(WB+

RA) (15)

crA crB

With

this, equation

14 becomes

F

I

B (1 + UW

+.B

R )-CA

(1 +

)(Yx

+

RB()

where u

=

v4/vB.

For numerical

purposes, it is of

advan-

tage to

transform quation

16 in a nondimensional

orm as

VF

A

)[

B(+ )

(1+U

[

:(

+#

R )]17

*1?u)[l+

~WA

?RB)

(17)

where cx

=

(SA(t)/SB(t))

s

the

issue

ratio at the confronta-

tion line

and ,B= (SA(H)/SB(H))

is the issue

ratio at the

horizon.

Simple

examples

Consider the

simple case where

vB

=

WA

=

WB

=

0, and

c

=

,B.Then equation

17 becomes

A

=

(RA/RA

+

RB)

which

confirmsthe obvious situationthat the confrontation ine

moves faster

in favour of actor

A if

there

is less resistance

RB

by actor

B or more resistance

by actor

A

against

B. If

actor

B does

not

resist

the issues of

A

then

A

=

1,

and the

confrontation

ine moves with

the same speed as

the issues

of

A

arrive at

the

front.

Figure

2

depicts

the relation between

the

power

ratio

C/Cl3

and

the

issue ratio

cx.

For this, let

WA

=

WB

=

0,

H/HB

1

?A

=

gB

=

0

=

1,

CA

=

CB

=

0.

Let

the

actors

be completely rational,

such that

their reactions

depend

on the power ratio C4/Ce.

Thus,

zA/zB

ac CA/CB,

(CA

_CB)

?

pB o

CA/CB and

(CB-CA) ? pA Oc

CB/CA

.

Then equation 17 becomes ? /+ 1qDf]-[1?[ + d=

(CA/CB)4

(2

+

(X)(X2/(l

+

2cc).

It

is interesting

o

note

that,

even for low issue

ratios

cc,

a

power

ratio

of

more

than 3 does

not warrant

he additional

resources

since

the

speed

of the confrontation

line is

nearing its maximum

of 1.

For

power

ratios less

than

1,

the confrontation

ine moves against

actor

A

despite

high

issue

ratio.

In

other

words,

actors

moving forward

issues

without the

backup of power,

will not be

taken seriously.

Figure

3

shows the relation between

the

speed

of

the

issues moving

to the confrontation

ine and

the

power

ratio.

1.0

x

a=2

0.5-

a4-1

0.0-

-0.5

-

0

1

2

C 3

CB

Figure

2

Influence

of issue density

ratio

cx.



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1.0

I-----

0.5

-05

O

1

2

CA

3

C

B

Figure 3

Influence

of

issue

speed

ratio u.

With above numerical

values and

o

=

1, equation

17

becomes

B

4-

1

CA 41-i

1?u[

(cA

1+u

[

\CB,/j

The

influence of

the

issue

speed

ratio u is most

pronounced

between

2

<

CA/GB

<

3

which is

in

the

region

of

transition

in

the

dominance of

power

between

the actors.

For

a

power ratio

greater

than

2,

u has

practically

no

influence on A and there

is no need to

increase

resources.

In

comparison

to

Figure

2,

Figure

4

depicts

an actor

who

is

submissive if the

power

ratio is balanced as shown in

Table

1

and even more

so

if

CA/CB

<

1. The

resistance

term

RA

becomes

negative

which indicates

cooperation

and

results

in a

high

speed

(less

than

-

1)

of

the confrontation

line

against

actor A. For

power

ratios

greater

than

1,

actor A

becomes increasingly aggressive similar to Table 2. The

graph

also

shows that the

decision of actor A occurs

in

leaps

and

bounds

according

to some

thresholds whereas

in

Figure

2 an

infinitesimal action

by

actor B

caused an

infinitesimal

reaction

by

actor A.

1X00

3

-2.0

02 1 2

0A

Figure

4

Influence of

issue

density

ratio

a

for a submissive

or

agressive

ctor.

The

Chechen-Russian conflict

Russia

regards the

Caucasus as a

vulnerable

flank vis-a-vis

the

neighbouring

countries Turkey,

Iran, and the

general

influx of

revolutionary ideas

from Islamic

countries give

Russia

ample reasons for

'protecting'

the region.

Equally

important

are the

economic reasons since the

region is rich

on mineral resources and oil.

Some of

the current

instability of the

Caucasus

originates

from the

Russian

colonial expansion

and the long Cauca-

sian War in the

last

century. Others stem

from

the

Stalinist

method of

splitting ethnic

groups through artificial

division

of

regions into

administrative entities.

For some

years, Russia has

tolerated the

secessionist

government of

Chechnya

but

in

1993

has

begun

to

take

military

steps to resolve the

impasse. The

reasons

for

the

resistance to

Chechnya's independence is

Russia's deter-

mination not to

relinquish

control

of the

region

since there

are fears

of a domino

effect

if

Chechnya separates.

These

fears are justified because there are several other candidates

for

separation

in the

Caucasus,

notably, the

Tartars, Kara-

chai, Lezgins, and

Ossetians,

to name a

few.

Chechens have a

long

reputation

for

opposing the

Russians, as was the

case in the

Caucasian War,

despite

having

a

much inferior

military force.

Such attitudes

are

reflected in

the

term

(CA

-

CB)

0pB

which can be

eval-

uated

by

developing

the

response matrices given

in

Tables

2 and

3. The

advantage of

such an

approach

lies

in

the

flexibility

in

describing

the influences

of

numerous rational

and

subjective

criteria on the

response

of an

actor.

Indeed,

it

would be

very difficult to develop

functional

relation-

ships for this purpose.

For

clarity

reasons, the

rating

scale

for

the

capabilities

and

resulting

response

is

chosen as

{0, A,

B,

.

.

.,

F)

where

A

denotes a low level of

issues

arising,

weak

capability,

or

little

response,

while

F

represents

the

opposite.

The

choice is

arbitrary

but

it is ideal to demonstrate

the

combinatorial rules of

the

algebra

0. In the

associated

computer

program, however,

these

alpha-numeric

ratings

are converted to an

arbitrary

numerical

scale,

as

for

example

A

=

1, ...,

F

=

6.

The

response

term

shows that

Chechnya, denoted

by

the

superscript A,

is not submissive

or

ready

for

concessions

even when faced with an overwhelming military imbalance

CA

-

CB

to Russian's

advantage

paired

with

high

Russian

pressure.

For

example,

if

the

capability

imbalance is

-

E

(large),

and Russia's

pressure

is E

(high),

the

response

(CA

_

CB)

&

pB

=

D

(medium

high)

at

which

Chechnya

s

resisting Russian

demands or issues.

Initially, from 1991 to

1995,

Russia

sought

to

resolve the

issues with

Chechyna through

political

and

economic

threats

and later

through

some

military

actions.

Chechnya

is

politically important

to Russia

and

this is

why

the

Russian

response

to

the

Chechen issues is

aggressive

as

seen

in

Table 3.



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Table 3 Russian response profile for 1991-1995

Chechen

pressure

pA

0

A B C D E F

Capability imbalances CA

-

CB

0 0 0 0

A B C D

A

0

0 0

A B C

D

B

0

0

0

B

C D

E

C O

0 A B C D

E

D

O 0

B C D

E

F

E

0

A B

C D

E F

F

0

B C D E F F

Since there is no external intervention WA

WB

=

0,

and equation

17

becomes

VF I_ I_

t+ D

(+

B

( + P(ucxp

-

(18)

where p

=

RA/RB. n 1990-91,

Russia

was preoccupiedby

other mattersandpaid little attention o the issues concern-

ing Chechnya. Thus,

vB

=

0, and

TB =

0

yielding RB

-

0

and A becomes

A

=

(p/I + p)

= (1 + (RB

(t)/RA4(t))j1

=

1.

First reactions by Russia occurredafter PresidentDuda-

jew (in October 1991) unilaterally declared independence

and,in retaliation,Russia declareda state of emergencyon

Chechnya, threateningpossible military action.

The

para-

meters were assumed as:

a

=

,B

=

1,

HA

=

HB

=

1,

SA(H)/SB(H)

=

1,

SA(t)/SB(H)

=

2,

SB(t)/SA(H)

=

1,

SA(t)/SB(H)

=

3,

SA(t)/SB(t)

=

3,

CB/CA

-

2,

v

=

1,

(CB CA) ?pA

=

B

C

=

B

=

2, (CA-CB) pB =

B0 C

=

C

=

3, and vB

=

0

since Russia applied only

passive resistance.With this, the response terms becomes

RA(t)

= 15

TA(t)

and

RB(t)

=

6TB(t)

which yields

A

=

(I

+

(8z (t)/5zA(t)))

The threat parameters were initially TA

=

0.5 and

TB

=

0.2 but

steadily increasing

until

in

August

1994 a

coup

was launched

by

the

opposition, also supported by

Russia.

This

caused a slowdown of the advance of the

conformation

ine as depicted

in

Figure 5

with the threat

parameters reaching

now TA

- TB -

0.6.

0.0

0

.

.........s\s.;s-...

eaA o g

.

=,;

?

L

.

optimistic

iew

C) D

........................

....... . ., .

; .

.

..

1991 1996 2000

year

Figure

5

Russia-Chechnyaconflict.

In

September

1994,

President

Dudajew

declared

a state

of

war and

in

November

1994

the

opposition

started an

attack on Grosny.

In

January 1995,

Russian

troops

moved

towards

Grosny

and

in

February

1995

the

capital

fell and

fighting spread to other

areas.

During

this

period

the threat

parameters were

steadily approaching

the values

of

TA

=

TB =

1,

with u = 1,

CB/CI =

3,

SA(t)/SB(t)

=-,

(CB

CA) ?pA

=

C

Q D

=

C

=

3,

(CA-CB) pB=

C

?

C

=

C

=

3,

and

equation

18

becomes

A

=

2

with

P

=

3

(_B(t))_

Figure 5 shows

that the

conformation

line

moves now

against

Chechnya,

an

indication

that Russia is

controlling

the conflict. Recent Russian

elections,

economic considera-

tions

including military

resources,

and

negative

public

opinion

forced Russia to reduce the crisis which could

lead to a

new,

more relaxed

response

matrix

given

in Table

4,

a

pre-requisite

for

negotiations.

In

September

1996,

a

cease

fire was announced but

a

solution of

the

conflicting

issues

has

yet

to be

addressed. Russia is reluctant to

permit

Chechnya

to become an

independent country

and thus

Chechnya

may

raise the issue

pressure

which could

result

in unrest

over the next few

years

as

shown in

Figure

5.

Using

Tables

2

and

4

yield

(CA-CB)

?

B

=-0

?

A

=

B

=

2

and

(CB

C)

0

pA

=

O

0

D

=

A

=

1

and

assuming

an

opti-

mistic and

pessimistic

threat

parameter

such that

p

(optimistic)

=2

and

p(pessimistic)

=

9 leads to a

slowly

increasing speed of the confrontation line driven by

unsettled issues.

The

Russian

response given

in

Table

4

and a moderate

threat

parameter

prevents

a

negative

A

namely,

a Russian advance but

encourages Chechnya

to

pursue

the

goal

of

independence.

External

pressure

External pressure

can

decrease the imbalance between the

actors A and

B

expressed by

the movement of the confron-

tation line.

An

improvement

special

case is the status

quo



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Table 4

Russian

response

profile

for

1996-2000

Chechen

pressure

pA

0

A B

C D

E

F

Capability

mbalance

CA

CB 0 0

0

0

0

A

B

B

A

0 0

0

0

A

B

C

B 0 0 0 0 B C C

C

O

0

0

A B

C

D

D

O 0

0 A

B

C

D

E

9

0

0 B

C D

E

F

0

0

G B

C D E

where

vF

=

0. The

necessary

minimuminterventions

?X(t)

and

gB(t)

can be

estimated

by using

equation

16

to

yield

u(WB? RA)- (WA +RB)

(19)

a

Ignoring

he

'voluntary'

withdrawal

atesw,

above

equation

becomes

?B[B

+

uuA(AB+1)]

-

[A?A+

H

(A

+

1)] +A-B=O

(20)

where

1~~~~~~~~

A=

1

U CA{ (CA

_

CB) & pB }

(2 + CA) (21)

and

B

=

TBUBH B(CB

_ CA) CpA(t)}(

)

1(2

+

CB).

(23)

There

are many

combinations

of

eA

and BB which

render

vF

=

0. However,

the

most desired solution

is not only

the

minimum

efforts

EAICA

and

EB/CB but also

considers

the

'political

correct'

solution including

factors

such as

legiti-

macy of

issues raised,

humanitarian

roblems,

social condi-

tions,

etc.

Thus,

an

external

actor has

to select

the

most

appropriate

ombinations

among

the set of possible

solu-

tions.

All

parameters

n

above

equations

are time

dependent

and,in orderto keep vf = 0 over a longer time period,the

efforts

EA

and

EB

have

to

be

readjusted

onstantly

o offset

changes

in

the

conflict

evolution.

In

the

special

case

where

an

external

actor

fully

suppresses

the

reaction

RA or RB

of

actors

A

and

B

by

BA

=

gB

=

1,

the

status

quo

is controlled by

the withdrawal

terms uuA (CB

+ 1)

-

(HAI/HB)UB(CA

+ 1)

=

0.

In

another special

case

(as,

for

example,

in the

Bosnian

conflict)

where an external

actor

pressures

only

one

of

the

actors,

equation

21

becomes eB

=

B

-

A/2 +

B

where

CA

=

O,

and

u

=

CA

=

CB

=

1

with the

admissible

solution

0

2,

it

becomes

increasingly

more

difficult

for actor

E to

make

an

impact

on A.

For high

power

ratios,

the involvement

ratio

EA/CA

must move

closer

to 1 before

A

changes significantly.

The

Bosnian

conflict

The decades

old subliminal

ethnic

conflicts

in

the socialist

Yugoslavia

surfaced

after

Solvenia

and

Croatia

declared

independence

in

1991.

In Spring

1992 fighting

started

in

Bosnia-Herzegovina

with

the declaration

of

a

Serbian

Republic

of Bosnia.

The

United

Nations

(UN)

placed

an economic

embargo

on

Serbia

n May

1992

with additional

sanctions

ntroduced

in

May

1993.

Nevertheless,

one year

later,

the Bosnian

Serb

forces had capturedmorethantwo-thirdsof the territoryof

Bosnia-Herzegovina.

The

confrontation

front against

the

Bosnian

Muslims

moved

very

swiftly

as

shown

in Figure

7.

The UN

peacekeeping

forces

stationed

in Bosnia

since

1992were

increasingly

unable

to provide

humanitarian

id.

The restraint

by

the

UN not

to

use

force was

taken by

the

Serbs

as

a

sign

of weakness

which hampered

UN

activities

and also

had

a

negative

influence

on

the

peace

negotiations.

Towards

the

end

of 1995

the UN

finally

did

use

air

strikes

to

enforce

a safety

zone

aroundSarajevo.

For

the

simulation,

the following

numerical

values

were

selected

with

the

indices

A and

B

denoting

the

Bosnian

Moslem

O.75 e

cZ

='=

---~..- ............

.........

?..

0.0-

_>__0A3

CA=4

CA=1

CA

=2

CA

=1/2

-0.75-

0

1

2

3

EA

4

Figure

6

Influence

of external

intervention.



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0

1

0

.poptimistic

view

2

-0-02

a)~ ~~~~~~~ya

coue7TeBsna

ofit

forces

and the

Bosnian

Serbian

forces,

..

......

esspecticviewy

c=2

-1 0

..

...

.

*....

1991

1996

2000

year

Figure

7

The Bosnian

onflict.

forces and

the Bosnian

Serbian

forces,

respectively:

a

=

/31,

u

=

I

CA1

C=

4, o=1,

CB

=2,

H4 H

B

l

zA

=1, z=

2,

= O, (CA CB)

QpB

3,

and

(CB

-

CA)

?pA

= 5.

The

military

force

applied

by

the

United

Nations

(UN)

in

a small

but

highly

sophisticated

operation

matched

completely

the Bosnian Serb forces, thus

gB =

1.

Using equations

12, 13,

and

17

yields

A

=

4

which

indicates

that

the issues

of the Bosnian

Serbs

lost

pressure

and

the confrontation

line retreated

as

depicted

in Figure

7.

These

events

marked

the

turnaround

in negotiations

and

in Spring

1996,

a peace

treaty

between

the

actors

had

been

negotiated

(the

Dayton

accord).

To bring

both

actors at

the

negotiation

table,

and

to reach

a status

quo

at A

=

0,

the

UN had

to

apply

the minimum

pressures

of

u4

=

4

and

eB

=

2. This

result

was

calculated

by

using

equations

20-22

with u

= cA

= c=

1,

and

(CA

_

cB)

?

pB

=

(CB

_

CA)

?$ pA

=

2.

The reached peace treaty is far from ideal but the two

actors

are

economically

ruined

and

this

is part

of the

reason

why

the aggression

coefficient

T may

be

small for

the

moment

and

thus reducing

the

chance

of a renewed

conflict

considerably.

To

estimate

a possible

confrontation

in

future,

a hypothe-

tical

case is

studied

where

it

is assumed

that

the UN

would

withdraw

but its

accumulated

impact

over

the past

years

would

not evaporate

completely.

Let the two

actors

more

or

less

restrain

themselves

about

the equilibrium

conditions

u

=

a

=

p

=

u,

with

initially

I,

1 but increasing

slowly

over

time in favour

of the

stronger

Bosnian

Serbs.

The

movement of the confrontation line is then given by

A

=

(1

-

p2)(1

+

[L)-2

which is

depicted

in

Figure

7

from

1996-2000.

The lower

bound

is

the

more

pessimistic

case

where

the

actors

begin

to be

dissatisfied

with some

terms

of

the peace

treaty

while

the upper

bound

denotes

with

more

or

less

satisfaction.

From

this,

it can

be concluded

that

no

serious

conflict

will arise in

the next

few

years.

Conclusions

Applications

show

that

the speed

of movement

of

the

confrontation

ine is

an excellent

indicator

of the

evolution

of conflict.

Clearly,

the higher

the speed, the

more

imbal-

ance

exists

between

the

actors and

the

more

likelihood

there

is

for the

outbreak

of war.

It was found,

that

the

analysis

of conflict

should

be done

in two

parts.

The

first

part simulates

past

events

in orderto

establish

benchmark

values

for

the

numerousparameters

n the equations.

In

a

second

part,

these parameters

hen

can be

varied

to

study

a

numberof hypotheticalfuture events in the realm of the

what if..

environment

o

obtain

optimistic

or

pressimistic

views.

In this

way, a catalogue

could

be created,

based

on

historical

event,

to associate

the speed

of the

confrontation

line with

the actual

magnitude

of a

conflict.

Thus,

the

gravity

of a future

scenario

can be

analyzed

fromthe

speed

of

the confrontation

ine

as the

events

evolve.

Also,

the hypothetical

question

can be

studied of

how

much

invervention

by

an international

organization

is

necessary

to establish

the status

quo

between

the actors.

This study

will

open additional

avenues

for conflict

resolu-

tions and thus will have a higher chance of success to

stabilize

volatile geopolitical

regions.

Appendix

A

Integration

of the

conflict

equations

To facilitate

the task

of integration

of

equations

10, let

F

be

only

time dependent.

Then the

withdrawal

of

some

of the

issues of

actor

A

from

the confrontation

line can

be

approximated

y:

pAdiv

UA

=pAWA(t)

+

PA(I

-W (t))?A

(Al)

where

0

8/9/2019 3010117

11/11

NGass-An

nalytical

odelor

onflictynamics 87

no issues at

the

confrontation line before

the

start

of

conflict. Likewise for actorB

pB(rB,

t)

=

p'

(Ho,

0)

+

(pBp(FO,

t) (A4)

Differentiating equation A3 and

observing

that

apo/aro

=

aplaFo

=

0, leads

to

A

ag~&B a(rA

-F0)

P

P

t

=p(FO,

t)

a(r

-F0)

at

=A(FO,

t

_PO)'P

(A5)

where

Fo

is the

speed

of the

conformation

ine

and

rO

s the

velocity

of

the

issues

moving

towards

the confrontation

line. It is

interesting

o

note

that,

similar to fluid

dynamics,

the

excerted

pressure

of

a

moving

fluid on a

moving rigid

body

increases

with

the increase

in the

speed

differential.

Likewise,

the

pressure

differential for actor B becomes

P,t

=

po

(FO,

t)(Fo

-O

B)(P,A

(A6)

Substituting equations

9

and A5 into

A2

yields

the

pressure

of

actor A

at the

confrontation

ine

F

pA(t)

(G-F4_p)/

?

{WA

+

(1-

WA)EA}I((t)

?t

(A(H),) \p(H)

/

-

TB{(cB

_

CA)

?pA}(1l

_-

?)

p

B(H) 9p(t)

AdrB (A7)

PI,(H)P

(H)HB,

F-HB

where

the

now

unnecessary subscript

0

has been

dropped,

thus,

pA(t)

=pAO(FO,

t),

and

pA(H)

=pAO(HOA,

)

which is the

initial pressure

at the

issue

horizon.

Similar,

for actor

B

(p(t)(

_

iB)A + {Bw + (I - WB)

B

I(

+

V(H)

r(t~v~1v8~( )l

-

CA{(CA

I

GB)

pB}(

_?)

1A(H)

?

pA(t)

rF+H

cA

=

(A8)

X

pB(H)

pB(H)HAJF

=PBr

0

(8

Equations

A7

and A8 are the

conflict equations of actors

A and B

and can

be further

evaluated

if the

effectiveness

functions

pA

and

pB

of the

response are chosen.

References

1

Bailey R and FergusonN (1995). Modelling Force

Require-

ments for Peace Support Operations. 12th

International

Symposium on MilitaryOperationalResearch,Royal Military

College of Science, Shrivenham.

2 KaufmTan (1996). Possible and Impossible Solutions to

Ethnic Civil Wars.International ecurity,20: 137-175.

3 BennettPG (edr) (1987). Analysing Conflictand

its Resolution.

ClarendonPress: Oxford.

4

Richardson

LF

(1956).

Mathematical

of

War and Foreign

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In:

Newman

JR

(ed). The World of

Mathematics,

Vol.

2.

Simon and Schuster:

New York.

5 M. Nicholson (1989). Formal Theories n InternationalRela-

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UK.

6 Gillespie JV and ZinnesDA (1977). MathematicalSystemsin

International Relations Research.

Praeger

Publishers: New

York,pp 221-248.

7

Fraser

NM

and Hipel

KW

(1984).

Conflict

Analysis. North-

Holland:Oxford.

8 Gass N (1994). Conflict analysis

in

the politico-military

environmentof a new world order.

J

Opl

Res Soc 45:

133-142.

9 Gass

N

(1993).

The

Application of

Chaos Theory

in the

Modelling of Conflicts. Canadian Department

of National

Defence, Operational

Research and

Analysis

Research Note

3/93.

10

Nicholson

M

(1992). Rationality

and the Analysis of Interna-

tional

Conflict.CambridgeUniversityPress, Cambridge,

UK.

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J. Opl.

Res. Soc. 48: 132-141.

Received October 1996;

accepted January 1997 after one revision

3010117

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