=Paper=
{{Paper
|id=Vol-2784/rpaper16
|storemode=property
|title=Increasing, Decreasing and Flat Strategies in Information Warfare
|pdfUrl=https://ceur-ws.org/Vol-2784/rpaper16.pdf
|volume=Vol-2784
|authors=Alexander Mikhailov,Alexander Petrov,Gennadi Pronchev,Olga Proncheva
|dblpUrl=https://dblp.org/rec/conf/ssi/MikhailovPPP20
}}
==Increasing, Decreasing and Flat Strategies in Information Warfare==
https://ceur-ws.org/Vol-2784/rpaper16.pdf
Increasing, Decreasing and Flat Strategies in Information
Warfare
Alexander Mikhailov[0000-0002-2730-1538], Alexander Petrov[0000-0001-5244-8286],
Gennadi Pronchev[0000-0003-3887-6748] and Olga Proncheva[0000-0002-0029-2475]
Keldysh Institute of Applied Mathematics, Miusskaya sq., 4, Moscow, 125047, Russia
apmikhailov@yandex.ru, petrov.alexander.p@yandex.ru,
pronchev@yandex.ru, olga.proncheva@gmail.com
Abstract. The paper studies the mathematical model of information warfare in
which each of belligerent parties has the option of choosing between an increas-
ing, decreasing and flat strategies of broadcasting. The broadcasting resource of
each of the parties is supposed to be limited. Increasing strategy refers to broad-
casting weakly at the start of the campaign and increase the intensity of broad-
casting gradually. Similarly, decreasing strategy means broadcasting with high
intensity at the start of the campaign and decrease it gradually. Flat strategy refers
to constant intensity of broadcasting. Each party faces the question of which strat-
egy is the mast advantageous. We address this problem by making computational
experiments with the model of information warfare. Combining three options of
the first party with three options of the second party we obtain 9 scenarios. For
each of them the numbers of each party's supporters at the end of the warfare is
calculated. Which strategy appears to be the best depends on parameters that
characterize the intensity of mouth-to-mouth spread of information and deactiva-
tion of parties' supporters.
Keywords: Mathematical Modeling, Computational Experiment, Information
Warfare, Increasing, Decreasing and Flat Strategies of Broadcasting.
1 Introduction
Information warfare has been an increasingly important topic for mathematicians as
well as social science scholars and practitioners. The constantly growing volume of
literature includes, among others, papers that introduce, develop and analyze mathe-
matical models.
Mathematical modeling of the dissemination of information in society stems from
rumor models that do not take into account broadcasting by mass media but consider
only the diffusion of information in interpersonal communications. The earliest models
of this kind [1, 2] were proposed back in 1964 and 1973. The model of competing rumors
[3] was proposed basing on their approach and can be considered the earliest model of
information war. It deals with the spread of two competing rumors, considering that if
the spreader of the first rumor meets the spreader of the second rumor, then they switch
Copyright © 2020 for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
202
to spread the second rumor. The idea is that the first rumor is false, and the second con-
tains convincing evidence of its falseness. In this approach, the “winner” of the infor-
mation war is set by the researcher by defining that the second rumor is stronger than the
first. An alternative approach [4] assumes that a supporter of the other party cannot be
convinced, thus an individual becomes once and forever a supporter of the party whose
information they internalize earlier. The winner of the confrontation is derived from the
analysis of the model: in [4] the so-called “victory condition” was analytically obtained,
i.e. inequality (containing system parameters), which determines which side of the con-
frontation has a greater number of supporters when t→∞. Among modern trends in this
area, we also note emphasis on social networks, and agent-based and game theory-based
models [5–10]. In models of this kind the network structure plays an important role as
well as reputations of agents (network nodes). Related empirical studies (for example,
[11–14]) can be used in constructing mathematical models of information warfare.
Approaches to mathematical modeling of information warfare include the model of
making a decision by individual as for which party to support. The latest versions of this
model [15–17] incorporates the ideas of the agenda-setting [18, 19].
In this paper we address the problem of choosing a strategy in information warfare.
Suppose two Parties, say X and Y, are engaged in information warfare. During this
extended process each party broadcasts its propaganda via mass media.
It is supposed that each party has a limited resource for broadcasting. A useful image
would be that a party has enough resource to broadcast, say, 100 units over a campaign
that lasts 25 days. The flat strategy is to release 4 units every day. However, maybe it's
more advantageous to broadcast weakly at the start of the campaign and increase the
intensity of broadcasting gradually. This would be an increasing strategy. Similarly,
decreasing strategies start from powerful broadcasting and decrease the intensity grad-
ually. The other party has the same types of strategies. The question here is which strat-
egy is to choose. We approach it with our mathematical model.
2 Model
Take the model of information warfare [4]
x t x X N X Y X ,
dX
dt
dY
dt
y t yY N X Y Y ,
X 0 X 0 , Y 0 Y 0 .
Here X and Y are numbers of parties' supporters, N is the total numbers of individuals
in the population. It comprises supporters of X, supporters of Y and unattached indi-
viduals. It is supposed that each party's messages are spread by broadcasting and via
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interpersonal communication. The intensity of broadcasting of Party X is given by func-
tion x t and the intensity of mouth-to-mouth relaying of their message is character-
ized by parameter. Similarly, y t and y refer to broadcasting and relaying via
interpersonal communication the message of Party Y. The terms X , Y describe
the individuals' shift from supporters to unattached, so that parameter characterizes
the intensity of this deactivation process. The system is considered at 0 t T .
For the sake of simplicity suppose x t , y t to be linear functions of time:
x t k x t lx ,
y t k yt l y .
In terms of these equations, increasing, decreasing and flat strategies of Party X mean,
respectively, kx 0 , kx 0 , kx 0 , and the same for Party Y. Combing various strat-
egies by two parties we get scenarios such as "increasing strategy of Party X vs flat
strategy of Party Y" and so on. In comparing various strategies of a party as for which
one is more advantageous given the strategy of the other party fixed, it is necessary to
control that the total broadcasting resource given by
T
Resourcex x t dt ,
0
is the same across all strategies of Party X, and ditto for Party Y.
3 Computational Experiment: One of the Parties Has a Greater
Broadcasting Resource
Experiment 1. Let the duration of the warfare be T=4. Parameters x y 1 and
0.1 . Let the strategies of Party X be
Increasing : x t t ,
Flat : x t 2,
Decreasing : x t 4 t ,
so that Resourcex 8 . Similarly, strategies of Party Y are
Increasing : x t 1.2t ,
Flat : x t 2.4,
Decreasing : x t 4.8 1.2t ,
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so that Resourcey 9.6 . The results are presented in Table 1. The dynamics for two
cases is shown in Fig. 1.
It follows from Table 1 that for each party the increasing strategy is dominated by
the flat strategy, which in turn is dominated by the decreasing strategy. In other words,
under given parameters the decreasing strategy is the most advantageous while the in-
creasing strategy is the worst one.
The explanation of this is that mouth-to-mouth spread of messages is more intensive
than the deactivation of partisans, that is x y . Given this condition, the idea
of effective campaigning is to take the lead at the start, that is to circulate your message
powerfully at the start of the campaign and let your supporters relay it further to their
interlocutors.
Table 1. Experiment 1: the final numbers of supporters.
x t t x t 2 x t 4 t
y t 1.2t X=45.409, X=98.417, X=98.698,
Y=54.491 Y=1.481 Y=1.196
y t 2.4 X=1.154, X=45.405, X=62.094,
Y=98.743 Y=54.486 Y=37.804
y t 4.8 1.2t X=0.958, X=29.698, X=45.398,
Y=98.942 Y=70.191 Y=54.478
Fig. 1. Dynamics for Experiment 1. Left: x t t , y t 1.2t . Right: x t 4 t ,
y t 2.4 .
This being so, one can suppose that in the case of the opposite relation, that is if
x y , then the most advantageous strategy would be the increasing one. This
conjecture gives us the idea of the following experiment.
Experiment 2. Put T=10, x y 0.1 and 1 . Let the strategies of Party X be
205
Increasing : x t t ,
Flat : x t 5,
Decreasing : x t 10 t ,
so that Resourcex 50 . Similarly, strategies of Party Y are
Increasing : x t 1.2t ,
Flat : x t 6,
Decreasing : x t 12 1.2t ,
so that Resourcey 60 . The results are presented in Table 2. The dynamics for two
cases is shown in Fig. 2.
Table 2. Experiment 2: the final numbers of supporters.
x t t x t 5 x t 10 t
y t 1.2t X=44.040, X=33.002, X=15.368,
Y=52.848 Y=63.356 Y=80.174
y t 6 X=55.105, X=43.343, X=21.949,
Y=41.118 Y=52.011 Y=71.944
y t 12 1.2t X=74.768, X=64.959, X=41.014,
Y=20.347 Y=28.532 Y=49.217
Fig. 2. Experiment 2. Left: x t 10 t , y t 6 . Right: x t t ,
y t 12 1.2t .
As expected, the best strategy for each party is the increasing one. The intuition behind
the conclusion is like this. Imagine a person who receives a party's message to become
a supporter of this party but gives up their partisanship before relaying the message to
someone else. In this situation there is no sense of wasting much resource to recruit
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many supporters at the start of the campaign. More rationally would be to keep your
broadcasting resource until the final burst. This means an increasing strategy.
An obvious question arises about the intermediate case. We have got that decreasing
strategies tend to be the most advantageous when x y whereas increasing
strategies tend to be the most advantageous when x y . Thus the intermediate
case is presumably represented by equality x y . This is the question of the fol-
lowing experiment.
Experiment 3. Put T=15, x y 1 . Let the strategies of Party X be
Increasing : x t t ,
Flat : x t 7.5,
Decreasing : x t 15 t ,
so that Resourcex 112.5 . Similarly, strategies of Party Y are
Increasing : x t 1.3t ,
Flat : x t 9.75,
Decreasing : x t 19.5 1.3t ,
so that Resourcey 146.25 . The results are presented in Table 3. The dynamics for two
cases is shown in Fig. 3.
Table 3. Experiment 3: the final numbers of supporters.
x t t x t 7.5 x t 15 t
y t 1.3t X=43.151, X=43.217, X=34.927,
Y=56.097 Y=55.986 Y=64.227
y t 9.75 X=44.795, X=43.103, X=36.230,
Y=54.361 Y=56.034 Y=62.826
y t 19.5 1.3t X=54.038, X=50.472, X=42.996,
Y=45.091 Y=48.575 Y=55.895
The important point here is that there is no dominating strategies in this experiment.
4 Computational Experiment: A Viral Message vs Greater
Resource
In this section we consider a situation where Party X circulates a viral message,
which is relayed by individuals more intensively than the message of party Y. On the
other side, Party Y has a greater broadcasting resource.
207
Fig. 3. Experiment 3. Top left: x t t , y t 19.5 1.3t . Top right: x t 7.5 ,
y t 19.5 1.3t . Bottom left: x t 7.5, y t 1.3t . Bottom right: x t t ,
y t 1.3t .
Experiment 4. Put T=10, x 0.24, y 0.2, 0.8 . Let the strategies of Party X be
Increasing : x t t ,
Flat : x t 5,
Decreasing : x t 10 t ,
so that Resourcex 50 . Similarly, strategies of Party Y are
Increasing : x t 1.2t ,
Flat : x t 6,
Decreasing : x t 12 1.2t ,
so that Resourcey 60 . The results are presented in Table 4. The dynamics for one of
the cases is shown in Fig. 4.
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Table 4. Experiment: the final numbers of supporters.
x t t x t 5 x t 10 t
y t 1.2t X=51.670, X=47.005, X=37.113,
Y=46.445 Y=50.947 Y=60.515
y t 6 X=54.856, X=52.590, X=43.486,
Y=43.063 Y=45.006 Y=53.616
y t 12 1.2t X=67.599, X=65.272, X=57.452,
Y=29.967 Y=31.833 Y=38.977
Fig. 4. Experiment 4. Top left: x t 10 t , y t 6 . Top right: x t 10 t ,
y t 1.2t . Bottom: x t t , y t 1.2t .
To sum up the computational results, the generic feature is that the choice of strategy
depends on relation between the intensity of relaying messages via interpersonal com-
munication and the intensity of deactivation of individuals that is between β and .
Relatively big values of β and small values of γ are favorable for decreasing strategies
and reverse.
The research was supported by Russian Science Foundation (project No. 20-11-
20059).
209
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