CC BY
19UDC 51-77 10.23947/2587-8999-2022-1-3-114-120
BLOTTO GAME IN A PROPAGANDA BATTLE*
O.G. Podlipskaia
Moscow Institute of Physics and Technology (National Research University), Dolgoprudny, Russia
Hpodlipskaya.og@phystech.edu
The model describes the following process. Two parties, called Left and Right, are involved in information warfare on two topics that play the role of battlefields. Each party has limited broadcasting resources for propaganda, which each allocates between these two topics. Each member of the population backs one of the parties for each topic. A situation is possible in which an individual backs different parties on different topics. In this case, the individual is considered a supporter of the party supported on a more salient topic. Party supporters participate in participatory propaganda, campaigning on the topic or two topics they support for their party. The saliency of a topic depends on the amount of media broadcasting and communication on it. The number of party supporters' changes over time under the influence of media and party propaganda. The problem is to determine the parties' best strategies.
Each party apportion its broadcasting resource between two topics, thereby choosing its strategy. Therefore, a Blotto game appears. The Blotto game is a two-player zero-sum game in which the players distribute limited resources over several battlefields. In this matrix game, payoffs of the parties are the numbers of their supporters at the end of the propaganda battle. Numerical experiments were conducted in which these payoffs were calculated numerically and the obtained game was solved.
Typically, the best strategies are those where the resource is allocated between the topics very unevenly. Moreover, often the best strategy is spending all the resource on one topic.
Keywords: Blotto game, propaganda battle, mathematical model, optimal strategy, numerical experiment.
Introduction. Political competition is essential to political life, and information warfare is essential to political competition. Information warfare often occurs simultaneously on several topics with various salience. Some of the topics may be more favorable for left-wing groups, and others are favorable for right-wing groups. Accordingly, when describing warfare, this paper considers the provisions of the theory of information agenda [1]. The competing parties direct their efforts to persuade population members to take their view on a certain political topic and make some topics more salient than others. In this paper, we restrict the case to two topics and two parties.
If a given individual backs a given party on both topics, the individual is a party supporter. However, someone may share the Left party's views on one topic and the Right party's views on another topic. In this case, we assume that the individual is a party supporter aligned with the more salient topic. Moreover, when communicating with other individuals, this individual campaigns for this party (let it be the Left party for the sake of definiteness) on this topic and refrains from criticizing the party on the other topic, as it would mean criticizing the individual's party.
How can the Right party lure this individual over to its side using media resources? A straightforward strategy is to agitate the individual on a more salient topic to impel the individual to
* The research was supported by Russian Foundation for Basic Research (project 20-01-00229).
change viewpoints. A roundabout strategy is to increase the salience of the second topic to alter partisanship without changing views on the topics. If the second topic becomes more salient, this individual becomes a supporter of the Right party, campaigning for it on the second topic and abstaining from the first topic. This approach to the analysis of persuasion in information warfare is possible only when considering several topics. The present model implements this approach.
The earliest models of information propagation in a population are rumor models. Modern literature includes papers on detecting rumors [2] and a model that mutual promotion of rumors [3]. Models of competing rumors [4] consider the spread of two competing rumors, one of which is stronger than the other. Another approach [5] considers information warfare between two or more parties and derives the relations between the parameters to determine which party obtains more supporters. Ample literature is focused on the spread of information in social networks, agent- and game theory-based models [6-9], empirical studies [10-12] the dynamics of public attention [13] and political mobilization due to spread of information [14]. Some studies have analyzed national laws and legal documents in the context of information threats [15].
Model. Following [16], we consider a propaganda battle between two parties over two topics in a population of individuals. We assume that a proportion m of the population is initially more
inclined to consider Topic 1 to be more important than Topic 2, and vice versa for the rest of the individuals. On each of the topics, an individual backs the position of either the Left or Right party. The parties broadcast their propaganda via affiliated mass media and supporters of the party campaign through interpersonal communication with other individuals.
The ith individual backs the Right party on the ith topic if ^ . (t) > 0 and the Left party
otherwise, where e(-<x>,ro) is the ith individual's predisposition on topic j. The variables ^ (t) and y2 (t) are the main variables in the model. If an individual backs different parties on different topics, the individual supports the Right party if g(t)(^1 (t)) + (l - g(t))(^2 (t))> 0; otherwise, the individual supports the Left party, where g(t) e (0;1). The vector jg(t) ,1- g(t)} is called the agenda. Say, g (t) > 0.5 relates to the situation where the first topic is more salient than the second one. Increase of the function g (t) means that the first topic becomes more salient.
Unlike in the previous model [16], we assume decreasing returns from media broadcasting (square roots from intensities of broadcasting). The model has the following form:
(1)
d
-A/b2T + 0"g)(1 "m)C(R -L2) ,
(2)
dt
dg
k [b 1T + b 1R+ gC (+ L1 )
dt
where biL and are the intensities of the parties' broadcasting on the i topic, and all parameters are positive. Here b1L + b1R is the both parties' volume of media propaganda on the first topic, g C (R1 + L1 ) is the both parties' volume of participatory propaganda on this topic.
The numbers of campaigners for the Right and Left parties on the first topic are given by:
Ri = JJN92)d91d^ L1 = JJN92)d91d92 ,
R1 LI
R2 = JJN(91.92)d9id92; L2 = JJN(9i.92)d9id92 ,
R 2 L 2
where N (9,9) is the distribution of the members of society by attitudes:
R1: g(9i + ¥1 ) + (l"g)(92 + ¥2)>9i + ¥1 >0. L1: g(91 +¥1 ) + (1-g)(92 +¥2)<0 91 +¥1 <0 R2: g(91 +¥1 ) + (1-g)(92 +¥2)<0. 91 +¥1 <0. L2: g(91 +¥1 ) + (1"g)(92 +¥2)<92 +¥2 < The Right party possesses a fixed broadcasting resource b1R + b2R, allocated between the two
topics. The Left party does the same. For simplicity, we assume that these decisions are integers. Thus, the Blotto game takes place, in which each party aims to maximize the number of supporters at time T by the end of the battle. Following the above, an individual with the attitude vector (9; 9)
supports the Right party if g(T+¥1 (T)) + (l - g(T))(92 +¥2 (T)) > 0, and supports the Left
party in the opposite case. Thus, the number of supporters for each party at time T is given by the following expressions:
R(T) = JJ N(91,92)d91d92, L(T)= JJ N(992)d9d9 ,
Qr (T) Ql (T)
where the areas of integration are as follows:
Qr (T): g (91 + ¥1 (T)) + (1 - g) (92 + ¥2 (T)) > 0,
Ql (T): g (91 + ¥1 (T)) + (1 - g) (92 + ¥2 (T)) < 0.
Numerical Experiments. We present here one of conducted experiments. The settings are as follows:
a = 1,C = 1,m = 0.6, k = 1, ¥i(0) = ¥2(0) = 0, g(0) = 0.5 .
This means that at time t = 0 the parties have equal number of supporters, and the topics are equally salient.
The distribution of individuals by attitudes is taken to be normal and unbiased in both dimentions:
N1 (9^ 92 )^T1Texp
2TCCT
f „2 A 2
91
V 2a2,
9 2
exp-2a2
where a = 2. This means that the majority of individuals are neither Left- nor Right-wingers, they are centrists.
Regarding the broadcasting resources, we consider the situation in which the Right party has a moderate advantage:
b1L + b2L = 5 b1R + b2R = 6.
The task for both parties is to determine the integer values b1R and b 2R or b1L and b 2L to
maximize the number of supporters at the end of the battle. So, there are six possible strategies for the Left party and seven strategies for the Right party, thus the experiment consists of 42 runs. The numerical calculations were conducted up to the moment T = 10. The results are given in Table 1.
Table 1. Blotto game: the computed variables at the end of the propaganda battle. Raws: strategies of
the Right party. Columns: strategies of the Left party
0+5 1+4 2+3 3+2 4+1 5+0
0+6 R=0.547, L=0.453, g=0.0 R=0.58, L=0.42, g=0.088 R=0.591, L=0.409, g=0.176 R=0.586, L=0.414, g=0.267 R=0.564, L=0.436, g=0.359 R=0.598, L=0.402, g=0.451
1+5 R=0.519, L=0.481, g=0.088 R=0.55, L=0.45, g=0.177 R=0.572, L=0.428, g=0.268 R=0.577, L=0.423, g=0.359 R=0.566, L=0.434, g=0.452 R=0.598, L=0.402, g=0.544
2+4 R=0.509, L=0.491, g=0.177 R=0.529, L=0.471, g=0.268 R=0.549, L=0.451, g=0.360 R=0.552, L=0.448, g=0.453 R=0.536, L=0.464, g=0.546 R=0.549, L=0.451, g=0.639
3+3 R=0.521, L=0.479, g=0.268 R=0.525, L=0.475, g=0.360 R=0.543, L=0.457, g=0.453 R=0.543, L=0.457, g=0.547 R=0.523, L=0.477, g=0.64 R=0.517, L=0.483, g=0.732
4+2 R=0.554, L=0.446, g=0.361 R=0.54, L=0.46, g=0.454 R=0.556, L=0.444, g=0.547 R=0.553, L=0.447, g=0.640 R=0.53, L=0.47, g=0.732 R=0.509, L=0.491, g=0.823
5+1 R=0.606, L=0.394, g=0.450 R=0.517, L=0.483, g=0.732 R=0.585, L=0.415, g=0.641 R=0.578, L=0.422, g=0.732 R=0.554, L=0.443, g=0.822 R=0.533, L=0.473, g=0.912
6+0 R=0.612, L=0.388, g=0.55 R=0.574, L=0.426, g=0.641 R=0.596, L=0.404, g=0.733 R=0.603, L=0.397, g=0.824 R=0.589, L=0.407, g=0.913 R=0.574, L=0.425, g=0.999
If the Right party plays «0+6» strategy, then the worst output for it occurs if the Left party plays «0+5». Then, the Right party has R=0.547 supporters art the end of the battle. Similarly, if the Right party plays «1+5» strategy, then the worst output is R=0.547, and so on. It follows from this that the best strategy for the Right party is «6+0». In the same way, by considering the greatest value of L in each column, we obtain that the best strategy for the Left party is «1+4», which provides the saddle point. Therefore, in this numerical experiment, the Right party wins with 57% support.
Numerical experiments with various parameters were conducted, and the most general conclusion is that the best warfare strategies are those that are far from allocating the resources evenly.
In the example given, the optimal strategies are «0+6» and «1+4», which appears to be typical. All or nearly all eggs should be put in the same basket.
Discussion. What data are needed for examining and using this theoretical approach? First, the intensity of broadcasting on different topics. The key task here is to assign a political topic to a given document. Topic modeling, although being a respectable technique in the field of natural language processing, is hardly applicable here because what we call «topic» in political science, is a prescribed thing. Thus, in [17] a different approach was employed. The purpose of the cited paper was to assign episodes of Russian political talk shows to political topics. The list of these topic was predetermined. A list of key words was made (and collocations) for each topic and collected the abstracts for each episode of the show from websites of TV companies. These abstracts were considered as documents that were allocated among topics basing on these lists of keywords and using tailored software. Thus, estimating the intensity of broadcasting on different topics, that is, estimating the vector jg (t) ,1 — g (t)j is a solvable task. In similar way, the topics of users' posts in social
media can probably be determined. However, the easier way to find out what people are interested in is to consider the data on search queries using Google Trends to obtain time series for public attention to different topics [13]. Thus, variables specific to presented approach can be retrieved by means of computational science.
References
1. McCombs, M., Stroud, N.J.: Psychology of agenda-setting effects: Mapping the paths of information processing. Review of Communication Research, 2(1), 2014, pp. 68-93.
2. Alkhodair, S. A., Ding, S. H., Fung, B. C., & Liu, J. Detecting breaking news rumors of emerging topics in social media. Information Processing & Management, 57(2), 2020, pp. 102018.
3. Wang, Z., Liang, J., Nie, H., Zhao, H. A 3SI3R model for the propagation of two rumors with mutual promotion. Advances in Difference Equations, 2020, 1, pp. 1-19.
4. Xiao, Y., Yang, Q., Sang, C., Liu, Y. Rumor diffusion model based on representation learning and anti-rumor. IEEE Transactions on Network and Service Management, 2020, 17(3), pp. 1910-1923.
5. A.P. Mikhailov, G.B. Pronchev, O.G. Proncheva. Mathematical Modeling of Information Warfare in Techno-Social Environments // Techno-Social Systems for Modern Economical and Governmental Infrastructures. IGI Global, 2019. pp. 174-210. DOI: 10.4018/978-1-5225-5586-5.ch008.
6. Chkhartishvili, A.G., Gubanov, D.A. and Novikov, D.A. Social Networks: Models of Information Influence, Control and Confrontation, Springer, 2018, Vol. 189.
7. I.V. Kozitsin, A general framework to link theory and empirics in opinion formation models, Scientific reports, 2022, Vol. 12, p. 5543.
8. I.V. Kozitsin, Formal models of opinion formation and their application to real data: evidence from online social networks, The Journal of Mathematical Sociology, 2022, 46(2), pp. 120-47.
9. A.G. Chkhartishvili, D.A. Gubanov, On the Concept of an Informational Community in a Social Network, Journal of Physics: Conference Series, 2021, Vol. 1864, No. 1, pp. 012052.
10. Boldyreva, A., Sobolevskiy, O., Alexandrov, M., Danilova,V. Creating collections of descriptors of events and processes based on Internet queries, In: Proc. of 14-th Mexican Intern. Conf. on Artif. Intell. (MICAI-2016), Springer Cham, LNAI, Vol. 10061, (chapter 26), 2016, pp. 303-314.
11. V. Danilova, S. Popova, M. Alexandrov, "Multilingual Protest Event Data Collection with GATE," Proc. of 21-th Intern. Conf. on Natural Languages and Data Base (NLDB-2016), Springer, Vol. 9612, 2016, pp. 115-126.
12. A.P. Petrov, S.A. Lebedev. Online Political Flashmob: the Case of 632305222316434 // Computational mathematics and information technologies, 2019, No 1, pp. 17-28. DOI: 10.23947/2587-89992019-1-1-17-28.
13. Podlipskaia O.G. (2022) Duration of the Public Attention to Terrorist Attacks in the United States: Does It Depend on Political Opinion? Monitoring of Public Opinion: Economic and Social Changes, No 1, pp. 5-21. https://doi.org/10.14515/monitoring.2022.L2019. (In Russ.)
14. A. Akhremenko, A. Petrov. Modeling the Protest-Repression Nexus // Proceedings of the Conference on Modeling and Analysis of Complex Systems and Processes, 2020, pp. 1-11. http://ceur-ws.org/Vol-2795/paper1.pdf
15. Pronchev G.B., Shisharina E.V., Proncheva N.G. Cyber threats for modern Russia in the context of the coronavirus pandemic // Political Science Issues, Vol. 11, pp. 26-34. DOI: 10.35775/PSI.2021.48.1.003.
16. Proncheva, O. A Model of Propaganda Battle with Individuals' Opinions on Topics Saliency.13th International Conference Management of largescale system development, 2020, pp. 1-4. DOI: 10.1109/MLSD49919.2020.9247796.
17. A. Petrov, O. Proncheva. Identifying the Topics of Russian Political Talk Shows // Proceedings of the Conference on Modeling and Analysis of Complex Systems and Processes, 2020, pp. 79-86. DOI: 10.6084/m9.figshare.15109338, http://ceur-ws.org/Vol-2795/short1.pdf.
Author:
Olga Podlipskaia, Moscow Institute of Physics and Technology, Dolgoprudny, Russia, Department of Higher Mathematics, Associate Professor, 9 Institutskiy per., Dolgoprudny, Moscow Region, 141701, Russian Federation, e-mail: podlipskaya.og@phystech.edu
УДК 51-77 10.23947/2587-8999-2022-1-3-104-120
ИГРА БЛОТТО В ПРОПАГАНДИСТСКОЙ БИТВЕ * О. Г. Подлипская
Московский физико-технический институт (национальный исследовательский университет), Долгопрудный, Московская область, Российская Федерация
Н podlipskaya.og@phystech.edu
В работе рассматривается следующий процесс. Две партии, вовлечены в информационную войну по двум темам, которые понимаются как поля сражений. Каждая сторона имеет ограниченный ресурс вещания для пропаганды, и распределяет свой ресурс между этими двумя темами. Каждый индивид поддерживает одну из сторон по каждой теме. Если индивид поддерживает разные партии по разным темам, то он считается сторонником партии, которую он поддерживает по более значимой теме. Сторонники партии участвуют в партиципаторной пропаганде, проводя кампанию по одной или обеим темам, по которым они поддерживают свою партию. Значимость темы зависит от объема вещания СМИ по ней. Число сторонников партии меняется с течением времени под влиянием средств массовой информации и партийной пропаганды. Задача заключается в том, чтобы определить наилучшие стратегии сторон.
Каждая сторона распределяет свой вещательный ресурс между двумя темами, тем самым выбирая свою стратегию. Таким образом, появляется игра Блотто, т.е. игра двух игроков с нулевой суммой, в которой игроки распределяют ограниченные ресурсы по нескольким полям сражений. В этой матричной игре выигрыш партий - это количество их сторонников в конце пропагандистской битвы. Были проведены численные эксперименты, в ходе которых эти выигрыши были численно рассчитаны. Как правило, наилучшими стратегиями являются те, в которых ресурсы распределяются между темами неравномерно. Более того, часто лучшая стратегия - потратить все ресурсы на одну тему.
Ключевые слова: игра Блотто, пропагандистская битва, математическая модель, оптимальная стратегия, численный эксперимент.
Авторы:
Ольга Геннадьевна Подлипская, Московский физико-технический институт (национальный исследовательский университет), г. Долгопрудный, Московская область, Российская Федерация. Кафедра высшей математики, доцент, (141701, Московская область, г. Долгопрудный, Институтский пер., 9), e-mail: podlipskaya.og@phystech.edu
* Исследование выполнено при финансовой поддержке РФФИ в рамках научного проекта № 20-01-00229.
