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 mathematical 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 to spread the second rumor. The idea is that the first rumor is false, and the second contains convincing evidence of its falseness. In this approach, the "winner" of the information 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 confrontation 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][6][7][8][9][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][12][13][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][16][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 gradually. The other party has the same types of strategies. The question here is which strategy is to choose. We approach it with our mathematical model.

Model

Take the model of information warfare [4]  , XY    describe the individuals' shift from supporters to unattached, so that parameter  characterizes the intensity of this deactivation process. The system is considered at 0 tT  . For the sake of simplicity suppose

    , xx dX t X N X Y X dt              , yy dY t Y N X Y Y dt             00 0 , 0 . X X Y Y   x t  ,   y t

 to be linear functions of time:

  x x x t k t l    ,   y y y t k t l    .

In terms of these equations, increasing, decreasing and flat strategies of Party X mean, respectively, 0

x k  , 0 x k  , 0

x k  , 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   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 increasing 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 xy      . 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.

    10 , 6 xy t t t      . Right:   x tt  ,   12 1.2 y tt    .

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 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 xy      whereas increasing strategies tend to be the most advantageous when xy      The important point here is that there is no dominating strategies in this experiment.

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.    

.

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 communication 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.