3.1. Information Security Investment Scenario in OSN
To understand the profitability and cost of security investment in the OSN scenario, we consider OSN service providers and online platforms as the “game players” representing security investors. OSN service providers are technological providers (e.g., Microsoft, Cisco, and Oracle) that provide support to secure information systems, databases, and software for OSN. Whether or not they sustain the security investment depends on the trade-off between profits and costs. Online platforms provide a social networking service on which users build social networks or social relations with other persons. The security state of online platforms is positively related to the level of security management of online platforms, which also must make a moderate investment. Therefore, the strategic choice of OSN service providers and online platforms influence each other, and the cooperative interactions can be viewed as a dynamic game process.
To formulate this scenario, we built a two-echelon security investment chain, consisting of OSN service providers (denoted by ) and online platforms (denoted by ), both of which have two strategies: “Invest” and “Not Invest”. Therefore, there are four possible combinations of the two strategies: (Not Invest, Not Invest), (Not Invest, Invest), (Invest, Not Invest), and (Invest, Invest). However, it is difficult to optimize security investment based on classical game theory because of incomplete information, complicated scenarios, and the bounded rationality of players.
Considering the decision problems of OSN security investment, this paper applies evolutionary game theory (EGT) to model such situations. We investigate the optimal strategies of security investment in an OSN context not only based on cost-benefit analysis, but also from an evolutionary perspective. The motivation of using evolutionary game theory can be concluded as follows:
Equilibrium solution refinement. The evolutionary game approaches provide a refined solution that ensures the stability of a strategy adopted by a population, where no small subgroup of deviants could successfully invade the whole population. Such a strategy is known as an evolutionary stable strategy (ESS) [
20,
34];
Bounded rationality. In traditional game theory, the game players are assumed as rational and the players believe that the other side is also rational throughout the game. However, this assumption is often unrealistic. This situation is avoided in evolutionary game, where players adopt dynamic strategies that lead them to sustain in the population without caring about instant profits maximization [
20]; and
Game dynamics. Since players in evolutionary game interact with each other for multiple rounds by adopting different strategies, the state of their interaction varies over time according to the replication games. Thus, the evolutionary game provides a natural way to introduce dynamics, where success strategies are imitated by others and propagate over interaction rounds.
3.3. Model Solutions
In the initial stage of the evolutionary game, we define as the population of OSN service providers making the strategic choice of “Invest”. In contrast, represents the population making the strategic choice of “Not Invest”. Similarly, represents the population of online platforms making the strategic choice of “Invest”, and represents the population making the strategic choice of “Not Invest”.
Based on the assumptions in
Section 3.2, we assume that
represents the expected payoff of OSN service providers that make the strategic choice of “Invest”,
represents the expected payoff of OSN service providers that make the strategic choice of “Not Invest”, and
represents the average expected payoff of OSN service providers. Therefore:
Thus, the average expected payoff of OSN service providers can be written as follows:
It is assumed that
represents the expected payoff of online platforms that make the strategic choice of “Invest”,
represents the expected payoff of online platforms that make the strategic choice of “Not Invest”, and
represents the average expected payoff of online platforms. Therefore:
According to the Malthusian dynamic equation [
36], the replicator equation of population
for OSN service providers is:
The replicator equation of population
for online platforms is:
When the replicator equation equals 0, an equilibrium point of the evolutionary game has been reached, and will no longer evolve. This results in five equilibrium points—that correspond to the equilibria of the dynamic system: (0, 0), (0, 1), (1, 0), (1, 1), . Note that is a mixed equilibrium point where , .
3.4. Stable Analysis of Equilibrium Points
The stability of equilibrium points can be analyzed using a Jacobian matrix [
37]. The Jacobian matrix can be defined as follows:
The stability of equilibrium points can be examined using the following conditions [
38]:
We can compute the values of the equilibrium points and classify them into different types that are shown in
Table 3. Note that
is not satisfied under the above condition because
. Other equilibrium points will be ESSs, whereas the values of related parameters are satisfied under different conditions. The propositions are analyzed as follows:
Proposition 1. When,and,, (0, 0) is an evolutionarily stable point. OSN service providers and online platforms will make the strategic choice of “Not Invest”.
Proof. We define
as the expected profits of OSN service providers if only they make the strategic choice of “Invest”, and
as the expected profits of online platforms if only they make the strategic choice of “Invest”. Therefore:
□
In this scenario, we find that the expected profits from security investment is lower than the profits if they make the strategic choice of “Not Invest”. Both players have no incentive to sustain the security investment due to the little profits.
Proposition 1 also presents the business implications from the perspective of evolutionary analysis. We assume there are several OSN service providers and online platforms in an OSN context. OSN service providers, , may choose “Invest” at first because of information asymmetry and bounded rationality. Then, finds (another OSN service provider) chooses “Not Invest” and can get higher profits. Therefore, will adjust and improve its choices by imitating the strategy of for profit maximization. We can conclude that the strategy of will impact on the strategic decision of . Moreover, the investment strategies of online platforms also have impacts on the strategic decision of OSN service providers. The interaction with each other will result in the evolution of the strategic choice.
Panel (a) in
Figure 1 displays the evolution of the dynamic model when the profit growth coefficients are small. We can find that the evolutionary model will eventually converge at (0, 0) no matter what strategies are initially taken by OSN service providers and online platforms. Therefore, (0, 0) is the evolutionarily stable point; (0, 1) and (1, 0) are saddle points; and (1, 1) is the unstable point. The ESS profile is (Not Invest, Not Invest).
Proposition 2. When,and, (0, 1) is the evolutionarily stable point. OSN service providers will make the strategic choice of “Not Invest”, and online platforms will make the strategic choice of “Invest”.
Proof. If the profit growth coefficients are satisfied under the above conditions:
□
We define
as the expected profits of OSN service providers, if both players make the strategic choice of “Invest”. The term,
, is the expected profits of online platforms if both players make the strategic choice of “Invest”. We find that:
From the perspective of evolutionary analysis, we assume online platforms, , may choose “Not Invest” at first because of investment costs. Then, finds (another online platform) chooses “Invest” and can get higher profits. Therefore, will improve its choices by imitating the strategy of . Moreover, the investment strategies of OSN service providers have no significant impacts on the strategic decision of hospitals because online platforms cannot free ride on the other side of game players.
Panel (b) in
Figure 1 depicts the dynamic evolution model. As shown, the model will eventually converge at (0, 1) no matter what strategies are initially taken by OSN service providers and online platforms. Therefore, (0, 1) is the evolutionarily stable point; (0, 0) and (1, 0) are saddle points; and (1, 1) is the unstable point. The ESS profile is (Not Invest, Invest).
Proposition 3. Whenand,, (1, 0) is an evolutionarily stable point. OSN service providers will make the strategic choice of “Invest”, and online platforms will make the strategic choice of “Not Invest”.
Proof. If the profit growth coefficients are satisfied under the above conditions:
□
From the perspective of evolutionary analysis, OSN service provider, , may choose “Not Invest” at first because of bounded rationality. Then, finds chooses “Invest” and can get higher profits. Therefore, will adjust its strategic choice by imitating the strategy of . Similarly, the investment strategies of online platforms have no significant impacts on the strategic decision of OSN service providers because OSN service providers cannot free ride on the other side of game players.
Panel (c) in
Figure 1 illustrates the evolution of the dynamic model. The figure shows it will eventually converge at (1, 0) no matter what strategies are initially taken by OSN service providers and online platforms. Therefore, (1, 0) is the evolutionarily stable point; (0, 0) and (0, 1) are saddle points; and (1, 1) is the unstable point. The ESS profile is (Invest, Not Invest).
Proposition 4. Whenand, (0, 1) and (1, 0) are evolutionarily stable points. Both OSN service providers and online platforms have two strategic choices: “Not Invest” and “Invest”.
Proof. If the profit growth coefficients are satisfied under the above conditions:
□
From the perspective of evolutionary analysis, OSN service provider, , and online platforms, , may choose “Invest” at first because of higher profits from security investment. Then, finds that it can get higher profits if it can free ride off . For example, if chooses “Invest”, there will be more users to use OSN APPs. Therefore, can get extra profits from a larger market, and without any investment costs. However, it is not the end of the evolution process. will also choose “Not Invest” and will want to free ride off . Therefore, and will always adjust their strategy by imitation for profit maximization.
Panel (d) in
Figure 1 depicts the evolution of the dynamic model. As shown, the model will eventually converge at (0, 1) or (1, 0). Therefore, (1, 0) and (0, 1) are the evolutionary stable points;
is the saddle point; and (1, 1) and (0, 0) are the unstable points. The ESS profiles are (Not Invest, Invest) and (Invest, Not Invest).
Proposition 5. Whenand, (1, 1) is an evolutionarily stable point. OSN providers and online platforms will choose (Invest, Invest).
Proof. If the profit growth coefficients are satisfied under the above conditions:
□
From the perspective of evolutionary analysis, or may choose “Not Invest” at first. Then, they will find that “Invest” can bring higher profits sooner or later. Therefore, both OSN service providers and online platforms will adjust its strategic choice by imitating others.
Panel (e) in
Figure 1 shows the evolution of the dynamic model. As shown, it will eventually converge at (1, 1) regardless of strategies initially taken by OSN service providers and online platforms. Therefore, (1, 1) is the evolutionarily stable point; (0, 1) and (1, 0) are saddle points; and (0, 0) is the unstable point. The ESS profile is (Invest, Invest).