6.3.1. Stackelberg Game Solution
shows a graphical representation of the Stackelberg Equilibrium solution of the game. The black lines symbolize the defender’s optimal patrolling strategy, i.e., the non-zero probabilities for each of the actions of the defender. Each line segment has an associated number representing the probability that the defender will take that action, which is not shown in the figure. For instance, at time 0, the defender will move to check-in area (
) with a probability of 0.129. Alternatively, the defender also has an option to stay at the airport entrance (
) for 60 s with a probability of 0.871.
An interesting result of the generated strategy is that is not patrolled at all. This target is covered by patrolling , which is close to . The area around is in the observation radius of the defender when she is in . Furthermore, is a more central target, and can, therefore, be reached faster from the other targets.
The attacker’s best response strategy is to attack the checkpoint area (
), entering the airport at a time between ten to 15 min, illustrated in Figure 6
as a red line. Please note that the red line only covers
for visualization simplicity. In reality, the attacker always enters the airport through
and takes some time to arrive at the target.
shows the agent-based model results associated with the patrol movements corresponding to the optimal patrol strategy. Only the patrol movements that lead to a defender–attacker interaction are shown. It is important to note that there is one movement for which the period does not coincide with the attacker entering time of 10 to 15 min. This occurs since the attacker takes time to reach his target destination in a crowded airport. All other movements that are part of the optimal strategy, but are not present in Table 4
, are those where there was no interaction between both players. The payoff associated with those movements is set to zero.
When the probability value and expected number of casualties associated with each movement (as outlined in Table 4
) are introduced in Equation (11
), the optimal reward values for the defender and attacker are obtained.
The attacker reward is the negation of the defender’s reward, i.e.,
. Figure 7
shows every attacker’s reward value associated with each attacker’s strategy against the defender optimal (probabilistic) patrolling strategy. These are computed similarly as the one illustrated in the equation above.
These results show that attacking the security checkpoint () between 5 and 20 min yields the highest reward for the attacker when compared to attacking other targets within the same time frame. This may be explained as follows. Passengers arriving in previous time intervals finished their check-in activity and are going towards the security checkpoint, leading to a higher density of people around that area. Thus, if the attack is successful, its impact would be large. This is not the case for all the other targets since there are passengers who did the check-in online and go straight to the which results in a lower concentration of passengers around those areas. Moreover, an attack within the first five minutes has a lower consequence since fewer people are at the airport terminal. The airport gets more crowded as time gets closer to the flight departure time.
It is also worth noticing that an attack on targets , and , at the latest time interval yields higher rewards for the attacker when compared to other periods. This is the case, as the number of people entering the airport considerably increases during that time interval which results in a higher concentration of people in those areas. This increase results from the fact that as time passes by, it gets closer to the flight departure time and therefore more people start entering the airport. As mentioned earlier, the latter increases the chances and consequences of a successful attack.
By comparing the results of Figure 6
and Figure 7
, the defender’s optimal strategy choice may be justified as follows. From Figure 7
it can be observed that the attacker reward by attacking
while entering the airport between five to ten minutes yields the second-highest value. Therefore, the defender favors the patrol of that area during the corresponding period. The latter observation may be the reason the defender’s optimal strategy does not contain additional movements that patrol the optimal attack target at the optimal attack time (between 10 to 15 min).
However, the optimal defender strategy does not coincide with the attacker target for the entire attack time interval. Specifically, the defender choice after leaving is to go either to or , and, eventually, staying there until a new patrol starts. These results can be explained by the fact that the attacker, in his path to , may be detected by the defender if she is either at check-in area 2 () or the airport entrance ().
These results show that the optimal security patrol gives special emphasis to high-impact areas, such as the security checkpoint, to reduce the total security risk. This is an improvement over the more simplistic strategies as shown in the work of Janssen et al. [11
6.3.2. Deterministic Patrolling Strategy
In the current patrolling practice, the security officer may follow a deterministic patrolling strategy. In a deterministic patrolling strategy, the probability that an action is taken is constrained to be either 0 or 1, rather than a probabilistic value between 0 and 1. To investigate this scenario, we follow the same procedure illustrated in Section 6.3.1
, but with the aforementioned constraint where the decision variables are either 0 or 1. Figure 8
illustrates the optimal strategy for both agents. The red line represents the attacker’s optimal strategy, while the black line denotes the defender’s best response. It is interesting to observe that for a fixed patrolling strategy, the attacker’s best response remains to be
, but changes the attacking time interval to a time range between five to ten minutes. This result shows that attacking
during the time interval between five and ten minutes yields a high payoff for the attacker. Therefore, it reinforces the defender’s patrol choice of covering that target during that time interval in the probabilistic patrol strategy, as discussed in Section 6.3.1
Results, as shown in Figure 8
, show that if the defender would follow the fixed patrolling route and the attacker plays his best response rewards for the defender and the attacker are −21.417 and 21.417 respectively. This shows that by randomizing over different movements at different times, the defender can generate strategies that are effective against a potential terrorist attack. These conclusions can help airport managers design security procedures.