Signaling Security Games with Attack Planner Deception
Abstract
:1. Introduction
- We model the attack initiator as the attack planner—the individual they hire to carry out the attack is the attacker—and we propose a defender–attack planner (D-AP) model to describe the interaction between them;
- We account for the interests of each attacker to ensure compliance, framing this as a multi-objective optimization issue;
- This paper also addresses the defender’s uncertainty to the attack planner’s payment and the particulars of the hired attackers, constructing a defender–attack planner game model with deception (D-APD).
- We articulate D-APD as a tri-level programming problem and propose a customized decomposition algorithm based on a genetic algorithm (DA-GA) to address it.
2. Related Work
2.1. Deception in Games
2.2. About Planner
3. Model
3.1. Defender–Attack Planner Game Model
3.1.1. Model Description
3.1.2. Strategy Expression
3.1.3. Utility Function
3.2. Defender–Attack Planner Game Model with Deception
3.2.1. The Attack Planner with Multiple Types
3.2.2. Attack Planner’s Deception
3.2.3. Game Process
3.2.4. Utility of Both Sides
3.2.5. The Equilibrium
3.3. Motivating Example
4. Equilibrium Computation
4.1. Modified Model
4.2. Tri-Level Programming
4.2.1. Signaling-Level Problem
4.2.2. Defense-Level Problem
4.2.3. Attack-Level Problem
4.3. Algorithm Description
4.3.1. Attack-Level Algorithm
- 0.
- Code: The planner’s attack scheduling strategy is a matrix with size , so we first need to encode it. Specifically, we encode each attack scheduling strategy as an -dimensional row vector with elements in set . For example, a strategy can be encoded as , which represents that the 1-th attacks the 1-th target, the 2-th attacks the 5-th target and the 3-th attacks the 3-th target. In fact, the number of columns in each row of the policy matrix with the value of 1 is taken as the encoding of the policy.
- 1.
- Determine an initial population P consisting of n individuals that meet the constraints;
- 2.
- Calculate the fitness of each individual in P;
- (1)
- For :
- Calculate the function value for each individual on each objective;
- For each individual p, calculate the number of individuals that dominate it and the set of individuals that it dominates;
- Set the of the individuals with to 1, and the set of these individuals to , then subtract the values of the individuals in by 1; then, set the of the individuals with at this time to 2, the set of these individuals to , and so on until the of each individual is calculated.
- (2)
- For , solve it for each individual in in separately:
- For each individual i in , let i’s crowding distance ;
- Calculate the m-th function value for each individual and sort them in ascending order according to these values;
- In order to give individuals on the edge a selection advantage, let ; that is, after sorting, let the crowding distance of the first and last individuals be infinity. Then for the other individuals
- The above three steps are repeatedly calculated for each objective function, and the crowding distance obtained when each individual corresponds to different objective functions is added up to obtain the final crowding distance, which is denoted as .
- 3.
- By selecting, crossing and mutating for P, a new population Q is generated;
- (1)
- Selection operator: The selection is based on fitness using the binary tournament method. Randomly select two individuals for comparison, select the better individual to enter the next generation and repeat this operation until we select enough n individuals where an individual p is better than an individual q, if, and only if, the of p is less than q or the of p is equal to q and the of p is greater than q. That is, compare the rank first; the individual with a lower rank is better, individuals with an equal rank have a comparative crowding distance, and individuals with larger crowding distances are better;
- (2)
- Crossover operator: For two individuals in the population, a location is randomly selected and the genes at that location are exchanged with a certain probability;
- (3)
- Mutation operator: In order to ensure the diversity of the population and avoid falling into the local optimal situation, the individuals in the population are randomly selected in a position and the genes at that position are changed.
- 4.
- According to step 2, calculate the fitness of each individual in ;
- 5.
- Selection* operator: Select the n individuals in as the offspring; if there is an l, that makes true, so is the offspring population. Otherwise, when l satisfies the conditions and , the individuals in are sorted in descending order of crowding distance , and then the first individuals are selected to form offspring P together with ;
- 6.
- Return to step 3 and loop until the termination condition is satisfied, and finally obtain a Pareto optimal solution set.
Algorithm 1: GA for AL problem |
4.3.2. Master Problem
4.3.3. Subproblem
Algorithm 2: GA for SP |
4.3.4. Customized Decomposition Algorithm Based on Genetic Algorithm
Algorithm 3: Customized DA-GA |
5. Experiment and Analysis
5.1. Algorithm Performance
5.2. The Impact of Defense Budgets
5.3. The Impact of Signaling Strategy
5.4. The Impact of Prior Probability
5.5. The Impact of Deception Cost
6. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Appendix A
Appendix B
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Symbol | Description |
---|---|
Sets: | |
The set of defender’s pure strategy | |
The set of attack planner’s pure strategy | |
The set of attack planner’s payment types | |
The set of attacker types | |
The set of signals about payment type | |
The set of signal about attacker type | |
The set of attack planner’s types | |
S | The set of signals |
The set of mixed signaling strategy about planner’s payoff type | |
The set of mixed signaling strategy about planner’s attacker type | |
Parameters: | |
N | The number of targets |
Attacker’s base salary vector | |
c | Defender’s cost vector |
B | Defender’s budget constraint |
n | The number of strong and weak attacker |
Defender’s valuation of targets | |
Attack planner’s valuation of targets | |
Defender’s pure strategy | |
Attack planner’s pure strategy | |
The probability that the target j is attacked successfully | |
i-th attacker’s base salary | |
The i-th attacker’s cost when they attacks target j | |
The ratio of the commission when attack is successful | |
The prior probability distribution about payment types | |
The target valuation of Attack planner with payment type | |
The prior probability distribution about attacker type | |
The posterior probability about the payment types | |
The posterior probability about the attacker type | |
Attack planner’s deception cost | |
The prior probability distribution of attack planner’s type | |
The probability that the attack planner’s type is after receiving the signal s | |
The utility of defender, planner and the i-th attacker without deception | |
The utility of defender, planner and the i-th attacker with deception | |
Decision variables: | |
The signaling strategy | |
Defender’s strategy scheme | |
Attack planner’s attack scheme |
Variable | Value |
---|---|
N | 3 |
m | 2 |
[0.5 0.5] | |
M | 2 |
[0.5 0.5] | |
0.01 | |
[5000 3000 2000] | |
[2000 5000 2000;5000 5000 3000] | |
[50 30] | |
[1 0;1 1] | |
50 | |
[30 50] | |
[2 1 3] | |
[1.5 2 1;1 3 2] | |
[3 3] | |
c | [0 20 30] |
B | 40 |
B = 400 | B = 600 | |||
---|---|---|---|---|
N | EA | DA-GA | EA | DA-GA |
3 | ||||
4 | ||||
5 |
Type | 1 | 2 | 3 | 4 | 5 | |
---|---|---|---|---|---|---|
= [0,50] | 1 | (1,1) | (2,1) | (1,1) | (2,1) | (2,1) |
2 | (1,2) | (4,2) | (5,2) | (5,2) | (1,2) | |
3 | (1,3) | (5,3) | (3,3) | (3,3) | (1,3) | |
= [10,50] | 1 | (3,1) | (5,1) | (2,1) | (5,1) | (5,1) |
2 | (1,2) | (1,2) | (3,2) | (4,2) | (5,2) | |
3 | (1,3) | (2,3) | (4,3) | (4,3) | (5,3) | |
= [20,50] | 1 | (1,1) | (5,1) | (5,1) | (4,1) | (5,1) |
2 | (1,2) | (2,2) | (3,2) | (4,2) | (5,2) | |
3 | (1,3) | (3,3) | (3,3) | (4,3) | (5,3) | |
= [30,50] | 1 | (1,1) | (1,1) | (3,1) | (4,1) | (5,1) |
2 | (1,2) | (2,2) | (3,2) | (4,2) | (5,2) | |
3 | (1,3) | (2,3) | (3,3) | (4,3) | (5,3) | |
= [40,50] | 1 | (1,1) | (5,1) | (3,1) | (4,1) | (5,1) |
2 | (1,2) | (2,2) | (3,2) | (4,2) | (5,2) | |
3 | (1,3) | (2,3) | (3,3) | (4,3) | (5,3) | |
= [50,50] | 1 | (1,1) | (5,1) | (3,1) | (4,1) | (5,1) |
2 | (1,2) | (2,2) | (3,2) | (4,2) | (5,2) | |
3 | (1,3) | (2,3) | (3,3) | (4,3) | (5,3) | |
= [60,50] | 1 | (1,1) | (2,3) | (3,1) | (4,1) | (5,1) |
2 | (1,2) | (2,2) | (3,2) | (4,2) | (5,2) | |
3 | (1,3) | (2,3) | (3,3) | (4,3) | (5,3) | |
= [70,50] | 1 | (1,1) | (2,3) | (3,2) | (4,1) | (5,1) |
2 | (1,2) | (2,2) | (3,2) | (4,2) | (5,2) | |
3 | (1,3) | (2,3) | (3,3) | (4,3) | (5,3) | |
= [80,50] | 1 | (1,1) | (2,2) | (3,1) | (4,2) | (5,3) |
2 | (1,2) | (2,2) | (3,2) | (4,2) | (5,2) | |
3 | (1,3) | (2,3) | (3,3) | (4,3) | (5,3) | |
= [90,50] | 1 | (1,1) | (2,3) | (3,1) | (4,1) | (5,1) |
2 | (1,2) | (2,2) | (3,2) | (4,2) | (5,2) | |
3 | (1,3) | (2,3) | (3,3) | (4,3) | (5,3) | |
= [100,50] | 1 | (1,1) | (2,3) | (3,1) | (4,1) | (5,1) |
2 | (1,2) | (2,2) | (3,2) | (4,2) | (5,2) | |
3 | (1,3) | (2,3) | (3,3) | (4,3) | (5,3) |
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He, S.; Li, M.; Zhang, R. Signaling Security Games with Attack Planner Deception. Mathematics 2024, 12, 2532. https://doi.org/10.3390/math12162532
He S, Li M, Zhang R. Signaling Security Games with Attack Planner Deception. Mathematics. 2024; 12(16):2532. https://doi.org/10.3390/math12162532
Chicago/Turabian StyleHe, Santing, Mingchu Li, and Runfa Zhang. 2024. "Signaling Security Games with Attack Planner Deception" Mathematics 12, no. 16: 2532. https://doi.org/10.3390/math12162532