# Game Theoretic Honeypot Deployment in Smart Grid

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## Abstract

**:**

## 1. Introduction

#### 1.1. Related Works & Motivation

#### 1.2. Contribution

#### 1.3. Structure

## 2. System Model

## 3. One-Shot Game

#### 3.1. Game Formulation

- The set of players $\mathcal{S}$, which includes the attacker and the defender, i.e., $\mathcal{S}=\{A,D\}$
- The set of actions for each player, i.e, ${\mathcal{A}}_{D}=\{\theta \in [0,1],N\in [0,{N}_{\mathrm{max}}]\}$ for the defender and ${\mathcal{A}}_{A}=\varphi \in [0,{\varphi}_{\mathrm{m}}]$ for the attacker.
- The payoff functions for each player, i.e., ${U}_{\mathrm{A}}$ and ${U}_{\mathrm{D}}$.

#### 3.2. Solution of Game 1

**Definition**

**1.**

**Lemma**

**1.**

**Proof.**

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

#### 3.3. Strategy Selection When NE Does Not Exist

## 4. Repeated Game with Uncertainty about the Type of Attacker

#### 4.1. Game Formulation

- (i)
- The set of players $\mathcal{S}$ that includes the attacker and the defender, i.e., $\mathcal{S}=\{A,D\}$.
- (ii)
- The set of states of nature, denoted by $\mathsf{\Omega}$.
- (iii)
- The types of the attacker, i.e., the set $(a,b)$.
- (iv)
- The set of actions for each player, i.e, ${\mathcal{A}}_{D}=\{\theta ,N\}$ for the defender and $({\mathcal{A}}_{{A}_{a}},{\mathcal{A}}_{{A}_{a}})=({\varphi}_{a},{\varphi}_{b})$ for the attacker of type a and b, respectively.
- (v)
- The expected payoff functions for each player, i.e., $\mathbb{E}\left[{U}_{\mathrm{A}}\right]$ and $\mathbb{E}\left[{U}_{\mathrm{A}}\right]$.
- (vi)
- The belief $\mu $ about the type of the attacker.
- (vii)
- The history ${h}^{t}$ of the game at the t-th round.

#### 4.2. Solution of Game 2 Given Updated Beliefs

**Lemma**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

#### 4.3. Update of Belief

## 5. Simulation Results & Discussion

#### 5.1. One-Shot Game

#### 5.2. Max-Min Solution in the One-Shot Game

#### 5.3. Repeated Game

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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Parameter | Definition |
---|---|

A | attacker |

D | defender |

${s}_{\mathrm{A},i}$ | strategy of the attacker for the i-th host |

${s}_{\mathrm{D},i}$ | strategy of the defender for the i-th host |

${N}_{\mathrm{r}}$ | number of real devices |

${N}_{\mathrm{max}}$ | total number of available hosts |

N | sum of connected real devices and honeypots |

${a}_{i}$ | different terms’ weights of attacker’s payoffs |

${d}_{i}$ | different terms’ weights of defender’s payoffs |

$\theta $ | portion of the number of hosts (N) that are honeypots |

$\varphi $ | portion of the number of hosts (N) that are attacked |

${\varphi}_{\mathrm{m}}$ | the maximum portion of the number of hosts (N) that are attacked |

${U}_{\mathrm{i}}$ | payoff of player i |

$f(\xb7)$,$g(\xb7)$,$\tilde{f}(\xb7)$,$\tilde{g}(\xb7)$ | functions of $(\xb7)$ |

$\mathcal{S}$ | set of players |

${\mathcal{A}}_{i}$ | set of actions for player i |

y, ${N}_{1}$, ${N}_{2}$ | auxiliary variables |

$\mathbb{E}[\xb7]$ | expected value of $[\xb7]$ |

$\mathbb{P}[\xb7]$ | probability of the event $[\xb7]$ |

a, b | the two types of attacker |

${A}_{j}$ | attacker of type j |

${a}_{i,j}$ | weight’s of attacker’s payoff when he is of type $j\in \{a,b\}$ |

${d}_{j}$ | weight’s of attacker’s payoff when he is of type $j\in \{a,b\}$ |

$\mu $ | belief that the attacker is of type a |

${\varphi}_{i}$ | probability of attacking each host for the attacker of type i. |

${\varphi}_{i,\mathrm{m}}$ | maximum value of the probability of attacking each host for the attacker of type i |

$\mathsf{\Omega}$ | states of the nature |

t | round of the game in a repeated game |

${G}_{i}$ | game i |

${h}^{t}$ | history of the game after t-th play |

${(\xb7)}^{*}$ | $(\xb7)$ belongs to the NE |

${C}_{i}$ | cost of under or over estimating the demand of the i-th device |

${f}_{\mathrm{R},i}$ | the probability density function of the actual energy consumption |

${\delta}_{i}$ | the mean energy demand of the i-th device |

${E}_{\mathrm{max}}$ | the maximum energy consumption |

${p}_{\mathrm{uc}}$ | energy price in the unit commitment stage |

${p}_{\mathrm{ed}}$ | energy price in the economic-dispatch stage |

Parameter | Value |
---|---|

${N}_{r}$ | 3 |

${N}_{\mathrm{max}}$ | 10 |

${\varphi}_{\mathrm{max}}$ | 1 |

${a}_{\{1,2,3\}}$ | $[0.76,0.01,0.10]$ |

${d}_{\{1,2,3,4\}}$ | $[0.03,0.40,0.45,0.01]$ |

Random solutions for $\theta $ | 2000 |

Random solutions for $\varphi $ | 2000 |

Parameter | Value |
---|---|

${N}_{r}$ | 3 |

${N}_{\mathrm{max}}$ | 10 |

${\varphi}_{\mathrm{max}}$ | 1 |

${a}_{\{1,2,3\}}$ | $[0.81,0.01,0.06]$ |

${d}_{\{1,2,3,4\}}$ | $[0.31,0.24,0.81,0.14]$ |

Random solutions for $\theta $ | 2000 |

Parameter | Value |
---|---|

Number of rounds | 50 |

${N}_{r}$ | 6 |

${N}_{\mathrm{max}}$ | 8 |

${\varphi}_{a,\mathrm{max}}$ | $0.6$ |

${\varphi}_{b,\mathrm{max}}$ | $0.2$ |

${a}_{a\{1,2,3\}}$ | $[0.48,0.46,0.10]$ |

${a}_{b\{1,2,3\}}$ | $[0.39,0.48,0.02]$ |

${d}_{a\{1,2\}}$ | $[0.70,0.04]$ |

${d}_{b\{1,2\}}$ | $[0.04,0.68]$ |

${d}_{3},{d}_{4}$ | $0.77,0.006$ |

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**MDPI and ACS Style**

Diamantoulakis, P.; Dalamagkas, C.; Radoglou-Grammatikis, P.; Sarigiannidis, P.; Karagiannidis, G.
Game Theoretic Honeypot Deployment in Smart Grid. *Sensors* **2020**, *20*, 4199.
https://doi.org/10.3390/s20154199

**AMA Style**

Diamantoulakis P, Dalamagkas C, Radoglou-Grammatikis P, Sarigiannidis P, Karagiannidis G.
Game Theoretic Honeypot Deployment in Smart Grid. *Sensors*. 2020; 20(15):4199.
https://doi.org/10.3390/s20154199

**Chicago/Turabian Style**

Diamantoulakis, Panagiotis, Christos Dalamagkas, Panagiotis Radoglou-Grammatikis, Panagiotis Sarigiannidis, and George Karagiannidis.
2020. "Game Theoretic Honeypot Deployment in Smart Grid" *Sensors* 20, no. 15: 4199.
https://doi.org/10.3390/s20154199