Remote Estimation Games with Random Walk Processes: Stackelberg Equilibrium

TL;DR

This work studies remote estimation of two stochastic random-walk processes under strategic sampling with information leakage, formulating a Stackelberg game where a leader commits to a sampling probability and a follower reacts under asymmetric information. By restricting to stationary probabilistic policies, the authors derive closed-form estimation-cost functions $J_1(p_1,p_2)$ and $J_2(p_1,p_2)$ and analyze the Stackelberg equilibrium as a function of $K_1=\frac{2(\alpha_2-\alpha\alpha_1)}{c_1}$ and $K_2=\frac{2(\alpha_1-\alpha\alpha_2)}{c_2}$, with distinct regimes for $K_2\le 0$ and $K_2>0$. They provide explicit SE characterizations, including the notable result that when $K_2>1$ and $K_1>0$, the equilibrium is $(p_1^*,p_2^*)=(0,1)$, illustrating leader-edge commitment can compel follower sampling. Numerical results corroborate the analytical findings and reveal how parameter choices drive equilibria toward sampling or non-sampling strategies, shedding light on privacy-aware remote monitoring in cyber-physical systems.

Abstract

Remote estimation is a crucial element of real time monitoring of a stochastic process. While most of the existing works have concentrated on obtaining optimal sampling strategies, motivated by malicious attacks on cyber-physical systems, we model sensing under surveillance as a game between an attacker and a defender. This introduces strategic elements to conventional remote estimation problems. Additionally, inspired by increasing detection capabilities, we model an element of information leakage for each player. Parameterizing the game in terms of uncertainty on each side, information leakage, and cost of sampling, we consider the Stackelberg Equilibrium (SE) concept where one of the players acts as the leader and the other one as the follower. By focusing our attention on stationary probabilistic sampling policies, we characterize the SE of this game and provide simulations to show the efficacy of our results.

Figures (2)

• Figure 1: The Markov chain where the states represent the timeliness of the information at player $P_1$ with a given probabilistic sampling policy. Here, $q_1$ is equal to $q_1 = 1-(1-p_1)(1-p_2)$.
• Figure 2: The plots of the objective function $J_1(p_1, BR(p_1))$ for (a) $K_2=1.25$ and $K_1>0$, (b) $K_2=0.1$ and $K_1>0$, (c)$K_2>0$ and $K_1=-0.5$, and (d) $K_2>0$ and $K_1 = -2$.