Bayesian Equilibrium: A Two-Player Stochastic Game

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Formally, a two-player stochastic game is a described as the following set (S; A1 ; A2 ; H; B1 ; B2 ; ρ) where S = {s0, s1 … st… sN } is the state set which is nonempty.
An = {α1n, α 2n … α tn … α Nn} is the action set of player n and α Nn is the action of player “n” at state sN.
The action set for player n at state st is a subset of An, or AnS ⊆ An and ⋃_(t=0)^N▒〖A_(s_t)^n=A^n 〗.
H:S×A^1×A^2×S→[0,1] is the state transition probability
B^n:S×A^1×A^2×S→R is the rewards or benefits function of player n
0 ) 〗=0.
A Perfect Bayesian Equilibrium is a set of strategies and beliefs that at any stage of the game we have an optimal strategy, conditional on the beliefs that are obtained from the game using Bayes rule. Perfect Bayesian equilibrium is always Nash equilibrium but not the other way around. Given the players beliefs, the strategies must be sequentially, i.e. at each
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There are many challenges involved when we have to compute the solution of the game or the equilibrium, because of computational problems, data availability, or practical implementation of the different stochastic games. Although all of the upper mentioned authors dedicated their research on formulating the problem as a strategic game, there are many challenges because of the difficulties in computing the equilibrium.

Sallhammar, et al. [] proposed an approach of integrating reliability and cyber security. They implemented a stochastic game to predict the hacker’s behavior. The basic idea is to evaluate the connection between hacker and administrator as a two-player zero-sum game. They were able to analyze each state and to model the relationship between the set of system states and the set of states in the stochastic game model. Then, they have calculated the transition rates of the Markov model of the network by solving the game

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