Thus, it should be proved that the Bayesian conditions is satisfied in MIDG.
Theorem 1 If the MIDG is defined as seven-tuple G in Definition 1, then the Bayesian condition is satisfied in MIDG.
Thus, the MIDG satisfies conditions 1 in Definition 2.
The purpose of MIDG is to formulate the equilibrium solution pi and q*, which can optimize the utility function of IDS.
In this section, the Bayesian equilibrium of MIDG is derived according to the characteristics of security weights in section 3.
Assume that ([P.sub.A], [Q*.sub.D]) is Bayesian equilibrium of MIDG. Noticing that sender S chooses the strategy that guarantees the maximum revenue of the utility function, the attack probability to sensor node i tends to zero in case of [mathematical expression not reproducible].
Considering the resource constraints of the participants, the equilibrium solution of MIDG is computed in three cases.
(1) When the resources of the participants are exhausted in MIDG, combined with formula (12) and (13), the equilibrium solution is calculated as follows:
The Bayesian equilibrium of MIDG is evaluated in multiple cases.
Based on the solution of Bayesian equilibrium ([P.sub.A], [Q*.sub.D]) and the parameters of MIDG, the optimal scheme of IDS in WSNs is designed in this section.
Once there is a timeout, the posterior probability in the tth stage is calculated for MIDG, which can be denoted as [P.sup.(t+1)] ([[theta].sub.S] = 1| [a.sub.S] (t +1)).