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Table 2 Notations and its description of systems

From: Selfish node detection based on hierarchical game theory in IoT

NotationDescription
NThe number of nodes
AkThe set of the actions of the nodes in round k
\( {\boldsymbol{A}}_{\boldsymbol{i}}^{\boldsymbol{k}} \)The action of node i in round k
ukSet of the utility function
\( {\boldsymbol{u}}_{\boldsymbol{i}}^{\boldsymbol{k}} \)The utility function of node i in round k
powiThe highest sending power required by node i
r0The constant value considered as a reward for node i
\( {\boldsymbol{n}}_{\boldsymbol{i}{\boldsymbol{CH}}_{\boldsymbol{i}}}^{\boldsymbol{k}} \)The total number of the received packets from cluster head CHi by node i
\( {\boldsymbol{n}}_{\boldsymbol{Rij}}^{\boldsymbol{k}} \)The total number of the forwarded packets of node j
\( {\boldsymbol{n}}_{\boldsymbol{i}}^{\boldsymbol{k}} \)The number of node i packet in round k
ciThe total energy required for sending the data packets
d(i, j)The distance between node i and node j
piThe probability of node i to run one strategy
PiThe set of probabilities for node i to run all the strategies
siOne strategy of node i
Si= {F, NF}The set of all strategies (forwarding and not forwarding)
pi(si)The probability for every pure strategy of si
(Si)Strategies by node i in a mixed game
\( {\boldsymbol{p}}_{\boldsymbol{i}}^{\ast} \)New probability for node i
∆pThe probability changes rate in each round
\( {\boldsymbol{u}}_{\boldsymbol{i}}\left({\boldsymbol{p}}_{\mathbf{1}}^{\ast},\dots, {\boldsymbol{p}}_{\boldsymbol{n}}^{\ast}\right) \)Payoffs function of node i in new probability
π1(F, NF)The probability distribution for node 1
πi(F, NF)The probability distribution for node i
\( {\boldsymbol{\pi}}_{\boldsymbol{i}}^{\boldsymbol{k}}\left(\boldsymbol{F},\boldsymbol{NF}\right) \)The probability distribution for node i in round k
\( {\boldsymbol{E}}_{{\boldsymbol{n}}_{\mathbf{1}}}\left(\boldsymbol{F}|{\boldsymbol{p}}_{\mathbf{2}}^{\ast},{\boldsymbol{p}}_{\mathbf{3}}^{\ast},\dots, {\boldsymbol{p}}_{\boldsymbol{n}}^{\ast}\right) \)The expectation value for node 1
\( {\boldsymbol{E}}_{{\boldsymbol{n}}_{\boldsymbol{i}}}\left(\boldsymbol{F}\right|{\boldsymbol{p}}_{-\boldsymbol{i}}\Big) \)The expectation value for node i for forwarding the packets
\( {\boldsymbol{E}}_{{\boldsymbol{n}}_{\boldsymbol{i}}}\left(\boldsymbol{NF}\right|{\boldsymbol{p}}_{-\boldsymbol{i}}\Big) \)The expectation value for node i for not forwarding the packets