Table 2 Notations and its description of systems

N

The number of nodes

A^k

The set of the actions of the nodes in round k

\( {\boldsymbol{A}}_{\boldsymbol{i}}^{\boldsymbol{k}} \)

The action of node i in round k

u^k

Set of the utility function

\( {\boldsymbol{u}}_{\boldsymbol{i}}^{\boldsymbol{k}} \)

The utility function of node i in round k

pow_i

The highest sending power required by node i

r₀

The 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 CH_i 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

c_i

The total energy required for sending the data packets

d(i, j)

The distance between node i and node j

p_i

The probability of node i to run one strategy

P_i

The set of probabilities for node i to run all the strategies

s_i

One strategy of node i

S_i = {F, NF}

The set of all strategies (forwarding and not forwarding)

p_i(s_i)

The probability for every pure strategy of s_i

∆(S_i)

Strategies by node i in a mixed game

\( {\boldsymbol{p}}_{\boldsymbol{i}}^{\ast} \)

New probability for node i

∆p

The 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

π_i(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