A Game-Theoretic Analysis of Bandwidth Allocation under a User-Grouping Constraint

Algorithm 1

The scheme to locate the user-grouping NEP.

(P1) Initiate the algorithm with a feasible 𝐫_𝑔 and set 𝑠=0.

(P2) (if current step is 𝑘) Perform a batch update for users in ℳ at step 𝑘+1

  by assigning 𝑟_𝑗^𝑘+1=𝑟^𝑘_𝑗  for all 𝑗∈𝒩⧵ℳ,

  to reach 𝑟_𝑖^𝑘+1=𝒜_𝑖𝑇_𝑖^𝑘+1,𝑇_𝑖^𝑔,𝑘+1  for all 𝑖∈ℳ.

(P3) (if current step is 𝑘) Perform an individual update at user 𝑖 in 𝒩⧵ℳ

  by assigning 𝑟𝑘_𝑗+1=𝑟𝑘_𝑗  for all 𝑗∈𝒩⧵{𝑖},

  to reach 𝑟𝑘_𝑖+1=𝐴_𝑖(𝑇𝑘_𝑖), and let 𝑘=𝑘+1.

  Sequentially implement this whole procedure till each user in 𝒩⧵ℳ

  is updated.

(P4) Repeat (P3) to complete a batch update at users in 𝒩⧵ℳ.

(P5) Set 𝑠=𝑠+1 and record the current 𝐫_𝑔 as ℛ(𝑠).

  If ‖ℛ(𝑠)−ℛ(𝑠−1)‖<𝜀 then stops, otherwise go to (P2).