This setting extends nested GVI models beyond the bilevel case, for which $\mathcal{O}(\delta^{-4})$ complexity bounds are known for any prescribed positive tolerance $\delta$, to a fully three-level hierarchical structure.
We analyze a projected averaged subgradient method combined with a Tikhonov-like regularization scheme.
Under compactness, maximal monotonicity, and a H\”older-type error bound assumption on the bottom-level problem, we derive explicit approximation bounds for the three levels of the hierarchy in terms of corresponding Minty gap functions, prove (subsequential) ergodic convergence, and establish iteration-complexity estimates. In the sharpest regime covered by our analysis, the method achieves complexity of $\mathcal{O}(\delta^{-8})$. We also discuss the underlying error bound condition and provide sufficient conditions for it to hold.