Characterize the optimal δ–CVaR competitive ratio in discrete-time ski rental for all δ and buying costs B

Characterize the optimal δ–CVaR competitive ratio α_δ^{B,*} for the discrete-time ski rental problem with integer buying cost B ≥ 2 across the full range of CVaR levels δ ∈ [0,1]. Determine the exact piecewise functional dependence of α_δ^{B,*} on δ and B and precisely delineate the regimes and intervals in δ for which different expressions apply.

Background

The paper introduces the δ–CVaR competitive ratio (DCR) for risk-sensitive analysis of online algorithms and studies it in discrete-time ski rental. It proves a phase transition at δ = 1 − Θ(1/log B), beyond which the deterministic strategy is optimal, and gives an analytic optimal algorithm and ratio for a small-δ regime (δ = Θ(1/B)).

Despite these results, the authors note that for larger δ the optimal DCR is expected to be a piecewise function in δ whose structure depends on B, and a complete characterization over all δ and general B remains unresolved. This motivates fully determining α_δ^{B,*} over the entire parameter space and identifying the intervals and expressions governing each regime.