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Worst-case risk dominates Expected Shortfall at every depth

Proved
CoherentRisk.es_le_worstCase

by viratkota · Sep 6, 2026 · Mathlib 0df444a (Lean v4.33.1)

operations-researchprobability

RETIRED. This statement is subsumed by CoherentRisk.es_antitone, which proves that Expected Shortfall is decreasing in the depth for every pair of depths. Taking the shallower depth to be 000 and combining with CoherentRisk.es_zero_eq_worstCase, which identifies Expected Shortfall at depth 000 with the worst-case measure, recovers exactly the bound below. It is retired as redundant rather than as incorrect: the statement is true and its proof stands. Prefer CoherentRisk.es_antitone for new work.


At every depth mmm, Expected Shortfall reports at most as much risk as the worst-case measure. Averaging the losses over a group of m+1m+1m+1 bad states can never exceed the single largest loss, so worstCase is the most conservative member of the family.

When this was published, full monotonicity in the depth was not yet proved, and the statement recorded explicitly that it was the weaker claim. That gap has since been closed by CoherentRisk.es_antitone.

As throughout CoherentRisk, the development counts states and refers to no probability measure, so mmm is an integer depth rather than a confidence level.

Preamble
import Definitions.Def_ExpectedShortfall

open CoherentRisk
Formal statement
namespace CoherentRisk

theorem es_le_worstCase {n : ℕ} (m : Fin (n+1)) (X : Fin (n+1) → ℝ) :
    ES X m ≤ worstCase X := by
  sorry

end CoherentRisk
Source
P. Artzner, F. Delbaen, J.-M. Eber and D. Heath, Coherent Measures of Risk, Mathematical Finance 9 (1999) 203-228, Section 3, on the worst-case measure as the most conservative coherent measure

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