Broaden conditions guaranteeing sequential price equality beyond current assumptions

Prove general sufficient conditions for price equality in sequential gamble spaces—specifically, conditions under which the sequentially consistent upper expectation equals the (restricted) game-theoretic upper expectation (equivalently, minimax duality holds)—even in cases where the current chain of price inequalities breaks down, such as when sequentially consistent and globally consistent measures do not coincide or when additional restrictions on gambles (e.g., bb(Z_∞) ∪ ⋃_{t∈N} Z_t) are not imposed.

Background

The paper develops a chain of sequential price inequalities and shows equality under specific assumptions (disintegrability, upward scalability, sequential normality, and a particular enlargement of the gamble set Z′ = bb(Z∞) ∪ ⋃_{t∈N} Z_t). However, examples demonstrate that sequential consistency need not imply global consistency, and the chain can fail even though price equality for the quantities of interest may still hold.

The author explicitly identifies the need for broader, more robust conditions ensuring equality of prices—hence minimax duality—in settings where these alignments are currently not guaranteed, calling this an interesting open question.