01 — CONSISTENCY OF BUYING PRICES

No sure loss.
But is it coherent?

A lower prevision is your supremum acceptable buying price for a gamble. Compare one-sided buying commitments with precise, two-sided prices. Set four prices, then discover whether they fit together.

Set your buying prices

Three mutually exclusive outcomes: A, B, C. Each ticket pays €1 if its event occurs, and €0 otherwise.

Buying one unit of f at price q produces the net payoff f(ω) − q. Nonnegative stakes mean you may buy any number of tickets.

THE PROBABILITY VIEW

Which distributions respect your prices?

Credal set M: all dominating distributions

Each point is a distribution (p_A, p_B, p_C). The shaded region satisfies E_p[f] ≥ P̲(f) for every assessed gamble.

THE GAMBLE VIEW

Try a portfolio

For coherence, also sell one unit of:

The search resets the sale to “Nothing” and normalizes total buying stakes to 1. A negative payoff in every state certifies sure loss.

THE PRICE VIEW

Coherence means every price is tight

Natural extension is the smallest expectation over M. A gap reveals a buying price already implied by your other commitments.

Why these are different requirements

Avoiding sure loss: every nonnegative buying portfolio has a nonnegative payoff in at least one state.

max_ω ∑ λᵢ[fᵢ(ω) − P̲(fᵢ)] ≥ 0
for all λᵢ ≥ 0

Coherence: even when you sell one assessed gamble at its quoted price, buying other gambles cannot force a strictly negative payoff in every state.

On this finite domain, coherence is equivalent to P̲(f) = min_p∈M E_p[f] for each assessed f. Avoiding sure loss only requires M ≠ ∅.

These prices are limiting buying prices. Strictly acceptable transactions can use prices just below them; strict sure-loss certificates persist under sufficiently small perturbations.

A two-minute experiment

Start with “Avoids sure loss only”: A and B each cost 0.30, but A ∪ B costs only 0.40. Buying A and B implies a price of 0.60 for their sum. Selling A ∪ B at 0.40 while buying A and B loses 0.20 in every state. This violates coherence, even though buying alone cannot force a sure loss.