What is a gamble worth to you?
A set of probabilities gives a range of expected payoffs. Explore its lower envelope, then see how it determines the prices at which you would buy or sell a gamble.
Your model
The unknown outcome is either A or B.
Choose your credal set
p(B) = 1 − p(A). Every probability in the interval is considered possible.
Set the gamble f
Payoffs are net gains in arbitrary units. A negative payoff is a payment.
Every belief, one expectation
The highlighted segment shows all expectations allowed by your credal set.
Would you trade at this price?
A cautious price, supported by every belief
Buying f for μ gives the net gamble f − μ. For every μ below the lower prevision, this net gamble has strictly positive expectation under every probability in M. The lower prevision is the supremum of these buying prices.
Selling f for μ gives μ − f. Prices above the upper prevision have strictly positive expected net gain under every probability in M. The upper prevision is the infimum of these selling prices.
At an endpoint, acceptance can depend on additional boundary commitments. Positive expectation under every model does not guarantee a positive realized payoff.
Combining gambles can improve the lower price
Keep your current f and combine it with g = (−2, 4). The same credal set evaluates f, g, and f + g.
Try “Mixed payoffs” with “Imprecise”: the least favorable probabilities differ between the two gambles. Then try “Precise” and the inequality becomes equality.
This app defines a coherent lower prevision on all two-outcome gambles using a nonempty closed interval of probabilities. Values are rounded to two decimals. Here the minimum is attained; for more general representations it is written as an infimum.