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What are the consequences of this action? The buyer will have a net gain of (1 − π )S, that is the stake S less the price π S, if θ = 1, or a net gain of −π S, if θ = 0. 1. We are also going to allow for negative stakes. 1 Payoffs for the actions corresponding to buying and selling bets at stake S on event θ at odds π : (1 − π ). 1, reverses the role of the buyer and seller compared to aS,θ . A stake of zero will represent abstaining from the bet. Also, action a−S,θ (selling the bet) has by definition the same payoff as buying a bet on the event θ = 0 at stake S and price π1−θ = 1 − πθ .

A related issue concerns events of probability zero more generally. Shimony (1955), for example, has criticized the coherence condition we discussed in this chapter as too weak, and prefers a stricter version that would not consider it rational to choose a bet whose return is never positive and sometimes negative. This version implies that no possible event can have probability zero—a requirement sometimes referred to a “Cromwell’s Rule” (Lindley 1982a). 3 Coherent conditional probabilities In this section we present a second Dutch Book theorem that applies to coherence of conditional probabilities.

De Montmort, in your letter of September 9, 1713 (problem 5, page 402). For the sake of simplicity I shall assume that A tosses a coin into the air and B commits himself to give A 1 ducat if, at the first throw, the coin falls with its cross upward; 2 if it falls thus only at the second throw, 4 if at the third throw, 8 if at the fourth throw, etc. (Bernoulli 1954, p. 33) What is the fair price of this game? , 2 4 8 that is infinitely many ducats. So a ticket that costs 20 ducats up front, and yields the outcome of this game, has a positive expected payoff.

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A Bayesian analysis of colonic crypt structure and coordinated response to carcinogen exposure incor by Morris J.S.

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