Dutch Book Arguments and Imprecise Probabilities

I have an urn that contains 100 marbles. 30 of those marbles are red. The remainder are yellow. What sort of bets would you be willing to make on the outcome of the next marble drawn from the urn? What odds would you accept on the event “the next marble will be yellow?”. A reasonable punter should be willing to accept any betting quotient up to 0.7. I define “betting quotient” as the ratio of the stake to the total winnings. That is the punter should accept a bet that, for an outlay of 70 cents, guarantees a return of £1 if the next marble is yellow. And the punter should obviously accept bets that cost less for the same return, but what we are really interested in is the most the punter would pay for a bet on an event. I am making some standard simplifying assumptions here: agents are risk neutral and have utility linear with money; the world of propositions contemplated is finite. The first assumption means that expected monetary gain is a good proxy for expected utility gain and that maximising monetary gain is the agents’ sole purpose. The second assumption is made for mathematical convenience. Now consider a similar case. This case is due originally to Daniel Ellsberg;1 this is a slightly modified version of it.2 My urn still contains 100 marbles, 30 of them red. But now the remainder are either yellow or blue, in some unknown proportion. Is it rational to accept bets on Yellow at 0.7? Presumably not, but what is the highest betting quotient the punter should find acceptable? Well, you might say, there are 70 marbles that could be yellow or blue; his evidence is symmetric so he should split the difference:3 a reasonable punter’s limiting betting quotient should be 0.35. Likewise for Blue. His limiting betting quotient for Red should be 0.3. What this suggests is that this punter considers Yellow more likely than Red, since he’s willing to paymore for a bet on it. So, as a corollary, he should