When Fair Betting Odds Are Not Degrees of Belief
The "Dutch Book" argument, tracing back to Rarnsey (1926) and deFinetti (1974), offers prudential grounds for action in conformity with personal probability. Under several structural assumptions about combinations of stakes (that is, assumptions about the combination of wagers), your betting policy is consistent (coherent) only if your fair-odds are probabilities. The central question posed here is the following one: Besides providing an operational test of coherent betting, does the "Book" argument
also provide for adequate measurement (elicitation) of the agent's degrees of beliefs? That is, are an agent's fair odds also histher personal probabilities for those events?
We argue the answer is "No!" The problem is created by state-dependent utilities. The methods of elicitation proposed by Rarnsey, by deFinetti and by Savage (1954), are inadequate to the challenge of state-dependent values.