• Coherence1 for previsions of random variables with generalized betting; • Coherence2 for probability forecasts of events with Brier score penalty; • Coherence3 probability forecasts of events with various proper scoring rules.

Pi(AS) = Pi(A)Pi(S) for i = 1, 2. But the Linear Pool created a group opinion P3 with positive dependence. P3(A|S) > P3(A).

Part 1 Background on de Finetti’s twin criteria of coherence: Coherence1: 2-sided previsions free from dominance through a Book. Coherence2: Forecasts free from dominance under Brier (squared error) score. Part 2 IP theory based on a scoring rule.

We extend de Finetti’s (1974) theory of coherence to apply also to unbounded random variables. We show that for random variables with mandated infinite prevision, such as for the St. Petersburg gamble, coherence precludes indifference between equivalent random quantities. That is, we demonstrate when the prevision of the difference between two such equivalent random variables must be positive. This result conflicts with the usual approach to theories of Subjective Expected Utility, where preference is defined over lotteries. In addition, we explore (...) similar results for unbounded variables when their previsions, though finite, exceed their expected values, as is permitted within de Finetti’s theory. In such cases, the decision maker’s coherent preferences over random quantities is not even a function of probability and utility. One upshot of these findings is to explain further the differences between Savage’s theory (1954), which requires bounded utility for non-simple acts, and de Finetti’s theory, which does not. And it raises a question whether there is a theory that fits between these two. (shrink)