In the generic case one would rather expect multiple fixed points, with no room for anything to influence, even probabilistically, which would be realized.
Systems were probabilistically almost always close to equilibrium.
Perhaps there is always a way to contrive decision models such that acts are intuitively probabilistically independent of states.
For one thing, in many real-world decision circumstances, it is hard to frame the decision model in such a way that states are intuitively probabilistically independent of acts.
Moreover, now we see that one of Savage's rationality constraints on preference the Sure Thing Principle is plausible only if the modelled acts are probabilistically independent of the states.
Thus, the true hypothesis hi probabilistically implies that as the amount of evidence, n, increases, it becomes highly likely (as close to 1 as you please) that one of the outcome sequences en will occur that yields a likelihood ratio P[en | hj·b·cn] / P[en | hi·b·cn] less than ε; and this holds for any specific value of ε you may choose.
For, likelihoods represent the empirical content of a hypothesis, what the hypothesis (together with background b) probabilistically implies about the evidence.
Whereas scientist α takes theory h1 to probabilistically imply that event e is highly likely, his colleague β understands the empirical import of h1 to say that e is very unlikely.
When this equality holds the individual bits of evidence are said to be probabilistically independent on the hypothesis.
My decision problem can be captured with the following matrix: Expected utility theory says that it is permissible for me to accept the deal accepting has expected utility of 0. (This is so on both the Jeffrey definition and the Savage definition, if we assume that how the dart lands is probabilistically independent of how you bet).
(We can expect Jeffrey's definition to agree with Savage's on the plausible assumption that, given the evidence in our possession, the hypothesis is probabilistically independent of whether we accept or reject it).
