Basically, they extended probability theory to include real numbers which may be negative or greater than 1. This necessitates an unconventional interpretation of probabilities, and for that, they used the "no Dutch book" interpretation of probability theory. For the "no Dutch book" interpretation to even make any sense, they have to introduce cybernetic agents, which they call Information Gathering and Utilizing Systems (IGUS). These agents can make bets, and a Dutch book against them is a collection of bets made with the agent such that individually, the agent would prefer to make the bet, but collectively, the agent will lose a fixed amount no matter what the outcome of the bet is. The condition that no Dutch book can be made against an agent is equivalent to the agent's belief system satisfying the axioms of classical probability theory. An agent with negative probabilities or a probability greater than one is susceptible to Dutch books.
Only bets on outcomes which can be recorded can be settled, but such outcomes aren't susceptible to Dutch books, or so they claim. Otherwise, don't bet. This already brings up a whole host of issues. Do the bets really have to happen, or is it enough to merely imagine them? Or is it enough that the agent is prepared to bet should the need arise? Or does the bet happen at the meta level? Is the agent free to choose whether or not to bet? Besides, an agent which always loses isn't necessarily an inconsistent agent, merely one which fails to satisfies its goals.
An agent can have a belief system not susceptible to Dutch books but is still lousy anyway. Another more knowledgeable agent can exploit their better knowledge to offer a series of bets, which while not guaranteeing a profit all the time, is likely to bring a profit most of the time, and only a loss rarely (whatever "most of the time" and "rarely" really mean).
The authors state that the preferred basis is the basis of the Feynman path integral, but what about S-dualities? Even without S-dualities, a quantum field theory can be described equivalently either as a path integral over field configurations, or a path integral over Feynman diagrams with edges and vertices. Which basis is preferred?
The fine-grained probability distribution can have negative probabilities, but there are coarse-grainings with only non-negative probabilities, and at least one such coarse-graining always exists; the coarse-graining with no resolution. A maximal coarse-graining is one associated with non-negative marginal probabilities for which any extension of it will contain negative probabilities. Such coarse-grainings are referred to as realms. Unfortunately, mutually incompatible realms are the norm, and not the exception.
No problem, they write. There "really" is a "real" fine-grained history, and whichever realm we choose, the "real" history is the one containing the "real" fine-grained history, which is unobservable. We may choose incompatible realms if we so wish, and whichever one we choose, there is a corresponding "real" history. But what is the probability distribution for the "real" fine-grained history? It can be negative or greater than 1! How can something with negative probability happen? So just how real is it anyway?
As they so clearly pointed out, the past history is unobservable. Only present records are observable. But there's always the possibility that the present record is misleadingly deceptive about the past history, assuming an actual past history even exists. When should we choose to measure the records? Now? When is "now"? Why isn't now one second ago or a year from now?
The records are only approximately recorded and approximately decoherent anyway. So, we can in principle make a Dutch book with exponentially small payoff against such agents.
Realms which are "typical" are preferred over realms which are not. Typicality means the log of the coarse-grained probability for the "real" coarse-grained history is much less than the Shannon entropy. But why is typicality preferred?