Okay I have finished Wallace's book. All I will say is I am very thankful that Mandolesi's papers exist, otherwise it would have taken me a long time! Wallace is a very good writer, but for discussing physics the mix of formal proofs with informal discussions is quite head wrecking in some places.
Just to be clear he doesn't exactly use Gleason's versions of non-contextuality or continuity, but closely related ideas (the decision theoretic analogues in a sense). His basic set up is:
(i) A Hilbert Space of microscopic states
(ii) A set of subspaces, ##\mathcal{M}##, which is the set of macrostates. Each ##M \in \mathcal{M}## is a subspace of states that are macroscopically indistinguishable
(iii) A set of events, ##\mathcal{E}##, with ##E \in \mathcal{E}## being a subspace of the Hilbert Space. In short ##\mathcal{M}## are "worlds" and ##\mathcal{E}## are branch structures, where multiple worlds are in superposition.
(iv) A set of rewards. These are coarsenings (although Wallace is never quite clear on this) of ##\mathcal{M}## as you might get the same reward in different worlds.
(v) A set of acts ##\mathcal{U}_{E}##, for each event, that represent bets or experiments.
He then looks at ##>^{\psi}##, a preference ordering on ##\mathcal{U}_{M}##. That is a preference ordering on acts available in a macroworld ##M##, given the microstate is ##\psi##. The idea is to prove that ##>^{\psi}## uses the Born Weights.
He has two sets of axioms. One set that ensures the set of experiments or bets ##\mathcal{U}_{E}## is rich enough without artificial restrictions (for example that it doesn't exclude composition of acts). The other set are conditions on the preference ordering ##>^{\psi}## that supposedly encode an ordering being rational.
He then proves both continuity and non-contextuality of ##>^{\psi}## (non-contextuality is Corolloray 1 on p.19 of Mandolesi's first paper) and then proceeds via a Gleason style argument.
Mandolesi, in his first paper, proves non-contexuality and a sequence of lemma's related to continuity. To be honest, I would read Mandolesi's proof rather than Wallace's, as he has redundant axioms and several points where the proof is not entirely clear. Also Mandolesi uses a simpler ordering of lemmas for the proof.
The end result is that ##>^{\psi}## is given by the Born Rule*.
Mandolesi has two classes of objections. Objections to the axioms and objections to the result. I'll discuss the latter first. Basically just because ordering of preferences for experiments/bets might use the Born Rule, does this imply the Born Rule as normally understood, i.e. would it mean within a given history records are expected to have Born Rule frequencies. One might have a way of acting rationally without this uniquely fixing physical behaviour. This will be the topic of his next paper.
Mandolesi's second paper contains the objections to the axioms**, which I will discuss in detail tomorrow. My basic assessment is that some of Mandolesi's objections are possible just a result of Wallace being unclear, or in need of tightening his statements. However others point to deep inconsistencies in the axioms or require the branching structure to have properties that are difficult to justify. In essence some of his axioms presume that branching has a small amount of the Born rule built in.
* There has been a disagreement about what the Born rule is here, that I haven't had time to read, apologies. I mean the squares of the branch coefficients in this case. I'll read that discussion tomorrow.
** Some of which are already found in Kent and others, but not as clearly or comprehensively.
