If the agents are considered free in their betting behaviour within the confines of the decision theoretical proofs of the born rule in everettianism; would this not be also analyzed (under their interpretation of decision theory) as the agent having made both bets
so that (1) the world branches according to a bet A world and a bet ~A world and another branching into a A world, and ~A world
(2) should these be considered entangled; or two distinct branching; and if entangled couldnt they simplify their derivation by saying that the world either in on of two ways; (A bet A &, ~A, bet ~A), or [(A bet~A & (~A bet A)); or perhaps both but it already settled that by a previous branching which of the two the agent is in, so that know now that had they bet the opposite way, they would have had the experience of being in the opposite outcome branch, and would achieved necessarily the same result (if they bet A and win, had they bet ~A they would have won) or the other way around.
Of course the argument would have to be rather 'if they founding themselves betting A, rather then betting A; but for all extents and purpose it would appear the same to the agent)- there are no bets which take place, that is prior to the branching but rather the bets and position of the bet occur in the same branching transaction, after branching and as result of the same same branching that results in up or down
(3) if the betting occurs before the branching into spin up and down however, and it branches along both bet options before hand; then its a bit more difficult to make the relevant counterfactual claims; as there are four branches, and the other two corresponding to the other possible bet (one where you win, spin up) and one where you lose spin down) are already distinct and have already branched off; and if you bet both ways or the world branches in either cases, it makes it more difficult to compare them with each other; and one could consider both spin up branches, or the spin up branch bet up as identical in some sense as the spin down branch spin down as they both exist as distinct entities (unless there is some self-worlds transporting bets where these world are correlated).
Either way there will still be four branches, and the state of the entire multiverse branch system will be identical
(3) are the agents already in one of the branching worlds at the moment of betting; I mean if the state is in superposition, would the branching have already occurred and they would already be in one of the worlds; or is the basis or supersposition conditional on betting or some that basis spin x where up and down are equiprobable, being such that it will be selected, but it has not been selected yet; so they have no branched.
Likewise I would have thought that if the everettians think that dominance is equivalent to monotonicity they would have used conditional logic dominance (qualitative dominance) which would not have the state supervenience assumption; of course there are structural or empirical requirement that have to be made; because even if they standard quantitative dominance (and not stochastic or higher utility dominance,or dominance of expected values) nearly amounts to expected value dominance, and is probabilistic in nature
. That is not to say that there is not a non-probabilistic dominance principle and that is qualiative dominance, which is equivalent to monotonicity, if it could be justified; the justification of it, is not that regardless of what the probabilities are, and we dont know them, given they are equal, whatever is the case the expected utility will be greater, buta one shot (pre-probabilistic) dominance conditional logic dominance which looks identical but the reasoning behind it is different. Its what one normally thinks about when one considers dominance, had i committed the other bet, I would have the utility (or have had a better utility). It has expected utility maximization as a consequence ; but the probabilities are trivially the same for both state as a consequence (as opposed to being used as a justification) of causal irrelevence; of a consequence that the truth values of the same state, are necessarily the same under both acts; and so they must have the same probability under both acts
But that is not the justification in the reasoning behind qualitative dominance, the justification is the fact that the truth values are the same on that single case, and not the equiprobability consequence the states, that is a consequence of it (the biconditional does not go in the other direction; the justification of equi-probability of both states does not in itself justify equal truth values, but it does conversely). The justification is that the same state would have on that single case occurred, under a different act; but where one act leads to a better or equal outcomes), and thus i must ascribe, given this, the same degrees/or greater degres of belief or utility to the dominating state); otherwise one contradicts oneself; and not in virtue of the consequence of maximized expected utility.
Whatever justification of these conditional claims would be structural (built into the physics or the metaphysics of chance or amplitude and not of rational degrees of belief) and thus would not be prescriptive in nature (or a rationality constraint); it would not be thus circular in that regard; it would either just be a false claim or a true claim (and I cant see how one would have to be accused of assuming that the rational degrees of belief in both states are teh same in justifying it; as it would not even be a sufficient justification or the right kind of justification in any case; a consequence of , and not a justification of, the conditional claim )
As an example, it can work even if lebesque non-measurable, as a qualitative rule, when not only is that one does not know the numerical probability values, but when the state cannot have numerical probability values; the comparative probabilites would be the same, and if measurable they would be the same as the states are considered logically equivalent under both acts)
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