Why does wanting to maintain indifference to shifting probability mass between (some) trajectories, imply that we care about ex-ante permissibility?
The ex-ante permissible trajectories are the trajectories that the agent lacks any strict preference between. Suppose the permissible trajectories are {A,B,C}. Then, from the agent’s perspective, A isn’t better than B, B isn’t better than A, and so on. The agent considers them all equally choiceworthy. So, the agent doesn’t mind picking any one of them over any other, nor therefore switching from one lottery over them with some distribution to another lottery with some other distribution. The agent doesn’t care whether it gets A versus B, versus an even chance of A or B, versus a one-third chance of A, B, or C.[1]
Suppose we didn’t have multiple permissible options ex-ante. For example, if only A was permissible, then the agent would dislike shifting probability mass away from A and towards B or C—because B and C aren’t among the best options.[2] So that’s why we want multiple ex-ante permissible trajectories: it’s the only way to maintain indifference to shifting probability mass between (those) trajectories.
[I’ll respond to the stuff in your second paragraph under your longer comment.]
The analogous case with complete preferences is clearer: if there are multiple permissible options, the agent must be indifferent between them all (or else the agent would be fine picking a strictly dominated option). So if n options are permissible, then u(xi)=u(xj)∀i,j∈Nn. Assuming expected utility theory, we’ll then of course have ∑ni=1u(xi)p(xi)=∑ni=1u(xi)p′(xi) for any probability functions p,p′. This means the agent is indifferent to shifting probability mass between the permissible options.
The ex-ante permissible trajectories are the trajectories that the agent lacks any strict preference between. Suppose the permissible trajectories are {A,B,C}. Then, from the agent’s perspective, A isn’t better than B, B isn’t better than A, and so on. The agent considers them all equally choiceworthy. So, the agent doesn’t mind picking any one of them over any other, nor therefore switching from one lottery over them with some distribution to another lottery with some other distribution. The agent doesn’t care whether it gets A versus B, versus an even chance of A or B, versus a one-third chance of A, B, or C.[1]
Suppose we didn’t have multiple permissible options ex-ante. For example, if only A was permissible, then the agent would dislike shifting probability mass away from A and towards B or C—because B and C aren’t among the best options.[2] So that’s why we want multiple ex-ante permissible trajectories: it’s the only way to maintain indifference to shifting probability mass between (those) trajectories.
[I’ll respond to the stuff in your second paragraph under your longer comment.]
The analogous case with complete preferences is clearer: if there are multiple permissible options, the agent must be indifferent between them all (or else the agent would be fine picking a strictly dominated option). So if n options are permissible, then u(xi)=u(xj)∀i,j∈Nn. Assuming expected utility theory, we’ll then of course have ∑ni=1u(xi)p(xi)=∑ni=1u(xi)p′(xi) for any probability functions p,p′. This means the agent is indifferent to shifting probability mass between the permissible options.
This is a bit simplified but it should get the point across.