I think you're confusing yourself. Watch something like Naom Brown's NIPS 2017 lecture on subgame solving in imperfect information games and you'll realize you should be very wary of someone who is trying to play around with solutions to imperfect information games which use state based reasoning - not information set based reasoning.
If you have a tree:
0.5
/ \
A 2A
In a perfect information game you can do this thing where your like "assume a" and think ahead and then "assume 2a" and think ahead. But in the the imperfect information game world you don't know whether you have a or 2a. So you have to "assume a 2a" and you have to deal with your solution space using counterfactual reasoning. Picture a chain coming out from the A to the 2A. They are married. They cannot be parted.
It is really counterintuitive, but the way I think about it is "you are playing every game at once" and "you are playing a game over information sets and strategy choices" not "state space and actions" so since you are playing every game at once already when they come in with that other game - well, you were already playing another game. They're overwriting that part of your solutions memory. Which in math terms declares an equality. In that case that equality is 2=1 because they say A = A/2 or that 2A=A depending on which framing you use to build the real game tree.
The actual recurrence relationship is obviously
R(KEEP) = {0.5: A, 0.5: 2A} = 3/2A.
R(SWITCH) = P(KEEP)*R(KEEP) + P(SWITCH)*R(SWITCH)
Solving the game from there is really trivial - you're only allowed one strategy selection per information set and you only one set the null set. You literally get to make only one decision as to a scalar between 0 and 1 to define P(KEEP) and P(SWITCH) which is going to be 1-P(KEEP).
`R(SWITCH)` becomes `R(KEEP)` because of the way the markov chain drains into it. Literally the only thing you can't pick is `P(SWITCH)=1`. Everything else is an equivalent EV solution. P(SWITCH) <= P(KEEP) because there exists one P(SWITCH=1) is undefined or else 0 (in practice we take the limit after a horizon by multiplying by some term like 0.9999999 to avoid the infinite regress).
And since the problem didn't make this clear, your policy __influences your expected value__ and so their reasoning process is just so broken at a fundamental level. They literally declared 2=1 and therefore undefined solution is the best answer despite infinite solutions that aren't undefined. Its... really bad reasoning. Don't listen to the idea that switching is better. It can't be. Look at the actual relationship. Look at the actual graph. They skip some really important steps, like using the same gametree for the entire calculation.