How does one determine the value represented by the ability to choose from many options, and to switch between them over a period of time? 

Intuitively, it makes sense that getting to choose from Choice A or Choice B over a time period t, is more valuable than selecting one of A or B and sticking with it for that time period. But how would one go about calculating that premium?

For example, owning a house presents the optionality of living in it, selling it, or renting it out. Imagine a world where you could move in to that house whenever, rent it out quickly for any period of time (AirBnb?), or sell it instantly (Opendoor?). 

Without having the choice and just selecting some option, you just calculate the value of house over a time period t as max(value(Live, t), value(Rent, t), value(Sell, t)).

What is the premium for having the choice between these options and the ability to switch between them, over the single decision?

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The value of optionality is defined by drawing out the decision tree for the scenarios with and without the option, doing backwards induction for the optimal strategy and estimating the value of each. (In financial option theory, you calculate the price of a literal option by simulating out all of the possible price trajectories and how you would respond to them, to figure out what would be a too cheap or too expensive price.) Because scenarios can be arbitrarily complex, no general answer is possible. If an option wouldn't be used at any state of the world, it might have a value of $0, for example, and this is automatically taken into account: the backwards induction will produce a policy that never invokes the option, and the difference in the value of the 2 scenarios = $0 option value.

For the house scenario, you would, say, define scenarios where each month you can sell/rent/live-in-it and there are random shocks (like Airbnb prices going up/down or housing prices going up/down, I guess), and a horizon of like 10 years and then do backwards induction to understand the value of being able to exploit decreases in Airbnb prices or to shelter in your house from Airbnb price surges.

Well, I’d probably start with some expected value stuff. What's the probability of a scenario, multiplied by the payoff. For housing specifically, I know the govt does a weird calculation for inflation in housing where they ask honeowners what they think they could rent their house for, something like that may help you make a framework for directly comparing