Scope bound prospect preference

Mar 1

An extension of permission function from action space is the decision theoretic preference function. The preference function maps one of each prospects (in our case an action from an action subspace) into a preference value that come from a totally ordered set of possible preference values. In the case of action space, it would appear that each preference function must be invoked with two parameters. $v(a , as)$ v is the value function, $a$ is the action and $as$ gives us a context of all actions being compared(the action subspace of consideration). This will make it easier to define the value function so that it can calibrate its outputs to give meaningful values that are ordered correctly according to true preference.

It is somewhat difficult to come up with an instantaneous decision that alters ordinal utility due to availability of another prospect, with everything else held equal. However one can certainly construct a situation where sequential decision making is available:

Dig for gold
Buy a shovel
Buy a sandwich

If $as$ contained only choices 2 and 3, one is hard pressed to chose 2 over 3. However when $as$ contain all three available actions (for the foreseeable future), suddenly 2 is preferred over 2, and over 1. One would certainly rather buy the shovel, then dig for gold over digging for gold barehand before or after eating a sandwich.

Holding all else equal is important. Some purported change in ordinal preference due to addition of choice is hidden behind other information changes

Eat apple
Don't eat apple
Eat just worm in the apple

Clearly if $as$ contains all three possibilities, $u(2)>u(1)>3u(3)$ . But without option 3, $u(1) > u(2)$ . The truth is that 3 should not affect the ordinal preference between 1 and 2. The knowledge of whether there is a worm in the apple or not is part of all else that we hold equal. Erroneous addition of 3 not only added an action prospect, but also added the knowledge that there is a worm in the apple. So when all else is really held equal, an addition of an action prospect should not affect ordinal preference between existing actions.

Cardinal preference is a different matter. A simple requirement we may impose is: utility value function always return non-negative reals and that sum of value among all choices add to 1. In this case additional prospect changes cardinal value of all actions due to the requirement that values sum to 1.