My use of ‘next’ need not be read temporally, though it could be. You might simply want to define a transitive preference relation for the agent over {A,A+,B,B+} in order to predict what it would choose in an arbitrary static decision problem. Only the incomplete one I described works no matter what the decision problem ends up being.
As a general point, you can always look at a decision ex post and back out different ways to rationalise it. The nontrivial task is here prediction, using features of the agent.
If we want an example of sequential choice using decision trees (rather than repeated ‘de novo’ choice through e.g. unawareness), it’ll be a bit more cumbersome but here goes.
Intuitively, suppose the agent first picks from {A,B+} and then, in addition, from {A+,B}. It ends up with two elements from {A,A+,B,B+}. Stated within the framework:
The set of possible prospects is X = {A,A+B,B+}×{A,A+B,B+}, where elements are pairs.
There’s a tree where, at node 1, the agent picks among paths labeled A and B+.
If A is picked, then at the next node, the agent picks from terminal prospects {(A,A+),(A,B)}. And analogously if path B+ is picked.
The agent has appropriately separable preferences: (x,y) (x’,y’) iff x x″ and y y″ for some permutation (x″,y″) of (x’y’), where is a relation over components.
Then (A+,x) (A,x) while (A,x) (B,x) for any prospect component x, and so on for other comparisons. This is how separability makes it easy to say “A+ is preferred to A” even though preferences are defined over pairs in this case. I.e., we can construct over pairs out of some over components.
In this tree, the available prospects from the outset are (A,A+), (A,B), (B+,A+), (B+,B).
Using the same as before, the (dynamically) maximal ones are (A,A+), (B+,A+), (B+,B).
But what if, instead of positing incomparability between A and B+ we instead said the agent was indifferent? By transitivity, we’d infer A B and thus A+ B. But then (B+,B) wouldn’t be maximal. We’d incorrectly rule out the possibility that the agent goes for (B+,B).
Nice! The standard method would be the BDM mechanism, and what’s neat is that it works even if people aren’t risk-neutral in money. Here’s the idea.
Have people report their disvalue Y. Independently draw a number z at random, e.g. from an exponential distribution with expectation 1/λ=X. Then offer to pay them z iff z≥Y.
Now it’s a dominant strategy to report your disvalue truthfully: reporting above your disvalue can never raise what you’re offered but it can risk eliminating a profitable trade, while reporting below it either makes no difference or gives you an offer you wouldn’t want to take.
This is the exact same logic behind the second-price auction.