I am reading a book about rational decision making [1]
There is an example of transitivity rule:
If you prefer fish to burgers and burgers to pudding, you should also prefer fish to pudding. However, we do not always adhere to this principle, as shown in the following example [...]
Anyway I am perplexed, because I suspect that transitivity rule can be applied to choices that map outcomes belonging to the same set of conditions (like, I gotta choose between two type of foods) but:
is transitivity still valid to describe emergent behaviour in complex systems ?
(like, I may like a recipe, result of the combination of two ingredients, in a different way that what i expected) ?
Consider this example:
If I like rice more than peas, and peas more than zucchini, a rational decision according to transitivity rule would be that I choose rice over zucchini.
In formal language, I would write:
a > b, b > c => a > c
Then, one should say that I choose rice with zucchini over rice with peas.
In formal language, I think that could equate to:
a > b, b > c => a + e > c + e
OR (interpreting "rice with zucchini" as "rise and zucchini" , to describe a "risotto" :
ae > ce
But that may be wrong:
indeed, in spite of the above conditions, I may choose "rise and peas" better than "rise and zucchini".
So, what is going on here ?
Is it a nuance of human language that I failed to interpret ? Or is it a case of non linearity ?
I thought to model this example as:
a > b, b > c (rice > peas, peas > zucchini)
I can then map a function to conditions, like:
ab => e (e is a "risotto" of rice and peas, that is a new element non in the set of conditions) ac => f (f is a "risotto" of rice and zucchini, another element non belonging to the set of conditions)
And I can have:
a + c > a + b (As distinctive elements, I like rice and zucchini better than rice and peas)
But I can have a contraddiction in that, if two elements "affects each other" (or in probabilistic terms, one is influenced by the other) :
ac > ab (I still rice and zucchini better that rice and peas)
but:
e > f (I prefer rice and peas better that rice and zucchini : I may choose a consequence that emerged from the interaction of the two ingredients)
Could you please help in understanding better how to apply transitivity logic to this mundane example ?
Is it correct to say transition rule may only apply to linear problems, that maps a set of consequences in the same set of conditions (E.g. choices between ingredients as combinations of single elements not influencing each other) ?
How to deal with emergence of complexity, where outcomes may emerge from a combination of elements and belong to a "wider" set (e.g. like a risotto is more that just rice + zucchini) ?
Also, kindly give a feedback if the model I made to represent this mundane example flaws, particularly in concluding that there is a contradiction ?
[1] Franz Eisenführ, Martin Weber, Thomas Langer - Rational Decision Making (2010, Springer) - https://www.springer.com/gp/book/9783642028502