# Decision theory is hard

Posted by Yuling Yao on Jun 04, 2021.One mental challenge is decision-making with more than two options. To simplify the dilemma that your humble author is encountering, assume that it is on a long haul flight and you are asked by a friendly cabin crew which of the following dish you would prefer

- chicken tikka masala,
- chicken madras,
- apple pie.

There is a limited supply and you are asked to order your preference, which is not necessarily honored. To be fair, I don’t think these dishes are on any actual menu, but the point of this example is that options (1) and (2) are nearly identical (from your humble author’s point of view).

## selection or mixing

One psychological confusion is the difficulty to distinguish between “selection” and “mixing”. The ability of “ordering your preference” refers to first generate a list of latent preferences and then order them. Assuming I have a sophisticated mind and I automatically self-normalize the latent preferences into $x_1, x_2, x_3$ such that $x_i\geq 0$ and $x_1+ x_2+ x_3=1 $.

But it is not clear how reliable our preference generation ability is. There are two orthogonal approaches to do so. First, one by one. We figure out how much utility we would have when having only chicken tikka masala and so on. Maybe I would prefer chicken overall much better than an apple pie, so I will have $x_1=0.46, x_2=0.44, x_3=0.1$.

Second, we can embed this discrete problem into a larger continuous problem: We imagine there is a tasting menu that mixes these three dishes, and we are considering the optimal mixing proportion. This time, because human generally has a convex utility function, the Jensen’s inequality vividly suggests I should not order two chicken curry dishes simultaneously. Then this optimal weight would inflate the preference on the third item, such as $x_1=0.3, x_2=0.3, x_3=0.4$. That is an order flip.

## sequential decision making

When it comes to sequential decision-making, it is even harder. Instead of an order of the list, we are now asked to bid for a dish one at a time. Also assume my actual preference is 0.5 0.1, 0.4. Because (1) and (2) are alike, my mental process might first distinguish between (1) and (2)— (1) is an easy win. It is like matching, if most coordinates match perfectly, ordering is easy. Then I will process curry dishes and apple pie, in which I might have some struggle: they are just very different two items, and I can typically make up reasons for both of them. But anyway, I find curry better than apple pie after some self-fighting. So I tell the flight crew I will order (1).

But then the flight crew checks the headcount in the kitchen, and dish (1) is sold out. So I am asked to choose between (2) and (3) again.

A good mental process should be consistent in some way: the behavior of

- choosing between (2) and (3) conditioning on (1) being not available
- choosing between (2) and (3) if (1) had not been brought out at all

should be the same. It is like multinomial classification with $K$ categories is equivalent to a $K-1$ binomial classifications. If that is the case, I should pick item (3) for $x_3=0.4$.

Except no, my mental process is often not a martingale. It is natural to be sad when learning (1) is not honored, and that will influence how I make my next stage decision: I might tend to pick (2), just due to its similarity to (1) and this similarity compensates for my disappointment/regretfulness. Is it necessarily irrational? Maybe, but the disappointment is a real feeling, and maximizing the utility of the whole process including the decision-making phase is also a sensible goal.