If it were the case that the mind had a big look-up table, specifying for each possible world our degree of belief in that world, then this transformation may not seem to add complexity and would preserve behavioural output. But that is not the way the mind works. My concern is that given the representational framwork we actually use, the operation of off-setting biases by modifying preferences may in fact increase complexity.

My explanation:
A male who sincerely believes “she wants me” sends signals that he is an attractive mate (since successful males are more likely to believe that: they have rationally come to expect success.), and these signals make his success more likely. Faking self-confidence is not easy.
Wanting her more, OTOH, does not make him more likely to succeed to the same degree. It could signal that he’s willing to give her more in exchange, but if he’s a low-status male, it’s likely that he’s incapable of giving her what she wants.

Of course, this bias creates a self-perpetuating “bubble” effect, which is why some seduction literature suggests a “fake it till you make it” approach. If enough people succeeded at this, the signals produced by these beliefs (like “she wants me”) would become unreliable, women might eventually notice (i.e. stop being attracted to such signals), and the biased beliefs causing them might eventually go away. But, until the day when people can exert perfect control over their beliefs, this seems astronomically unlikely.

Alice lives in some world w from the unit line. She has a belief function B(x) for which world she is in. She has a utility of U0(x) for moving left and U1(x) for moving right. If I[f(x)] means the integral of f(x), she’ll move left if

I[B(x)*U0(x)] / I[B(x)*U1(x)] > 1

Now Alice accepts a piece of evidence with a likelihood function E(x). This is learning if the evidence is true, or bias if the evidence is not necessarily true. She will now move left if:

I[B(x)*E(x)*U0(x)] / I[B(x)*E(x)*U1(x)] > 1

Trivially this is the same effect you’d have from multiplying her utility functions by the same likelihood function. Cancellation is usually but not always possible by dividing the utility functions by the likelihood function.

Self deception isn’t just for the young.

Curtis, perhaps you could explain your reasoning by applying it to some simple example, say a hypothetical bias for thinking that objects are green more often then they really are. What is the simple change in preferences that would combine with this bias to produce standard behaviour?

Robin, I’m not sure about the complexity being constant. The complexity of the input-output mapping being computed might be constant, but not necessarily the complexity of the process computing this mapping. It might be like achieving a simple task either in the ordinary way or by means of a Rube Goldberg Machine. (Btw, I’m not suggesting biased belief is in general simpler than accurate belief plus context dependent desire – in fact I think it’s usually the other way around, but with some possible exceptions.)

In a very simple example, the complexity might be the same. Consider a simple agent Alice who lives in the unit interval, and who perfers to be near to one of the endpoints of this interval. Normally, one would think the agent would get some clues about where she is, and then she decides to move in the direction of whichever pole she estimates (unbiased-ly) that she is closest to. Now conside agent Bob. He gets the same clues as Alice but he is biased towards thinking he is near Zero. If we adjust his preferences so that he likes (to the right degree) being close to One even more than he likes being close to Zero, then he will behave just like Alice.

But let’s make the example only a tiny bit more complicated. Let us change Bob’s bias to the following: he now overestimates the relevance of the latest clue he has obtained relative to clues he obtained earlier. How might we change his preferences to ensure that he will still behave the same way as Alice? It seems that we will have to make his preferences time-dependent and extend them to cover facts about what evidence he has obtained. Something like: The strength of his desire to be close to One depends on exactly how strong his most recent clues suggesting that he is near One are; etc. Already this is beginning to get complex. If one adds some futher dimensions to the problem, it would seem that the complexity of the necessary adjustments would quickly spiral out of control.

If this idea is correct, it would suggest that the default design would be accurate beliefs + relatively simple desires. Sometimes it might be useful to bias beliefs, and perhaps to complicate the desires to partially offset the effects of the belief bias. But this would be the exception from the rule – something that would require a special explanation.

The whole business seems obvious to me since I’ve become accustomed to working with betting ratios [Pr(A)/Pr(B)] rather than normalized probabilities directly as they are mathematically equivalent (in the sense that you can normalize out the probabilities from a set of consistent betting ratios at any point without ever having worked them out from earlier betting ratios) but way easier to work with since you dodge renormalization. The fact that they are mathematically equivalent is known and published although it’s apparently ignored since it got caught up in an arcane philosophical debate. If my claim is not obvious on a little reflection I could write up something.

In term of which system gets altered I must immediately point out that both desires and inborn priors seem to vary genetically. It would be very hard to set up a metric of which changes more. On thinking about it, I actually like the model that the two systems are lumped together and the brain just keeps track of the expected utility for each outcome [Pr(A)*U(A)], bumps it up and down based on either probability or utility estimate changes, and makes decisions by comparing expected utility. I figure evolution hates renormalizing probabilities as much as I do.