self-control and meta-rationality

Ideal theories of rationality assume perfect self-control. Thus, long-term plans obtained by optimizing utility over all possible plans are only good to the extent that I have control over my future self. Can you trust your future self to do the right thing? Or are you like a werewolf, prone to bouts of temporary insanity?

People with compulsions, tics (and to a lesser extent drug addicts) know that there exist shades of grey between the voluntary and the involuntary. Some call these shades "semi-voluntary": you *can* resist your compulsion, but only temporarily, and only by enduring a great deal of psychological pain, i.e. your mind coerces you to do what you don't want to do.

People who want to quit smoking sometimes establish a system by which they are forced to pay a large fine for each cigarette that they smoke. Is this rational? Obviously, if they just wanted to quit cold turkey, and could trust their future self, then they wouldn't bother making the system.

If I know that I may not be rational in the future (or my goals may become different than my present goals), then the realistically optimal thing to do (i.e. "realistically" = given how much self-control I have), may be a plan that would be suboptimal for an ideal agent.

Formalizing this in terms of game theory, your "future self" is a different player, whose goals are sometimes similar to yours, sometimes not; who may or may not be more short-sighted than you. Not trusting your future self implies that the rational strategy is to play it safe, i.e. a maximin strategy.

algebraic mind

There is a cognitive style I like to call algebraic-mindedness. jcreed calls it "analogical mindedness".

I had just explained Landsburg's account of price discrimination in books: hard-cover vs soft-cover. Even though they cost the same to produce, and *everyone* prefers hard-cover, the producer has an interest in making soft-cover books, so that they can charge more from those who are willing to more pay more, while not losing those customers who are not willing to pay more. (This only works because copyright laws gives the producer a monopoly. I predict that ending copyright laws would drastically reduce the number of soft-cover books produced.)

Labour unions are the "dual" of corporate collusion to keep the salaries low (supposing this exists).

So he asked the analogical question: "what is the dual of price discrimination?"

There is no economic reason for asking this question, other than perhaps as a way to find out how to fight back against price discrimination.

The trait of algebraic-mindedness tends to lead one away from the original goal of the investigation, and towards improved understanding of the situation (a tighter knowledge network). The algebraic thinker is interested in general results: when looking at a specific situation, he wants to identify the minimum set of reasons that lead to the observed result. The algebraic thinker is the programmer who, when given a debugging task, insists on refactoring the code first. It is the physicist who will try to resolve paradoxes (philosophical progress) before making "ordinary" progress.

While making one slow at individual tasks, a reasonable amount of algebraic-mindedness tends to pay off on the long term: the algebraic thinker uses his numerous analogical bridges to "see" things that ordinary people can't. Math makes your smarter because it produces chunks of purely logical knowledge, and pure logic is extremely reusable: theorems can be instantiated on anything exhibiting the same mathematical structure. The algebraic thinker is the person who tries to extract the mathematical content out of ordinary situations, thereby creating this general knowledge.

While discussing topic A, algebraic thinkers T are known for bringing up apparently unrelated topics (let's call it B) which they know more about, because (1) A and B are analogically-linked in T's mind (2) T's knowledge of B lets him draw conclusions about A for free (or, in some cases, not for free (if the analogy is incomplete and needs to be verified), but still cheaper).

Questions algebraic thinkers tend to ask:
* if A and B are similar, why does A exhibit property X while B doesn't?

hmm... I'm thinking of splitting algebraic thinking into:
* analogical thinking
* mathematical thinking

The difference between these two kinds of thinking is that analogical thinking is about mappings between specific instances, while mathematical thinking is about mapping between abstract and concrete instances.

The mathematical idea of a situation X corresponds to the equivalence class of all possible situations analogous to X. (This type of analogy is an equivalence relation)

Model Counting: A New Strategy for Obtaining Good Bounds

Carla P. Gomes, Ashish Sabharwal, Bart Selman - Model Counting: A New Strategy for Obtaining Good Bounds tackles the interesting problem of counting the number of assignments satisfying an instance of SAT (the instance is satisfiable if this is >0).

On page 11, they suggest a coin-flipping strategy: each variable assignment flips a coin until there only 1 remaining, and the estimate will be 2^#coin-flips. But they don't explain how one can check that there is only one left. (Also, what if you go from 2 to 0? I imagine that for precision, they probably stop once they get a number < 256... but again, it's not clear how they are counting in the first place)


On page 14, they suggest a way to make assignments flip coins:

Use XOR/parity constraints

...

Which XOR constraint X to use? Choose at random!

Two crucial properties:

* Use XOR/parity constraints
* For every truth assignment A, Pr [ A satisfies X ] = 0.5
* For every two truth assignments A and B, “A satisfies X” and “B satisfies X” are independent

However, what we want is the above probability and independence above to be conditional on A and B being satisfying assignments, otherwise the coin-flips will be biased. I do not see this addressed in the slides.