Utility Functions and Limited Rationality

Brian Caplan writes "I agree that we do not 'need to examine' whether people optimize. Treat it as a tautology, I won't demur." Well, I mostly disagree; it might be a tautology in the limit of arbitrarily brilliant rational actors, but in the world of limited computational resources, it seems to depend on design tradeoffs.

There's no particular reason that it there needs to be a utility function behind people's decision patterns. Mathematically there's no reason that a directed field must correspond to the gradient of a potential field, and a utility function seems to be entirely analogous to a potential. I don't think this is entirely a mathematical curiosity, either; in real systems that are simple enough for us to understand (not people, but simple-minded agents working toward some goal) sometimes there isn't. Thinking of the actions that people take in any situation as a potential gradient in some space...the potential may not exist. Particular odd consequences like Dutch books may seem uncommon, but many milder odd behaviors could be common.

Moreover, although I will grant that a supremely rational idealized being would tend to have decision patterns consistent with a potential function, real actors in the real world have limited brainpower. Given the limitations on how expensive it is to compute things, even for brains, just because a behavior would be optimal if computation were perfect and free doesn't mean that we can deduce that it must be present in successful organisms.

One example that I know more than my share about is in the construction of software to play games, especially the game of Go. In some games, like Chess, the most successful programs often embrace the idea of a utility-like function wholeheartedly, and deviations from utility-function-driven behavior are relatively subtle. But in Go, many successful programs deviate fairly strongly from actions which can be described by a gradient of a potential. I can illustrate this with a very simple example that David Fotland (the author of a strong Go-playing program) has used to illustrate an unrelated point. (Fotland's point is that tactics are overwhelmingly important in the game, in a way that people, including him, usually take a while to realize, as they begin by puttering around with grand strategy. His example is a trivially simple program which can tear a clever program to shreds, even if the programmer put many man-months into the usual big-picture stuff like "influence functions" that programmers are usually drawn to initially, if the clever program doesn't pay some serious attention to tactics.)

    1) If you have a single stone, one liberty 
       group, add a stone to it.
    2) Among opponent blocks without two eyes, 
       find the one with the least number of 
       liberties, and fill the liberty with the 
       most second order liberties.  If there is 
       a tie, pick one at random.
    3) Make a random move that doesn't fill one 
       of your own eyes.
    4) Pass.

Note that in this algorithm no potential appears; it's a set of simple rules like "if the hand is painfully hot, yank it back" rather some sophisticated combination of a utility function "I don't like being burned" combined with a world model where the probability of burningly hot things diminishes strongly as one retreats toward the body. In fact, not only does no potential appear explicitly, I believe that it is impossible to construct a potential such that this algorithm's behavior follows from the gradient of the potential.

(Why do I believe that no potential exists? Well, everything involving "eyes" is a mess to try to prove things about, so I could be in error because of that. But if we consider a simplified algorithm "if you have a single stone one-liberty group, add a stone to it; otherwise, make a random move" then it seems easy to show that no potential exists, and intuitively, I'd expect that the difficulty would continue in the full algorithm. In the simplified algorithm, on a one-dimensional board with stones represented as "1" in binary numbers, from 10000 we could choose moves to give 11000, 10100, 10010, or 10001, but we prefer 11000, so a 11000 outcome would need to have higher utility than 10100. From 01000 we can choose moves to give 11000, 01100, 01010, or 01001, and we choose randomly, so they must all have equal utility. From 00100 we can choose moves to give 10100, 01100, 00110, or 00101, and again we choose randomly, so they must have equal utility. But then 11000 would need higher utility than 10100, 11000 would need equal utility to 01100, and 01100 would need equal utility to 10100, which is impossible.)

A serious program to play a good game of Go is much more complicated than that, of course, and only a few of them have revealed much about how they work. But the successful programs that I know of are not particularly well described by utility functions. Their success can be interpreted as something like rationality, so I think the lack of utility functions is an interesting datapoint.

Outside the field of computer Go, to what extent a model of the world, and/or a utility function on it, is necessarily involved in "rational" behavior (in real robots and organisms, as opposed to philosopher kings) seems to be something of an open question in general. I have seen bits and pieces of the debate elsewhere, although choosing good moves in games is the only bit that I know very much about.

And, to move away from my narrow computer geekery back to broad, renaissance economics geekery, I wonder whether in markets something else might kick in, a sort of central limit theorem. That is, even when rationality and utility functions are a fairly poor approximation, if you lump a bunch of actors together "properly" in a market, the market might tend to act like a collection of considerably more rational actors. This would be a central limit theorem analogous to the usual theorem of statistics, where a collection of not-too-skewed random variables combined properly acts very much like a collection of unskewed Gaussian variables. For the real central limit theorem, the meaning of "properly" is well-understood. For this hypothetical conjectured limit, I'm not sure quite how to state the analogous conditions. "Not-too-irrational actors?" And "independent actors?" Are actors still usefully independent if they do the typical human thing of watching the price and adjusting their behavior based on guessed trends? But it seems as though there should be something there -- quite possibly something in the economics literature that I just haven't heard of.:-| I'm also not sure how to capture the time dependence in markets where actors are allowed to get richer by being rational, so that presently the really-irrational actors can be marginalized by the now-wealthy very-rational actors.

Also I will veer back into computer and math geekery for a final remark. This central-limit-ish behavior, where ensembles of many idiosyncratic individuals acting much like ensembles of less-interesting individuals, also arises in combinatorial game theory. In CGT, the analysis of games gives a "mean" score somewhat analogous to a mean of a random variable, and then an arbitrarily messy description of all the other hairiness around that. When you combine many games into one game, the means add up, and the other messiness tends to get washed out, because it adds up much less simply constructively than the means do. (Furthermore, the way to combine games to get this behavior is a bit reminiscent of an open market: in effect you set all the subgames side-by-side on a long table, then on any turn in the combined supergame you get to make only one move in only one subgame, and you choose the subgame which is most attractive to you.)


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This page contains a single entry by published on April 19, 2005 6:43 PM.

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