## April 22, 2007

### Response to Vassar on thoughtless equilibrium

Michael Vassar writes in response to my post on rationality as an optimzation for equilibriums that emerge from thoughtless wandering:

So, how should we compare the equilibriums in question if not rationally?

I can’t tell whether the statement “that all rationality is an optimization, which lets us get much faster to the same place we’d end up after sufficiently extended thoughtless wandering of the right sort.” is trivial or trivially wrong, which is probably a bad sign. The statement invokes clear opinions about math, evolution, and computer science, but verbalizing them seems neither easy nor necessary. At any event, one of the major classes of findings that interest me in financial economics are those which refute the idea that a mix of rational and irrational agents necessarily produce an efficient equilibrium, such as

this paper.

The standard neo-classical proofs that a market will produce optimal equilibria require assumptions of unbounded, costless computational power, omniscience, etc. Arrow and Debreu got their Nobel prize for those proofs. Presumably they would have preferred to use weaker, more realistic assumptions but didn’t see how. So proving that we can approximate those equilibria with much weaker thoughtless wandering seems far from trivial.

I’m not sure why Michael thinks this idea could be trivially wrong — the papers I reference seem pretty conclusive. Perhaps the phrase “thoughtless wandering” is too informal, but the papers show that these equilibria can be approximated by populations of entities that have no ability to anticipate consequences or plan, so they’re pretty thoughtless.

Certainly we can always come up with thoughtless wandering of the wrong sort, which will lead to equilibria that don’t optimize the function we want, or perhaps to systems that don’t converge to any equilibrium at all. But this is actually one of the big advantages of viewing rationality as an optimization of thoughtless wandering. It lets us ask specifically what sorts of thoughtless wandering do and don’t approximate the equilibria that we find valuable, or conversely, what interesting failure modes arise in a given class of thoughtless wandering.

The paper by DeLong et al on noise traders that Michael references is a good example of the kind of insights we can gain by stepping back from rationality. Analyzing a simple stochastic regime, the authors show that even in competition with rational traders, noise traders can capture a significant share of wealth in a market, at the cost of most of them going bankrupt. In effect, the small fraction of lucky survivors have been so lucky that they got very, very rich.

However the assumptions the authors have to make indicate the difficulties of this enterprise. Specifically, to make their analysis tractable, the authors assume that these very wealthy noise traders have no effect on prices, even though they dominate the market (!). So we don’t know what noise traders would actually do to the equilibrium. Unfortunately, analyzing this kind of stochastic system is hard — but it is very worthwhile.

Finally, Michael’s question of how we should compare equilibria isn’t answered by any concept of optimality — rational, stochastic, evolutionary, or otherwise. To optimize we always have to specify an objective function, and the objective function is exogenous — it comes from somewhere outside the optimization process itself. Typically in economics the objective function is the (weighted) vector of utilities of all the consumers, for example. Economics doesn’t have any intrinsic way to say that consumers have “irrational” utilities.

Objective functions may be subject to critiques based on internal inconsistencies, observations that other “nearby” objective functions lead to much higher optima on some dimensions, etc. I conjecture that generally these critiques can be understood in the “thoughtless wandering” perspective in terms of the dynamics of the system — it may fail to converge at all if an objective function is inconsistent, for example. Also, while “rationality intensive” neoclassical economic equlibria are very fragile — they don’t hold up under perturbation — the “thoughtless wandering” approximations are much more robust since they are stochastic to begin with, so they are less likely to produce bad results due to small problems with initial conditions.

Filed by Jed at 12:02 am under Economics,Social order

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