Peter Boettke wrote:Analytically, the biggest difference between the Austrians and their mainstream brethren is a focus on processes of adjustment and changing conditions, as opposed to static or equilibrium states of affairs. In a supply and demand curve, a standard economist would focus on the price and quantity vector that would clear the market. The Austrians want to talk about all the exchanges and activity that take place that results in that vector being discovered and the market being cleared.

Peter Boettke wrote:Most standard economics assumes that the relationships we are trying to understand can be captured by a continuous function that’s smooth and twice differentiable. What the Austrian analytics suggests is that life is not actually a continuous and smooth function that’s twice differentiable, but instead a lumpy function, a discrete function, in which there are all kinds of difficulties in the ability for us to model them the way our standard approach does. So, instead, what we engage in is discursive reasoning.

Peter Boettke wrote:...Austrians want to talk about the institutional environment within which economic activity takes place. They want to talk about cultural frames of reference that form the priors that rational actors have. They want to talk about the fact that we each have different priors, because we’re diverse individuals who have different perspectives on the world.

Joshua Epstein wrote:4.9. Incompleteness (attainability at all) and complexity (attainability on time scales of interest) in social science

As background, in mathematical logic, there is a fundamental distinction between a statement’s being true and its being provable. I believe that in mathematical social science there is an analogous and equally fundamental distinction between a state of the system (e.g., a strategy distribution) being an equilibrium and its being attainable (generable). I would like to discuss, therefore, the parallel between the following two questions: (1) Is every true statement provable? and (2) Is every equilibrium state attainable?

In general, we are interested in the distinction between satisfaction of some criterion (like being true, or being an equilibrium) and generability (like being provable through repeated application of inference rules, or being attainable through repeated application of agent behavioral rules). Now, mathematico-logical systems in which every truth is provable are called complete.8 The great mathematician David Hilbert, and most mathematicians at the turn of the Twentieth Century, had assumed that all mathematical systems of interest were complete, that all truths statable in those systems were also provable in them (i.e., were deducible from the system’s axioms via the system’s inference rules). A major objective of the so-called Hilbert Programme for mathematics was to prove precisely this. It came as a tremendous shock when, in 1931, Kurt Godel proved precisely the opposite: all sufficiently rich9 mathematical systems are incomplete. In all such systems, there are true statements that are unprovable! Indeed, he showed that there were true statements that were neither provable nor refutable in the relevant systems—they were undecidable10 [See Godel (1931), Smullyan (1992), Hamilton (1988)].

Now, truth is a special criterion that a logical formula may satisfy. For example, given an arbitrary formula of the sentential calculus, its truth (i.e., its tautologicity) can be evaluated mechanically, using truth tables. Provability, by contrast, is a special type of generability. A formula is provable if, beginning with a distinguished set of “starting statements” called axioms, it can be ground out—attained, if you will–by repeated application of the system’s rule(s) of inference.

Equilibrium (Nash equilibrium, for example) is strictly analogous to truth: it too is a criterion that a state (a strategy distribution) may satisfy. And the Nash “equilibriumness” of a strategy configuration (just like the truth of a sentential calculus formula) can be checked mechanically.

I venture to say that most contemporary social scientists—analogous to the Hilbertians of the 1920s—assume that if a social configuration is a Nash equilibrium, then it must also be attainable. In short, the implicit assumption in contemporary social science is that these systems are complete.

However, we are finding that this is not the case. Epstein and Hammond (2002) offer a simple agent-based game almost all of whose equilibria are unattainable outright. More mathematically sophisticated examples of incompleteness include Prasad’s result, basedon the unsolvability of Hilbert’s 10th problem:

For n-player games with polynomial utility functions and natural number strategy sets the problem of finding an equilibrium is not computable. There does not exist an algorithm which will decide, for any such game, whether it has an equilibrium or not ... When the class of games is specified by a finite set of players, whose choice sets are natural numbers, and payoffs are given by polynomial functions, the problem of devising a procedure which computes Nash equilibria is unsolvable. [Prasad (1997)]

Other examples of uncomputable (existent) equilibria include Foster and Young (2001), Lewis (1985, 1992a, 1992b), and Nachbar (1997). Some equilibria are unattainable outright.

A separate issue in principle, but one of great practical significance, is whether attainable equilibria can be attained on time scales of interest to humans. Here, too, we are finding models in which the waiting time to (attainable) equilibria scales exponentially in some core variable. In the agent-based model of economic classes of Axtell et al. (2001), we find that the waiting time to equilibrium is exponential in both the number of agents and the memory length per agent, and is astronomical when the first exceeds 100 and the latter 10. Likewise, the number of time steps (rounds of play) required to reach the attainable equilibria of the Epstein and Hammond (2002) model was shown
to grow exponentially in the number of agents.

One wonders how the core concerns and history of economics would have developed if, instead of being inspired by continuum physics and the work of Lagrange and Hamilton [see Mirowski (1989)]—blissfully unconcerned as it is with effective computability—it had been founded on Turing. Finitistic issues of computability, learnability, attainment of equilibrium (rather than mere existence), problem complexity, and undecidability, would then have been central from the start. Their foundational importance is only now being recognized. As Duncan Foley summarizes,

The theory of computability and computational complexity suggest that there are two inherent limitations to the rational choice paradigm. One limitation stems from the possibility that the agent’s problem is in fact undecidable, so that no computational procedure exists which for all inputs will give her the needed answer in finite time. A second limitation is posed by computational complexity in that even if her problem is decidable, the computational cost of solving it may in many situations be so large as to overwhelm any possible gains from the optimal choice of action. [See Albin (1998).]

For fundamental statements, see Simon (1982, 1987), Hahn (1991), and Arrow (1987). Of course, beyond these formal limits on canonical rationality, there is the body of evidence from psychology and laboratory behavioral economics that homo sapiens just doesn’t behave (in his decision-making) like homo economicus.

Now, the mere fact that an idealization (e.g., homo economicus) is not accurate in detail is not grounds for its dismissal. To say that a theory should be dismissed because it is “wrong” is vulgar. Theories are idealizations. There are no frictionless planes, ideal gases, or point masses. But these are useful idealizations in physics. However, in social science, it is appropriate to ask whether the idealization of individual rationality in fact illuminates more than it obscures. By empirical lights, that is quite clearly in doubt.

This brings us to the issue of generality. The entire rational choice project, if you will, is challenged by (1) incompleteness and outright uncomputability, by (2) computational complexity (even of computable equilibria), and by (3) powerful psychological evidence of framing effects and myriad other systematic human departures from canonical rationality.

8 Sometimes the terms adequate or analytical are used.

9 For a punctilious characterization of precisely those formal systems to which the theorem applies, see Smullyan (1992).

10 Importantly, he did so constructively, displaying a (self-referential) true statement that is undecidable; that is, neither it nor its negation are theorems of the relevant system.

Wikipedia wrote:..is critical of econometrics and the application of empirical research in economic theory, which are commonly used in mainstream economics.

Sigillum Militum wrote:Sure but I suspect that a lot of econometrics is voodoo though.