Prelude to Chapter 3

EQUILIBRIUM, EXPLANATION,
AND GAUSS'S TOMBSTONE

 

 

 

COUNTLESS ARTICLES purport to explain social phenomena by furnishing a Game in which the phenomenon is proved to be an equilibrium, typically a Nash equilibrium or some refinement thereof. To be sure, there are many cases in which the attempt to demonstrate that an important social regularity is Nash is deeply revealing. But there are at least three cases¹ where it won't be:

Case 1.	The phenomenon of interest is a nonequilibrium dynamic.
Case 2.	Equilibrium is attainable in principle, but not on acceptable time scales.
Case 3.	Equilibrium exists but is unattainable outright.

Every model in this volume falls into one of those three categories. I hope the book demonstrates that, nevertheless, agent-based modeling can be explanatory where “the equilibrium approach,” if you will, is either infeasible (Case 1) or lacking in explanatory significance (Cases 2 and 3).²

Case 1

The Anasazi and Smallpox models are designed and empirically calibrated to reproduce observed spatiotemporal dynamics, not to arrive at an equilibrium. I see them as examples of empirical nonequilibrium social science. The Retirement model, too, is concerned with explaining the dynamics of a shock-induced long-term transition from one distribution of retirement ages to another. The Norms and Civil Violence models both exhibit classic features of complex systems:³ local conformity, global diversity, and punctuated (not stable) equilibrium. Similarly, the Demographic Prisoner's Dilemma model exhibits persistent nonequilibrium cycling. The Organization model, while more exploratory than explanatory, is concerned with structural adaptation in dynamic environments, and finds that, in the main case discussed, optimal adaptation involves a time-varying, rather than an equilibrium, structure.

Case 2

The Classes model is of the second type. There is indeed an asymptotic equilibrium—a kind of fairness norm. But it is subject to astronomical waiting times; these are shown to scale exponentially in a number of core variables (including the number of agents). The equilibrium—the equity norm—is not attainable on practical time scales, and the system spends most of its time far from the equitable equilibrium.

Case 3

As discussed in the preceding two chapters, there are a number of deep papers demonstrating that equilibria can be uncomputable in a strict mathematical sense. However, this literature, while extremely important, is also quite sophisticated technically. The next chapter, entitled, “Non-Explanatory Equilibria,” offers an extremely simple game most of whose equilibria are unattainable in principle (Case 3) and in which the time to attain those equilibria that are attainable grows exponentially in the number of players (Case 2). It demonstrates the essential distinction between existence and attainability (both in principle and in practice) in a very accessible way. And in so doing, I hope it provokes further and deeper thought regarding the explanatory significance of equilibrium.

Incompleteness

Also as noted in the preceding chapter, the model can be seen as an example of incompleteness in mathematical social science. If “being true” is taken as the analog of “being an equilibrium” and “being provable” is the analogue of “being attainable,” then the mapping is clear: Propositions that are true but unprovable from the axioms (à la Gödel) correspond to model states that are equilibria but are unattainable from any permissible initial conditions.

Now, even incompleteness is a special case of the general distinction between satisfaction of some mechanically checkable condition (like being a tautology or being a Nash equilibrium) and generability (like being deducible from axioms under rules of inference or being attainable from initial conditions under rules of agent interaction). And this brings us to Gauss's tombstone.

Gauss's Tombstone

Why is the tombstone of Carl Friedrich Gauss (1777-1855) engraved with a regular 17-gon? Because he was worried about the very same distinction. But, the story actually begins in Greece in roughly 500 BC. The Greeks knew perfectly well what it would mean for a geometrical shape to satisfy the following criterion: to be a regular n-gon, a polygon with n sides of equal length. And they also knew what it would mean to be constructible—that is, generable—by a sequence of specific operations with only a straightedge ruler and a compass. In essence, the Greeks asked: For what values of n is the regular n-gon constructible? Good question!

It took roughly twenty-two centuries to arrive at the answer. A complete (and very beautiful) theory of constructibility was finally achieved by Evariste Galois (1811-32). En route, Gauss proved that the regular 17-gon is, in fact, constructible. This singular achievement had a profound effect on the young Gauss, convincing him to pursue mathematics over philology. And, indeed, it accounts for his unusual tombstone. Now, Gauss actually went a good deal farther, correctly stating that The regular n-gon is constructible if and only if n equals some nonnegative power of 2, multiplied by a product of distinct Fermat primes;⁴ these are prime numbers of the form 2^2k + 1. Notice that 2⁰(2²² + 1) = 17.

Now, one might presume that there is no upper limit on the number of regular n-gons that can be constructed. From Gauss's result, it is clear that this set will indeed be infinite if the set of distinct Fermat primes is itself infinite. So, is it? That, too, is a good question. To this day, the answer is not known!

What is known is that, for roughly 2,500 years, worrying about the distinction between satisfaction and generability has been extremely fruitful in logic and mathematics, stimulating profound researches of Gödel, Gauss, Galois, and many others. I think it will be fruitful in the social sciences as well.

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¹ The possibility of multiple equilibria raises further issues.

² Here, I agree entirely with Brian Skyrms: “The explanatory significance of equilibrium depends on the underlying dynamics.” Brian Skyrms, “Stability and Explanatory Significance of Some Simple Evolutionary Models,” Philosophy of Science 67 (March 2000): 94.

³ H. Peyton Young, Individual Strategy and Social Structure: An Evolutionary Theory of Institutions (Princeton: Princeton University Press, 1998).

⁴ Gauss proved sufficiency and asserted that he had a proof of necessity. On the history and mathematics of Galois Theory, see Charles R. Hadlock, Field Theory and Its Classical Problems (Washington, DC: Mathematical Association of America, 1978); and I. Stewart, Galois Theory (London: Chapman and Hall, 1973).