Introduction

This chapter presents a formal and concise account of the process of designing a decentralized mechanism that realizes a given goal function. The presentation here differs from the one in Chapter 2 in that this process is set-theoretic, whereas in Chapter 2 (and Chapter 4), sets and relations are represented by equations. The set-theoretic formulation is more general, and covers cases in which the set of environments is a finite or discrete set. A merit of the set-theoretic formulation is that it helps make clear the essential logic and properties of our procedures for designing decentralized mechanisms. On the other hand, the formulation in which sets and relations are represented by equations permits the components of a mechanism to be expressed in terms of algebraic expressions that hold across the set of environments, rather than pointwise, as is the case with the set-theoretic formulation. The process of mechanism design is “algorithmic” in both formulations, in the sense that the design process consists of a sequence of prescribed steps that starts with a specified goal function, and results in an informationally efficient decentralized mechanism that realizes the given goal function. Both approaches use the axiom of choice to prove the existence of a transversal. However, the equations approach may in specific instances derive a transversal by algebraic means, or through the use of calculus. In the approach that uses smooth equations to characterize sets and functions, an analysis may require solution of systems of nonlinear equations of high degree.

Introduction

Our objective in this chapter is to provide a somewhat informal description of systematic procedures for constructing informationally efficient decentralized mechanisms that realize a given goal function.

A goal function represents outcomes or actions deemed desirable in each possible environment under consideration. An environment is specified by the values of finitely many parameters that together form the parameter space. Their values define feasibility of allocations and preferences of the agents. A goal function has as its domain a factored parameter space (in this chapter usually Euclidean or finite), and a Euclidean or finite outcome space as its range.

A mechanism is a triple consisting of (i) a message space, (ii) a system of decentralized equilibrium relations (correspondences, equations), and (iii) the outcome function that translates equilibrium messages into outcome choices. A mechanism models communication through messages, their verification by agents, and the outcomes associated with equilibrium messages.

DEFINITION. We say that a mechanism realizes the goal function if

1. (i) (existence) for every parameter point θ, there exists a corresponding equilibrium point m in the message space.

2. (ii) (optimality) if m is an equilibrium message for θ, then the outcome z specified for m by the outcome function is desirable according to the goal function – F-optimal.

As was noted earlier, the traditional conception of what we call mechanical computation or computation by machine relies on a purely kinematical conception of mechanics. It entirely omits any notion of force and focuses attention only on abstract states and motion between them. In this it follows a trend in mechanical formalism that moved away from considering forces and spatial motions to considering mainly Hamiltonian motion through abstract spaces, with no mention of either the central notion of force or the key notion of mass (cf. Hermann 1990, Sussman & Wisdom 2001).

This disconnect between mechanical computation and mechanics comes closest to being bridged in the related field of information theory, in which some authors have viewed information content as a type of mass measure (Manthey & Moret 1983) and have produced formal relations between information content and thermostatic theories of entropy (Chaitin 1975). These ropes tossed across the gap lack tether to the notion of force and still leave the crossing perilous.

Let us now reconsider the notion of computation from the mechanical point of view, to treat “mechanizability”—viewed in terms of machines—as mechanizability—viewed in terms of mechanics. We seek to understand the notion of effectiveness as involving not just abstract kinematics but also those fundamental concepts that distinguish mechanics from geometry, especially the concepts of rate of motion limited by limits on force and bounds on the rate of work.

Discussions in previous chapters have touched on these ideas already.

The preceding treatment of reasoning indicates how we can interpret psychological rationality in terms of mechanical processes. Let us now look at the ways in which mechanical concepts enter into characterizing forms of economic rationality.

Limits on rationality

The difficulty and slowness with which real agents change their mental state constitutes one of the most evident limitations on rationality. As noted earlier, we can see reflections of the mechanical connection between momentum and force in “the more you need to change, the more you have to force yourself,” “the more you know, the harder it is to change your mind,” and other truisms of popular psychology. We can read the first of these truisms as stating a monotonicity relation between the size of changes and the size of the required forces and work done, and the second as stating a monotonicity relation between the mass and the force required for given changes. Notions of monotonicity and proportionality among the numerical magnitudes of momentum and force are familiar in traditional mechanics, but how do these apply in the discrete mechanical setting?

A mechanical interpretation of thinking also naturally relates slowness of change to inertia. From the same perspective, the unreality of ideal rationality appears because when we determine actions by finding the maxima of an expected utility function generated by instantaneous beliefs and desires, large changes can come from small impulses.

The preceding chapters presented the beginnings of a mathematical and mechanical theory of mind.

We began by examining the curious divorce between mechanical understandings of mind and nature that occurred when natural philosophy developed mathematical techniques useful in characterizing physical mechanics but inapplicable to mental mechanics. The mathematical study of mental materials developed separately, but with the key mathematical theories of logical and economic rationality lacking any connection to mechanics. The mechanical reconciliation of mind and nature began to take shape only when the development of artificial computers enabled construction of artificial minds precise and concrete enough to relate to a new rational mechanics broad enough to encompass mental as well as physical materials. The reconciliation promises not only to open traditional philosophical questions to new forms of technical analysis, but also to provide a new formal vocabulary for describing agents of limited rationality and for engineering computational and social systems based on such agents.

We then examined two sides of the reconciliation of physical and mental mechanics. On the physical side, we recast the axioms of modern rational mechanics so as to cover discrete mechanical systems and their hybrids with physical mechanical systems.

The preceding development of mental mechanics does not require determinism of mechanical systems. It instead requires only that motion satisfy mechanical relationships independent of determinism requirements.

The preceding chapters also illustrated several sources of possible indeterminacy. Reasoning, whether habitual or deliberate, can produce indeterminism when several reasons apply at the same instant, requiring serialization or conflict resolution. In addition, rational deliberation can result in several possible self-constructions from reasoning rules; conservative update in response to reasoned changes can follow multiple resolutions; and volition can encompass multiple choices of action on the basis of the same desires and intentions. These sources of mechanical indeterminism complement the forms of indeterminism acknowledged in traditional mechanics, including situations of indeterministic collapse and bifurcation considered in continuum mechanics and the pervasive indeterminacy of quantum physics. All of these forms of indeterminacy represent theoretical allowances of multiple possibilities that stand separate from uncertainties arising from the practicalities of measurement connected with repeatability and resolution of measuring apparati.

From the viewpoint of psychology, mechanical indeterminism generates what one can call a “kinematical” notion of uncertainty, in which one seeks to measure the amount of indeterminism, or degree of uncertainty about predictions introduced by indeterminism. In the simplest terms, qualities of motion shared by all possible histories represent certain predictions about motion, while qualities exhibited by some histories but not by others represent uncertain predictions about motion. The kinematic conception of uncertainty provides means for comparing these degrees of certainty and uncertainty in quantitative terms.

Understanding psychology and economics in mechanical terms requires looking at specific concepts of psychology and economics from the mechanical point of view. If we look to the literature, however, we find that the cognitive sciences study a wide range of possible or hypothesized psychological organizations as explanations of human thought. For example, the ideally rational agents of economics have one kind of mind, a kind very different from almost all known human minds. But even among humans, individual minds have very different characters, exhibiting different levels of intelligence at different tasks, different temperaments, different degrees of adaptability, and so on. The well-known Myers–Briggs test (Myers & Myers 1980), to give another example, sorts minds into sixteen well-populated classes. These classes correspond to recognizable and common types of personal character, types that give some insight and enable reasonable, though not perfect, predictions of individual behavior.

It does not take deep reflection to realize that if we are already on page 225 and just starting the mechanical examination of psychology and economics, we cannot hope to examine all the concepts of all hypothesized mental organizations in this book, no matter how long, without exhausting all patience. I therefore undertake to examine the structure and mechanical nature of some special kinds of minds that serve to illustrate the mechanical nature of thinking, in part to open the special classes to mechanical investigation, and in part to suggest ways of understanding other kinds of minds in mechanical terms.