The Tower of Hanoi Problem
Procedural rationality
Although the prase "bounded rationality" was originated by Herbert Simon, it has been taken over by more conventional economists and redefined to cohere with mainstream economic theory. For Simon, bounded rationality implied limited information-processing and computational capacities. I do not know for certain who first redefined it, but certainly Williamson [21] claimed to encompass bounded rationality as limited access to information in his introduction of transactions cost economics. The difference is that, for Simon, bounded rationality entails the availability of more information than can be taken into account by decision-makers while for economists such as Williamson bounded rationality entails the paucity of information which is therefore a constraint in optimization procedures.
The effect of Williamson's redefinition of bounded rationality was to leave in tact the underlying precept of mainstream economic theory that agent behaviour can be represented by some constrained optimization algorithm. The compulsion of economists to adhere to constrained optimization as the defining characteristic of human behaviour is manifest also in Sargent's specification of artificially intelligent agents. Genetic algorithms in the hands of Sargent or Arifovic to represent behaviour are different but they remain optimizing algorithms which arguably misspecify the nature of human cognition.
The argument that these implementations of genetic algorithms misspecify human cognition starts from the implication from cognitive science that cognition takes the form of exploitation of what we know and a highly directed exploration of our environment which is focused by our knowledge. This implications follows from the distinction in cognitive science between procedural and declarative knowledge.
Procedural knowledge is knowledge about how to do something and this knowledge is held by individuals in a way which does not allow it to be communicated directly to other individuals. Declarative knowledge is knowledge of what is true and can be communicated directly to other individuals. For example, an Englishman may have both procedural and declarative knowledge about the game of cricket. He can explain the rules of the game and describe or show a novice how to stand at the wicket or where to stand if he is to play off-stump or what to do if he is the wicket-keeper or the necessity of keeping the bowling arm straight at the elbow. All of this knowledge is declarative. To hit the ball successfully and place it where the batsman wants the ball to land or to spin-bowl so that the ball hits the ground and bounces so as to hit the wicket without coming into the range of the bat require abilities that can only be attained by practice. However well a person might know the rules and be able to describe the practices of cricket, that person will not be able actually to play cricket without acquiring substantial procedural knowledge.
Independently, this same distinction has been made by historians of business, the organization and technological change who demonstrated its relevance to these areas of economic activity. Edith Penrose [16] in her seminal (1959) analysis of the direction of the growth of the firm called the two types of knowledge objective and subjective. But her distinction between the two was couched in the same terms as Anderson's [2] discussion of the distinction between procedural and declarative knowledge. Similar distinctions -- though not quite so explicit as in Penrose -- are found in Chandler's (e.g. [5]) work on the development of organizational structures and Rosenberg's (1975, 198x) discussions of the determinants of the direction of technical change.
Because we cannot know everything, a reasonable assumption is that what declarative knowledge we do have comes from the activities in which we engage. How we use this declarative knowledge follows from our experience and, to the extent that experience is necessary to use declarative knowledge effectively, its use is governed by procedural knowledge. In other words, we start from what we know and develop new ideas and perceptions only by extending our experience.
Genetic algorithms, by contrast, search the whole of the environmental information space randomly and, if well constructed, evenly at the outset and concentrate increasingly on the parts of the information space that yield the best results. In the language of the field, genetic algorithms explore the search space and then exploit the subspaces described by classifiers that yield the greatest fitnesses. For cognitive scientists, human cognition takes the form of exploitation of what we know and a highly directed exploration which is focused by our procedural knowledge.
The difference could hardly be more important for how economists practice their discipline. Either agents are global optimizers and genetic algorithms can be used to represent that optimization in conditions of constrained information-processing and computational capacities or they can at best exploit their existing procedural and declarative knowledge in the hope of gaining some local (though possibly large) improvement in their circumstances. The substance of this difference is that the assumption of global optimization can evidently support the construction of models with no concern for the procedures by means of which agents actually go about collecting declarative knowledge and then developing their procedural knowledge where as the Simon approach requires the specification of precisely those aspects of cognitive behaviour.
This brings us back to Simon's earlier distinction between substantive and procedural rationality.
Procedural rationality concerns the choice or development of procedures for taking decisions when the decision-maker has effectively limited capacities to process information and calculate appropriate outcomes. Certainly, procedural rationality entails satisficing. Our concern here is to find a representation of satisficing that will support models of decision-making in a macroeconomic environment.
The particular representation reported in this paper is drawn from several cognitive theories that have been implemented as computer software architectures designed to replicate data from psychological experiments. These architectures are Soar [9] and ACT-R [2]. Both of these architectures are based around the concept of a problem space architecture which itself is a tree structure of goals and sub-goals. The original specification of this goal and subgoal structure was by Newell and Simon (1972) as a planning algorithm. The sort of situation it might be used for in the Newell-Simon version was planning a trip from an office at MIT to an office in Berkeley. If the goal were to make that journey, a subgoal would be to fly from the nearest airport to MIT to the nearest airport to Berkeley. The subgoal of making that flight would be to get from the MIT office to the airport which would be undertaken by (say) car or taxi. To take the car would entail the subgoal of getting from the MIT office to the car by walking.
A classic problem on which to test artificial-intelligence algorithms is the Tower of Hanoi problem. This involves moving a set of discs of graded size from one peg to another, using a third peg as an intermediate step. The five-disc Tower of Hanoi problem is illustrated below. The discs can be moved one at a time and it is not permitted to place a larger on a smaller disc. The problem-space architecture for this problem, as specified by Simon [19] is to specify a subgoal of getting the top four discs onto peg B so that the largest disk and be placed on peg C and then to execute the next subgoal of moving the four discs on peg B to peg C. That move entails a subgoal of moving the top three discs to peg A so that the remaining disc cam be placed on the largest disc which is already on peg C. There is then a similar subgoal to get the three-disc tower onto peg A, and so on. Anderson [2] developed a program in ACT-R to learn to solve the Tower of Hanoi problem and compared the results of that program with the results of experiments with human subjects. He found that the students did indeed learn use a goal stack in the same way as ACT-R. The actual movements of the discs and the setting of goals and subgoals were accomplished in ACT-R by production (if-then) rules.
Figure 1:
The Tower of Hanoi Problem
Three points about the ACT-R representation of cognition are relevant here. The first is that the results obtained from ACT-R programs can be compared with the results of psychological experiments to verify the accuracy of a program as a representation of cognitive behaviour in particular circumstances. Secondly, ACT-R is an encoding of an underlying theory of cognition. Thirdly, the representation of the problem space architecture as rules for moving up and down the goal tree and the rules for performing tasks to achieve each goal can, in principle, be obtained by the standard knowledge-elicitation techniques used for building expert systems.
Taking the first two of these points together, we have a means of encoding procedural knowledge about how agents learn which is informed and justified by a particular theoretical structure and discipline which is independent of the domain of application in economics or the management sciences. Discussions or arguments about the appropriateness of that particular encoding are not likely to be influenced by the results desired for economics models. The third point allows us to develop independent evidence to support a particular encoding of agents' procedural and declarative knowledge.
In the rest of this paper, I will report a pilot model of a transition economy. This model will be used to investigate the characteristics of procedures for learning and decision making which are validated in relation to cognitive science and verified in relation to economic time-series data. These procedures take for granted bounded rationality in the sense of Simon. These limitations preclude the assumption of optimizing behaviour. Encoding the process of goal formation, learning and declarative knowledge about the environment in a manner which corresponds to encodings in the cognitive sciences, we are able to determine whether procedures for forming expectations and perceptions about the environment are rational in the sense that action based on those perceptions is increasingly likely and, in any case, not less likely to further the attainment of agents' goals.
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