Date of this Version
Published in Transactions of the Nebraska Academy of Sciences, Volume 2 (1973).
The Purpose of this paper is to apply a branch of mathematics which has recently come to the fore in investigations in the social sciences, namely automata theory. This application will show how automata theory may be used in the formal description of a wide variety of social phenomena.
My basic plan will be to start by describing a broad class of social systems in terms of game-theoretical notions, especially the notions of strategy and strategy mixture. Then I will define automata and show how, under a certain translation of terms, every social system of the class I have taken turns out to be an automaton.
Let us first introduce certain notions of game theory: we take a certain set of external states Gtm which are potentially relevant to the behavior of any participant in a social system at any time t, where m is the number of such states. We will consider the set of states to contain a record of all past behavior of all participants in the social system, so that any particular participant can consider this in making his future decisions. We let In be the set of n participants and Si the strategy set of the ith participant. The strategy set includes all possible actions that a participant might undertake under the various circumstances he might observe to hold and which are elements of the set Gtm of external states.
Now as is well known, in many social situations, it is preferable to randomize one's choice of actions so as not to let those with whom one might be in competition be able to predict perfectly one's own actions, since the opponents might then take advantage of their perfect knowledge. If, for example, a batter in a baseball game knew exactly what sequence of curves and fastballs a pitcher were going to throw, he would certainly raise his batting average. Now the problem of randomizing strategies involves the introduction of probability in order to at least describe the proportion of the kinds of actions one chooses, even if their exact sequences be random. Even though an observer could not predict any individual actions the participant might undertake, he would still know their proportions and therefore the probability that any particular action will be undertaken. The set of probabilities describing the actions of participant i is called a strategy mixture and is designated by 'σik,t,, which is a function of external state or circumstance k and time t.