Procedural Rationality and Inductive Learning I: Towards a Theory of Subjective Games

This paper investigates the situation of strategic conflict in which players have limited prior knowledge about the objective game, that is, they do not know their true, objective payoff functions, and therefore, have to formulate their own payoff functions based on their past experiences in a subjective way. In distinction with the objective game, we will define the subjective game by the combination of these subjective payoff functions and the sets of actions. Most of real economic situations are complex and not even well-structured. A real economic agent spends most time to visualize and perceive the situation. Formulating the subjective game would be regarded as the most important step for an actual agent in reaching a decision. Despite its unquestionable importance, the investigation of the subjective game is at this time very immature. Especially, applied game theorists in the 1970's and 1980's have never dealt with the question of how players formulate the subjective game by assuming that the objective game is common knowledge among the players and assuming that players are ideally rational. Since actual players are boundedly rational as Herbert A. Simon has stressed, they might formulate the subjective game which is essentially different from the objective game and fail to achieve a Nash equilibrium of the objective game. The main purpose of this paper is to give clear answers to several substantial questions such as what are the characteristics of the subjective games and the choices of actions. A player is modeled as an inductive learning procedure in a dynamic decision making which translates past experiences into subjective evaluations and decisions, and is mainly motivated by the maximization of the subjective expected payoffs. We will assume that, in every period, a player is never convinced that the situation is recurrent, and therefore, she can not establish a firm experience-based belief about the uncertain situation in which she will seldom waver when observing unlikely events. By requiring a couple of plausible conditions on inductive learning procedures, we can derive the following drastic results in a wide class of environments including various recurrent an non-recurrent situations: The subjective game formulated in the long run belongs to an extremely restricted class of simplified games which are called trivial games. In a trivial game, there always exists the unique action profile which is both strictly dominant and Pareto-efficient among the set of pure action profiles. Moreover, players need not to be strategically sophisticated, because this strict dominance property holds irrespective of the details of its extensive form. Zero-sum games, prisoner-dilemma games, coordination games, stag-hunt games, and hawk-dove games are not trivial games, and therefore, are never perceived as subjective games. This strictly dominant action profile in the subjective game is neither a Nash equilibrium nor Pareto-efficient in the objective game. Of particular importance is that it is always equal to the maximin action profile in the objective game.