Finding the (lost) beings, stories and narratives
In order to grasp the sense of the mathematical concepts, the reader may imagine that game theory deals with ideal agents endowed with unlimited computational powers, as well as with abiding utility functions.
These agents, moreover, have perfect knowledge of the strategies available to the other players, as well as of the probability spaces induced by them, even though not necessarily perfect memory; that is to say, in a given stage of the game, information about the precise moves that the other players have actually made. The theory pretends that the social equilibria found in reality are determined by some strategic equilibrium, which consists of a system (profile) of choices of strategies of the agents such that any of them is a best response to the others. In other words, the agents behave as they do because they have chosen respective behavioral strategies that jointly form an equilibrium.I will prove in a detailed and rigorous way that behavioral strategies determine a probability measure μ over the space of all possible histories of the game, making some histories more probable than others. On the other hand, the empirical observation of the actual behavior of the agents is to be represented by an histogram or Pearson curve ν, over the same space of possible histories, that exhibits some of them as being more probable than others. It is important to stress that μ is obtained out of the assumption of a system (profile) of strategies of the agents, whereas ν is empirically obtained and is logically independent of μ. The fundamental law of the theory expresses precisely that both are related in a certain way, to wit, that ν must be equal or approximately equal to μ. That is to say, the observed behavior can be rationalized as strategic behavior in equilibrium. That is why it might be said that μ is subjectively determined, while ν is obtained out of empirical data. The law of the theory might be interpreted as expressing that any observed histogram is the result of strategic rational behavior (that possibly takes into account probable states of nature).
It might be thought that the constraint (which is a generalization of Samuelson- Houthakker’s strong axiom of revealed preference) is too rigid in insisting that the utility function of any of the agents must be assumed identical throughout every application of the theory to the same agents. Nevertheless, this condition is a condition of possibility of comparative statics, for it is required in order to obtain the function of Walrasian demand. It must be considered as an implicit definition of ‘same agent’.
8.3