Calculi with monotone deductions and their economic interpretation

Abstract

Let the possible ways of development of some system from the initial state X₀ be given by the deductive system <α;X₀> (X₀ is an axiom, the algorithm α defines the relation of deducibility in one step). Let Y₁,..., Y_e be all states directly derivable from X [i.e., α (X)={y₁,..., Y_e]. Let α be an algorithm assigning for each X transition probabilities p₁,..., p _l , where$p = i - \sum\limits_{i = 1}^\ell {Pi}$ is the transition probability to the special state STOP.

defines a probability measure on the set of all deductions. We define the information in the pair < α;X₀> by the forla:

where px is the probability of being in X directly before STOP. We consider α, assigning a fixed p for each X and satisfying the condition p₁= ...=p _l . Then the information in < α; X₀> becomes a function <α;X₀> of one p. The essential characteristic of the system <α;X > is given by the asymptotic behavior of <α;X₀> as p → 0. This characteristic corresponds well with the intuitive notion of the relative “power” of calculi. Now we consider <α,X>(p) as a function of X. For many types of systems there is a useful strategy for maximizing this function (the strategy of increasing freedom of choice); we consider in this connection the simplest systems of economic character. Let X, Y, Z be n-dimensional vectors with nonnegative components (the components are interpreted as resources and products of a certain economic system, α gives the technological possibility of transformations of the resources). Let