مدل سازی ذهنی از عرضه و تقاضا __ حداقل راه حل اطلاعات فیشر

Two of the present authors have put forward a projective geometry based model of rational trading that implies a model for subjective demand/supply profiles if one considers closing of a position as a random process. We would like to present the analysis of a subjectivity in such trading models. In our model, the trader gets the maximal profit intensity when the probability of transaction is ∼0.5853∼0.5853. We also present a comparison with the model based on the Maximum of Entropy Principle. To the best of our knowledge, this is one of the first analyses that show a concrete situation in which trader profit optimal value is in the class of price-negotiating algorithms (strategies) resulting in non-monotonic demand (supply) curves of the Rest of the World (a collective opponent). Our model suggests that there might be a new class of rational trader strategies that (almost) neglects the supply–demand profile of the market. This class emerges when one tries to minimize the information that strategies reveal.

Conscious and rational economic activity requires, among others, an optimization of profit in given economic conditions and, usually, during more or less definite (basic) intervals. Usually, these intervals are chosen so that they contain a certain characteristic economic cycle (e.g. one year, an insurance period or a contract date). Often, it is possible (and always risky!) to make a prognosis for a more or less distant future of an undertaking by extrapolation from the already known facts, e.g. by various sorts of “statistical analyses”. The quantitative description of an undertaking is extremely difficult when the duration of the intervals in question is itself a random variable (denoted by ττ in the following). The profit gained during the specific period, represented as a function of ττ, also becomes a random variable and that hardly can be used as an acceptable measure of the quality of the undertaking. To investigate activities that might have different periods of duration we define, following queuing theory [1], the profit intensity as a measure of this economic category [2]. A model of this type, although simple and elegant, has several drawbacks from the theoretical point of view. Besides, such phenomena perceived as games do not have any natural “quantum-like” version.1 Such a possibility would be welcome because, for example, the appearance of non-Gaussian probability distribution functions suggest the existence of the so called Giffen goods [11]. Obstacles in constructing quantum-like versions of such models can be overcome by replacing the maximum Boltzmann/Shannon entropy principle with the requirement that the Fisher information gets its minimum under certain additional conditions (a discussion of the connection between the principle of the minimum of Fisher information and equations of quantum theory can be found in Ref. [12]). In this way, a simple method of quantum-like reformulation in terms of game theory that stem from statistical considerations is possible. Moreover, this approach allows for analysis of subjectivity in strategy selection. This paper is in some sense an extension and continuation of our previous work [13] and is organized as follows. First we describe the merchandizing mathematician model (MM) put forward in Ref. [2] and quote the relevant definitions. Then we argue for the use of logarithmic quotations and define the logarithmic rate of return and briefly describe its scaling invariance [14]. The results showing the usefulness of Fisher information as a criterion for strategy selection and estimation of probabilities of making profits will be given in Section 3. Finally we will point out some issues that still need to be addressed.

We have considered a wide class of models with a bounded on one side supply/demand that is reconstructed from historical transactions via statistics based on a single estimator. Such estimation is, by definition, subjective and agent dependent. We have shown that Fisher information can be effectively used for optimal strategy selection (the probability of optimal transactions). The optimization based on Fisher information gives a smaller probability of optimal transaction than that based on Boltzmann/Shannon [13]. Both models, (Fisher and Boltzmann/Shannon) have scale invariance with respect to the rate qq. Therefore only the product αψαψ can be optimized—both models suggest only how to choose optimal strategies (PP). Rates ψψ should be fitted “experimentally”. Such models can be tested on real markets! Quantum decision theory could be useful for taking the subjectivity of market analysis into account [10].