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Edward W. Piotrowski

Publications and source records attributed to Edward W. Piotrowski.

At least 19 recordsLinked to original sources
arXiv2022

Transactional Interpretation for the Principle of Minimum Fisher Information

The principle of minimum Fisher information states that in the set of acceptable probability distributions characterizing the given system, it is best done by the one that minimizes the corresponding Fisher information. This principle can be applied to transaction processes, the dynamics of which can be interpreted as the market tendency to minimize the information revealed about itself. More information involves higher costs (information is physical). The starting point for our considerations is a description of the market derived from the assumption of minimum Fisher information for a strategy with a fixed financial risk. Strategies of this type that minimize Fisher information overlap with the well-known eigenstates of a the quantum harmonic oscillator. The analytical extension of this field of strategy to the complex vector space (traditional for quantum mechanics) suggests the study of the interference of the oscillator eigenstates in terms of their minimization of Fisher information. It is revealed that the minimum value of Fisher information of the superposition of the two strategies being the ground state and the second excited state of the oscillator, has Fisher information less than the ground state of the oscillator. Similarly, less information is obtained for the system of strategies (the oscillator eigenstates) randomized by the Gibbs distribution. We distinguish two different views on the description of Fisher information. One of them, the classical, is based on the value of Fisher information. The second, we call it transactional, expresses Fisher information from the perspective of the constant risk of market strategies. The orders of the market strategies derived from these two descriptions are different. From a market standpoint, minimizing Fisher information is equivalent to minimizing risk.

q-fin.GN↗
arXiv2018

Schrödinger type equation for subjective identification of supply and demand

The present authors have put forward a quantum game theory based model of market prices movements. By using Fisher information, we present a construction of an equation of Schrödinger type for probability distributions for relationship between demand and supply. Various analogies between quantum physics and market phenomena can be found.

econ.TH↗
arXiv2015

Generalization of the Aoki-Yoshikawa sectoral productivity model based on extreme physical information principle

This paper presents a continuous variable generalization of the Aoki-Yoshikawa sectoral productivity model. Information theoretical methods from the Frieden-Soffer extreme physical information statistical estimation methodology were used to construct exact solutions. Both approaches coincide in first order approximation. The approach proposed here can be successfully applied in other fields of research.

q-fin.GN↗
arXiv2015

Do transitive preferences always result in indifferent divisions?

The transitivity of preferences is one of the basic assumptions used in the theory of games and decisions. It is often equated with rationality of choice and is considered useful in building rankings. Intransitive preferences are considered paradoxical and undesirable. This problem is discussed by many social and natural sciences. The paper discusses a simple model of sequential game in which two players in each iteration of the game choose one of the two elements. They make their decisions in different contexts defined by the rules of the game. It appears that the optimal strategy of one of the players can only be intransitive! (the so-called \textsl{relevant intransitive strategies}.) On the other hand, the optimal strategy for the second player can be either transitive or intransitive. A quantum model of the game using pure one-qubit strategies is considered. In this model, an increase in importance of intransitive strategies is observed -- there is a certain course of the game where intransitive strategies are the only optimal strategies for both players. The study of decision-making models using quantum information theory tools may shed some new light on the understanding of mechanisms that drive the formation of types of preferences.

quant-ph↗
arXiv2015

The intensity of the random variable intercept in the sector of negative probabilities

We consider properties of the measurement intensity $ρ$ of a random variable for which the probability density function represented by the corresponding Wigner function attains negative values on a part of the domain. We consider a simple economic interpretation of this problem. This model is used to present the applicability of the method to the analysis of the negative probability on markets where there are anomalies in the law of supply and demand (e.g. Giffen's goods). It turns out that the new conditions to optimize the intensity $ρ$ require a new strategy. We propose a strategy (so-called $\grave{a}$ rebours strategy) based on the fixed point method and explore its effectiveness.

q-fin.GN↗
arXiv2014

When "I cut, you choose" method implies intransitivity

There is a common belief that humans and many animals follow transitive inference (choosing A over C on the basis of knowing that A is better than B and B is better than C). Transitivity seems to be the essence of rational choice. We present a theoretical model of a repeated game in which the players make a choice between three goods (e.g. food). The rules of the game refer to the simple procedure of fair division among two players, known as the "I cut, you choose" mechanism which has been widely discussed in the literature. In this game one of the players has to make intransitive choices in order to achieve the optimal result (for him/her and his/her co-player). The point is that an intransitive choice can be rational. Previously, an increase in the significance of intransitive strategies was achieved by referring to models of quantum games. We show that \textit{relevant intransitive strategies} also appear in the classic description of decision algorithms.

cs.GT↗
arXiv2014

Parameter estimation by fixed point of function of information processing intensity

We present a new method of estimating the dispersion of a distribution which is based on the surprising property of a function that measures information processing intensity. It turns out that this function has a maximum at its fixed point. We use a fixed-point equation to estimate the parameter of the distribution that is of interest to us. We illustrate the estimation method by using the example of an exponential distribution. The codes of programs that calculate the experimental values of the information processing intensity are presented.

physics.data-an↗
arXiv2011

Decisions in elections --- transitive or intransitive quantum preferences

Our preferences depend on the circumstances in which we reveal them. We will introduce a dependency which allows us to illustrate the relation between the possibility of winning of particular candidates in a quantum election and the type of preference. It occurs that if voters start to clearly prefer one of the candidates, the significance of intransitive preferences in the quantum model decreases. This dynamic change cannot be observed in the case of the classical model.

quant-ph↗
arXiv2011

Transitivity of an entangled choice

We describe a quantum model of simple choice game (constructed upon entangled state of two qubits), which involves the fundamental problem of transitive - intransitive preferences. We compare attainability of optimal intransitive strategies in both classical and quantum models with the use of geometrical interpretation.

quant-ph↗
arXiv2008

Inconsistency of the judgment matrix in the AHP method and the decision maker's knowledge

In this paper we propose a method for a quantitative estimation of the decision maker's knowledge in the context of the Analytic Hierarchy Process (AHP) in cases, where the judgment matrix is inconsistent. We show that the matrix of deviation from the transitivity condition corresponds to the rate matrix for transaction costs in the financial market. For the quantitative estimation of the decision maker's professionalism, we apply the Ising model and thermodynamics tools.

q-fin.ST↗
arXiv2008

A model of subjective supply-demand: the maximum Boltzmann/Shannon entropy solution

We investigate activities that have different periods of duration. We define the profit intensity as a measure of this economic category. The profit intensity in a repeated trading has a unique property of attaining its maximum at a fixed point regardless of the shape of demand curves for a wide class of probability distributions of random reverse transaction (ie closing of the position). This type of market games is often considered in the research aiming at finding an algorithm that maximizes profit of a trader who negotiates prices with the Rest of the World (a collective opponent) that posses a definite and objective supply profile. Such idealization neglects the sometimes important influence of an individual trader on the demand/supply profile of the Rest of the World and in extreme cases questions the very idea of demand/supply profile. Therefore we put forward a trading model in which the demand/supply profile of the Rest of the World induces the (rational) trader to (subjectively) presume that he/she lacks (almost) all knowledge concerning the market but his/hers average frequency of trade. This point of view introduces maximum entropy principles into the model and broadens the range of economics phenomena that can be perceived as a sort of thermodynamical system. As a consequence, the profit intensity has a fixed point:the profit in tensity reaches its maximum when the probability of transaction is given by the Golden Ratio rule $\frac{\sqrt{5}-1}{2}$.

q-fin.TR↗
arXiv2008

Deterministic definition of the capital risk

In this paper we propose a look at the capital risk problem inspired by deterministic, known from classical mechanics, problem of juggling. We propose capital equivalents to the Newton's laws of motion and on this basis we determine the most secure form of credit repayment with regard to maximisation of profit. Then we extend the Newton's laws to models in linear spaces of arbitrary dimension with the help of matrix rates of return. The matrix rates describe the evolution of multidimensional capital and they are sensitive to both quantitative changes of individual elements and flows between them. This allows us for simultaneous analysis of evolution of complex capital in both continuous and discrete time models.

q-fin.RM↗
arXiv2007

Projective Market Model Approach to AHP Decision-Making

In this paper we describe market in projective geometry language and give definition of a matrix of market rate, which is related to the matrix rate of return and the matrix of judgements in the Analytic Hierarchy Process (AHP). We use these observations to extend the AHP model to projective geometry formalism and generalise it to intransitive case. We give financial interpretations of such generalised model and propose its simplification. The unification of the AHP model and projective aspect of portfolio theory suggests a wide spectrum of new applications such extended model.

q-fin.PM↗
arXiv2007

Reinforcement learning in market games

Financial markets investors are involved in many games -- they must interact with other agents to achieve their goals. Among them are those directly connected with their activity on markets but one cannot neglect other aspects that influence human decisions and their performance as investors. Distinguishing all subgames is usually beyond hope and resource consuming. In this paper we study how investors facing many different games, gather information and form their decision despite being unaware of the complete structure of the game. To this end we apply reinforcement learning methods to the Information Theory Model of Markets (ITMM). Following Mengel, we can try to distinguish a class $Γ$ of games and possible actions (strategies) $a^{i}_{m_{i}}$ for $i-$th agent. Any agent divides the whole class of games into analogy subclasses she/he thinks are analogous and therefore adopts the same strategy for a given subclass. The criteria for partitioning are based on profit and costs analysis. The analogy classes and strategies are updated at various stages through the process of learning. This line of research can be continued in various directions.

q-fin.TR↗
arXiv2007

The relativistic velocity addition law optimizes a forecast gambler's profit

We extend the projective covariant bookmaker's bets model to the forecasting gamblers case. The probability of correctness of forecasts shifts probabilities of branching. The formula for the shift of probabilities leads to the velocity addition rule of the special theory of relativity. In the absence of information about bookmaker's wagers the stochastic logarithmic rates completely determines the optimal stakes of forecast gambler.

physics.data-an↗
arXiv2006

Geometry of Financial Markets -- Towards Information Theory Model of Markets

Most of parameters used to describe states and dynamics of financial market depend on proportions of the appropriate variables rather than on their actual values. Therefore, projective geometry seems to be the correct language to describe the theater of financial activities. We suppose that the object of interest of agents, called here baskets, form a vector space over the reals. A portfolio is defined as an equivalence class of baskets containing assets in the same proportions. Therefore portfolios form a projective space. Cross ratios, being invariants of projective maps, form key structures in the proposed model. Quotation with respect to an asset X (i.e. in units of X) are given by linear maps. Among various types of metrics that have financial interpretation, the min-max metrics on the space of quotations can be introduced. This metrics has an interesting interpretation in terms of rates of return. It can be generalized so that to incorporate a new numerical parameter (called temperature) that describes agent's lack of knowledge about the state of the market. In a dual way, a metrics on the space of market quotation is defined. In addition, one can define an interesting metric structure on the space of portfolios/quotation that is invariant with respect to hyperbolic (Lorentz) symmetries of the space of portfolios. The introduced formalism opens new interesting and possibly fruitful fields of research.

physics.soc-ph↗
arXiv2006

The matrix rate of return

In this paper we give definitions of matrix rates of return which do not depend on the choice of basis describing baskets. We give their economic interpretation. The matrix rate of return describes baskets of arbitrary type and extends portfolio analysis to the complex variable domain. This allows us for simultaneous analysis of evolution of baskets parameterized by complex variables in both continuous and discrete time models.

physics.soc-ph↗