arXiv subjects

Misha Perepelitsa

Publications and source records attributed to Misha Perepelitsa.

At least 19 recordsLinked to original sources

arXiv2026

On a PDE model for Learning in Stochastic Market Entry Games

We study a continuum model for stochastic reinforcement learning in repeated market entry games. Starting from a discrete-time microscopic learning rule, we derive a Fokker--Planck-type equation for the distribution of agents' propensities and, using a kinetic closure, obtain a nonlinear one-particle equation of a mean-field type. For the resulting Cauchy problem, we prove existence and uniqueness of solutions and analyze their long-time behavior. The PDE captures two key phenomena observed in market entry dynamics: aggregate learning (the average number of entrants approaches market capacity) and sorting (propensities concentrate near extreme behaviors). The model also yields explicit characteristic time scales, showing that aggregate learning occurs faster than sorting, in agreement with experimental and computational evidence.

math.AP↗

arXiv2024

Role of Non-Exponential Reversal times in Aggregation Models of Bacterial Populations

In this paper, we consider 1D agent-based and kinetic models of aggregation with reversals. In particular, we fit a Gamma distribution to represent the run times in myxobacteria and analyze numerically the importance of non-exponential reversal times. We demonstrate that non-exponential reversal times aid aggregation and result in tighter aggregates. We compare and contrast the behavior of agent-based and kinetic models, and also consider kinetic models with aggregation driven by chemotaxis. Thus, incorporating non-exponential reversal times into models of aggregation can be particularly important for reproducing experimental data, such as aggregate persistence and dispersal.

physics.bio-ph↗

arXiv2024

Breakdown of Boltzmann-type Models for the Alignment of Self-propelled Rods

Studies in the collective motility of organisms use a range of analytical approaches to formulate continuous kinetic models of collective dynamics from rules or equations describing agent interactions. However, the derivation of these kinetic models often relies on Boltzmann's hypothesis of "molecular chaos", that correlations between individuals are short-lived. While this assumption is often the simplest way to derive tractable models, it is often not valid in practice due to the high levels of cooperation and self-organization present in biological systems. In this work, we illustrated this point by considering a general Boltzmann-type kinetic model for the alignment of self-propelled rods where rod reorientation occurs upon binary collisions. We examine the accuracy of the kinetic model by comparing numerical solutions of the continuous equations to an agent-based model that implements the underlying rules governing microscopic alignment. Even for the simplest case considered, our comparison demonstrates that the kinetic model fails to replicate the discrete dynamics due to the formation of rod clusters that violate statistical independence. Additionally, we show that introducing noise to limit cluster formation helps improve the agreement between the analytical model and agent simulations but does not restore agreement completely. These results highlight the need to both develop and disseminate improved moment-closure methods for modeling biological and active matter systems.

physics.bio-ph↗

arXiv2024

Existence of weak solutions for the kinetic models of motion of myxobacteria with alignment and reversals

In this paper, we consider three non-linear kinetic partial differential equations that emerge in the modeling of motion of rod-shaped cells such as myxobacteria. This motion is characterized by nematic alignment with neighboring cells, orientation reversals from cell polarity switching, orientation diffusion, and transport driven by chemotaxis. Our primary contribution lies in establishing the existence of weak solutions for these equations. Our analytical approach is based on the application of the classical averaging lemma from the kinetic theory, augmented by a novel version where the transport operator is substituted with a uni-directional diffusion operator.

math.AP↗

arXiv2022

Elementary Bitcoin economics: from production and transaction demand to values

In this paper we give an elementary analysis of economics of Bitcoin that combines the transaction demand by the consumers and the supply of hashrate by miners. We argue that the decreasing block reward will have no significant effect on the exchange rate (price) of Bitcoin and thus the network will be transitioning to a regime where transaction fees will play a bigger part of miners' revenue. We consider a simple model where consumers demand bitcoins for transactions, but not for hoarding bitcoins, and we analyze market equilibrium where the demand is matched with the hashrate supplied by miners. Our main conclusion is that the exchange rate of Bitcoin cannot be determined from the market equilibrium and so our arguments support the hypothesis that Bitcoin price has no economic fundamentals and is free to fluctuate according to the present demand for hoarding and speculation. We point out that increasing fees bear the risk of Bitcoin being outcompeted by its main rival Ethereum, and that decreasing revenues to miners depreciate the perception of Bitcoin as a medium for store value (hoarding demand) which will have effect its exchange rate.

econ.GN↗

arXiv2022

Investing in crypto: speculative bubbles and cyclic stochastic price pumps

The problem of investing into a cryptocurrency market requires good understanding of the processes that regulate the price of the currency. In this paper we offer a view of a cryptocurrency market as an environment for realization of a self-organized speculative scheme that results in a formation of a characteristic price bubble as a transient phenomenon. We use microscale, agent-based models to simulate the system behavior and derive macroscale ODE models to estimate such parameters as the return rate and the market value of investments. We provide the formula for the total risk of the system as a sum of two independent components, one being characteristic of the price bubble and the other of the investor behavior.

q-fin.TR↗

arXiv2022

A mean-field model for nematic alignment of self-propelled rods

In this paper we develop a model for nematic alignment of self-propelled rods interacting through binary collisions. We avoid phenomenological descriptions of rod interaction in favor of rigorously using a set of microscopic-level rules. Under the assumption that each collision results in a small change to a rod's orientation, we derive the Fokker-Planck equation for the evolution of the kinetic density function. Using analytical and numerical methods, we study the emergence of the nematic order from a homogeneous, uniform steady-state of the mean-field equation.

physics.bio-ph↗

arXiv2022

General-purpose cooperativeness and altruism in humans: elements of the mathematical framework for the Interdependence Hypothesis

We propose a decision-making model for joint intentionality by interpreting it as group-mindedness at the microlevel. We apply this model to give a formal justification of the first part of the Interdependence Hypothesis due to Tomasello et al. [Current Anthropology, 2012] which asserts that the emergence of joint intentionality evolved due to the challenges of difficult collaborative foraging practices among early humans, and that its evolution led to robust collaboration and some form of altruism. In another application of the microlevel group-mindedness we consider the problem of establishing cooperation in high-risk-of-defection strategic conflicts and we show that the emergence of cooperation in such situations can be explained in the context of cultural group selection as the result of adaptive learning.

q-bio.PE↗

arXiv2021

Psychological dimension of adaptive trading in cryptocurrency markets

In this paper we extend the analysis of an agent-based model for adaptive trading, called asynchronous stochastic price pump (ASPP) introduced by Perepelitsa and Timofeyev (2019), to the model with heterogeneous distribution of psychological parameters of speculative optimism and pessimism across the population of traders. We show that the new model has a range of qualitatively different dynamics when the correlation between those factors ranges from low negative to large positive values. A statistical parameter estimation suggests a heterogeneous ASPP with negative correlation as a model of price variations of Bitcoin.

q-fin.TR↗

arXiv2021

Reaction-diffusion models for morphological patterning of hESCs

In this paper we consider mathematical modeling of the dynamics of self-organized patterning of spatially confined human embryonic stem cells (hESCs) treated with BMP4 (gastruloids) described in recent experimental works. In the first part of the paper we use the activator-inhibitor equations of Gierer and Meinhardt to identify 3 reaction-diffusion regimes for each of the three morphogenic proteins, BMP4, Wnt and Nodal, based on the characteristic features of the dynamic patterning. We identify appropriate boundary conditions which correspond to the experimental setup and perform numerical simulations of the reaction-diffusion (RD) systems, using the finite element approximation, to confirm that the RD systems in these regimes produce realistic dynamics of the protein concentrations. In the second part of the paper we use analytic tools to address the questions of the existence and stability of non-homogeneous steady states for the reaction-diffusion systems of the type considered in the first part of the paper. We find sufficient conditions on the data of the problem under which the system has an universal attractor.

math.AP↗

arXiv2020

Self-sustained price bubbles driven by Bitcoin innovations and adaptive behavior

We show that infinite divisibility of a trading commodity leads to a self-sustained price bubble when traders use adaptive investment strategies. The adaptive strategy can be viewed as a psychological response of a trader to the situation when the trader's estimation of future prices does not match the actual, realized price. We use a multi-agent model to illustrate the price bubble formation and to quantify its main statistical properties such as the return, the volatility, and the systematic risk of the price bubble to crash. We discuss the plausibility for bubbles to drive prices of digital currencies.

q-fin.TR↗

arXiv2020

Small dispersion approximation of shock wave dynamics

We introduce a dispersion approximation of weak, entropy solutions of multidimensional scalar conservation laws using variational kinetic representation, where equilibrium densities satisfy the Gibb's entropy minimization principle for a piecewise linear, convex entropy. For such solutions, we show that small scale discontinuities, measured by the entropy increments, propagate with characteristic velocities, while the large scale, shock-type discontinuities propagate with speeds close to the speeds of classical shock waves. In the zero-limit of the scale parameter, approximate solutions converge to a unique, entropy solution of a scalar conservation law.

math.AP↗

arXiv2020

A model of cultural evolution in the context of strategic conflict

We consider a model of cultural evolution for a strategy selection in a population of individuals who interact in a game theoretic framework. The evolution combines individual learning of the environment (population strategy profile), reproduction, proportional to the success of the acquired knowledge, and social transmission of the knowledge to the next generation. A mean-field type equation is derived that describes the dynamics of the distribution of cultural traits, in terms of the rate of learning, the reproduction rate and population size. We establish global well-posedness of the initial-boundary value problem for this equation and give several examples that illustrate the process of the cultural evolution for some classical games.

q-bio.PE↗

arXiv2019

A model of discrete choice based on reinforcement learning under short-term memory

A family of models of individual discrete choice are constructed by means of statistical averaging of choices made by a subject in a reinforcement learning process, where the subject has short, k-term memory span. The choice probabilities in these models combine in a non-trivial, non-linear way the initial learning bias and the experience gained through learning. The properties of such models are discussed and, in particular, it is shown that probabilities deviate from Luce's Choice Axiom, even if the initial bias adheres to it. Moreover, we shown that the latter property is recovered as the memory span becomes large. Two applications in utility theory are considered. In the first, we use the discrete choice model to generate binary preference relation on simple lotteries. We show that the preferences violate transitivity and independence axioms of expected utility theory. Furthermore, we establish the dependence of the preferences on frames, with risk aversion for gains, and risk seeking for losses. Based on these findings we propose next a parametric model of choice based on the probability maximization principle, as a model for deviations from expected utility principle. To illustrate the approach we apply it to the classical problem of demand for insurance.

econ.EM↗

arXiv2019

Adaptive Learning in Large Populations

We consider the adaptive learning rule of Harley (1981) for behavior selection in symmetric conflict games in large populations. The rule uses organisms' past, accumulated rewards as the predictor for the future behavior, and can be traced in many life forms from bacteria to humans. We derive a partial differential equation (PDE) that describes the stochastic learning in a population of agents. The equation has simple structure of the `conservation of mass'-type equation in the space of stimuli to engage in a particular type of behavior. We analyze the solutions of the PDE model for typical 2x2 games. It is found that in games with small residual stimuli, adaptive learning rules with faster memory decay have an evolutionary advantage.

q-bio.PE↗

arXiv2019

RPS(1) Preferences

We consider a model for decision making based on an adaptive, k-period, learning process where the priors are selected according to Von Neumann-Morgenstern expected utility principle. A preference relation between two prospects is introduced, defined by the condition which prospect is selected more often. We show that the new preferences have similarities with the preferences obtained by Kahneman and Tversky (1979) in the context of the prospect theory. Additionally, we establish that in the limit of large learning period, the new preferences coincide with the expected utility principle.

econ.TH↗

arXiv2019

Learning by Fictitious Play in Large Populations

We consider learning by fictitious play in a large population of agents engaged in single-play, two-person rounds of a symmetric game, and derive a mean-filed type model for the corresponding stochastic process. Using this model, we describe qualitative properties of the learning process and discuss its asymptotic behavior. Of the special interest is the comparative characteristics of the fictitious play learning with and without a memory factor. As a part of the analysis, we show that the model leads to the continuous, best-response dynamics equation of Gilboa and Matsui (1991), when all agents have similar empirical probabilities.

cs.GT↗

arXiv2018

A model of adaptive, market behavior generating positive returns, volatility and system risk

We describe a simple model for speculative trading based on adaptive behavior of economic agents.The adaptive behavior is expressed through a feedback mechanism for changing agents' stock-to-bond ratios, depending on the past performance of their portfolios.The stock price is set according to the demand-supply for the asset derived from the agents' target risk levels. Using the methodology of agent-based modeling we show that agents, acting endogenously and adaptively, create a persistent price bubble. The price dynamics generated by the trading process does not reveal any singularities, however the process is accompanied by growing aggregated risk that indicates increasing likelihood of a crash.

q-fin.TR↗