Mathematical Game Theory and Applications

Maximization of efficiency of advertising expenditure

2026-08-24T15:11:16+03:00

The paper models the dynamics of a firm’s communication activities. The aim is to determine an advertising policy that maximizes the firm’s efficiency index, defined as the ratio of ''output'' to ''input'' weighted values. The ''output'' measures include the final level of goodwill and the cumulative sales achieved by the firm over the entire selling period. The ''input'' measures consist of the firm’s expenditures, namely marginal costs and advertising costs. We describe the structure of optimal controls and propose an algorithm for computing optimal trajectories.

Optimal behavior in a finite-horizon gambler's ruin problem

2026-08-24T15:11:16+03:00

A multi-stage model with a finite horizon related to the gambler's ruin problem is considered. Two players compete against each other at each of the $n$ steps of the game. The probability that the player wins at the next step depends on the win/lose ratio in previous steps. The player's payoff is determined at the end of the game and is equal to the difference between wins and losses. The player's payoff also accounts for the cost $c$, which the player incurres at each step. It is assumed that the players are asymmetric and one of them can stop the game if the number of her losses exceeds the number of her wins by an amount $k$. The payoff of the player using such strategy is obtained. The numerical simulations of the player's payoffs for different values of the problem parameters are provided.

Resource management problem with different reproduction regimes

2026-08-24T15:11:16+03:00

The paper considers dynamic resource management problem with different reproduction regimes. The planning horizon is assumed to be divided into equal intervals with their own resource natural growth function. The optimal control for the problems with finite and infinite planning horizon is obtained in general form. Nash equilibrium for game-theoretic setting of the problem with infinite planning horizon and interchanging reproduction regimes is constructed. The results of numerical modeling are presented.

The growth model with a logistic hazard rate of the resource exhaustion

2026-08-24T15:11:16+03:00

The paper studies a probabilistic interpretation of the dynamic model of a resource-dependent economy with phase constraints. It is shown that the main model factor is one of the important indicators of the reliability theory, which is a hazard rate determined by a random time moment when the model constraints are violated. If a hazard function is given, then the distribution of the random variable can be uniquely established. In addition, all the main factors of the model can be calculated through the hazard rate and its integral analogue. The paper proposes to consider a logistic function as a candidate for the hazard function and shows the ways for identifying its unknown parameters. The obtained results are illustrated by numerical experiments.

Nontransitive Penney game, its generalizations, applications in finance and fractal Brownian motion

2026-08-24T15:11:16+03:00

The paper investigates a nontransitive paradox which is known as the <<Penney game>>. An algorithm for calculating theoretical winning probabilities is given, which makes it possible to generalize the game to binary sequences of arbitrary length in which 0 and 1 are independent and uniformly distributed. Based on a numerical experiment, it is established that for a sequence constructed from the dynamics of stock prices, this nontransitive effect persists. A generalization of the game to fractal Brownian motion is provided: a model is constructed and, using the results of numerical experiment, the parameter values are found at which nontransitivity disappears in the game. The values of the parameter at which the winning probabilities in the <<Penney game>> for fractal Brownian motion most accurately correspond to the game for stock price dynamics are investigated.

Using Yandex Maps data for the Wardrop model

2026-08-24T15:11:17+03:00

In practical problems, including related to modeling and analyzing transportation systems, one of the most pressing issues is collecting initial data. Model parameters are determined based on this data, and it is important to make the model as close to reality as possible. This paper describes the experience of collecting data for the Wardrop transportation model using available tools including the Yandex Maps API.