HSE Researchers Prove the Existence of Nash Equilibrium for a New Class of Problems in Game Theory
Researchers at HSE University's St Petersburg School of Economics and Management have been exploring methods for the efficient allocation of resources in systems involving multiple players. The scientists have proven the existence of strategies for optimal decision-making in competition for limited, discrete resources in four different cases. The developed mathematical model can be applied in various fields, ranging from education and medicine to managing networks and computing power. The paper has been published in Games and Economic Behaviour.
There is ongoing competition for various resources in the world, and individuals must choose the optimal strategy to acquire them. Prospective students must choose which school to apply to, companies compete for the best projects, and gym-goers rush to claim an available machine.
Scientists strive to find methods that help systems with many participants operate efficiently, avoiding overloads and the irrational use of resources. These problems can be translated into a mathematical model where agents make decisions based on potential benefits, available resources, and their capabilities. In game theory, such models are analysed through the lens of Nash equilibrium—a state where no participant wishes to change their choice, as doing so would not lead to increased benefits.
Researchers at HSE University's St Petersburg School of Economics and Management have proven the existence of Nash equilibrium in certain types of games involving indivisible (ie discrete) resources, such as airplane seats, apartments, library books, or school placements. These resources are considered discrete because they cannot be divided—for instance, you cannot take half a window seat and half an aisle seat on a plane.
To demonstrate the existence of Nash equilibrium, the researchers developed a mathematical model in which each participant (agent) selects one of the available resources. Participants aim to gain the greatest benefit and maximise their share of the resource. This creates a system where an agent's choice is influenced not only by the resource itself, but also by the number of other participants selecting the same resource.
In the model, each participant is assigned a certain weight, which plays a significant role. For example, when prospective students select a school, they consider factors such as the quality of the educational programme, the school's accessibility, its reputation, and more. They assess the value of the resource and their ability to acquire it. Weight represents the agent's own capabilities in relation to a specific resource.
The researchers proved that a Nash equilibrium, beneficial for all agents, exists in four cases. The first case occurs when the agent's weight is independent of the resource, while the values of the resources vary. For example, when an applicant with the highest USE score chooses from several schools with varying rankings.
In the second case, the agent's weight depends on the resource, but the values of all resources are the same, eg an applicant wins an internal olympiad at a top school and gains priority for admission.
In the third case, an agent's weight depends on the resource, the values of the resources vary, but there are only two resources: eg an applicant can easily enrol in a lesser-known school by winning an internal olympiad, or they can compete for a spot at a prestigious school on equal terms with others.
The fourth case is special. If the number of participants in the system is large—more than 20 agents—the system tends to approach equilibrium but does not fully reach it. This is an asymptotic result that accurately reflects reality. In such cases, each participant acts nearly optimally. Even if the strategy is changed, the situation cannot be significantly improved.
Vasily Gusev
'We did not immediately find a proof for the fourth case; it was suggested by a student participating in an internship programme run by our laboratory. His name is Mikhail Reshetov, and he became our co-author,' says Vasily Gusev, Senior Research Fellow at the International Laboratory of Game Theory and Decision Making at HSE Campus in St Petersburg and co-author of the paper.
According to the researchers, the developed mathematical model opens up opportunities for improving resource allocation systems across various fields, from education and medicine to the distribution of computing power and management of large networks.
Alexander Nesterov
'We have shown that competition for resources does not lead to chaos, but instead stabilises without external interference. We have mathematically proven that if agents are given the freedom of choice, each will find a profitable outcome for themselves,' comments Alexander Nesterov, Head of the International Laboratory of Game Theory and Decision Making at HSE Campus in St Petersburg and co-author of the paper.
See also:
Resource Race and Green Transition: Three Unexpected Conclusions from Foresight Centre’s Research on Climate and Poverty
Beneath the surface of green energy—which most people associate with solar panels, electric vehicles, and reduced CO2 emissions—lies a complex web of geopolitical interests, international inequality, and resource constraints. Researchers from the Laboratory for Science and Technology Studies (LST) at the HSE ISSEK Foresight Centre have published a series of articles in leading international journals on hidden and overt conflicts surrounding critically important metals and minerals, as well as related processes in the energy sector.
Immersion in Second Language Environment Influences Bilinguals’ Perception of Emotions
Researchers at the Cognitive Health and Intelligence Centre at the HSE Institute for Cognitive Neuroscience have discovered how bilingual individuals process emotional words in their native (first) and non-native (second) languages. It was found that the link between word meaning and bodily sensations is weaker in a second language than in a first language. However, the more a person is immersed in a language environment, the smaller this difference becomes. The article has been published in Language, Cognition and Neuroscience.
Researchers Find More Effective Approach to Revealing Majorana Zero Modes in Superconductors
An international team of researchers, including physicists from HSE MIEM, has demonstrated that nonmagnetic impurities can help more accurately reveal Majorana zero modes—quantum states considered promising building blocks for quantum computing. The researchers found that these impurities shift the energy levels that typically obscure the Majorana signal, while leaving the mode itself largely unaffected, thereby making its spectral peak more distinct. The study has been published in Research.
New Development by HSE Scientists Helps Design Reliable Electronics Faster at a Lower Cost
Scientists from HSE MIEM have developed a new approach to modelling electrothermal processes in high-power electronic circuits on printed circuit boards (PCB). The method allows engineers to quickly and accurately predict how electronic components heat up during operation, helping prevent overheating and potential failures. The results have been published in Russian Microelectronics.
The Future of Cardiogenetics Lies in Artificial Intelligence
Researchers from the AI and Digital Science Institute at the HSE Faculty of Computer Science have developed a program capable of analysing regions of the human genome that were previously inaccessible for accurate interpretation in genetic testing. The program adapts large generative AI (GenAI) models for cardiogenetics to predict how specific mutations affect the function of individual genes.
HSE Researchers: Young Russians Have Sufficient Knowledge About Money but Lack Money Management Skills
Adolescents and young adults in Russia today are well versed in financial terminology: they know what bank cards, loans, interest rates, and online payments are. However, as researchers at HSE University have found, real money-management skills remain poorly developed among most young people. The study ‘Financial Literacy, Financial Culture, and Financial Autonomy of Youth’ has been published in Monitoring of Public Opinion: Economic and Social Changes.
Why Weaker Competitors Give Up—and How to Keep Them in the Game
Anastasia Antsygina, Assistant Professor at HSE University’s Faculty of Economic Sciences, has developed a prize distribution model that maximises competitor engagement. She proposed revising the traditional ‘winner-takes-all’ approach and, in certain cases, offering a small reward even to those who have lost. According to her, this could increase participant motivation and make the competition more intense. The findings of her research were published in the Economic Theory journal.
HSE Researchers Compile Scientific Database for Studying Children’s Eating Habits
The database created at HSE University can serve as a foundation for studying children’s eating habits. This is outlined in the study ‘The Influence of Age, Gender, and Social-Role Factors on Children’s Compliance with Age-Based Nutritional Norms: An Experimental Study Using the Dish-I-Wish Web Application.’ The work has been carried out as part of the HSE Basic Research Programme and was presented at the XXVI April International Academic Conference named after Evgeny Yasin.
New Foresight Centre Study Identifies the Most Destructive Global Trends for Humankind
A team of researchers from the HSE International Research and Educational Foresight Centre has examined how global trends affect the quality of human life—from life expectancy to professional fulfilment. The findings of the study titled ‘Human Capital Transformation under the Influence of Global Trends’ were published in Foresight.
Scientists Develop Algorithm for Accurate Financial Time Series Forecasting
Researchers at the HSE Faculty of Computer Science benchmarked more than 200,000 model configurations for predicting financial asset prices and realised volatility, showing that performance can be improved by filtering out noise at specific frequencies in advance. This technique increased accuracy in 65% of cases. The authors also developed their own algorithm, which achieves accuracy comparable to that of the best models while requiring less computational power. The study has been published in Applied Soft Computing.