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:
Physicists Discover What Happens Inside a Stable Vortex
Large vortices with characteristic spiral arms are often observed in the atmosphere and the ocean. Physicists from HSE University have explained how these structures form and why they retain their shape. The researchers found that velocities at points located along the same vortex arc remain correlated even over long distances. At the same time, this correlation weakens rapidly with increasing distance from the vortex centre. These differences help explain the formation of spiral arms and may improve models of atmospheric and oceanic currents. The findings have been published in Physical Review Fluids.
‘The Peak of Stupidity’ and ‘The Valley of Despair’: HSE Economists Propose an Explanation for the Dunning–Kruger Effect
The Dunning–Kruger effect, which describes a sharp surge in self-confidence among beginners followed by an equally rapid decline as they gain experience, can be explained by the nature of the learning process and the acquisition of new knowledge. This conclusion was reached by Andrey Vorchik of the HSE Faculty of Economic Sciences together with independent researcher Murat Mamyshev. They developed a mathematical model of learning and demonstrated how subjective confidence is formed and changes as knowledge accumulates, as well as how teachers can reduce the ‘valley of despair’ experienced by learners.
Toffee and Risk: Scientists Discover Why People Who Crave Sweets Make More Impulsive Choices
Having a sweet tooth may be linked not only to eating habits but also to the way people make decisions. Researchers at HSE University have found that people with a preference for sweet foods tend to behave more impulsively—not because they want immediate rewards, but because they are less willing to tolerate uncertainty. These findings may help improve treatments for addiction. The study findings have been published in Frontiers in Psychology.
Physicists Find a Way to Model Ion Parameters in Plasma in Seconds
Researchers from HSE University and the Moscow Institute of Physics and Technology (MIPT) have developed a set of simple analytical methods for calculating the properties of heavy ions in helium under the influence of a strong electric field. The new approach speeds up calculations of ion mobility and ion–molecule reaction rates by thousands of times while maintaining sufficient accuracy for plasma jet modelling. The findings have been published in the journal Physica Scripta.
Two Years of Growth or Decline: How to Choose an Investment Strategy
Economists from HSE University, together with colleagues from international universities, have analysed stock market movements over almost a century and proposed an investment strategy that could have delivered returns nearly twice as high as the market average. Their research suggests following a momentum strategy during periods of sustained market growth and switching to a value strategy after prolonged market declines. The study has been published in the Journal of Banking and Finance.
Researchers Reveal Link Between Attention and Communication Difficulties in Autism
Researchers at HSE University have examined how communication difficulties in children with autism are related to brain function. The findings show that not only language networks but also attention networks play an important role. The weaker the connections involved in maintaining focus and switching attention, the more pronounced communication difficulties were. The study has been published in European Child & Adolescent Psychiatry.
Scientists Discover Why Some People Wore Masks During COVID-19 While Others Did Not
Why do some people voluntarily follow new rules while others ignore them? Researchers at HSE University have found that the answer lies not so much in people's willingness to cooperate, as previously believed, but in their ability to empathise with others. Empathy proved to be the strongest predictor of whether people chose to wear face masks voluntarily during the COVID-19 pandemic. The findings have been published in Frontiers.
Physicists at HSE University and FIAN Discover Way to 'Photograph' Sound for Testing Materials Used in 6G Communications
Researchers at HSE University, in collaboration with colleagues from the Lebedev Physical Institute of the Russian Academy of Sciences (FIAN), have developed a method for rapidly determining how firmly a film is bonded to a substrate. This is important for the creation of ultrahigh-frequency acoustic filters, which are key components of next-generation 5G and 6G communications. For the first time, researchers have succeeded in measuring the lateral rigidity of the bond between a two-dimensional material film and a substrate in this way. The study results have been published in Applied Physics Letters.
Scientists Create Open Dataset for Studying Concentration
A team of Russian researchers, including scientists from HSE University–St Petersburg, has developed the first open multimodal dataset containing recordings of brain activity, heart function, and video observations to help researchers understand what happens in the human brain during deep concentration. In the future, the dataset could accelerate the development of neural interfaces, rehabilitation technologies, and AI systems. The article has been published in Scientific Data.
Scientists Propose Method for More Efficient Resource Use in Machine Learning
An international group of researchers, including mathematicians from the AI and Digital Science Institute at the HSE Faculty of Computer Science, has provided a theoretical justification for a simple and computationally efficient method of estimating uncertainty in Stochastic Gradient Descent (SGD). The paper has been published on the scientific preprint server arXiv.org and presented at AISTATS 2026.