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:
HSE University Scholars Uncover E-Learning Preferences of Top Students
HSE University experts have analysed students’ digital footprints and shown for the first time that final grades depend on one’s personal approach to an online course. Balanced students have proven to be more successful than those who follow a more traditional and practical approach. The findings from this study will help create a more adaptive and personalised educational system. This research has been published in the journal The Internet and Higher Education.
HSE Scientists Develop Method to Stabilise Iodine in Solar Cells
Scientists at HSE MIEM, in collaboration with colleagues from China, have developed a method to improve the durability of perovskite solar cells by addressing iodine loss from the material. The researchers introduced quaternary ammonium molecules into the perovskite structure; these molecules form strong electrostatic pairs with iodine ions, effectively anchoring them within the crystal lattice. As a result, the solar cells retain more than 92% of their power after a thousand hours of operation at 85°C. The study has been published in Advanced Energy Materials.
HSE Researchers Create Genome-Wide Map of Quadruplexes
An international team, including researchers from HSE University, has created the first comprehensive map of quadruplexes—unstable DNA structures involved in gene regulation. For the first time, scientists have shown that these structures function in pairs: one is located in a DNA region that initiates gene transcription, while the other lies in a nearby region that enhances this process. In healthy tissues, quadruplexes regulate tissue-specific genes, whereas in cancerous tissues they influence genes responsible for cell growth and division. These findings may contribute to the development of new anticancer drugs that target quadruplexes. The study has been published in Nucleic Acids Research.
Mathematician from HSE University–Nizhny Novgorod Solves Equation Considered Unsolvable in Quadratures Since 19th Century
Mathematician Ivan Remizov from HSE University–Nizhny Novgorod and the Institute for Information Transmission Problems of the Russian Academy of Sciences has made a conceptual breakthrough in the theory of differential equations. He has derived a universal formula for solving problems that had been considered unsolvable in quadratures for more than 190 years. This result fundamentally reshapes one of the oldest areas of mathematics and has potential to have important implications for fundamental physics and economics. The paper has been published in Vladikavkaz Mathematical Journal.
Scientists Reveal How Language Supports Complex Cognitive Processing in the Brain
Valeria Vinogradova, a researcher at HSE University, together with British colleagues, studied how language proficiency affects cognitive processing in deaf adults. The study showed that higher language proficiency—regardless of whether the language is signed or spoken—is associated with higher activity and stronger functional connectivity within the brain network responsible for cognitive task performance. The findings have been published in Cerebral Cortex.
HSE AI Research Centre Simplifies Particle Physics Experiments
Scientists at the HSE AI Research Centre have developed a novel approach to determining robustness in deep learning models. Their method works eight times faster than an exhaustive model search and significantly reduces the need for manual verification. It can be applied to particle physics problems using neural networks of various architectures. The study has been published in IEEE Access.
Scientists Show That Peer Influence Can Be as Effective as Expert Advice
Eating habits can be shaped not only by the authority of medical experts but also through ordinary conversations among friends. Researchers at HSE University have shown that advice from peers to reduce sugar consumption is just as effective as advice from experts. The study's findings have been published in Frontiers in Nutrition.
HSE University Develops Tool for Assessing Text Complexity in Low-Resource Languages
Researchers at the HSE Centre for Language and Brain have developed a tool for assessing text complexity in low-resource languages. The first version supports several of Russia’s minority languages, including Adyghe, Bashkir, Buryat, Tatar, Ossetian, and Udmurt. This is the first tool of its kind designed specifically for these languages, taking into account their unique morphological and lexical features.
HSE Scientists Uncover How Authoritativeness Shapes Trust
Researchers at the HSE Institute for Cognitive Neuroscience have studied how the brain responds to audio deepfakes—realistic fake speech recordings created using AI. The study shows that people tend to trust the current opinion of an authoritative speaker even when new statements contradict the speaker’s previous position. This effect also occurs when the statement conflicts with the listener’s internal attitudes. The research has been published in the journal NeuroImage.
Language Mapping in the Operating Room: HSE Neurolinguists Assist Surgeons in Complex Brain Surgery
Researchers from the HSE Center for Language and Brain took part in brain surgery on a patient who had been seriously wounded in the SMO. A shell fragment approximately five centimetres long entered through the eye socket, penetrated the cranial cavity, and became lodged in the brain, piercing the temporal lobe responsible for language. Surgeons at the Burdenko Main Military Clinical Hospital removed the foreign object while the patient remained conscious. During the operation, neurolinguists conducted language tests to ensure that language function was preserved.