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.
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