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Dr Alpár Mészáros from our department of Mathematical Sciences sheds light on inequality through the lens of math. He is teaming up with Dr Mauro Bambi of Durham University Business School to solve inequality's code using the surprising key of partial differential equations (PDEs).

In our daily lives, we engage in continuous interactions with others, making game theory a valuable tool for understanding most societal activities. This is because these activities typically involve intricate interactions between individual agents, who collaborate to achieve common goals or compete with one another. The collective behaviours of these large groups of agents can significantly impact and shape individual decisions.

For instance, the value of a company’s stock on the stock market is influenced not only by the company’s profitability but also by its perceived popularity and reputation among shareholders. Similarly, when students select their academic modules, their choices are heavily influenced by the experiences and opinions of previous students who have taken these modules in the past. One way to analyse these scenarios is by examining the concept of equilibria within game theory. Roughly speaking, we can define a Nash equilibrium as a situation where no agent has any incentive to unilaterally deviate from their current choices.

In models that involve a vast number of agents, often in the millions or billions, and where the impact of any individual agent is negligible when compared to the entire agent population, rather than attempting to track the actions of each agent, it becomes more practical to describe the model by focusing on the evolution of the density or distribution of these agents. This can be achieved by utilising the powerful tools of partial differential equations (PDEs).

Partial differential equations, originally introduced by Euler nearly 300 years ago to model perfect fluids within the framework of classical Newtonian mechanics, have proven to be indispensable for comprehending complex phenomena in our world. Consequently, PDEs serve as a unique lens through which various models in fields such as physics, biology, chemistry, economics, as well as social sciences and machine learning, can be enriched with new perspectives.

The concept of mean field games (MFGs), which emerged around 17 years ago through the work of two groups, Lasry—Lions and Huang—Malhamé--Caines, is designed to address scenarios involving an exceedingly large number of agents, potentially tending to infinity. Much like how PDEs allow us to describe the behaviour of a fluid not by tracking the motion of each individual particle but by analysing the particle density, PDEs are at the core of MFGs.

This relatively recent theory has proven to be an exceptionally powerful tool in the search for Nash equilibria within large systems of competitive agents. Its applications span a wide range of fields, including finance, cybersecurity, population dynamics, and macroeconomics, among others. One such specific macroeconomic model concerns with the distribution of income and wealth within a society. While models of this nature were initially introduced by Aiygari and Krussel--Smith in the 1990s, they have been recently revisited from the perspective of MFG, as demonstrated by the work of Achdou—Han—Lasry—Lions—Moll in 2022. Their fresh perspective has yielded intriguing insights, including “an analytical characterisation of the consumption and savings behaviour of the poor, particularly their marginal propensities to consume.”

These models, which aim to understand wealth and income distribution, hold tremendous untapped potential, primarily due to the inherent complexity of real-world socio-economic systems. To explore this potential, Dr Mauro Bambi from Durham Business School and I have launched an ambitious research programme. We are set to lead a major project at Durham IAS titled ‘The Many Facets of Social Inequality’, commencing in Epiphany 2025.

This project's objectives are multifaceted, reflecting our collaboration with colleagues from all four faculties at Durham University. We intend to delve into the underlying drivers of social inequality, incorporating novel elements into these models. These elements encompass behavioural factors, societal habits, trust dynamics, and notably, the impact of ‘big shocks’, such as the recent pandemic.

Our aspiration is that, through the fusion of our mathematical and macroeconomic expertise, alongside interdisciplinary collaborations with social scientists, psychologists, anthropologists, and philosophers, we can shed new light on the complexities of these critical questions.

We are confident that Durham IAS will offer an intellectually invigorating environment conducive to the success of this project.

Find out more:

Our Department of Mathematical Sciences combines world-leading research with a dedication to the learning experience of our students. Ranked 4th in the UK in The Complete University Guide 2023, we offer a unique blend of high-quality teaching and research across a wide range of disciplines and provide practical experience to support future careers and employment prospects.

Feeling inspired? Visit our Mathematical Sciences webpages to learn more about our postgraduate and undergraduate programmes.

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