A statistical physics perspective on fairness in shared expenses: The bar bill analogy

In social contexts where individuals consume varying amounts, such as shared meals or bar gatherings, splitting the total bill equally often yields surprisingly fair outcomes. In this work, we develop a statistical physics framework to explain this emergent fairness by modeling individual consumption as stochastic variables drawn from a realistic distribution, specifically the gamma distribution. Introducing a Boltzmann-like weighting factor, we derive exact analytical expressions for the partition function, average consumption, variance, and entropy under economic or social penalization constraints. Numerical simulations, performed using the Marsaglia-Tsang algorithm, confirm the analytical results with high precision. Drawing a direct parallel between individual consumption and ideal gas particle energy in the canonical ensemble, we show how the law of large numbers, mutual compensation, and the effective ordering induced by penalization combine to make equal cost-sharing statistically robust and predictable. These findings reveal that what appears to be an informal social convention is, in fact, grounded in the same fundamental principles that govern the collective behavior of particles in thermodynamic systems, highlighting the interdisciplinary power of statistical physics.