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A Statistical Physics perspective on fairness in shared expenses: The bar bill analogy

Nuno Crokidakis, Lucas Sigaud

TL;DR

The paper addresses why equal-cost sharing often feels fair despite heterogeneous individual consumption. It casts bar-bill division as a canonical ensemble with a Boltzmann-like penalty $e^{-${\alpha}x}$, using a gamma prior $\rho(x)$ for expenditures and deriving closed-form expressions for $Z_{\alpha}$, $\langle x\rangle_{\alpha}$, $\sigma_{\alpha}^2$, and $S_{\alpha}$; validation is provided via Marsaglia-Tsang gamma sampling. Key contributions include showing that increasing $\alpha$ reduces both mean expenditure and dispersion, and that the relative fluctuation of the group-averaged bill scales as $N^{-1/2}$, indicating robust fairness in larger groups. The findings offer a concrete thermodynamic mechanism for a common social convention and provide a quantitative framework for fairness in collective spending with potential implications for sociophysics and econophysics.

Abstract

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.

A Statistical Physics perspective on fairness in shared expenses: The bar bill analogy

TL;DR

The paper addresses why equal-cost sharing often feels fair despite heterogeneous individual consumption. It casts bar-bill division as a canonical ensemble with a Boltzmann-like penalty {\alpha}x}\rho(x)Z_{\alpha}\langle x\rangle_{\alpha}\sigma_{\alpha}^2S_{\alpha}\alphaN^{-1/2}$, indicating robust fairness in larger groups. The findings offer a concrete thermodynamic mechanism for a common social convention and provide a quantitative framework for fairness in collective spending with potential implications for sociophysics and econophysics.

Abstract

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.

Paper Structure

This paper contains 6 sections, 15 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Statistical model and Thermodynamic analogy
  3. The maximum relative entropy method: A natural way of explanation
  4. Canonical analysis of statistical quantities
  5. Simulation results
  6. Discussion and conclusions

Figures (3)

  • Figure 1: Normalized histogram (squares), constructed by dividing the number of occurrences in each bin by the total number of generated values, for gamma-distributed random numbers $x$ obtained numerically as described in the text. For comparison, we plot the analytical gamma distribution given by Eq. \ref{['eq7']} (solid line). The parameters are $k=2.0$ and $\theta=10.0$, and $10^{7}$ random values were generated.
  • Figure 2: Comparison between analytical (solid lines) and numerical (squares) results for three key thermodynamic quantities as functions of the penalization parameter $\alpha$: (a) the canonical average consumption $\langle x\rangle_{\alpha}$, (b) the canonical variance $\sigma^{2}_{\alpha}$, and (c) the canonical entropy $S_{\alpha}$. Numerical results were obtained from simulations using $10^{6}$ samples per value of $\alpha$, with error bars smaller than the symbol sizes. The excellent agreement between analytical predictions and numerical estimates confirms the validity of the theoretical framework and the robustness of the computational implementation. The group size is $N=10$.
  • Figure 3: Relative deviation $\delta_{N}$, given by Eq. \ref{['eq16']}, vs group size $N$, for typical values of $\alpha$, plotted in the log-log scale. As a guide to the eye, a straight line with slope $-0.5$ is shown. Each point represents an average over $10^{6}$ independent simulations.