Models of wealth distribution

The aim of this literature review is to understand better how wealth is distributed under different models: wealth exchange, proof of stake and related.
This helps understand how governance tokens, and thus governance power, might be distributed.

A useful intuition-building toy model for wealth exchange is the Yard Sale model:

  • This article deals about it in context of econophysics.
  • A basic simulation of which can be found here.

A very fruitful toy model for PoS and other proportional accumulation schemes is the Polya Urn:

  • A complete and easy-to-use simulator of which can be found here.
  • To understand better the Beta distribution, look here.
  • A basic simulation of which can be found here.
  • Another simulation with comparison with target distributions (Dirichlet, Beta) can be found here.

Cross-paper remarks:

  • Interestingly, between [Degado et al, 2022] (an ABM approach) and [Bouchaud et al, 2000] (a mathematical econophysics approach), wealth inequality is proven to be reached by very different means.
  • [Fanti et al, 2018] uses a Beta distribution which is not in line with [Roşu et al, 2020] and other Polya Urn models. [I couldn’t make clear where the surprising discrepancy with other papers and simulations stems from and how it might be justified. I tend to favor [Roşu et al, 2020]'s model which is a bit more detailed and easier to follow].

Literature review: models of wealth distribution

Agent Base Models

Agent-based Model of Initial Token Allocations: Evaluating Wealth Concentration in Fair Launches

Econophysics review: II. Agent-based models

  • Source:
  • Authors: Chakraborti, A., Toke, I. M., Patriarca, M., & Abergel, F.
  • Year: 2011
  • Description: Review of econophysics models, including kinetic models and ABMs
  • Relevance: Useful and complete review of ABMs, a staple modeling framework. The review of Kinetic Weatlh Exchange Models points to, without savings, an equilibrium wealth distribution with an exponential tail (pp. 13, 14). Hence, a simple form of exchange produces an exponential wealth distribution. This statement holds notably for governance tokens.

Wealth condensation in a simple model of economy

  • Source: [cond-mat/0002374] Wealth condensation in a simple model of economy
  • Authors: Jean-Philippe Bouchaud, Marc Mezard
  • Year: 2000
  • Description: Introduces a simple but powerful model of wealth exchange and redistribution that hints at concentration by default under exchanges
  • Relevance: Seminal paper in econophysics, useful both as an inspiration and in terms of its conclusions. Imports physics concepts, like phase-transformation from fluid dynamics, in context of wealth dynamics. Concludes that wealth concentrates (condensates) towards a Pareto (power) law (in real world, this is a Pareto-tail, or Pareto “near the top”). More exchange produces a less unequal distribution (in context of this paper, unequal means very broadly distributed along the wealth line). Tweaking exchange and redistribution rates can change the equilibrium, akin to a phase transformation.

Evolution of wealth distribution in Proof of Stake

Evolution of Shares in a Proof-of-Stake Cryptocurrency

  • Source:
  • Authors: Roşu, I., & Saleh, F.
  • Year: 2020
  • Description: Studies the evolution of shares of stakers under a Polya Urn-based model
  • Relevance: In short, staking rewards do not produce concentration. Under a buy-and-hold strategy, investor shares are stable in the long run and converge to Dirichlet distributions with parameters the initial distribution. This also holds when trading is enabled between agents but under some conditions (delta-neutral). This type of result can be extended to other forms of dividend bearing tokens as long as rewards are proportional to stake and do not grow too fast.

Compounding of Wealth in Proof-of-Stake Cryptocurrencies

  • Source: [1809.07468] Compounding of Wealth in Proof-of-Stake Cryptocurrencies
  • Authors: Giulia Fanti, Leonid Kogan, Sewoong Oh, Kathleen Ruan, Pramod Viswanath, Gerui Wang
  • Year: 2018
  • Description: Introduces the Equitability concept, studies evolutions of stakers shares under Polya Urn model, concludes there is compounding and thus concentration of wealth, and suggests the Geometric Reward Function as a a solution
  • Relevance: In short, constant staking rewards do produce concentration, because of randomness of rewards and stake compounding (which is considered true by default in PoS). This conclusion is partly based on a disconcerting [to me] usage of the Beta(⅓, ⅔) distribution as a limit distribution of shares, whereas this distribution suggests that the initial distributions are below the reward unit. This conclusion is further used as a basis to develop a more complex reward scheme. Nevertheless, the concept of Equitability, the power for an investor to amplify her stake, can prove very useful.