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1. A tale of two (and more) altruists

   https://doi.org/10.1088/1742-5468/ac2906  

   De Bruyne, B; Randon-Furling, J; Redner, S (October 2021, Journal of Statistical Mechanics: Theory and Experiment)

   Abstract We introduce a minimalist dynamical model of wealth evolution and wealth sharing among N agents as a platform to compare the relative merits of altruism and individualism. In our model, the wealth of each agent independently evolves by diffusion. For a population of altruists, whenever any agent reaches zero wealth (that is, the agent goes bankrupt), the remaining wealth of the other N − 1 agents is equally shared among all. The population is collectively defined to be bankrupt when its total wealth falls below a specified small threshold value. For individualists, each time an agent goes bankrupt (s)he is considered to be ‘dead’ and no wealth redistribution occurs. We determine the evolution of wealth in these two societies. Altruism leads to more global median wealth at early times; eventually, however, the longest-lived individualists accumulate most of the wealth and are richer and more long lived than the altruists. 

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2. Wealth Dynamics Over Generations: Analysis and Interventions

   https://doi.org/10.1109/SaTML54575.2023.00013  

   Acharya, Krishna; Arunachaleswaran, Eshwar Ram; Kannan, Sampath; Roth, Aaron; Ziani, Juba (February 2023, IEEE Conference on Secure and Trustworthy Machine Learning (SaTML))

   We present a stylized model with feedback loops for the evolution of a population's wealth over generations. Individuals have both talent and wealth: talent is a random variable distributed identically for everyone, but wealth is a random variable that is dependent on the population one is born into. Individuals then apply to a downstream agent, which we treat as a university throughout the paper (but could also represent an employer) who makes a decision about whether to admit them or not. The university does not directly observe talent or wealth, but rather a signal (representing e.g. a standardized test) that is a convex combination of both. The university knows the distributions from which an individual's type and wealth are drawn, and makes its decisions based on the posterior distribution of the applicant's characteristics conditional on their population and signal. Each population's wealth distribution at the next round then depends on the fraction of that population that was admitted by the university at the previous round. We study wealth dynamics in this model, and give conditions under which the dynamics have a single attracting fixed point (which implies population wealth inequality is transitory), and conditions under which it can have multiple attracting fixed points (which implies that population wealth inequality can be persistent). In the case in which there are multiple attracting fixed points, we study interventions aimed at eliminating or mitigating inequality, including increasing the capacity of the university to admit more people, aligning the signal generated by individuals with the preferences of the university, and making direct monetary transfers to the less wealthy population. 

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3. Dynamic size counting in population protocols

   David Doty and Mahsa Eftekhari (January 2022, SAND 2022: 1st Symposium on Algorithmic Foundations of Dynamic Networks)

   James Aspnes and Othon Michail (Ed.)

   The population protocol model describes a network of anonymous agents that interact asynchronously in pairs chosen at random. Each agent starts in the same initial state s. We introduce the *dynamic size counting* problem: approximately counting the number of agents in the presence of an adversary who at any time can remove any number of agents or add any number of new agents in state s. A valid solution requires that after each addition/removal event, resulting in population size n, with high probability each agent "quickly" computes the same constant-factor estimate of the value log2n (how quickly is called the *convergence* time), which remains the output of every agent for as long as possible (the *holding* time). Since the adversary can remove agents, the holding time is necessarily finite: even after the adversary stops altering the population, it is impossible to *stabilize* to an output that never again changes. We first show that a protocol solves the dynamic size counting problem if and only if it solves the *loosely-stabilizing counting* problem: that of estimating logn in a *fixed-size* population, but where the adversary can initialize each agent in an arbitrary state, with the same convergence time and holding time. We then show a protocol solving the loosely-stabilizing counting problem with the following guarantees: if the population size is n, M is the largest initial estimate of logn, and s is the maximum integer initially stored in any field of the agents' memory, we have expected convergence time O(logn+logM), expected polynomial holding time, and expected memory usage of O(log2(s)+(loglogn)2) bits. Interpreted as a dynamic size counting protocol, when changing from population size nprev to nnext, the convergence time is O(lognnext+loglognprev). 

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4. Time-Space Trade-offs in Population Protocols

   https://doi.org/10.1137/1.9781611974782.169  

   Alistarh, Dan; Aspnes, James; Eisenstat, David; Gelashvili, Rati; Rivest, Ronald L. (January 2017, Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms)

   Population protocols are a popular model of distributed computing, in which randomly-interacting agents with little computational power cooperate to jointly perform computational tasks. Inspired by developments in molecular computation, and in particular DNA computing, recent algorithmic work has focused on the complexity of solving simple yet fundamental tasks in the population model, such as leader election (which requires convergence to a single agent in a special “leader” state), and majority (in which agents must converge to a decision as to which of two possible initial states had higher initial count). Known results point towards an inherent trade-off between the time complexity of such algorithms, and the space complexity, i.e. size of the memory available to each agent. In this paper, we explore this trade-off and provide new upper and lower bounds for majority and leader election. First, we prove a unified lower bound, which relates the space available per node with the time complexity achievable by a protocol: for instance, our result implies that any protocol solving either of these tasks for n agents using O(log log n) states must take Ω(n/polylogn) expected time. This is the first result to characterize time complexity for protocols which employ super-constant number of states per node, and proves that fast, poly-logarithmic running times require protocols to have relatively large space costs. On the positive side, we give algorithms showing that fast, poly-logarithmic convergence time can be achieved using O (log2 n) space per node, in the case of both tasks. Overall, our results highlight a time complexity separation between O (log log n) and Θ(log2 n) state space size for both majority and leader election in population protocols, and introduce new techniques, which should be applicable more broadly. 

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5. Allocating Public Goods via Dynamic Max-Min Fairness: Long-Run Behavior and Competitive Equilibria

   https://doi.org/10.1145/3711695  

   Onyeze, Chido; Banerjee, Siddhartha; Fikioris, Giannis; Tardos, Éva (March 2025, Proceedings of the ACM on Measurement and Analysis of Computing Systems)

   Dynamic max-min fair allocation (DMMF) is a simple and popular mechanism for the repeated allocation of a shared resource among competing agents: in each round, each agent can choose to request or not for the resource, which is then allocated to the requesting agent with the least number of allocations received till then. Recent work has shown that under DMMF, a simple threshold-based request policy enjoys surprisingly strong robustness properties, wherein each agent can realize a significant fraction of her optimal utility irrespective of how other agents' behave. While this goes some way in mitigating the possibility of a 'tragedy of the commons' outcome, the robust policies require that an agent defend against arbitrary (possibly adversarial) behavior by other agents. This however may be far from optimal compared to real world settings, where other agents are selfish optimizers rather than adversaries. Therefore, robust guarantees give no insight on how agents behave in an equilibrium, and whether outcomes are improved under one. Our work aims to bridge this gap by studying the existence and properties of equilibria under DMMF. To this end, we first show that despite the strong robustness guarantees of the threshold based strategies,no Nash equilibrium existswhen agents participate in DMMF, each using some fixed threshold-based policy. On the positive side, however, we show that for the symmetric case, a simple data-driven request policy guarantees that no agent benefits from deviating to a different fixed threshold policy. In our proposed policy agents aim to match the historical allocation rate with a vanishing drift towards the rate optimizing overall welfare for all users. Furthermore, the resulting equilibrium outcome can be significantly better compared to what follows from the robustness guarantees. Our results are built on a complete characterization of the steady-state distribution under DMMF, as well as new techniques for analyzing strategic agent outcomes under dynamic allocation mechanisms; we hope these may prove of independent interest in related problems. 

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