The persistence of disparity is thus attributed to classifier policy.

The persistence of disparity is thus attributed to classifier policy.

Institutions play a critical role in enabling communities to manage common-pool resources and avert tragedies of the commons. However, a fundamental issue arises: Individuals typically perceive participation as advantageous only after an institution is established, creating a paradox: How can institutions form if no one will join before a critical mass exists? We term this conundrum the institution bootstrapping problem and propose that misperception, specifically, agents' erroneous belief that an institution already exists, could resolve this paradox. By integrating well-documented psychological phenomena, including cognitive biases, probability distortion, and perceptual noise, into a game-theoretic framework, we demonstrate how these factors collectively mitigate the bootstrapping problem. Notably, unbiased perceptual noise (e.g., noise arising from agents' heterogeneous physical or social contexts) drastically reduces the critical mass of cooperators required for institutional emergence. This effect intensifies with greater diversity of perceptions. We explain this counter-intuitive result through asymmetric boundary conditions: proportional underestimation of low-probability sanctions produces distinct outcomes compared to equivalent overestimation. Furthermore, the type of perceptual distortion, proportional versus absolute, yields qualitatively different evolutionary pathways. These findings challenge conventional assumptions about rationality in institutional design, highlighting how "noisy" cognition can paradoxically enhance cooperation. Finally, we contextualize these insights within broader discussions of multi-agent system design and collective action. Our analysis underscores the importance of incorporating human-like cognitive constraints, not just idealized rationality, into models of institutional emergence and resilience.

This paper presents a physics-informed deep learning approach for predicting the replicator equation, allowing accurate forecasting of population dynamics. This methodological innovation allows us to derive governing differential or difference equations for systems that lack explicit mathematical models. We used the SINDy model first introduced by Fasel, Kaiser, Kutz, Brunton, and Brunt 2016a to get the replicator equation, which will significantly advance our understanding of evolutionary biology, economic systems, and social dynamics. NTRODUCTION Game theory helps to understand how strategic behaviours evolve and persist in biological, social, and economic systems where individuals interact. It also helps in how complex social behaviours and strategies can evolve and persist in diverse contexts.

It has long been posited that there is a connection between the dynamical equations describing evolutionary processes in biology and sequential Bayesian learning methods. This manuscript describes new research in which this precise connection is rigorously established in the continuous time setting. Here we focus on a partial differential equation known as the Kushner-Stratonovich equation describing the evolution of the posterior density in time. Of particular importance is a piecewise smooth approximation of the observation path from which the discrete time filtering equations, which are shown to converge to a Stratonovich interpretation of the Kushner-Stratonovich equation. This smooth formulation will then be used to draw precise connections between nonlinear stochastic filtering and replicator-mutator dynamics. Additionally, gradient flow formulations will be investigated as well as a form of replicator-mutator dynamics which is shown to be beneficial for the misspecified model filtering problem. It is hoped this work will spur further research into exchanges between sequential learning and evolutionary biology and to inspire new algorithms in filtering and sampling.

We introduce a mathematical model that combines the concepts of complex contagion with payoff-biased imitation, to describe how social behaviors spread through a population. Traditional models of social learning by imitation are based on simple contagion -- where an individual may imitate a more successful neighbor following a single interaction. Our framework generalizes this process to incorporate complex contagion, which requires multiple exposures before an individual considers adopting a different behavior. We formulate this as a discrete time and state stochastic process in a finite population, and we derive its continuum limit as an ordinary differential equation that generalizes the replicator equation, the most widely used dynamical model in evolutionary game theory. When applied to linear frequency-dependent games, our social learning with complex contagion produces qualitatively different outcomes than traditional imitation dynamics: it can shift the Prisoner's Dilemma from a unique all-defector equilibrium to either a stable mixture of cooperators and defectors in the population, or a bistable system; it changes the Snowdrift game from a single to a bistable equilibrium; and it can alter the Coordination game from bistability at the boundaries to two internal equilibria. The long-term outcome depends on the balance between the complexity of the contagion process and the strength of selection that biases imitation towards more successful types. Our analysis intercalates the fields of evolutionary game theory with complex contagions, and it provides a synthetic framework that describes more realistic forms of behavioral change in social systems.

Evolution describes how distributions change. Specifically, evolution provides a model for how a population's distribution of traits or strategies (hereafter hypotheses) changes over time as an environment modulates reproduction rates (i.e., of individuals or of hypotheses; Lloyd, 2020): Hypotheses that have higher fitness are "selected" by the environment and, in expectation, become more popular with time. The replicator equation is a formal, analytic model of evolution and is indispensable to biology (Sinervo and Calsbeek, 2006; Queller, 2017). In the replicator equation, the absolute fitness (in this paper, the negative loss L) of hypotheses h H is identified with its rate of replication: exponential growth (or decline) in a population where different hypotheses compete for relative frequency ρ(h) [0, 1]. For probability distributions over hypothesis space H, this equation induces replicator dynamics, selecting hypotheses with lower than average loss.

Motivated by our recent work on rooted tree matching, in this paper we provide a solution to the problem of matching two free (i.e., unrooted) trees by constructing an association graph whose maximal cliques are in one-to-one correspondence with maximal common subtrees. We then solve the problem using simple replicator dynamics from evolutionary game theory. Experiments on hundreds of uniformly random trees are presented. The results are impressive: despite the inherent inability of these simple dynamics to escape from local optima, they always returned a globally optimal solution.