When Does Population Diversity Matter? A Unified Framework for Binary-Choice Dynamics

Abstract

We propose a modeling framework for binary-choice dynamics in which agents update their states using two mechanisms selected based on individual preference drawn from an arbitrary distribution. We compare annealed dynamics, where preferences change over time, and quenched dynamics, where they remain fixed. Our framework bridges gaps between existing models and provides a systematic approach to assess when individual-level diversity affects collective dynamics and when it can be effectively ignored. We identify a constraint on transition probabilities that makes annealed and quenched dynamics equivalent. We show that when this condition is satisfied, the quenched dynamics reduces to a one-dimensional system, ruling out oscillatory behavior that may otherwise emerge.

Figures (5)

• Figure 1: Structure of the modeling framework. Agent $i$ must choose between two options, $A$ and $B$. The choice is made by using one of two mechanisms $X$ or $Y$. The mechanisms are defined by transition probabilities, which give the likelihood of changing the option from $A$ to $B$ ($AB$) and from $B$ to $A$ ($BA$). The transition probabilities can be arbitrary functions of the actual fraction of agents with option $A$ in the system, denoted by $a$. The functions shown are only examples. Agent $i$ has a personal preference towards mechanism $X$ represented by probability $p_i$ of choosing this mechanism. With complementary probability, $1-p_i$, mechanism $Y$ is chosen. The preference, $p_i$, is assigned to agent $i$ from the preference distribution $\phi(p)$, which is the same for all the agents. In the annealed case, preferences change in time, while in the quenched case, they stay fixed, as illustrated in Fig. \ref{['fig:que-ann-vis']}.
• Figure 2: (a) Annealed dynamics: Agent $i$ is assigned a new preference towards mechanism $X$, $p_i(t)$, at each time step $t$ ($t_0$ and $t_n$ are highlighted in the plot as examples). This preference is drawn independently from the same distribution $\phi(p)$ for each agent and at each time step. As a result, the preferences change in an uncorrelated way. (b) Quenched dynamics: At the beginning of the dynamics, $t_0$, agent $i$ is assigned a preference, $p_i(t_0)$, which stays fixed throughout time. This preference is drawn independently from the same distribution $\phi(p)$ for each agent.
• Figure 3: Time evolution of the $q$-voter model with anticonformity Abr:Paw:Szn:19 for different distributions of preferences with the same mean: $\bar{p}=0.5$. Panels (a) and (b) correspond to the model parametrization that satisfies the balancing condition ($\alpha=\beta=6$), while panels (c) and (d) show the case that does not satisfy the condition ($\alpha=6$, $\beta=2$). The four distributions of preferences are: (e) uniform distribution on [0, 1], (f) normal distribution centered at $p=0.5$ with variance $\sigma^2=1/800$, (g) mixture of two normal distributions centered at $p=0$ and $p=1$, both with variance $\sigma^2=1/800$, and (h) beta distribution with parameters: $\alpha_\text{B}=\beta_\text{B}=2$. Each distribution is represented by a unique color and symbol consistently across all panels. For the parametrization satisfying the balancing condition, the trajectories for (a) annealed and (b) quenched dynamics coincide for all distributions with the same mean, and any heterogeneous population can be mapped into a homogeneous one with that mean. When the balancing condition is not satisfied, only (c) annealed dynamics depends on the mean, while (d) quenched dynamics depends on the full distribution shape. In this case, heterogeneous populations cannot be reduced to homogeneous ones, and they may exhibit oscillations.
• Figure 4: Fixed-point diagrams under the quenched dynamics for the noisy threshold $q$-voter model Vie:etal:20 ($q=10$, $q_0=9$), which does not satisfy the balancing condition. Each panel corresponds to a different distribution of preferences: (a) sliding one-point distribution, (b) expanding uniform distribution, and (c) Bernoulli distribution. Each distribution leads to qualitatively different fixed-point diagram.
• Figure 5: Fixed-point diagrams under the quenched dynamics for the dynamical system model of decision-making Yan:etal:21 ($q=1.5$ and $m=0.6$), which satisfies the balancing condition. Each panel corresponds to a different distribution of preferences: (a) sliding one-point distribution, (b) expanding uniform distribution, and (c) Bernoulli distribution. Despite differences in shape, all the distributions lead to the same fixed-point diagram.