Consensus Dynamics in a non-deterministic Naming Game with Shared Memory

In the naming game, individuals or agents exchange pairwise local information in order to communicate about objects in their common environment. The goal of the game is to reach a consensus about naming these objects. Originally used to investigate language formation and self-organizing vocabularies, we extend the classical naming game with a globally shared memory accessible by all agents. This shared memory can be interpreted as an external source of knowledge like a book or an Internet site. The extended naming game models an environment similar to one that can be found in the context of social bookmarking and collaborative tagging sites where users tag sites using appropriate labels, but also mimics an important aspect in the field of human-based image labeling. Although the extended naming game is non-deterministic in its word selection, we show that consensus towards a common vocabulary is reached. More importantly, we show the qualitative and quantitative influence of the external source of information, i.e. the shared memory, on the consensus dynamics between the agents.

💡 Research Summary

This paper extends the classic naming game—a minimal model of how a population of agents can converge on a shared vocabulary—by introducing a globally shared memory that all agents can read with a certain probability. In the original naming game, agents interact pairwise: a speaker selects a word from its local inventory (or invents a new one if empty) and transmits it to a hearer; successful interactions lead to the removal of all competing words from both inventories. The extended model adds two stochastic mechanisms governed by a parameter λ (0 ≤ λ ≤ 1). First, when a speaker’s local memory is empty, it either draws a word from the shared memory with probability λ or invents a new word with probability 1 − λ. Second, after a successful transmission, the two agents consult the shared memory with probability λ; if the transmitted word is present there, they purge all other words, otherwise they simply keep only the transmitted word. The shared memory is initialized with C distinct words (C ∈ {1, 5, 10, 50, 100, 500}) and remains static throughout each simulation run.

The authors implement the model on a fully connected network of N = 100 agents, varying λ from 0.0 to 1.0 in steps of 0.1 and testing each (λ, C) pair over 1,000 independent runs. Three standard observables are recorded as functions of discrete time t: (i) the total number of word tokens in the system, N_w(t); (ii) the number of distinct word types, N_d(t); and (iii) the success rate S(t), defined as 1 for a successful interaction and 0 otherwise. Averaged curves reveal a characteristic disorder‑to‑order transition: initially N_w(t) rises as many failures introduce new words, N_d(t) quickly reaches a peak, and S(t) remains low. As successful exchanges become more frequent, N_w(t) and N_d(t) attain their maxima, after which both decline while S(t) accelerates toward 1, indicating convergence to a consensus state (all agents hold the same single word).

Key quantitative findings include:

1. Effect of λ on lexical diversity. For any fixed C, the peak value of N_d(t) decreases monotonically with increasing λ. Higher λ means agents more often copy from the shared memory instead of inventing novel words, thus suppressing the emergence of many distinct local inventions.

2. Impact on total word load N_w(t). When λ is small, the peak of N_w(t) is modest because most agents quickly accumulate the same few locally invented words. At intermediate λ (≈ 0.3–0.6) the peak becomes pronounced: the population simultaneously harbors words drawn from the shared memory and locally invented ones, leading to a temporary overload of tokens.

3. Convergence time T_conv. With λ = 0 (the classic game) the average convergence time is about 2,500 interaction steps, matching prior literature. For C = 1, T_conv decreases monotonically as λ grows, reflecting the fact that a single shared word dominates the dynamics when agents frequently consult the external source. For larger C, T_conv exhibits a non‑monotonic dependence on λ: it first rises, reaching a maximum at an intermediate λ, then falls as λ approaches 1. This reflects a trade‑off between the diversity introduced by many shared words and the accelerating alignment when agents heavily rely on the shared source.

4. Probability that the consensus word originates from the shared memory. The authors define P_shared as the fraction of runs in which the final consensus word belongs to the initial shared set. Empirically, P_shared ≈ 1 for λ > 0.5, regardless of C. For λ < 0.5, P_shared declines with decreasing λ and with increasing C, indicating that local invention can dominate the final outcome when agents rarely consult the external source.

5. Temporal ordering of peaks. The time at which N_d(t) reaches its maximum (t_max_d) shortens as λ increases, while the time at which N_w(t) peaks (t_max_w) lengthens with λ. This asymmetry underscores that higher λ accelerates the saturation of distinct types but delays the accumulation of total tokens because shared words spread more uniformly before local overload occurs.

The discussion highlights several avenues for future work. The current study assumes a mean‑field (fully connected) topology, yet prior research shows that network structure (e.g., scale‑free, small‑world) can dramatically affect consensus dynamics. Extending the model to heterogeneous topologies could reveal how community structure or degree heterogeneity modulates the influence of the shared memory. Moreover, allowing λ to vary across agents (heterogeneous propensity to consult external sources) or granting a privileged entity write access to the shared memory would enable the exploration of controlled manipulation of consensus—a scenario reminiscent of media influence or algorithmic recommendation systems. Finally, dynamic updates of the shared memory (e.g., adding or removing words over time) could model evolving cultural or informational environments.

In conclusion, the paper demonstrates that even when agents have only read access to a global repository, the presence of that repository significantly shapes the trajectory toward linguistic consensus. The extended naming game still guarantees eventual agreement, but the path—characterized by the speed of convergence, the peak lexical diversity, and the likelihood that the final word originates externally—is tunable via the parameters λ and C. These insights bridge theoretical models of language emergence with practical systems such as collaborative tagging platforms, social bookmarking services, and human‑in‑the‑loop image labeling games, where both local peer interaction and shared reference resources jointly drive the formation of common vocabularies.