I came across this paper in the bibliography section of signals. Based on the titles of his work Kevin J. S. Zollman seems not only a solid researcher but also interested in many of the questions I find fascinating.
I decided to review this one as it investigates how augmenting the Lewis signaling alters its stability.
Stability is not the first concept that comes to mind when thinking about game theory. I am usually interested in understanding what behaviors are possible in a game and how changing incentives might alter these. (Not all games have equilibria.)
Languages are known to change over time. when it comes to Agent based modeling time can pass much faster then in the real world. We can actually see the evolution a dynamic system like a languages in real time. If the language is hard to interpret we may not be happy if it also keeps changing. The stability of equilibria may also be of value to an algorithm designer looking for unique behaviors that exists in unique equilibria. If these are hard to get by chance then we may need better algorithms to find/learn them.
The main research questions being explored in this paper are:
- What is the effect of adding spatial structure to communication games?
- looks at adding spatial structure to the Lewis Signaling game and a modified Stag Hunt game.
- What happens when both communication and spatial arrangement are combined in these models?
- The author highlight that these two modifications have largely been studied individually, and their paper aims to study the combined models. This leads to the question of whether combining them yields new insights.
- Does increasing the complexity of the model by adding spatial structure and/or communication add explanatory value for social cooperation?
- The author questions whether more complex models behave analogously to simpler ones or if new features emerge that can explain other types of social behavior. More specifically he want to determine if these richer models provide equally good or better explanations for the evolution of social behavior than prior models.
- Does increasing the realism of the model by adding these complexities generally assist in the evolution of cooperative behavior?
- The authors explicitly aim to assess the “general lesson” that the probability of cooperation increases as models become more realistic. They also explore whether this holds true when communication and spatial structure are combined, acknowledging the possibility that these modifications might interfere with each other. The paper investigates two types of cooperation in this context: achieving the Pareto optimum payoff and the achievement of meaning.
- How does the addition of spatial structure affect the emergence and nature of meaning in signals?
- The paper analyzes the status of meaning of signals in both the Sender-Receiver game and the Stag Hunt game in the spatial context. They specifically examine if signals have meaning (provide more information than prior beliefs) and how this meaning evolves in a spatially structured population, leading to the concept of “regional meaning”. They compare the nature of meaning in the spatial model to that observed in non-spatial models using replicator dynamics.
In essence, the overarching research goal is to understand how making game-theoretic models more realistic by incorporating spatial interaction and communication influences the evolution of cooperation and the emergence of meaning.
Here is a light hearted Deep Dive into the paper:
Abstract
In seeking to explain the evolution of social cooperation, many scholars are using increasingly complex game-theoretic models. These complexities often model readily observable features of human and animal populations. In the case of previous games analyzed in the literature, these modifications have had radical effects on the stability and efficiency properties of the models. We will analyze the effect of adding spatial structure to two communication games: the Lewis Sender-Receiver game and a modified Stag Hunt game. For the Stag Hunt, we find that the results depart strikingly from previous models. In all cases, the departures increase the explanatory value of the models for social phenomenon. — (Zollman 2005)
Glossary
This paper has a number of big words that seem technical and may seem impediments to understanding it. Here are some key definitions to help us navigate the content:
- Replicator Dynamics
- A mathematical model describing how the frequencies of different strategies in a population change over time based on their relative payoffs.
- Nash Equilibrium
- A state in a game where no player can improve their payoff by unilaterally changing their strategy, assuming the other players’ strategies remain the same.
- Payoff Dominant Equilibrium
- A Nash equilibrium where all players receive higher payoffs compared to any other equilibrium.
Babbling Equilibrium: An equilibrium in a communication game where signals convey no information about the state of the world or the sender’s intentions.
- Evolutionarily Stable Strategy (ESS)
- A strategy that, if adopted by a population, cannot be invaded by any alternative strategy under the replicator dynamics.
- Spatial Structure
- The arrangement of individuals in a population, often modeled as a grid or network, influencing who interacts with whom.
- Imitate-the-Best Dynamics
- A strategy updating rule in spatial models where individuals adopt the strategy of their most successful neighbor.
- Secret Handshake
- A signal used by a subpopulation to coordinate their actions and achieve cooperative outcomes by identifying each other.
- Polymorphic Equilibrium
- A stable state in a population where multiple strategies coexist at specific frequencies.
- Regional Meaning
- The phenomenon where a signal’s meaning becomes localized to a specific area within a population due to spatial clustering of signaling systems.
- Aumann Stag Hunt
- A variation of the Stag Hunt game where Hare Hunters would prefer their partner to hunt Stag, even though they choose to hunt Hare themselves.
Outline
Here is an outline summary of the paper “Talking to Neighbors: The Evolution of Regional Meaning”:
- Introduction
- The paper investigates the evolution of social cooperation using game-theoretic models by adding complexities that model observable features of human and animal populations.
- Traditional equilibrium analysis is often inadequate to explain observed social practices, leading to the increasing use of dynamic models like replicator dynamics and myopic best response.
- The paper focuses on the effect of adding spatial structure and communication to two communication games: the Lewis Sender-Receiver game and a modified Stag Hunt game.
- Previous literature suggests that increasing the reality of models by adding communication or spatial arrangement often helps cooperation, but this may be an oversimplification as new pitfalls can be introduced.
- Studying the combined effects of communication and spatial arrangement can reveal if increased model complexity adds explanatory value and test the general lesson that realism enhances cooperation.
- The paper examines two types of cooperation: achieving mutually beneficial outcomes and the emergence of meaning in signals.
- The Sender-Receiver Game
- David Lewis proposed that meaning can be explained by communicators using strategies in repeated cooperation games.
- A simple Sender-Receiver game with two states, acts, and signals has two high-payoff equilibria and babbling equilibria where signals lack meaning.
- Standard equilibrium analysis requires high cognitive capabilities, which may not be realistic for humans or animals.
- While factors like natural salience and focal points have been suggested for the emergence of signaling systems, they have limitations in explaining all cases, especially in creatures with low cognitive capacities.
- Blume et al. (1998) found that humans can converge to a signaling system even with meaningless signals.
- Skyrms (1996) used evolutionary game theory with replicator dynamics to show that signaling systems can evolve from various starting points and are evolutionarily stable.
- To address concerns about the assumptions of replicator dynamics, the paper introduces a spatial model with 10,000 players on a torus, each interacting with eight neighbors and updating strategies by imitating more successful neighbors.
- Simulations of this spatial model show the emergence of both possible signaling systems, leading to a neutrally stable state unlike the unstable equilibrium in Skyrms’s model.
- A small proportion of starting points in the spatial model can lead to populations of babblers, which can be invaded by signalers.
- Similar results of coexisting conventions in spatial populations have been found with different dynamics like myopic best-reply (Berninghaus and Schwalbe, 1996) and in slightly different games (Grim et al., 2001).
- In the spatial model, signals acquire regional meaning, where the signal provides perfect information about the state of the world if the receiver knows the region of the sender.
- Replacing replicator dynamics with a spatial model strengthens the explanation for the evolution of meaning and explains the emergence of different signaling systems in different locations.
- The Stag Hunt
- The Stag Hunt game models situations where individuals must risk cooperation for a higher payoff. It has two Nash equilibria: all hunt stag (payoff dominant) and all hunt hare (less risky).
- Replicator dynamics analysis suggests that stag hunting takes over only if initially present in more than three-quarters of the population.
- The paper examines the Stag Hunt with communication and spatial structure, both individually and combined.
- Stag Hunt with Communication: Aumann (1990) raised concerns about costless preplay communication, arguing signals might be meaningless. However, Skyrms (2002) found that signals can significantly promote cooperative (stag hunting) behavior using replicator dynamics.
- Spatial Stag Hunt: Ellison (1993) found that with best-reply dynamics on a circle, hare hunting usually dominates. Lee and Valentinyi (2000) found a similar result on a two-dimensional lattice with best-reply dynamics and no mutations. Skyrms (2004) found that with imitate-the-best dynamics on a two-dimensional lattice, stag hunting prevails in 99% of cases. On a circle with imitate-the-best, neither equilibrium is contagious without a larger imitation neighborhood.
- Spatial Stag Hunt with Communication: The paper uses the same spatial structure and imitate-the-best dynamics as in the Sender-Receiver game, with strategies represented by a 3-tuple (Signal, Act if signal 1, Act if signal 2).
- Simulations show that almost all populations evolve to a state with six of the eight strategies coexisting, and everyone hunting stag. This stability relies on inertia in the model (players only switch if a neighbor does better).
- Even with relaxed inertia or variations in payoffs and mutation rates, stag hunting remains prevalent. Aumann Stag Hunts take longer to stabilize but still mostly result in stag hunting.
- Populations of Hare Hunters with an unused signal can be invaded by two neighboring Nationalists using a “secret handshake”. This invasion is possible under certain payoff conditions or with a larger imitation neighborhood.
- The polymorphic equilibrium of Individualists found in Skyrms’s replicator dynamic model can be invaded by Stag Hunters in the spatial model.
- Signals in the spatial Stag Hunt with communication contribute beyond spatial structure alone. Invading an all-hare-hunting population requires only two non-simultaneous mutations compared to at least six in the spatial Stag Hunt without communication.
- Signals in the combined model acquire more global meaning, indicating a higher probability of a sender being a Stag Hunter or Nationalist, and provide evidence of a player’s propensity to cooperate with those sending the same signal. They also exhibit regional meaning.
- Conclusion
- The combination of signaling and spatial arrangement leads to new results not observed in simpler models. For the Sender-Receiver game, it explains the emergence of different signaling systems within a population. For the Stag Hunt, it results in a radically different population structure.
- The general lesson that increasing model complexity helps cooperation is partially confirmed.
- Achieving the Pareto optimum payoff (highest mutual benefit) is generally enhanced in both games with increased complexity.
- Population-wide meaning is harmed by spatial structure but a weaker form of regional meaning emerges.
- Overall, increasing the reality of the model assists the evolution of cooperative behavior in the Sender-Receiver and Stag Hunt games, suggesting simpler models do not fully capture the dynamics of cooperation and spatial structure.
The Review
So adding spatial structure to the Lewis signaling game and the Stag Hunt game has some interesting effects. The paper shows that the spatial structure can lead to the emergence of different signaling systems within a population. This is a significant departure from the traditional equilibrium analysis of these games.
The paper
Citation
@online{bochman2025,
author = {Bochman, Oren},
title = {🗣️ {Talking} to {Neighbors:} {Evolution} of {Regional}
{Meaning} in {Communication} {Games}},
date = {2025-03-13},
url = {https://orenbochman.github.io/reviews/2005/zolman-talking-to-neighbours/},
langid = {en}
}