More Than Opinions: The Role of Values in Shaping Fairness and Status in the Ultimatum Game within Structured Societies

TL;DR

Problem: how do values and observable status shape fairness in resource division under the Ultimatum Game within dynamically growing/shrinking networks? Approach: a co-evolutionary model where thresholds, value weights, and status estimates evolve with birth/death and network rewiring; proposer probability is $p_{ij} = s_i/(s_i+s_j)$ with $s_i = sum_{k=1}^V v_k^i q_k^i$, energy dynamics $E_i^{n} = E_i^{n-1} d - E_{cost} + \Pi_i^n$, and reproduction probability $p_{repro}=0.1$. Key contributions: demonstrate emergence of diverse sharing norms from the UG framework, including greedy to generous behavior; identify how reproduction timing, network wiring, and subjective hierarchies shape thresholds and inequality (Gini index); show that neighbor-based linkage can foster highly generous offers and clan-like substructures, while energy-led values modulate inequality in certain regimes. Significance: provides a minimal, interpretable framework linking hierarchy, value evolution, and UG outcomes in structured populations, with implications for understanding fairness in real-world social systems and guiding future extensions to richer social dynamics.

Abstract

Asymmetric evolutionary games, such as the Ultimatum Game, provide keys to understanding the emergence of fairness in social species. Building on this framework, we explore the evolution of social value systems and the operational role that social status plays in hierarchically organised societies. Within the asymmetric Ultimatum Game paradigm, where "proposers" suggest terms for resource distribution, and "responders" accept or reject these terms, we examine the assignment of roles between players under a subjective social order. This order is grounded in an emergent status hierarchy based on observable player attributes (such as age and wealth). The underlying rules for constructing such a hierarchy stabilise over time by inheritance and family ties. Despite their subjective nature these (often sub-conscious) value systems have operative meaning in controlling access of individuals to resources and decision making. We demonstrate these effects using a simple but sufficiently complex model with dynamical population size and network structure, where division of resources (prey) is carried out according to the principles of the Ultimatum Game. We focus on the emerging proposer and responder thresholds under distinct social hierarchies and interaction networks and discuss them in relation to the extensive body of Ultimatum Game experiments conducted across a wide range of cultural contexts. We observe the emergence of diverse sharing norms, ranging from unfair to highly generous, alongside the development of various social norms.

Figures (9)

• Figure 1: Evolved thresholds of offers (coloured) and rejections (grey) in simulations with a structured network and a minimal reproductive age of zero (left) and 180 month (right). Different colors represent various wiring configurations: blue (0) indicates that a child node is connected to both parents and the remaining six links are assigned randomly from the entire population. Red (2) represents simulations with two of the six remaining links selected from the parents' neighbours, and so on until the case where the child node is connected to parents and their neighbours only (6). Each box represents $20$ means, where each mean is calculated from the population distribution in a single run. The x-axis displays different value systems, while the y-axis shows boxplots of mean offer thresholds (coloured) and acceptance thresholds (grey).
• Figure 2: Evolved value systems for different settings. The upper plots show energy importance in the mixed systems of Age + Energy. The plots below show results for the All system values. On the left, we show results for minimal reproductive age of zero, on the right for age $180.$ Different colors refer to distinct wiring scenarios (see the legend).
• Figure 3: Network structure of the case with All value system and a minimal reproductive age $180.$ Different colors represent different offer thresholds in society (left) and different levels of energy value strength (right). The plots above have one link connected to a random individual in the entire population and the rest to neighbours (purple wiring (5)) and the plots below represent the wiring case where all links come form parents' neighbours (green wiring (6)).
• Figure 4: Age (a), energy (b), and degree (c) probability density estimates are shown for different wiring scenarios in the setups with a minimal reproductive age of $180$ and the All value system. A realistic age distribution is observed in panel (a). The energy distribution (b) differs noticeably in the green wiring scenario (6), with a much narrower distribution and lower average energy values compared to other scenarios. The degree distribution (c) is quite similar across wiring scenarios, with an average degree of approximately $9.9$.
• Figure 5: Gini index of the energy distribution in the evolved populations under different wiring scenarios and value systems. On the left, the results for a minimal reproductive age of zero are shown, and on the right, those for a minimal reproductive age of 180. Different colors represent various wiring scenarios, while the x-axis displays different value systems.