Prejudice driven spite: A discontinuous phase transition in ultimatum game

TL;DR

The paper investigates how prejudice, encoded as prejudicity $e$ in a symmetric, one-shot Ultimatum Game, can drive spiteful behavior in evolving populations. Using Moran dynamics with mutation (short-term) and the Imhof--Fudenberg--Nowak mutation-selection framework (long-term), it shows a first-order phase transition in spite frequency at the critical prejudicity $e_c = (h-l)/2$, with a discontinuous jump as $e$ crosses $e_c$. The transition is robust across population size $N$, selection strength, and persists in infinite-population replicator dynamics, with the jump magnitude increasing with $N$ and the payoff difference $(h-l)$ rather than absolute values. The work highlights a universal effect of prejudice, discusses extensions to network-structured populations and memory-based interactions, and emphasizes the potential real-world need to manage prejudice to prevent abrupt emergence of spiteful social behavior.

Abstract

In a mix of prejudiced and unprejudiced individuals engaged in strategic interactions, the individual intensity of prejudice is expected to have effect on overall level of societal prejudice. High level of prejudice should lead to discrimination that may manifest as unfairness and, perhaps, even spite. In this paper, we investigate this idea in the classical paradigm of the ultimatum game which we theoretically modify to introduce prejudice at the level of players, terming its intensity as prejudicity. The stochastic evolutionary game dynamics, in the regime of replication-selection, reveals the emergence of spiteful behaviour as a dominant behaviour via a first order phase transition -- a discontinuous jump in the frequency of spiteful individuals at a threshold value of prejudicity. The phase transition is quite robust and becomes progressively conspicuous in the limit of large population size where deterministic evolutionary game dynamics, viz., replicator dynamics, approximates the system closely. The emergence of spite driven by prejudice is also found to persist when one considers long-term evolutionary dynamics in the mutation-selection dominated regime.

Figures (10)

• Figure 1: Histograms illustrating the frequency of spite strategies $(x_{\text{spite}})$ and level of prejudice $(x_{\text{prej}})$ in the short-term evolution of the finite population. We arrange the subplots in $3\times3$ grid corresponding to three selection strengths $w \in \{0.05, 0.5, 0.95\}$ and three population sizes $N\in\{10, 20, 100\}$. The light green and dark green bars, respectively, correspond to prejudicity below $(e = 0.15 < e_c)$ and above $(e = 0.25 > e_c)$ the critical prejudicity ($e_c=0.2$).
• Figure 2: A discontinuous phase transition of spite. Plot exhibits in the frequency of spite ($x_{\text{spite}}$) and the level of prejudice ($x_{\text{prej}}$) at the point of critical prejudicity $e=e_c$ for three population size $N\in\{10,20,100\}$. The teal color indicates the frequency of spite, while the black color represents the level of prejudice. Dashed, dotted, and solid lines represent three population sizes: $N=10$, $N=20$, and $N=100$.
• Figure 3: Emergence of spite beyond threshold prejudicity: Plot showcases the frequency of spite strategies over time for three population sizes $N\in\{10,20,100\}$ in the Imhof--Nowak--Fudenberg process. The frequency of spite means how many times the spiteful monomorphic population appears up to time $t$. Dashed and solid lines, respectively, correspond to the prejudicity $e=0.15<e_c$ and $e=0.25>e_c$. Blue, red, and green colors represent the population sizes $N=10$, $N=20$, and $N=100$, respectively. We keep the parameter selection strength $\beta=0.95$ and mutation rate $\mu_{s_i s_j} = 10^{-3}$.
• Figure 4: Histogram illustrating the frequency of spiteful strategies $(x_{\text{spite}})$ and level of prejudice $(x_{\text{prej}})$ in the long-term evolution of a finite population. The light green and the dark green bars, respectively, represent the scenarios with prejudicity below $(e = 0.15 < e_c)$ and above $(e = 0.25 > e_c)$ the critical prejudicity ($e_c$). We fix the population size $N=100$, selection strength $\beta=0.95$ and mutation rate $\mu_{s_is_j} = 10^{-3}$.
• Figure 5: The histogram depicts the frequencies of four strategies: fairness, altruism, spite, and unfairness. For each strategy, two bars are shown---light green and dark green---representing cases where prejudicity is below $(e = 0.15 < e_c)$ and above $(e = 0.25 > e_c)$ the critical prejudicity threshold $(e_c=0.2)$, respectively. Subplots (a)--(i), generated from the Moran process, are arranged in a $3 \times 3$ grid corresponding to three selection strengths $w \in \{0.05, 0.5, 0.95\}$ and three population sizes $N \in \{10, 20, 100\}$. Subplot (j) illustrates the results from replicator dynamics in an infinite population, while subplot (k) shows the frequencies of all strategies in the the Imhof--Fudenberg--Nowak process.