Conditions for Altruistic Perversity in Two-Strategy Population Games

TL;DR

The paper addresses when altruistic agents can degrade social welfare in two-strategy population games. It develops a general model with heterogeneous altruistic and selfish types and introduces the perversity index ${ m PI}(G)$ to compare welfare at heterogeneous Nash equilibria against all-selfish equilibria, linking perversity to the convexity of the welfare function $W(u)$. The key result establishes that altruistic perversity can occur only when $W(u)$ is convex and the altruist mass is sufficiently large; if $W(u)$ is strictly concave, perversity cannot occur, and altruism tends to improve welfare. A Prisoner’s Dilemma case study provides explicit, piecewise expressions for ${ m PI}({ m PD}(p_a))$ under convex and concave welfare, illustrating how altruist behavior can both improve and degrade welfare depending on parameters. Overall, the work connects the structural properties of agent interactions to potential welfare outcomes and motivates further analysis for more general population games and dynamics.

Abstract

Self-interested behavior from individuals can collectively lead to poor societal outcomes. These outcomes can seemingly be improved through the actions of altruistic agents, which benefit other agents in the system. However, it is known in specific contexts that altruistic agents can actually induce worse outcomes compared to a fully selfish population -- a phenomenon we term altruistic perversity. This paper provides a holistic investigation into the necessary conditions that give rise to altruistic perversity. In particular, we study the class of two-strategy population games where one sub-population is altruistic and the other is selfish. We find that a population game can admit altruistic perversity only if the associated social welfare function is convex and the altruistic population is sufficiently large. Our results are a first step in establishing a connection between properties of nominal agent interactions and the potential impacts from altruistic behaviors.

Figures (1)

• Figure 1: Fig. \ref{['subfig:alt_payoff']} characterizes the payoff functions and possible Nash equilibria for altruists in an example game where the welfare function is convex: $R = 21$, $S = 1$, $T = 22$, $P = 20$. The stars represent the Nash equilibria available to altruists when $p_{\rm{a}} = 1$, and the shaded area contains feasible sub-population states for a given altruistic population. Fig. \ref{['subfig:convex_PI']} represents the perversity index as a function of the altruistic population, ${\rm{PI}}(p_a)$, for the same example game. Here, the perversity index is a piecewise constant function since, if their population is too small, altruists choose to defect just like selfish agents. If their population exceeds $u_{\rm{a}}^*$, altruists may choose the mixed Nash equilibrium, which results in the worst-case welfare. In this example, altruistic perversity can significantly degrade welfare, resulting in a $20\%$ drop in performance. Fig. \ref{['subfig:concave_PI']} represents ${\rm{PI}}(p_a)$ for an example game where the welfare function is concave: $R = 3$, $S = 1$, $T = 6$, $P = 2$. Here, the perversity index is continuous because the behavior of the altruistic payoffs is unlike that of Fig. \ref{['subfig:alt_payoff']}; altruists cooperate until the population is large enough to choose the mixed Nash equilibrium, resulting in the best-case welfare.

Theorems & Definitions (4)

• Definition 1
• Theorem III.1
• Proposition IV.1
• Lemma V.1