Competition between self- and other-regarding preferences in resolving social dilemmas

TL;DR

The paper investigates how self- versus other-regarding preferences shape cooperation in social dilemmas by extending the spatial public goods game to four profiles $(C_0,C_1,D_0,D_1)$ with productivity parameters $r_0$ and $r_1$ and learning rules that mix self- and other-regarding updates. It shows that other-regarding preferences promote cooperation more effectively than self-regarding ones at equal productivities and reveals a novel three-profile coexistence phase $(C_0+C_1+D_0)$ enabling dynamic invasions between classic solutions, a finding that persists in well-mixed populations and is robust to noise. When $r_0$ and $r_1$ differ, a phase near the diagonal allows $C_1$ to invade even if $r_0>r_1$, highlighting the stabilizing role of other-regarding learning. The results suggest that other-regarding behavior can evolve and stabilize cooperation without additional mechanisms, offering insight into the observed prevalence of altruistic tendencies in human and animal societies.

Abstract

Evolutionary game theory assumes that individuals maximize their benefits when choosing strategies. However, an alternative perspective proposes that individuals seek to maximize the benefits of others. To explore the relationship between these perspectives, we develop a model where self- and other-regarding preferences compete in public goods games. We find that other-regarding preferences are more effective in promoting cooperation, even when self-regarding preferences are more productive. Cooperators with different preferences can coexist in a new phase where two classic solutions invade each other, resulting in a dynamical equilibrium. As a consequence, a lower productivity of self-regarding cooperation can provide a higher cooperation level. Our results, which are also valid in a well-mixed population, may explain why other-regarding preferences could be a viable and frequently observed attitude in human society.

Figures (6)

• Figure 1: Panel (a): phase diagram on the parameter plane of productivity factor $r_0=r_1\equiv r$ and other-regarding rate $u$. The full defection and cooperation phases are separated by a mixed phase, conceptually similar to the traditional PGG model. Panel (b): the fraction of cooperation $\rho_C$ as a function of productivity $r_0=r_1\equiv r$ at different other-regarding rate $u$. An increase in other-regarding rate $u$ supports general cooperation.
• Figure 2: Phase diagram on the parameter plane of self-regarding productivity ($r_0$) and other-regarding productivity ($r_1$). The other-regarding rate is $u=1$. The white dashed line marks $r_0=r_1$. When $r_0$ and $r_1$ are largely different, we get back to the pattern in simplified two-profile models. Interestingly, other-regarding cooperation can conquer self-regarding cooperation even if $r_0 > r_1$. Furthermore, when $r_0$ slightly exceeds $r_1$, a new phase emerges composed of $C_0$, $C_1$, and $D_0$. The nature of this solution is discussed in the main text.
• Figure 3: Horizontal and vertical cross-sections of the phase diagram of Fig. \ref{['fig_phase2D_u1']} showing the stationary fractions of profiles. Panel (a): fractions as a function of $r_1$ at $r_0=3.8$. There is a discontinuous phase transition between $C_0+D_0$, $C_1+D_0$ followed by a continuous phase transition to the $C_1$ phase. Panel (b): fractions as a function of $r_1$ at $r_0=4$. The $C_0+D_0$ phase is replaced by $C_0+C_1+D_0$ and followed by the full $C_1$ phase, and the phase transitions are continuous. Panel (c): fractions as a function of $r_0$ at $r_1=3$. The transition between the $C_1+D_0$ and $C_0+D_0$ phases is discontinuous. Panel (d): fractions as a function of $r_0$ at $r_1=3.5$. The fraction of defection gradually increases through the $C_0+C_1+D_0$ phase as $r_0$ increases.
• Figure 4: Two solutions forming a new one. (a) When $D_1$ dies out, the remaining three profiles form a stationary solution, as their time evolution suggests. (b) A typical snapshot of the dynamic equilibrium in the $C_0+C_1+D_0$ phase on a $100\times 100$ square lattice. Both $C_0+D_0$ and $C_1+D_0$ solutions are stationary alone as the ellipses mark. However, there are permanent and mutual invasions between these phases. The fraction of $D_0$ players differ in these solutions. $C_1$ can better suppress $D_0$ even with low productivity, thus squeezing in from large defection cracks in $C_0+D_0$ regions, transforming $C_0+D_0$ into $C_1+D_0$, shown by white arrows. $C_0$, due to high productivity, invade $C_1$, transforming $C_1+D_0$ into $C_0+D_0$, shown by a black arrow on the top. The mini diagram of these processes is on the bottom. The arrows represent the direction of invasions. Parameters: $r_0=4.0$, $r_1=3.5$, $u=1$.
• Figure 5: Phase diagram on the $r_0$-$r_1$ parameter plane obtained at other-regarding rate $u=0.5$. The $C_0+C_1+D_0$ phase still exists but is significantly smaller. The white dashed diagonal marks $r_0=r_1$.