Balancing selfishness and norm conformity can explain human behavior in large-scale prisoner's dilemma games and can poise human groups near criticality

Cooperation is central to the success of human societies as it is crucial for overcoming some of the most pressing social challenges of our time; still, how human cooperation is achieved and may persist is a main puzzle in the social and biological sciences. Recently, scholars have recognized the importance of social norms as solutions to major local and large-scale collective action problems, from the management of water resources to the reduction of smoking in public places to the change in fertility practices. Yet a well-founded model of the effect of social norms on human cooperation is still lacking. Using statistical-physics techniques and integrating findings from cognitive and behavioral sciences, we present an analytically tractable model in which individuals base their decisions to cooperate both on the economic rewards they obtain and on the degree to which their action complies with social norms. Results from this parsimonious model are in agreement with observations in recent large-scale experiments with humans. We also find the phase diagram of the model and show that the experimental human group is poised near a critical point, a regime where recent work suggests living systems respond to changing external conditions in an efficient and coordinated manner.

Figure 1

Dynamical regimes of human cooperation. Colormap representation of the surfaces of discontinuous transitions defined by a function $A_d(x_0,y_0)$ that returns the transition value of $A$ for each value of the effective parameters $x_0$ and $y_0$, defined in Eqs. (16) and (17), respectively. Each (color) gray level encodes a level curve $A_d(x_0,y_0)=A$, which partitions the $x_0-y_0$ plane into three regions corresponding to different long-term dynamical regimes: Inside the sharp triangularlike curve (left) the system is bistable; inside the paraboliclike region (right) the system never reaches equilibrium (see also Fig. 3 in Appendix pp2). Outside these two regions the system is monostable (see Appendix pp2). The cloud of densely packed (green) dots around the plus and the cloud of sparsely scattered (yellow) little triangles around the cross represent a projection on the $x_0-y_0$ plane of the posterior population of parameters $\alpha , A, x_0$, and $y_0$ inferred here from the experiments performed in [19] on a heterogeneous network (HN) and on a lattice (L), respectively. We show the parameters estimated (HN, diagonal cross; L, vertical cross) and quantify their relative distance (HN, 3%; L, 11%) to the closest point (HN, square; L, circle) on the critical lines (dashed) (see Appendix pp5).

Figure 2

(a) and (b) Balance of selfishness and conformity to social norms explains human behavior in large-scale prisoner's dilemma games. Shown is a comparison of the dynamics of the global cooperation level observed in the laboratory experiment conducted in [19] (circles) on (a) a heterogeneous network and (b) a square lattice with that predicted by Eq. (12) (line) with the corresponding parameters inferred from the same experiments (see the caption of Fig. 1). The model and experimental dynamics are in agreement even in the transient regime. Also shown is the empirical probability for a representative agent to cooperate in a generic round based on whether she cooperated (“after $C$,” squares) or defected (“after $D$,” circles) and on the number of neighbors who cooperated in the previous round obtained in the same experiments [19] on (c) a heterogeneous network and (d) a lattice. The experimental data are compared with the values predicted by Eq. (19) (“after $C$,” red upper solid lines; “after $D$,” blue lower dash-dotted lines) with the corresponding parameters inferred from the same experiments (see the caption of Fig. 1). We can see that the assumption of linearity is valid and that our model agrees with the experimental values to a large extent. We include the linear fits (dashed straight lines) directly obtained from experimental data [19] for comparison.

Figure 3

Level curve of the surface $A_d(x_0,y_0)$ of discontinuous transitions [see Eqs. (B3) and (B4)], i.e., $A_d(x_0,y_0)=A$ (here $A=5$). Inside the (green) shaded triangularlike region (left) there are two stable fixed points. Inside the (blue) shaded paraboliclike region (right) there are no stable fixed points. In all the remaining white area there is one stable fixed point. These regions terminate on critical points (red squares). If the value of $A$ changes, these regions shift and so do the corresponding critical points along the critical lines (red dashed nonhorizontal lines crossing from the bottom left and bottom right to the top center). The points $R1,...,R_5$ on the (black) dashed horizontal line show examples of the five regions described in the text (see Appendix pp2). These correspond to parameter values $A=5, y_0=-1$, and $x_0=-0.90,0.00,0.35,0.80,1.62$, respectively.

Figure 4

Graph of $f$ [see Eq. (14)] corresponding to points $R_1,...,R_5$ in Fig. 3 in Appendix pp2. Notice that $f^{\prime }\left(P_1\right)=f^{\prime }\left(P_2\right)=1$ and $f^{\prime }\left(P_3\right)=f^{\prime }\left(P_4\right)=-1$ (green crosses). The fixed points of $f$ are its intersections with the identity function (dashed line); these are stable (blue circles) if $|f^{\prime }\left(x\right)|<1$ and unstable (black squares) if $|f^{\prime }\left(x\right)|>1$. (a) For small $x_0$ (e.g., point $R_1$ in Fig. 3), there is only one stable fixed point $x_1$. (b) Increasing $x_0$ (e.g., point $R_2$ in Fig. 3) until point $P_1$ touches the identity function, we enter region 2 where there are two stable fixed points $x_1$ and $x_2$ and an unstable one $x_u$ in between. (c) Increasing $x_0$ (e.g., point $R_3$ in Fig. 3) until $x_1$ hits point $P_2$ to then disappear, we enter region 3 where only the stable fixed point $x_2$ survives. (d) Increasing $x_0$ (e.g., point $R_4$ in Fig. 3) until $x_2$ hits the point $P_3$ to become unstable, we enter region 4 where there are no stable fixed points. (e) Increasing $x_0$ (e.g., point $R_5$ in Fig. 3) until $x_u$ hits point $P_4$ to become stable, we enter region 5 where there is only one stable fixed point $x_1$.