Imitation versus payoff: Duality of the decision-making process demonstrates criticality and consensus formation

We consider a dual model of decision making, in which an individual forms its opinion based on contrasting mechanisms of imitation and rational calculation. The decision-making model (DMM) implements imitating behavior by means of a network of coupled two-state master equations that undergoes a phase transition at a critical value of a control parameter. The evolutionary spatial game, being a generalization of the prisoner's dilemma game, is used to determine in objective fashion the cooperative or anticooperative strategy adopted by individuals. Interactions between two sources of dynamics increases the domain of initial states attracted to phase transition dynamics beyond that of the DMM network in isolation. Additionally, on average the influence of the DMM on the game increases the final observed fraction of cooperators in the system.

Figure 1

Decision-making process in a society presented as a two-layer network. The lower layer models the imitation behavior, with positive (+1) and negative $\left(-1\right)$ opinions held by an individual of the decision-making model being represented by black and white nodes, respectively. The upper layer models the rational behavior of the evolutionary game, with nodes selecting between two strategies: being a protagonist [P, blue (light gray) circle] or antagonist [A, red (dark gray) circle]. The color of the arrows in the lower layer [P, protagonist, red (dark gray); A, antagonist, blue (light gray)] indicates that strategy adopted in the upper layer influences the character of the imitating behavior.

Figure 2

The phase-transition property of a predominantly protagonist DMM lattice is indicated with an increasing number of antagonists. The black dots denote an all protagonist lattice, the dots receding from the first calculation depict 1%, 3%, 5%, and the squares depict 10%, 15%, and 20% antagonists in sequence.

Figure 3

The equilibrium fraction of protagonists present in the EGM lattice of $N=20\times 20$ nodes is plotted for the game parameters $R=1$ and $P=0$. The initial fraction of protagonists was $50\%$, randomly distributed over the lattice. Periodic boundary conditions are considered. The region $T>1,S\le 0$ locates the prisoner's dilemma (PD) game; the stag hunt (SH) game is found in the domain $0\le T\le 1,S\le 0;$ the snowdrift (SD) game is bounded by $T\ge 1,0\le S\le 1;$ the leader (LD) game is confined by $T=1$ and to the right of the $S=T$ diagonal; and the battle of the sexes (SX) is above the diagonal $S=T$ and bordered on the left by $T=1$.

Figure 4

The ratio $D$ of the final fraction of protagonists observed in the DMM-EGM system to the fraction of protagonists observed in the EGM network in isolation is color coded. Uniform gray area depicts the range of parameter values whose DMM-EGM network dynamics give rise to phase transition. X depicts $S$ and $T$ values for which fraction of protagonists in the isolated EGM network is smaller than $90\%$.

Figure 5

Typical equilibrium dynamical behavior for the global variable is depicted in four regions of $\left(S,T\right)$ space. The blue squares denote the equilibrium fraction of protagonists and the red dots denote value of the global variable in the two-layer system. The dashed line marks the equilibrium solution of the EGM in isolation. (a) $T\le 1,-1\le S\le 3$ global variable undergoes a phase transition at a $K_c\approx 2.2;$ (b) $T\ge 1,S\le 1$ (narrow band of Fig. 4 with no phase transition), the level of protagonists is below that of EGM in isolation; (c) the predominantly green region in Fig. 3 with no phase transition and the fraction of protagonists the same as for the isolated EGM; (d) the PD segment of Fig. 3 has no phase transition and a low fraction of protagonists that is above the isolated EGM level.

Figure 6

Scaling of the global-order parameter (top left panel, middle row) and the susceptibility (top right panel, bottom row) in the vicinity of the phase-transition point. Top panel illustrates the scaling present in an isolated DMM layer with critical control parameter $K_C=1.70$. In middle and bottom rows, the columns correspond to the dual-layer configuration with $T=0,S=2,T=0.5,S=0.5$, and $T=-0.5,S=-0.5$ for the left, middle, and right column, respectively. Lattice size is $N=20\times 20$, and scaling exponents are $\beta =1/8$ and $\gamma =-7/4.$ Red dots correspond to the values of the global-order parameter in the vicinity of the phase transition, blue squares and green diamonds correspond to the values of the susceptibility below and above phase transition, respectively. Dashed lines denote corresponding power law dependence of the global-order parameter or susceptibility on the value of the normalized coupling constant.

Figure 7

(a) Here the global variable $\xi (K,t)$ is depicted as a function of time for two values of $K$ and a sequence of $S$ and $T$ values. (b) Plotted is the difference between the fraction of nodes whose strategy was changed from the EGM to the DMM layer minus that changed from the DMM to the EGM layer at each time of the simulation.