Individual perception dynamics in drunk games

We study the effects of individual perceptions of payoffs in two-player games. In particular we consider the setting in which individuals' perceptions of the game are influenced by their previous experiences and outcomes. Accordingly, we introduce a framework based on evolutionary games where individuals have the capacity to perceive their interactions in different ways. Starting from the narrative of social behaviors in a pub as an illustration, we first study the combination of the Prisoner's Dilemma and Harmony Game as two alternative perceptions of the same situation. Considering a selection of game pairs, our results show that the interplay between perception dynamics and game payoffs gives rise to nonlinear phenomena unexpected in each of the games separately, such as catastrophic phase transitions in the cooperation basin of attraction, Hopf bifurcations and cycles of cooperation and defection. Combining analytical techniques with multiagent simulations, we also show how introducing individual perceptions can cause nontrivial dynamical behaviors to emerge, which cannot be obtained by analyzing the system at a macroscopic level. Specifically, initial perception heterogeneities at the microscopic level can yield a polarization effect that is unpredictable at the macroscopic level. This framework opens the door to the exploration of new ways of understanding the link between the emergence of cooperation and individual preferences and perceptions, with potential applications beyond social interactions.

Figure 1

T-S space. The four classes of two-player two-strategy symmetric games within the standard $T-S$ parameter space ($R=1,P=0$). Prisoner's Dilemma (PD), Snow Drift (SD), Stag-Hunt (SH), and Harmony Game (HG). Colors show the level of cooperation obtained by the replicator dynamics at the stable equilibrium starting from an initial fraction of cooperators $x_0=0.5$.

Figure 2

The Pub Dilemma. Coupling of the Prisoner's Dilemma ($G_1$: $S_{\mathrm{PD}}=-0.5,T_{\mathrm{PD}}=1.5$) and the Harmony Game ($G_2$: $S_{\mathrm{HG}}=0.5,T_{\mathrm{HG}}=0.5$) using $\kappa =1,\mu =0.5$. The field diagram illustrates how the proportion of cooperators $x$ and the average value of $\alpha$ in the population evolves and contains a number of fixed points indicated by circle markers, which are either stable (black), unstable (white), or saddle points (gray). When $\alpha \in \{0,1\}$ we recover the game dynamics and fixed points of the original games (HG, top; PD, bottom).

Figure 3

The Drunk Prisoner. Coupling of a Harmony Game ($G_1$) and a Prisoner's Dilemma ($G_2$) displays a Hopf bifurcation regulated by the payoffs $S_{\mathrm{HG}}$ and $T_{\mathrm{HG}}$ ($S_{\mathrm{PD}}=-1,T_{\mathrm{PD}}=2,\kappa =1,\mu =0.5$ for each panel). Two trajectories (red and blue curves) for arbitrary initial conditions are shown in each example. (a) $S_{\mathrm{HG}}=T_{\mathrm{HG}}<0.5$, the Drunk Prisoner contains no stable fixed points and displays an unstable spiral originating at $x=\mu =0.5$. All spirals converge to a trajectory that follows the boundaries of the $(x,\alpha )$ plane such that $0<\alpha <1$. (b) The onset of the Hopf bifurcation occurs at $S_{\mathrm{HG}}=T_{\mathrm{HG}}=0.5$ and we observe the same unstable fixed point at $x=\mu =0.5$ surrounded by an infinite set of closed cyclic trajectories. (c) $S_{\mathrm{HG}}=T_{\mathrm{HG}}>0.5$, the cycles collide over the fixed point at $x=\mu =0.5$ which becomes stable. In this setting a spirals appear such that any initialization of the system, except those on the boundaries, converge to the same interior fixed point where half of the population are cooperators and the other half defectors.

Figure 4

Generalized Pub Dilemma. Attractiveness of the cooperation basin for the generalized Pub Dilemma: coupling the PD ($G_1$: $S_1=-1,T_1=2$) with another game $G_2$ having parameters $T_2$ and $S_2$. The $(S_2,T_2)$ space indicates the probability of converging on the cooperative fixed point ($x=1$ and $\alpha =1$), i.e., the proportion of cooperation of the coupled games, when $\kappa =0.1$ (a), $\kappa =1$ (b), and $\kappa =10$ (c). As $\kappa$ increases we can see an overall increase of the attractiveness of cooperation.

Figure 5

The Drunken Battle of Coordination. Coupling of a Snow Drift ($G_1$: $T_{\mathrm{SD}}=2,S_1=S_{\mathrm{SD}}$) and the Stag Hunt ($G_2$: $T_{\mathrm{SH}}=0.5,S_{\mathrm{SH}}=-0.5$). (a) $S_{\mathrm{SD}}=0.25$ and $\kappa =1$, all trajectories converge on either the stable fixed point of SD or the stable fixed point of SH at full cooperation, with the exception of points initialized at a fixed point or on the orange arrow. (b) $S_{\mathrm{SD}}=0.5$ and $\kappa =1$, a first-order phase transition occurs and a line of fixed points emerge at $x=0.5$. (c) $S_{\mathrm{SD}}=0.75$ and $\kappa =1$, all trajectories converge on the full cooperation fixed point at the top right. (d) Attractiveness of the cooperation basin for all possible $S_1\in [0,1]$; cases (a)–(c) are marked with $\kappa =1$. Increasing $\kappa$ increases the overall attractiveness of cooperation. A first-order phase transition occurs at $S_1=0.5$ such that when $S_1>0.5$ all initial conditions lead to full cooperation.

Figure 6

$\alpha$-heterogeneity in the Drunk Prisoner game. The effect of initializing the Drunk Prisoner game with different payoffs, $S_{\mathrm{HG}}=T_{\mathrm{HG}}$ ($x$ axis) and levels of $\alpha$-heterogeneity, ${\mathrm{\Delta }}_{\alpha }^{\left(0\right)}$ ($y$ axis). (a) Heatmap of the distance of the population average from the interior fixed point. Increasing the initial $\alpha$-heterogeneity increases the stability of the interior fixed point and the Hopf bifurcation occurs at lower payoffs $S_{\mathrm{HG}}$ and $T_{\mathrm{HG}}$. (b) Heatmap of the change in $\alpha$-heterogeneity, ${\mathrm{\Delta }}_{\alpha }^{\left(t\right)}-{\mathrm{\Delta }}_{\alpha }^{\left(0\right)}$, at time $t={10}^4$. Three distinct behaviors are observed according to whether the strategies, $x$, and perceptions, $\alpha$ coalesce or polarize. The cyan markers indicate the individual trajectories shown in Fig. 7: circle (a), star (b), and pentagon (c).

Figure 7

ABM simulations of the Drunk Prisoner game. Each of the panels (a)–(c) displays four plots describing the dynamics of the proportion of cooperators $x$ and average perception $\alpha$ over $t=5\times {10}^3$ rounds. The left plots show the trajectory through the $(x,\alpha )$-plane starting from the star. The top plot shows two trajectories for each of the $\alpha$ initial values, while the bottom plot show the mean over the whole population. The plots on the right show the evolution of $\alpha$ (top) and $x$ (bottom) over time. (a) $S_{\mathrm{HG}}=T_{\mathrm{HG}}=0.4$ and ${\mathrm{\Delta }}_{\alpha }^{\left(0\right)}=0.04$ (circle marker in Fig. 6), (b) $S_{\mathrm{HG}}=T_{\mathrm{HG}}=0.8$ and ${\mathrm{\Delta }}_{\alpha }^{\left(0\right)}=0.04$ (star marker in Fig. 6), and (c) $S_{\mathrm{HG}}=T_{\mathrm{HG}}=0.4$ and ${\mathrm{\Delta }}_{\alpha }^{\left(0\right)}=0.4$ (pentagon marker in Fig. 6). We see that small amounts of initial heterogeneity in the population can result in very different outcomes relative to the mean-field results.