Adaptive dynamics of eco-evolutionary repeated games: Effect of reward and punishment

TL;DR

The paper develops an eco-evolutionary framework where reactive, memory-half strategies in a repeated two-action game coevolve with a renewable common resource, analyzed through adaptive dynamics. It derives a coupled p–q–n system, identifies fixed points and possible Hopf-induced limit cycles, and contrasts long-term dynamics with short-term replicator dynamics, showing that TFT-like strategies can avert ToC. The Donation Game is used to study state-dependent rewards and punishments, revealing conditions under which complete ToC aversion emerges and illustrating rich dynamics, including Hopf bifurcations and continuums of limit cycles, dependent on initial strategies and incentive parameters. The results demonstrate that institutional incentives, especially when applied in the depleted state, can robustly sustain cooperation over evolutionary timescales, with implications for managing shared resources. The work also points to avenues for future research on evolutionary branching and more general eco-evolutionary feedback mechanisms.

Abstract

Long-term evolutionary processes can strongly influence common-pool resource conservation by generating new traits or behaviours that modify the feedback between population strategies and the resource state. Here we develop an eco-evolutionary framework in which individuals repeatedly interact with the same opponent and follow direct reciprocity through reactive strategies. The strategic dynamics is coupled to a renewable common resource and analyzed using adaptive dynamics. After our exhaustive non-linear dynamical analysis of $2\times2$ strategic games, we focus on comparative and combined usefulness of institutional incentives in the form of rewards and punishments in preventing the Tragedy of the Commons even when defection dominates in the replete resource state. We also report possibility of robust stable oscillations -- emerging via Hopf bifurcation -- in resource state and population strategies.

Figures (5)

• Figure 1: An exhaustive depiction of parameter space detailing the stability of fixed-points. Panels $(A)$ and $(B)$ correspond to fixed-point lines arising from the intersections of the surfaces $n=0$ and $F(p,q,n)=0$, and $n=1$ and $F(p,q,n)=0$, respectively. The right-hand subpanels (a)–(o) illustrate the nature of stability of internal fixed-point structures corresponding to coloured parameter regions. Stable fixed-point curves appear in green, unstable fixed-point curves in red, and saturated fixed-point lines in blue. The dotted parameter region exhibits a distinctive feature, where the fixed-point line is partially stable and partially unstable. Panel $(C)$ shows the stability behaviour of the internal fixed-point line corresponding to the intersection of the surfaces $p+q=1$ and $F(p,q,n)=0$. Here $\lambda_1\lambda_2$ is product of to non-zero eigenvalues (see Eq. \ref{['eq:conjugate']}). The dot-dashed line corresponds to $\beta_{1}+\gamma_{1}=\frac{\alpha_1}{\alpha_0}(\beta_{0}+\gamma_{0})$ and in subpanel $a$, the slope $=\frac{|A|}{B}=-\frac{|\alpha_0 + \beta_{0} +\gamma_{0}|}{\alpha_1 + \beta_{1} +\gamma_{1}}$. Twelve games different classes of distinct games, corresponding to different parameter regimes, are depicted in panels $(A)$ and $(B)$. They are labeled by $(i)$ to $(xii)$: Prisoner’s dilemma (i), Chicken games (ii), Leader games (iii), Battle of sexes (iv), Stag-hunt (v), Harmony I (vi), Harmony II (vii), Deadlock II (viii), Coordination I (ix), Coordination II (x), Harmony III (xi), and Deadlock I (xii).
• Figure 2: A unified comparison of short- and long-term evolutionary dynamics across the $\beta_{0}$--$\gamma_{0}$ parameter space. The accompanying seven subplots (taken from ref. weitz2016pnas) illustrate the short-term replicator dynamics mediated eco-evolutionary dynamics, with the $x$-axis representing the fraction of cooperators and the $y$-axis representing the fraction of the common resource. The $x$--$n$ phase-plane trajectories depict the stability of the isolated fixed point: open circles denote unstable fixed points, while filled colored circles represent stable ones. The parameter region in which the ToC is averted is labeled 'Averted'. These should be compared with Figure \ref{['fig:boat4']} in its entirety; nevertheless, for the same of illustration we have put the backdrop of Figure \ref{['fig:boat4']}(A) so that one can do comparison with the long-term adaptive dynamics with $n=0$ fixed point in focus.
• Figure 3: ToC and its prevention in the reward $(\rho_0^+)$--punishment $(\rho_0^-)$ parameter space. Subplot (a) shows the stability properties of the fixed points on the $n=0$ plane. The green region corresponds to parameter values for which the fixed points are either completely unstable or do not exist; consequently, for any initial choice of reactive strategy, complete ToC is avoided in this region. In contrast, this is not possible in the light-red shaded region, where a stable fixed point exists on the $n=0$ plane. Subplots (b) and (c) show the stability properties of internal fixed points over the admissible range of reward and punishment. The light-blue region represents internal fixed points that are partially stable and partially unstable, while the dark-blue region corresponds to completely stable internal fixed points. The gray and magenta regions denote saddle and completely unstable internal fixed points, respectively. The corresponding dynamical outcomes in these regions are also indicated.
• Figure 4: Limit cycles through a Hopf bifurcation in the presence of reward and punishment. Subplot (a) shows the Hopf bifurcation scenario resulting from variation of the parameter $\rho^+_{0}$ while keeping $\rho^-_{0}=1.6$ fixed. For $\rho^+_{0}>\rho^-_{0}$, the system stabilizes at a fixed point, indicating steady-state behaviour. As $\rho^+_{0}$ crosses the critical threshold $\rho^+_{0}=\rho^-_{0}$, a Hopf bifurcation occurs, leading to the emergence of sustained oscillations in $n$ for $\rho^+_{0}<\rho^-_{0}$. Subplots (b) and (c) correspond to the specific parameter value $\rho^+_{0}=1.5$. Subplot (b) illustrates the occurrence of a continuum of limit cycles in the $n$--$p_0$ space, where $n$ denotes the evolving resource state and $p_0$ represents the initial reciprocity, while the initial generosity is fixed at $q_0=0.4$. The gray solid curve represents a continuous set of unstable fixed points. Subplot (c) visualizes the limit cycles for two different initial conditions, $(p=0.14,\,q=0.66,\, n=0.46)$ and $(p=0.12,\,q=0.77,\, n=0.85)$. The trajectories, shown by green and blue solid curves, converge to a closed orbit (red curve) surrounding the gray solid unstable fixed-point curve. The remaining parameters used in these plots are $\rho^-_{0}=1.6$, $\rho^+_{1}=\rho^-_{1}=0.5$, $b_1=2.2$, $b_0=2.0$, and $c=1$.
• Figure 5: Summary of the sign of the eigenvalue $\lambda_2$: Subplot (a) summarizes the sign of the eigenvalue $\lambda_2$ in the $\beta_0$--$\gamma_0$ parameter plane for the fixed-point curve arising from the intersection of the surfaces $n=0$ and $F(p,q,n)=0$. In this plot, blue and red colors indicate regions where $\lambda_2$ is negative and positive, respectively. The magenta region denotes parameter values for which $\lambda_2$ is negative along one portion of the fixed-point curve and positive along another portion. The white region represents parameter values for which no fixed-point curve exists. Subplot (b) corresponds to the magenta region and illustrates representative trajectories approaching or departing from the fixed-point curve, demonstrating that one portion of the curve is stable while the remaining portion is unstable.