Noisy information channel mediated prevention of the tragedy of the commons

TL;DR

This work integrates information theory with stochastic evolutionary game theory to study how noisy perception of a dynamically changing resource impacts cooperation and the prevention of the tragedy of the commons. Using a two-state resource model, memory-1 and reactive strategies, and a binary information channel with noise parameters $(n_1,n_2)$, the authors analyze the resulting Markov dynamics and define cooperation via the long-run rate $\ar{\\gamma}$ and an efficacy $E=\\log_{10}(I/K)$, where $I$ is mutual information and $K$ is channel capacity. They find that noise can promote cooperation for reactive strategies (notably under transition vectors $\bm{q_{00}}$ and $\bm{q_{11}}$) and that cooperation may rise even as information-processing efficacy falls, due to an emergent error-correcting effect where misperceived states shift TFT toward forgiving ALLC behavior. The results illuminate a counterintuitive mechanism by which imperfect information can stabilize cooperative behavior, offering potential avenues for mitigating the tragedy of the commons and suggesting extensions to alternating moves and other forms of informational noise.

Abstract

Synergy between evolutionary dynamics of cooperation and fluctuating state of shared resource being consumed by the cooperators is essential for averting the tragedy of the commons. Not only in humans, but also in the cognitively-limited organisms, this interplay between the resource and the cooperation is ubiquitously witnessed. The strategically interacting players engaged in such game-environment feedback scenarios naturally pick strategies based on their perception of the environmental state. Such perception invariably happens through some sensory information channels that the players are endowed with. The unfortunate reality is that any sensory channel must be noisy due to various factors; consequently, the perception of the environmental state becomes faulty rendering the players incapable of adopting the strategy that they otherwise would. Intriguingly, situation is not as bad as it sounds. Here we introduce the hitherto neglected information channel between players and the environment into the paradigm of stochastic evolutionary games with a view to bringing forward the counterintuitive possibility of emergence and sustenance of cooperation on account of the noise in the channel. Our primary study is in the simplest non-trivial setting of two-state stochastically fluctuating resource harnessed by a large unstructured population of cooperators and defectors adopting either memory-1 strategies or reactive strategies while engaged in repeated two-player interactions. The effect of noisy information channel in enhancing the cooperation in reactive-strategied population is unprecedented. We find that the propensity of cooperation in the population is inversely related to the mutual information (normalized by the channel capacity) of the corresponding information channel.

Figures (11)

• Figure 1: Schematic illustrating the noisy-channel–mediated stochastic game:(a) depicts a self-renewing, multi-state resource that is coarse-grained into a two-state representation on adopting a threshold $m_{\rm thr}$. (b) shows two possible states of the stochastic donation game. The green and the orange matrices, respectively, represent the more beneficial and the less beneficial states of the stochastic game. The transition from state $s_{i}$ to the other state $s_{1}$ happens with probability $q_{a\tilde{a}}^{i}$. The frequency of the true beneficial state is $\alpha_{\rm in}$. (c) depicts the noisy information channel $(n_{1},n_{2})$ through which players perceive state $s_{i}$ as $s_{\bar{i}}$ erroneously with probability $n_{i}$. (d) highlights the changed probability distribution, $(\alpha_{\rm out}, 1-\alpha_{\rm out})$, over the two states due to wrong perception of the true distribution, $(\alpha_{\rm in}, 1-\alpha_{\rm in})$.
• Figure 2: Symmetric noisy channel mediated evolution of cooperation: The first row (a-c) depicts the evolution of rate of cooperation, $\gamma(t)$, averaged over an ensemble of paths starting from unconditionally defecting resident population. The second row (d-f) exhibits corresponding evolutions of the frequency, $\alpha_{\rm in}$, of beneficial state. The first, the second, and the third columns, respectively, correspond to the outcomes of transition vectors $\bm{q_{00}}$, $\bm{q_{10}}$, and $\bm{q_{11}}$. In these plots, solid and dashed curves represent outcomes for populations with memory-$1$ and memory-$\frac{1}{2}$, respectively. The red and the grey colored curves in the first two rows, respectively, represent outcomes for cases where the information channel is maximally noisy $(n=0.5)$ and minimally noisy $(n=0)$. We observe that for the transition vectors $\bm{q_{00}}$ and $\bm{q_{11}}$, a noisy channel is beneficial for a population of memory-$\frac{1}{2}$ strategies, contrary to the case of the transition vector $\bm{q_{10}}$. This observation is quantified in the third row (g-i) that depicts the long-run time-averaged enhancement of cooperation, $\Delta\hat{\gamma}$ (blue curves), and of the probability of being in the most beneficial state, $\Delta\hat{\alpha}$ (green curves), for all possible symmetric noisy channels. For illustration purpose, we have fixed $N=100$, $b_{1} = 2.0$, $b_{2}=1.2$, $c=1.0$, $\epsilon=10^{-3}$, and $\beta=10$.
• Figure 3: Asymmetric noisy channel mediated evolution of cooperation and efficacy: The first, second, and third columns represent the outcomes for transition vectors $\bm{q_{00}}$, $\bm{q_{10}}$, and $\bm{q_{11}}$, respectively, for all possible binary asymmetric channels. In the first row (a-c), plots depict the long-run time-averaged cooperation rate, $\hat{\gamma}$; in the second row (d-f), plots depict the long-run time-averaged efficacy, $\hat{E}$; and in the third row (g-i), plots depict the maximally recurrent strategies $\bm{p}_{\rm m}$ and their long run frequencies of recurrence $w_{\bm{p}_{\rm m}}$ in the population. The dashed line $n_{1}=n_{2}$ represents subset corresponding to the symmetric channels. For illustration purpose, we have fixed $N=100$, $b_{1} = 2.0$, $b_{2}=1.2$, $c=1.0$, $\epsilon=10^{-3}$, and $\beta=10$.
• Figure 4: Only three reactive strategies decide sustenance of the cooperation: In the plot of $b_{1}$ vs. $n_{2}$, the region above the analytically found yellow dashed line is where payoff of $({\rm TFT};{\rm ALLC})$ against $({\rm TFT};{\rm ALLC})$ is greater than the payoff of $({\rm ALLD};{\rm ALLD})$ against $({\rm TFT};{\rm ALLC})$; and the region below the analytically found green dashed line is where payoff of $({\rm ALLC};{\rm TFT})$ against $({\rm ALLC};{\rm TFT})$ is greater than the payoff of $({\rm ALLD};{\rm ALLD})$ against $({\rm ALLC};{\rm TFT})$. Note that the numerically found common (blue) region of cooperation is in line with the analytical prediction. Here, $N=100$, $b_{2}=1.2$, $c=1.0$, $n_{1}=0.1$, $\epsilon=10^{-3}$, and $\beta=10$.
• Figure 5: Representative sequences of the state in the stochastic game, action $a$ of the focal player with strategy $\bm{p}$, and action $\tilde{a}$ of the opponent with strategy $\tilde{\bm{p}}$, with simultaneous (a) and alternating (b) move schemes: For both schemes, the Markov chain starts from state $\omega_0 = (s_1, C, C)$. After two rounds, the game reaches $\omega_2 = (s_2, D, C)$. The probability that the state of the stochastic game is $s_1$, regardless of the move scheme, is given by $P(\omega_3 = (s_i = s_1, a, \tilde{a}) \mid \omega_2) = q^2_{DC}$. Upon transitioning to state $s_1$, the focal player observes her last action as $D$ and the opponent's as $C$ (gray contour), leading them to cooperate with probability $p^1_{DC}$. Similarly, the opponent, observing her last action as $C$ and the focal player's as $D$ (gray contour), cooperates with probability $\tilde{p}^1_{CD}$. In the alternating move scheme, the focal player, acting as the leader, observes her last action as $D$ and the opponent's as $C$ (blue contour), and thus cooperates with probability $p^1_{DC}$. The opponent, seeing her last action as $C$ and the focal player's as $C$ (green contour), since the opponent acts after the leader, cooperates with probability $\tilde{p}^1_{CC}$.