Cooperative bots exhibit nuanced effects on cooperation across strategic frameworks

TL;DR

This paper investigates cooperative bots in hybrid human-bot populations playing the prisoner's dilemma, extending prior work from discrete to continuous and mixed strategies across well-mixed and lattice structures. It shows cooperative bots promote ordinary-player cooperation under weak imitation for all strategy forms, while under strong imitation they have nuanced, structure- and strategy-specific effects—disrupting cooperation in discrete and continuous frameworks but promoting it in mixed strategies. The findings highlight that the benefits of AI-assisted cooperation depend critically on how humans update strategies and which strategic framework they employ, with implications for deploying cooperative bots in real-world mixed-agent settings. By integrating fixed-behavior bots with diverse strategy updates, the work broadens evolutionary game theory modeling toward more realistic human-AI interactions and suggests avenues for future research involving richer social dynamics and more sophisticated AI behavior.

Abstract

The positive impact of cooperative bots on cooperation within evolutionary game theory is well documented; however, existing studies have predominantly used discrete strategic frameworks, focusing on deterministic actions with a fixed probability of one. This paper extends the investigation to continuous and mixed strategic approaches. Continuous strategies employ intermediate probabilities to convey varying degrees of cooperation and focus on expected payoffs. In contrast, mixed strategies calculate immediate payoffs from actions chosen at a given moment within these probabilities. Using the prisoner's dilemma game, this study examines the effects of cooperative bots on human cooperation within hybrid populations of human players and simple bots, across both well-mixed and structured populations. Our findings reveal that cooperative bots significantly enhance cooperation in both population types across these strategic approaches under weak imitation scenarios, where players are less concerned with material gains. However, under strong imitation scenarios, while cooperative bots do not alter the defective equilibrium in well-mixed populations, they have varied impacts in structured populations across these strategic approaches. Specifically, they disrupt cooperation under discrete and continuous strategies but facilitate it under mixed strategies. These results highlight the nuanced effects of cooperative bots within different strategic frameworks and underscore the need for careful deployment, as their effectiveness is highly sensitive to how humans update their actions and their chosen strategic approach.

Figures (7)

• Figure 1: In well-mixed populations, regardless of the strategic approaches adopted, introducing cooperative bots can enhance the cooperation levels among ordinary players. Shown are the cooperation levels of ordinary players within well-mixed populations independent of the dilemma strength $r$ under the different proportions $\rho$ of bots. We examined three strategic scenarios: discrete (left panel), continuous (middle panel), and mixed (right panel), with $\rho$ varying from 0 to 0.5 and the $r$ ranging from 0 to 0.2. Imitation strength was fixed at $\kappa^{-1}=10$.
• Figure 2: The effect of cooperative bots on promoting cooperation in well-mixed populations is limited to scenarios with weak imitation strength across all three strategic approaches. Shown are the cooperation levels of ordinary players within a well-mixed population independent of the imitation strength $\kappa^{-1}$ under the different proportions $\rho$ of bots. From left panel to right panel, we consider three strategic approaches: discrete, continuous, and mixed. The shaded areas indicate the variance between different simulation results. The dilemma strength was fixed at $r=0.1$.
• Figure 3: Unlike a well-mixed population, bots under discrete and continuous strategic approaches in a network population suppress the level of cooperation among ordinary players, but the mixed strategic approach still promotes it. Shown are the cooperation levels of ordinary players within a networked population independent of the dilemma strength $r$ under the different proportions $\rho$ of bots. We examined three strategic scenarios: discrete (left panel), continuous (middle panel), and mixed (right panel), with $\rho$ varying from 0 to 0.5 and the $r$ ranging from 0 to 0.2. Imitation strength was fixed at $\kappa^{-1}=10$.
• Figure 4: In a regular lattice, under discrete and continuous strategic approaches, introducing bots affects cooperation levels among ordinary players differently depending on imitation strengths. However, under the mixed strategic approach, introducing bots almost always enhances cooperation among ordinary players regardless of imitation strength. Shown are the average cooperation fractions of ordinary players, obtained from averaging over 100 simulations, as a function of the imitation strength within a networked population. From left panel to right panel, we consider three strategic approaches: discrete, continuous, and mixed. The solid lines of different colors represent the different proportions of bots. The shaded areas indicate the variance between different simulation results. All other parameters are consistent with Figure \ref{['fig2']}.
• Figure 5: Mixed strategic approach allows ordinary players with high probabilities of choosing cooperation (high values of strategies) to opt for defection, thereby achieving higher payoffs. Shown in the top row are the time evolution diagrams of the frequency for ordinary players with different strategy values. The middle and bottom rows, respectively, show the frequency distribution and cumulative frequency of the average social payoffs for bots, ordinary players with a strategy value of 0, and ordinary players with a strategy value of 0.9 during the total time steps. Moving from left to right, we employ discrete, continuous, and mixed strategic approaches. The parameter settings are as follows: the proportion of bots in the hybrid group was fixed at $\rho=0.5$; the dilemma strength was fixed at $r=0.1$; and the imitation strength was fixed at $\kappa^{-1}=10$.