Abstract
The positive impact of cooperative bots on cooperation within evolutionary game theory is well-documented. However, prior studies predominantly use discrete strategic frameworks with deterministic actions. This article explores continuous and mixed strategic approaches. Continuous strategies use intermediate probabilities for varying degrees of cooperation and focus on expected payoffs, while mixed strategies calculate immediate payoffs from actions taken within these probabilities. Using the prisoner's dilemma game, this study examines the effects of cooperative bots on human cooperation in both well-mixed and structured populations across these strategic approaches. Our findings reveal that cooperative bots significantly enhance cooperation in both population types under weak imitation scenarios, where players are less concerned with material gains. Conversely, under strong imitation scenarios, cooperative bots do not alter the defective equilibrium in well-mixed populations but have varied impacts in structured populations. 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.