  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.
