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.

From Wall Street to politics, quantitative analysts (or quants) are revolutionizing much of the world. Nowadays, that even includes horse racing. By using computers to identify hidden patterns in past racing data and arcane mathematics to optimize every aspect of their betting strategies, horse racing quants can confidently wager staggering amounts. At first, that may seem good: more money in the pot means the house and the winners take more home. Still, their trades have been blamed for (among other things) driving away other bettors and shrinking prizes for everyone over time.