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.

翻译：合作机器人在演化博弈论中对合作行为的积极影响已有充分记载；然而，现有研究主要采用离散策略框架，聚焦于概率固定为1的确定性行为。本文将研究拓展至连续与混合策略方法。连续策略采用中间概率来传递不同程度的合作意图，并关注期望收益；而混合策略则基于这些概率范围内特定时刻所选行动计算即时收益。本研究以囚徒困境博弈为模型，考察了在人类玩家与简单机器人混合群体中，合作机器人对人类合作行为的影响，研究覆盖均匀混合与结构化两种群体类型。研究发现，在弱模仿情景（玩家较少关注物质收益）下，合作机器人能显著提升两种群体类型在所有策略方法中的合作水平。然而，在强模仿情景下，合作机器人虽未改变均匀混合群体中的背叛均衡，但在结构化群体中其影响因策略方法而异：具体而言，在离散与连续策略下会破坏合作，而在混合策略下则促进合作。这些结果揭示了合作机器人在不同策略框架中作用的微妙差异，并强调其部署需审慎考量，因为其有效性对人类行动更新方式及所选策略方法高度敏感。