Large language models (LLMs) are increasingly deployed as autonomous agents that negotiate, coordinate, and act on behalf of users. Whether they cooperate in such settings is no longer just an academic question, but a central issue for AI governance. We approach it from a strategic-behaviour angle, asking how two everyday levers - the size of what is at stake, and the language in which the interaction is described - shape the strategies LLMs adopt in a repeated Prisoner's Dilemma. Rather than reading cooperation off raw action counts, we train supervised classifiers to recognise the canonical strategies of repeated games (always cooperate, always defect, Tit-for-Tat, Win-Stay-Lose-Shift) and use them as a lens onto LLM behaviour. To know what the strategy distribution should look like under the same payoffs, we derive an evolutionary game theory (EGT) baseline and compare it with the LLM data. The two outcomes disagree in a revealing way: as stakes grow, evolutionary theory predicts that defection should take over the population, yet LLMs move in the opposite direction, becoming more cooperative - a signature, we argue, of alignment training and the human-like reasoning patterns LLMs inherit from their training data. We further show that this picture is not particular to frontier-scale, proprietary models: it also occurs with three open-weight smaller LLMs. Overall, our analysis highlights that payoff design and linguistic framing are powerful but under-explored levers for steering LLM behaviour, with direct implications for evaluating, aligning, and governing multi-agent AI systems deployed in high-stakes, multilingual environments.