Are Large Language Models (LLMs) a new form of strategic intelligence, able to reason about goals in competitive settings? We present compelling supporting evidence. The Iterated Prisoner's Dilemma (IPD) has long served as a model for studying decision-making. We conduct the first ever series of evolutionary IPD tournaments, pitting canonical strategies (e.g., Tit-for-Tat, Grim Trigger) against agents from the leading frontier AI companies OpenAI, Google, and Anthropic. By varying the termination probability in each tournament (the "shadow of the future"), we introduce complexity and chance, confounding memorisation. Our results show that LLMs are highly competitive, consistently surviving and sometimes even proliferating in these complex ecosystems. Furthermore, they exhibit distinctive and persistent "strategic fingerprints": Google's Gemini models proved strategically ruthless, exploiting cooperative opponents and retaliating against defectors, while OpenAI's models remained highly cooperative, a trait that proved catastrophic in hostile environments. Anthropic's Claude emerged as the most forgiving reciprocator, showing remarkable willingness to restore cooperation even after being exploited or successfully defecting. Analysis of nearly 32,000 prose rationales provided by the models reveals that they actively reason about both the time horizon and their opponent's likely strategy, and we demonstrate that this reasoning is instrumental to their decisions. This work connects classic game theory with machine psychology, offering a rich and granular view of algorithmic decision-making under uncertainty.