This paper uses category theory to develop an entirely new approach to approximate game theory. Game theory is the study of how different agents within a multi-agent system take decisions. At its core, game theory asks what an optimal decision is in a given scenario. Thus approximate game theory asks what is an approximately optimal decision in a given scenario. This is important in practice as -- just like in much of computing -- exact answers maybe too difficult to compute or even impossible to compute given inherent uncertainty in input. We consider first "Selection Functions" which are functions and develop a simple yet robust model of approximate equilibria. We develop the algebraic properties of approximation wrt selection functions and also relate approximation to the compositional structure of selection functions. We then repeat this process successfully for Open Games -- a more advanced model of game theory.

What is the best compromise in a situation where different people value different things? The most commonly accepted method for answering this question -- in fields across the behavioral and social sciences, decision theory, philosophy, and artificial intelligence development -- is simply to add up utilities associated with the different options and pick the solution with the largest sum. This ``utilitarian'' approach seems like the obvious, theory-neutral way of approaching the problem. But there is an important, though often-ignored, alternative: a ``contractualist'' approach, which advocates for an agreement-driven method of deciding. Remarkably, no research has presented empirical evidence directly comparing the intuitive plausibility of these two approaches. In this paper, we systematically explore the proposals suggested by each algorithm (the ``Utilitarian Sum'' and the contractualist ''Nash Product''), using a paradigm that applies those algorithms to aggregating preferences across groups in a social decision-making context. While the dominant approach to value aggregation up to now has been utilitarian, we find that people strongly prefer the aggregations recommended by the contractualist algorithm. Finally, we compare the judgments of large language models (LLMs) to that of our (human) participants, finding important misalignment between model and human preferences.