The motility-induced phase separation (MIPS) phenomenon in active matter has been of great interest for the past decade or so. A central conceptual puzzle is that this behavior, which is generally characterized as a nonequilibrium phenomenon, can yet be explained using simple equilibrium models of thermodynamics. Here, we address this problem using a new theory, statistical teleodynamics, which is a conceptual synthesis of game theory and statistical mechanics. In this framework, active agents compete in their pursuit of maximum effective utility, and this self-organizing dynamics results in an arbitrage equilibrium in which all agents have the same effective utility. We show that MIPS is an example of arbitrage equilibrium and that it is mathematically equivalent to other phase-separation phenomena in entirely different domains, such as sociology and economics. As examples, we present the behavior of Janus particles in a potential trap and the effect of chemotaxis on MIPS.