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Graphite may store lithium or potassium, but not sodium, in its interlayer space under ambient conditions. It is, however, unclear whether binary alkali alloys of Li-Na, Li-K, and Na-K may substitute pure Li or K to form binary alkali alloy-graphite intercalation compounds. We investigate thermodynamics of the binary alloy-graphite intercalation compounds using density functional theory with van der Waals density functionals. We find Li-rich co-intercalation compounds and K-rich ones are associated with negative formation energies, and the Na-K alloy has the broadest domain of co-intercalation (approximately up to 36% Na). Because of convexity of the formation-energy functions, these compounds are metastable and tend to decompose even when formation energies are negative. Na metal is among the decomposition products. Binary Li-K alloys in graphite form segregated phases of LiC6 and KC8, and this allows one to fabricate Li-K mixed-ion batteries using graphite anodes, whereas Li-Na and Na-K alloys are thermodynamically unfavorable. The study highlights the importance of convexity of formation-energy functions in thermodynamics of alloy-graphite intercalation compounds. [ABSTRACT FROM AUTHOR]