<p>This paper identifies the maximal domain of transferable utility games on which population monotonicity (no player is worse off when additional players enter the game) and egalitarian core selection (no other core allocation can be obtained by a single transfer from a richer to a poorer player) are compatible, which is the class of games with an egalitarian population monotonic allocation scheme. On this domain, which strictly includes the class of convex games, efficiency and population monotonicity together imply egalitarian core selection and characterize the Dutta-Ray solution. We relate the class of games with an egalitarian population monotonic allocation scheme to several other classes of games.</p>