We introduce a modification of Sprumont (Games Econ Behav 2:378–394, 1990) population monotonic allocation scheme (PMAS), called monotonic core allocation path (MCAP) for assignment games, which is a sequence of allocations along an order on the set of players satisfying that (1) each allocation is in the core of the subgame of the corresponding players at that step, and (2) the payoffs for each player are non-decreasing through the sequence. The notion of MCAP preserves the population monotonicity of PMAS while avoids the difficulty that PMAS does not exist in many market games. We show that for every assignment game, there is an order of players along which a MCAP exists. The terminals of MCAP form a refinement of the core. We also show that the terminals of MCAP coincide with the extreme core allocations in two subclasses of assignment games: gloves games and Böhm-Bawerk games. The strong connection of MCAP with extreme core allocations suggests some conflict between the stability of a coalition formation process and the fairness of the resulting outcomes.
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