The maximum and the addition of assignment games

In the framework of two‐sided assignment markets, we first consider that, with several markets available, the players may choose where to trade. It is shown that the corresponding game, represented by the maximum of a finite set of assignment games, may not be balanced. Some conditions for balancedness are provided and, in that case, properties of the core are analyzed. Secondly, we consider that players may trade simultaneously in more than one market and then add up the profits. The corresponding game, represented by the sum of a finite set of assignment games, is balanced. Moreover, under some conditions, the sum of the cores of two assignment games coincides with the core of the sum game.