Two examples in a market with two types of indivisible good

We consider an extension of the ‘permutation game’ of Tijs et al. in which players are endowed with and ultimately wish to consume one unit of each of two types of good (i.e., a house and a car). We present two examples. The first is a case where even though the ‘corresponding linear program’ (CLP) does not solve with integers, the core of the market is not empty. The second example is a case with additively separable preferences in which there is a core vector in the market which does not correspond to any optimal dual solution of the CLP. Both examples demonstrate possible behavior that is impossible in many of the ‘standard’ matching games.