Fair allocation with (semi-single-peaked) preferences over location and quantity

We consider the problem of dividing and allocating a perfectly divisible heterogeneous good where agents have a preference for location and quantity. We assume that preferences are single-peaked in quantity, i.e., semi-single-peaked which can be represented by continuous indifference curves (ICs). We show existence of envy-free and Pareto efficient allocation rules, and characterize the set of all such rules using the notion of a balanced IC. We define the balanced-curve allocation (BCA) which uses the region between the two balanced ICs to obtain feasible allocations. We show that an allocation rule is envy-free and Pareto efficient if and only if it is in the set specified by the BCA rule. We show that there is no strategy-proof, envy-free and Pareto efficient allocation rule. We provide some insights into the problem when there are more than 2 agents.