The Price of Fairness for a Small Number of Indivisible Items

We consider the price of fairness for the allocation of indivisible goods. For envy-freeness as fairness criterion it is known from the literature that the price of fairness can increase linearly in terms of the number of agents. For the constructive lower bound a quadratic number of items was used. In practice this might be inadequately large. So we introduce the price of fairness in terms of both the number of agents and items, i.e., key parameters which generally may be considered as common and available knowledge. It turns out that the price of fairness increases sublinearly if the number of items is not too much larger than the number of agents. For the special case of conformity of both counts, exact asymptotics are determined. Additionally, an efficient integer programming formulation is given.