Achieving Equitability with Subsidy

Abstract: We study the fair allocation problem of indivisible items with subsidies. In this paper, we mainly consider the notion of fairness - equitability (EQ), which requires that items be allocated such that all agents value the bundle they receive equally. Firstly, we study the upper bounds of the required subsidy to achieve EQ in different settings of items and provide the corresponding lower bounds. Secondly, we consider the bounded subsidy for achieving EQ and another popular notion of fairness - envy-freeness (EF) and give a characterization of the allocations that can achieve both EQ and EF. Finally, we analyze the bounds of subsidy of the allocations achieving fairness and efficiency (utilitarian social welfare or Nash welfare), and design several polynomial-time algorithms to compute the desired allocation.