Abstract
We study the fair division problem of allocating m indivisible goods to n agents with additive personalized bi-valued utilities. Specifically, each agent i assigns one of two positive values a_i > b_i > 0 to each good, indicating that agent i's valuation of any good is either a_i or b_i. For convenience, we denote the value ratio of agent i as r_i = a_i / b_i. We give a characterization to all the Pareto-optimal allocations. Our characterization implies a polynomial-time algorithm to decide if a given allocation is Pareto-optimal in the case each r_i is an integer. For the general case (where r_i may be fractional), we show that this decision problem is coNP-complete. Our result complements the existing results: this decision problem is coNP-complete for tri-valued utilities (where each agent's value for each good belongs to {a,b,c} for some prescribed a>b>c≥0), and this decision problem belongs to P for bi-valued utilities (where r_i in our model is the same for each agent). We further show that an EFX allocation always exists and can be computed in polynomial time under the personalized bi-valued utilities setting, which extends the previous result on bi-valued utilities. We propose the open problem of whether an EFX and Pareto-optimal allocation always exists (and can be computed in polynomial time).