On Pareto-Optimal and Fair Allocations with Personalized Bi-Valued Utilities

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 $a>b>c\geq0$), 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).