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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2507.18251 |
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| _version_ | 1866912653400604672 |
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| author | Jin, Jiarong Tao, Biaoshuai |
| author_facet | Jin, Jiarong Tao, Biaoshuai |
| contents | 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). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_18251 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Pareto-Optimal and Fair Allocations with Personalized Bi-Valued Utilities Jin, Jiarong Tao, Biaoshuai Computer Science and Game Theory 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). |
| title | On Pareto-Optimal and Fair Allocations with Personalized Bi-Valued Utilities |
| topic | Computer Science and Game Theory |
| url | https://arxiv.org/abs/2507.18251 |