Towards optimal subsidy bounds for envy-freeable allocations

Yasushi Kawase, Kazuhisa Makino, Hanna Sumita, Akihisa Tamura, Makoto Yokoo

Abstract: We study the fair division of indivisible items with subsidies among n agents, where the absolute marginal valuation of each item is at most one. Under monotone nondecreasing valuations (where each item is a good), Brustle et al. [9] demonstrated that a maximum subsidy of 2(n−1)<math><mn is="true">2</mn><mo stretchy="false" is="true">(</mo><mi is="true">n</mi><mo linebreak="badbreak" linebreakstyle="after" is="true">−</mo><mn is="true">1</mn><mo stretchy="false" is="true">)</mo></math> and a total subsidy of 2(n−1)2<math><mn is="true">2</mn><msup is="true"><mrow is="true"><mo stretchy="false" is="true">(</mo><mi is="true">n</mi><mo linebreak="badbreak" linebreakstyle="after" is="true">−</mo><mn is="true">1</mn><mo stretchy="false" is="true">)</mo></mrow><mrow is="true"><mn is="true">2</mn></mrow></msup></math> are sufficient to guarantee the existence of an envy-freeable allocation. In this paper, we improve upon these bounds, even in a wider model. Namely, we show that, given an EF1 allocation, we can compute in polynomial time an envy-free allocation with a subsidy of at most n−1<math><mi is="true">n</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">−</mo><mn is="true">1</mn></math> per agent and a total subsidy of at most n(n−1)/2<math><mi is="true">n</mi><mo stretchy="false" is="true">(</mo><mi is="true">n</mi><mo linebreak="badbreak" linebreakstyle="after" is="true">−</mo><mn is="true">1</mn><mo stretchy="false" is="true">)</mo><mo stretchy="false" is="true">/</mo><mn is="true">2</mn></math>. Moreover, when the valuations are monotone nondecreasing, we provide a polynomial-time algorithm that computes an envy-free allocation with a subsidy of at most n−1.5<math><mi is="true">n</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">−</mo><mn is="true">1.5</mn></math> per agent and a total subsidy of at most (n2−n−1)/2<math><mo stretchy="false" is="true">(</mo><msup is="true"><mrow is="true"><mi is="true">n</mi></mrow><mrow is="true"><mn is="true">2</mn></mrow></msup><mo linebreak="badbreak" linebreakstyle="after" is="true">−</mo><mi is="true">n</mi><mo linebreak="badbreak" linebreakstyle="after" is="true">−</mo><mn is="true">1</mn><mo stretchy="false" is="true">)</mo><mo stretchy="false" is="true">/</mo><mn is="true">2</mn></math>.

External IDs:dblp:journals/ai/KawaseMSTY25