How to obtain an equitable optimal fair division

Legut, Jerzy

doi:10.1007/s10479-018-3053-2

Cited by 1 publication

(1 citation statement)

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“…A problem closely related to our infinite-dimensional Fisher-market setting is the cake-cutting or fair division problem. There, the goal is to efficiently partition a "cake"often modeled as a compact measurable space, or simply the unit interval [0, 1] -among n agents so that certain fairness and efficiency properties are satisfied (Weller 1985;Brams and Taylor 1996;Cohler et al 2011;Procaccia 2013;Cohler et al 2011;Brams et al 2012;Chen et al 2013;Aziz and Ye 2014;Aziz and Mackenzie 2016;Legut 2017Legut , 2020. See (Procaccia 2016) for a survey for the various problem setups, algorithms and complexity results.…”

Section: Introductionmentioning

confidence: 99%

Infinite-Dimensional Fisher Markets: Equilibrium, Duality and Optimization

Gao

Kroer

2021

AAAI

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This paper considers a linear Fisher market with n buyers and a continuum of items. In order to compute market equilibria, we introduce (infinite-dimensional) convex programs over Banach spaces, thereby generalizing the Eisenberg-Gale convex program and its dual. Regarding the new convex programs, we establish existence of optimal solutions, KKT conditions, as well as strong duality. All these properties are established via non-standard arguments, which circumvent the limitations of duality theory in optimization over infinite-dimensional vector spaces. Furthermore, we show that there exists a pure equilibrium allocation, i.e., a division of the item space. Similar to the finite-dimensional case, a market equilibrium under the infinite-dimensional Fisher market is Pareto optimal, envy-free and proportional. We also show how to obtain the (a.e. unique) equilibrium prices and a pure equilibrium allocation from the (unique) equilibrium utility prices. When the item space is the unit interval [0,1] and buyers have piecewise linear utilities, we show that approximate equilibrium prices can be computed in polynomial time. This is achieved by solving a finite-dimensional convex program using the ellipsoid method. To this end, we give nontrivial and efficient subgradient and separation oracles. For general buyer valuations, we propose computing market equilibrium using stochastic dual averaging, which finds approximate equilibrium prices with high probability.

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