# A Complementary Pivot Algorithm for Competitive Allocation of a Mixed Manna

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## Abstract

We study the fair division problem of allocating a mixed manna under additively separable piecewise linear concave (SPLC) utilities. A mixed manna contains goods that everyone likes and bads (chores) that everyone dislikes as well as items that some like and others dislike. The seminal work of Bogomolnaia et al. argues why allocating a mixed manna is genuinely more complicated than a good or a bad manna and why competitive equilibrium is the best mechanism. It also provides the existence of equilibrium and establishes its distinctive properties (e.g., nonconvex and disconnected set of equilibria even under linear utilities) but leaves the problem of computing an equilibrium open. Our main results are a linear complementarity problem formulation that captures all competitive equilibria of a mixed manna under SPLC utilities (a strict generalization of linear) and a complementary pivot algorithm based on Lemke’s scheme for finding one. Experimental results on randomly generated instances suggest that our algorithm is fast in practice. Given the PPAD-hardness of the problem, designing such an algorithm is the only non–brute force (nonenumerative) option known; for example, the classic Lemke–Howson algorithm for computing a Nash equilibrium in a two-player game is still one of the most widely used algorithms in practice. Our algorithm also yields several new structural properties as simple corollaries. We obtain a (constructive) proof of existence for a far more general setting, membership of the problem in PPAD, a rational-valued solution, and an odd number of solutions property. The last property also settles the conjecture of Bogomolnaia et al. in the affirmative. Furthermore, we show that, if the number of either agents or items is a constant, then the number of pivots in our algorithm is strongly polynomial when the mixed manna contains all bads.
Original language English (US) 1630-1656 27 Mathematics of Operations Research 48 3 https://doi.org/10.1287/moor.2022.1315 Published - Aug 2023

## Keywords

• LCP formulation
• competitive equilibrium
• complementary pivot algorithm
• fair division
• mixed manna

## ASJC Scopus subject areas

• General Mathematics
• Computer Science Applications
• Management Science and Operations Research

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