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Arctic Auctions, Linear Fisher Markets, and Rational Convex Programs

Vijay V. Vazirani

TL;DR

This paper addresses efficient allocation in Arctic Auctions and linear Fisher markets with differentiated goods. It develops a $Rational ext Convex rac{Program}{RCP}$ formulation for Arctic Auction equilibria and derives the first combinatorial polynomial-time algorithm via a primal-dual balanced-flow framework. It also extends the approach to a linear Fisher market with constant marginal costs, showing the problem remains an $RCP$ and offering a simple greedy algorithm. By linking market design with convex optimization, the work provides exact, scalable algorithms that are practically relevant for central-bank liquidity operations and complex substitution-enabled markets.

Abstract

This paper unifies two foundational constructs from economics and algorithmic game theory, the Arctic Auction and the linear Fisher market, to address the efficient allocation of differentiated goods in complex markets. Our main contributions are showing that an equilibrium for the Arctic Auction is captured by a Rational Convex Program, and deriving the first combinatorial polynomial-time algorithm for computing Arctic Auction equilibria.

Arctic Auctions, Linear Fisher Markets, and Rational Convex Programs

TL;DR

This paper addresses efficient allocation in Arctic Auctions and linear Fisher markets with differentiated goods. It develops a formulation for Arctic Auction equilibria and derives the first combinatorial polynomial-time algorithm via a primal-dual balanced-flow framework. It also extends the approach to a linear Fisher market with constant marginal costs, showing the problem remains an and offering a simple greedy algorithm. By linking market design with convex optimization, the work provides exact, scalable algorithms that are practically relevant for central-bank liquidity operations and complex substitution-enabled markets.

Abstract

This paper unifies two foundational constructs from economics and algorithmic game theory, the Arctic Auction and the linear Fisher market, to address the efficient allocation of differentiated goods in complex markets. Our main contributions are showing that an equilibrium for the Arctic Auction is captured by a Rational Convex Program, and deriving the first combinatorial polynomial-time algorithm for computing Arctic Auction equilibria.

Paper Structure

This paper contains 11 sections, 14 theorems, 52 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Related Works
  3. Rational Convex Program for Arctic Auction
  4. An Efficient Primal-Dual Algorithm for Arctic Auction
  5. Balanced Flow in Network $\mathcal{N}(p, r)$
  6. Description of Algorithm \ref{['alg.one']}
  7. Proof of Running Time
  8. Linear Fisher Market with Constant Marginal Costs
  9. Efficient Greedy Algorithm
  10. Discussion
  11. Acknowledgements

Key Result

Lemma 1

Equilibrium allocations and prices for the Arctic Auction satisfy the following conditions for each buyer $i \in B$.

Theorems & Definitions (32)

  • Definition 1
  • Definition 2
  • Lemma 1
  • Lemma 2
  • proof
  • Theorem 1
  • proof
  • Definition 3
  • Theorem 2
  • proof
  • ...and 22 more