## Abstract

We extend the notion of Combinatorial Walrasian Equilibrium, as defined by Feldman et al. [2013], to settings with budgets. When agents have budgets, the maximum social welfare as traditionally defined is not a suitable benchmark since it is overly optimistic. This motivated the liquid welfare of [Dobzinski and Paes Leme 2014] as an alternative. Observing that no combinatorial Walrasian equilibrium guarantees a non-zero fraction of the maximum liquid welfare in the absence of randomization, we instead work with randomized allocations and extend the notions of liquid welfare and Combinatorial Walrasian Equilibrium accordingly. Our generalization of the Combinatorial Walrasian Equilibrium prices lotteries over bundles of items rather than bundles, and we term it a lottery pricing equilibrium. Our results are two-fold. First, we exhibit an efficient algorithm which turns a randomized allocation with liquid expected welfare W into a lottery pricing equilibrium with liquid expected welfare 3-√5/2 W (≈ 0.3819-W). Next, given access to a demand oracle and an α-approximate oblivious rounding algorithm for the configuration linear program for the welfare maximization problem, we show how to efficiently compute a randomized allocation which is (a) supported on polynomially-many deterministic allocations and (b) obtains [nearly] an α fraction of the optimal liquid expected welfare. In the case of subadditive valuations, combining both results yields an efficient algorithm which computes a lottery pricing equilibrium obtaining a constant fraction of the optimal liquid expected welfare.

Original language | English |
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Title of host publication | EC 2016 - Proceedings of the 2016 ACM Conference on Economics and Computation |

Publisher | Association for Computing Machinery, Inc |

Pages | 401-418 |

Number of pages | 18 |

ISBN (Electronic) | 9781450339360 |

DOIs | |

State | Published - 21 Jul 2016 |

Externally published | Yes |

Event | 17th ACM Conference on Economics and Computation, EC 2016 - Maastricht, Netherlands Duration: 24 Jul 2016 → 28 Jul 2016 |

### Publication series

Name | EC 2016 - Proceedings of the 2016 ACM Conference on Economics and Computation |
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### Conference

Conference | 17th ACM Conference on Economics and Computation, EC 2016 |
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Country/Territory | Netherlands |

City | Maastricht |

Period | 24/07/16 → 28/07/16 |

### Bibliographical note

Publisher Copyright:© Copyright 2016 ACM.

## Keywords

- Budgets
- Combinatorial auctions
- Envy-free
- Lotteries
- Walrasian equilibrium