# Public Projects, Boolean Functions, and the Borders of Border's Theorem

@article{Gopalan2015PublicPB, title={Public Projects, Boolean Functions, and the Borders of Border's Theorem}, author={Parikshit Gopalan and Noam Nisan and Tim Roughgarden}, journal={Proceedings of the Sixteenth ACM Conference on Economics and Computation}, year={2015} }

Border's theorem gives an intuitive linear characterization of the feasible interim allocation rules of a Bayesian single-item environment, and it has several applications in economic and algorithmic mechanism design. All known generalizations of Border's theorem either restrict attention to relatively simple settings, or resort to approximation. This paper identifies a complexity-theoretic barrier that indicates, assuming standard complexity class separations, that Border's theorem cannot be…

## 25 Citations

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## References

SHOWING 1-10 OF 33 REFERENCES

### The Complexity of Optimal Mechanism Design

- Computer ScienceSODA
- 2014

It is shown that a revenue-optimal auction in multi-item settings cannot be found and implemented computationally efficiently, unless ZPP ⊇ P#P, which is true even for a single additive bidder whose values for the items are independently distributed on two rational numbers with rational probabilities.

### An algorithmic characterization of multi-dimensional mechanisms

- EconomicsSTOC '12
- 2012

The results provide a characterization of feasible, Bayesian Incentive Compatible mechanisms in multi-item multi-bidder settings, and give a proper generalization of Border's Theorem for a broader class of feasibility constraints, including the intersection of any two matroids.

### Designing and learning optimal finite support auctions

- Computer ScienceSODA '07
- 2007

This paper demonstrates that a Myerson-style auction can be constructed in time polynomial in the number of bidders and the size of the support sets and shows that the optimal mechanism is an order-based auction and is used to prove the correctness of the learning algorithm as well as to bound its running time.

### Barriers to Near-Optimal Equilibria

- Economics2014 IEEE 55th Annual Symposium on Foundations of Computer Science
- 2014

The results imply that the simultaneous first-price auction format is optimal among all "simple combinatorial auctions" in several settings, and this framework gives three families of lower bounds on the quality of equilibria, each motivated by a different set of problems.

### Why prices need algorithms

- EconomicsSECO
- 2016

This note surveys the main result from [Roughgarden and Talgam-Cohen 2015], which shows that the existence of pricing equilibria is inextricably connected to the computational complexity of related optimization problems: demand oracles, revenue- maximization and welfare-maximization.

### Optimal auctions with correlated bidders are easy

- Computer ScienceSTOC '11
- 2011

It is shown that in the oracle model introduced by Ronen and Saberi [FOCS'02], there exists a polynomial time truthful in expectation mechanism that provides a (1.5+ε)-approximation to the revenue achievable by an optimal truthful-in-expectation mechanism, and a poynomial time deterministic truthful mechanism that guarantees 5/3 approximation.

### The Complexity of Optimal Multidimensional Pricing

- BusinessSODA
- 2014

It is shown that the problem of computing a revenue-optimal pricing can be solved in polynomial time for distributions of support size 2 and its decision version is NP-complete for distributionsof support size 3.

### Optimal Multi-dimensional Mechanism Design: Reducing Revenue to Welfare Maximization

- Economics2012 IEEE 53rd Annual Symposium on Foundations of Computer Science
- 2012

The results are computationally efficient for all multidimensional settings where the bidders are additive, or can be efficiently mapped to be additive, albeit the feasibility and demand constraints may still remain arbitrary combinatorial.

### Optimal Auctions with Risk Averse Buyers

- Economics
- 1984

THIS PAPER STUDIES AUCTIONS designed to maximize the expected revenue of a seller facing risk averse bidders with unknown preferences.2 Although we concentrate on auctions where a seller sells a…

### Bayesian optimal auctions via multi- to single-agent reduction

- EconomicsEC '12
- 2012

An abstract optimal auction problem for selecting a subset of self-interested agents to whom to provide a service, and shows that interim feasible allocation rules can be optimized over and implemented via a linear program that has quadratic complexity in the sum of the sizes of the individual agents' type spaces.