A Constant Bound for the Periods of Parallel Chip-firing Games with Many Chips

Paul Myer Kominers; Scott Duke Kominers

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Article | Archiv der Mathematik

A Constant Bound for the Periods of Parallel Chip-firing Games with Many Chips

by Paul Myer Kominers and Scott Duke Kominers

Abstract

We prove that any parallel chip-firing game on a graph G with at least 4|E(G)| − |V(G)| chips stabilizes, i.e., such a game has eventual period of length 1. Furthermore, we obtain a polynomial bound on the number of rounds before stabilization. This result is a counterpoint to previous results which showed that the eventual periods of parallel chip-firing games with few chips need not be polynomially bounded.

Keywords: chip-firing; Parallel chip-firing; stabilization;

Format: Print

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Citation:

Kominers, Paul Myer, and Scott Duke Kominers. "A Constant Bound for the Periods of Parallel Chip-firing Games with Many Chips." Archiv der Mathematik 95, no. 1 (July 2010): 9–13.

About the Author

Scott Duke Kominers

MBA Class of 1960 Associate Professor of Business Administration

Entrepreneurial Management

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How to End the Plasma Shortage for Coronavirus Patients

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Kominers, Scott Duke. "How to End the Plasma Shortage for Coronavirus Patients." Bloomberg Opinion (May 11, 2020). View Details

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Working Paper | HBS Working Paper Series | 2020

Paying It Backward and Forward: Expanding Access to Convalescent Plasma Therapy Through Market Design

Scott Duke Kominers, Parag A. Pathak, Tayfun Sönmez and M. Utku Ünver

COVID-19 convalescent plasma (CCP) therapy is currently a leading treatment for COVID-19. At present, there is a shortage of CCP relative to demand. We develop and analyze a model of centralized CCP allocation that incorporates both donation and distribution. In order to increase CCP supply, we introduce a mechanism that utilizes two incentive schemes, respectively based on principles of “paying it backward" and “paying it forward." Under the first scheme, CCP donors obtain treatment vouchers that can be transferred to patients of their choosing. Under the latter scheme, patients obtain priority for CCP therapy in exchange for a future pledge to donate CCP if possible. We show that in steady-state, both principles generally increase overall treatment rates for all patients—not just those who are voucher-prioritized or pledged to donate. Our results also hold under certain conditions if a fraction of CCP is reserved for patients who participate in clinical trials. Finally, we examine the implications of pooling blood types on the efficiency and equity of CCP distribution.

Citation:
Kominers, Scott Duke, Parag A. Pathak, Tayfun Sönmez, and M. Utku Ünver. "Paying It Backward and Forward: Expanding Access to Convalescent Plasma Therapy Through Market Design." Harvard Business School Working Paper, No. 20-116, May 2020. View Details

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Article | Journal de Théorie des Nombres de Bordeaux | 2019

Configurations of Extremal Type II Codes via Harmonic Weight Enumerators

Noam D. Elkies and Scott Duke Kominers

We prove configuration results for extremal Type II codes, analogous to the configuration results of Ozeki and of the second author for extremal Type II lattices. Specifically, we show that for n ∈{8,24,32,48,56,72,96} every extremal Type II code of length n is generated by its codewords of minimal weight. Where Ozeki and Kominers used spherical harmonics and weighted theta functions, we use discrete harmonic polynomials and harmonic weight enumerators. Along the way we introduce "t12-designs" as a discrete analog of Venkov's spherical designs of the same name.

Keywords: Mathematical Methods;

Citation:
Elkies, Noam D., and Scott Duke Kominers. "Configurations of Extremal Type II Codes via Harmonic Weight Enumerators." Journal de Théorie des Nombres de Bordeaux 31, no. 3 (2019): 679–688. View Details

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A Constant Bound for the Periods of Parallel Chip-firing Games with Many Chips

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