Better ways to cut a cake

November 06, 2006

Providence, RI -- Suppose a cake is to be divided between two people, Alice and Bob. A fair procedure is to have Alice cut the cake and then have Bob choose whichever piece he prefers. Alice has an incentive to cut the cake exactly in half, since she will be left with whichever piece Bob does not take.

This "you cut, I choose" method, known since time immemorial, has been used in dispute resolutions ranging from land division in the Bible to children's squabbles over birthday cake. An article to appear in the December 2006 issue of the Notices of the AMS draws on the power and precision of mathematics to show there are even better ways to cut a cake.

The three authors of the article, Steven J. Brams, Michael A. Jones, and Christian Klamler, point out that the cut-and-choose method has the desirable property of "envy-freeness": Neither person envies the other, because each knows he has gotten at least half the cake. But the method lacks another desirable property, that of equitability: The subjective value that the two people place on the pieces they get might not be the same. For example, suppose one half of the cake is frosted with vanilla icing and the other with chocolate icing, and suppose Alice values chocolate icing twice as much as vanilla. It is possible that Alice's valuation of the piece she gets will be less than Bob's valuation of his piece, making these the two valuations inequitable.

Brams, Jones, and Klamler describe a new method for cake-cutting, which they call SP (for "Surplus Procedure"). Using SP, the cake can be cut in such a way that the value Alice puts on her piece is approximately the same as the value Bob puts on his---so both feel, for example, that they are getting about 65% of what they want! For cake division among 3 people, there is an extension of SP, called EP (for "Equitability Procedure"), that ensures all three get, say, 40% of what they want. However, it is not always possible to ensure both equitability and envy-freeness for divisions among 3 or more people. One desirable property that SP and EP share is that they are "strategy-proof": The players cannot assuredly manipulate these procedures to their advantage.

Potential uses of SP include fair division of land. For example, land bordering on water might be more valuable to Alice, while land bordering on forest might be more valuable to Bob. The SP method shows how to divide the land in such a way that the both parties place approximately the same value on the parcels of land they get.

Brams is well known for his work on fair-division algorithms. His 1999 book, "The Win-Win Solution: Guaranteeing Fair Shares to Everybody", written jointly with Alan D. Taylor, describes a host of contexts in which such algorithms apply, from the Camp David peace accords to the divorce of Donald and Ivana Trump. One of the algorithms described in the book, called "adjusted winner", has actually been patented by New York University, where Brams is a professor of politics.

With SP and EP, Brams says, "We are proposing a new, more scientific approach to dispute resolution. Even if it is not directly applicable, the reasoning that goes into fair-division algorithms is valuable. It shows how mathematics can contribute to making dispute resolution more rigorous and precise."
-end-
The article "Better Ways to Cut a Cake," by Brams, Jones, and Klamler, will be posted to the URL: http://www.ams.org/notices/200611/fea-brams.pdf on November 6, 2006.

Founded in 1888 to further mathematical research and scholarship, the more than 30,000-member American Mathematical Society fulfills its mission through programs and services that promote mathematical research and its uses, strengthen mathematical education, and foster awareness and appreciation of mathematics and its connections to other disciplines and to everyday life.

American Mathematical Society
201 Charles Street
Providence, RI 02904
401-455-4000

American Mathematical Society

Related Algorithms Articles from Brightsurf:

A multidisciplinary policy design to protect consumers from AI collusion
Legal scholars, computer scientists and economists must work together to prevent unlawful price-surging behaviors from artificial intelligence (AI) algorithms used by rivals in a competitive market, argue Emilio Calvano and colleagues in this Policy Forum.

Students develop tool to predict the carbon footprint of algorithms
Within the scientific community, it is estimated that artificial intelligence -- otherwise meant to serve as a means to effectively combat climate change -- will become one of the most egregious CO2 culprits should current trends continue.

Machine learning takes on synthetic biology: algorithms can bioengineer cells for you
Scientists at Lawrence Berkeley National Laboratory have developed a new tool that adapts machine learning algorithms to the needs of synthetic biology to guide development systematically.

Algorithms uncover cancers' hidden genetic losses and gains
Limitations in DNA sequencing technology make it difficult to detect some major mutations often linked to cancer, such as the loss or duplication of parts of chromosomes.

Managing data flow boosts cyber-physical system performance
Researchers have developed a suite of algorithms to improve the performance of cyber-physical systems - from autonomous vehicles to smart power grids - by balancing each component's need for data with how fast that data can be sent and received.

New theory hints at more efficient way to develop quantum algorithms
A new theory could bring a way to make quantum algorithm development less of an accidental process, say Purdue University scientists.

AI as good as the average radiologist in identifying breast cancer
Researchers at Karolinska Institutet and Karolinska University Hospital in Sweden have compared the ability of three different artificial intelligence (AI) algorithms to identify breast cancer based on previously taken mammograms.

Context reduces racial bias in hate speech detection algorithms
When it comes to accurately flagging hate speech on social media, context matters, says a new USC study aimed at reducing errors that could amplify racial bias.

Researchers discover algorithms and neural circuit mechanisms of escape responses
Prof. WEN Quan from School of Life Sciences, University of Science and Technology of China (USTC) of the Chinese Academy of Sciences (CAS) has proposed the algorithms and circuit mechanisms for the robust and flexible motor states of nematodes during escape responses.

Lightning fast algorithms can lighten the load of 3D hologram generation
Tokyo, Japan - Researchers from Tokyo Metropolitan University have developed a new way of calculating simple holograms for heads-up displays (HUDs) and near-eye displays (NEDs).

Read More: Algorithms News and Algorithms Current Events
Brightsurf.com is a participant in the Amazon Services LLC Associates Program, an affiliate advertising program designed to provide a means for sites to earn advertising fees by advertising and linking to Amazon.com.