PowerPoint Presentation

Topics in Algorithmic Game Theory

28 NOV 2012

Ahuva Mualem envy free cake cutting

, Divide and Choose)"):

, v1 (I) = v1 (J)

max (v2 (I1), v2 (I

I j max (v2(I), v2(J)) > min (v2(I), v2(J)) , ( ) " " n > 2.

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https://www.youtube.com/watch?v=mtZTIwSIuGw Sperners Lemma

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> 1 - Sperners Lemma

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The standard simplex in R3

Rental Harmony Problem

: n > 2 n . 1000 , 1000 .

'Rental Harmony' ' ',

https://www.youtube.com/watch?v=lAop4Su5Uag

Sperner + Simmons + Su How to Solve the Envy Free Cake Cutting and Rental Harmony Problems

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v1 (I1) = v1 (I2)

max (v2 (I1), v2 (I

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, n > 2

[[Dubins-Spanier:A knife is slowly moved at constant speed parallel. At each instant the knife is poised so that it could cut a unique slice of the cake. At time goes by the potential slice increases monotonely from nothing until it becomes the entire cake. The first person to indicate satisfaction with the slice then determined by the position of the knife receives that slice and is eliminated from further distribution of the cake. (If two or more participants simultaneously indicate satisfaction with the slice, it is given to any of them.) The process is repeated with the other n 1 participants and with what remains of the cake. The last player gets the remainder of the cake.

, n > 2

[[Dubins-Spanier:

The knife is passed over the cake from the left end to the right. A player says stop when he thinks 1/n of the cake is to the left of the knife, the cake is cut and he get that piece. Repeat with the remaining cake and players.The last player gets the remainder of the cake.

: 1, 2, ..., i 1/n . i+1, i+2, ..., n (n-i)/n , .

BibliographyFrancis Edward Su. Rental harmony: Sperner's lemma in fair division. Amer. Math. Monthly, 106(10):930942, 1999 Chandra Chekuri, CS 573: Algorithmic Game Theory, 2008Lecture Notes, , Wikipedia