1. WASTE MAKES HASTE: Erel Segal-Halevi, Avinatan Hassidim, Yonatan Aumann BOUNDED-TIME PROTOCOLS FOR ENVY-FREE CAKE CUTTING WITH FREE DISPOSAL

2. n agents with different tastes “I want lots of flowers” “I love white decorations” “No writing on my piece at all!”

3. Fairness Proportional Each agent gets a piece worth to it at least 1/ n Envy Free: No agent prefers a piece allotted to someone else

4. 2 agents: Cut and Choose  Alice cuts to equal pieces  Bob chooses better piece BobAlice Proportional Envy free

5. n agents: General Case ProportionalEnvy-Free 2 agents 2 queries 3+ agents Θ(n log n) Queries (Even&Paz, 1984; Edmonds, 2006; Woeginger, 2007)

6. ProportionalEnvy-Free 2 agents 2 queries 3+ agents Θ(n log n) Queries (Even&Paz, 1984; Edmonds, 2006; Woeginger, 2007) n agents: General Case Θ(infinity) Queries ! (Su, 1999; Stromquist, 2008)

7. Previous Work Disconnected pieces:  General valuations: Unbounded time (Brams and Taylor 1995, Robertson and Webb 1997)  Piecewise-constant: O(poly(input)) time (Kurokawa Lai and Procaccia 2013) Connected pieces:  ε -approximate envy-free: Θ(1/ε n ) (Su 1999; Deng and Saberi 2012)

8. This work

9. n=3 agents. Value ≥ 1/4  Alice cuts 3 equal pieces.  Bob trims best to equalize to 2 nd -best.  Carl then Bob then Alice choose pieces. Bob Alice Carl Based on Selfridge-Conway 1960.

10. n agents: General Case  Alice cuts 2 n-2 +1 equal pieces.  Bob trims to make 2 n-3 +1 equal pieces. Using a linear-time algorithm by Reitzig and Wild, 2015. ...  Carl trims to make 2 equal best pieces.  Agents choose in inverse order. Value ≥ 1/2 n -1

11. Envy-Free and Proportional With Waste: Envy-Free Proportional. Can we find in bounded time a division:  Envy-Free  Proportional (Value ≥ 1/ n):  Connected pieces? For n=3: Yes! Details in paper. ? For n ≥ 4: Open question.

12. Acknowledgements  Financial supporters: BIU Doctoral Fellowships of Excellence Program, ISF grant 1083/13, ISF grant1241/12 and BSF grant 2012344.  Insightful discussions: Raphael Reitzig, Sebastian Wild, Abhishek Bansai, Pal Gronas Drange, Shaull Almagor, Luke Mathieson, David Richerby, Rick Decker, Instructed A, Frank W, Babou, Tom van der Zanden and the stackexchange.com community. Collaborations are welcome! erelsgl@gmail