1. TRUTH, JUSTICE, AND CAKE CUTTING Yiling Chen, John K. Lai, David C. Parkes, Ariel D. Procaccia (Harvard SEAS) 1

2. Truth, justice, and cake cutting  Division of a heterogeneous divisible good  The cake is the interval [0,1]  Set of agents N={1,...,n}  Each agent has a valuation function V i over pieces of cake  Additive: if X  Y=  then V i (X)+V i (Y) = V i (X  Y)   i  N, V i (0,1) = 1  Find an allocation A 1,...,A n 2

3. Truth, justice, and cake cutting  Proportionality:  i  N, V i (A i )  1/n  Envy-freeness:  i,j  N, V i (A i )  V i (A j )  Assuming free disposal the two properties are incomparable  Envy-free but not proportional: throw away cake  Proportional but not envy-free 1/3 1/2 1/6 1 1 1 1 3

4. Truth, justice, and cake cutting  Previous work considered strategyproof cake cutting [Brams, Jones & Klamler 2006, 2008]  Their notion: agents report the truth if there exist valuations for others s.t. agent does not gain by lying  Truthful algorithm = truthfulness is a dominant strategy 4

5. Deterministic algorithms  Goal: design truthful, envy free, proportional, and tractable cake cutting algorithms  Requires restricting the valuation functions  Lower bounds for envy-free cake cutting [Procaccia, 2009]  Valuation V i is piecewise uniform if agent i is uniformly interested in a piece of cake  Theorem: assume that the agents have piecewise uniform valuations, then there is a deterministic alg that is truthful, proportional, envy-free, and polynomial-time  Related to work in econ on the random assignment problem [Bogomolnaia & Moulin 2004] 5

6. Randomized algorithms 6  A randomized alg is universally envy-free (resp., universally proportional) if it always returns an envy-free (resp., proportional) allocation  A randomized alg is truthful in expectation if an agent cannot gain in expectation by lying  Looking for universal fairness and truthfulness in expectation  Does it make sense to look for fairness in expectation and universal truthfulness?

7. Nobody’s perfect 7  A partition X 1,...,X n is perfect if for every i,k, V i (X k )=1/n  Algorithm: 1. Find a perfect partition X 1,...,X n 2. Give each player a random piece  Observation (see also [Mossel&Tamuz 2010]): alg is truthful in expectation, universally EF and universally proportional  Proof: if agent i lies it may lead to a partition Y 1,...,Y n, but  k (1/n)V i (Y k ) = (1/n)  k V i (Y k ) = 1/n  It is known that a perfect partition always exists [Alon 1987]  Lemma: if agents have piecewise linear valuations then a perfect partition can be found in poly time

8. Discussion 8  Conceptual contributions  Truthful cake cutting  Restricted valuation functions and tractable algorithms  Current work with Ioannis Caragiannis and John Lai: piecewise uniform with a minimum  Envy freeness and system performance?  Cake cutting is awesome!

9. Thank You!