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Truth, Justice, and Cake Cutting

Truth, Justice, and Cake Cutting. Yiling Chen, John K. Lai, David C. Parkes, Ariel D. Procaccia (Harvard SEAS). Truth, justice, and cake cutting. Division of a heterogeneous divisible good The cake is the interval [0,1] Set of agents N={1,...,n}

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Truth, Justice, and Cake Cutting

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  1. Truth, Justice, and Cake Cutting Yiling Chen, John K. Lai, David C. Parkes, Ariel D. Procaccia (Harvard SEAS)

  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 Vi over pieces of cake • Additive: if XY= then Vi(X)+Vi(Y) = Vi(XY) • iN, Vi(0,1) = 1 • Find an allocation A1,...,An

  3. Truth, justice, and cake cutting • Proportionality:iN,Vi(Ai)  1/n • Envy-freeness: i,jN, Vi(Ai)  Vi(Aj) • 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 1/6 1

  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

  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 Vi 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]

  6. Randomized algorithms • 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 • A partition X1,...,Xnis perfect if for every i,k,Vi(Xk)=1/n • Algorithm: • Find a perfect partition X1,...,Xn • 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 Y1,...,Yn, butk (1/n)Vi(Yk) = (1/n) k Vi(Yk) = 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 • Conceptual contributions • Truthful cake cutting • Restricted valuation functions and tractable algorithms • Current work with IoannisCaragiannis and John Lai: piecewise uniform with a minimum • Envy freeness and system performance? • Cake cutting is awesome!

  9. Thank You!

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