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Thou Shalt Covet Thy Neighbor’s Cake

Thou Shalt Covet Thy Neighbor’s Cake. Ariel D. Procaccia (Microsoft). Cake cutting. A cake must be divided between several children The cake is heterogeneous Each child has different value for same piece of cake How can we divide the cake fairly? What is “fairly”?

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Thou Shalt Covet Thy Neighbor’s Cake

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  1. Thou Shalt Covet Thy Neighbor’s Cake Ariel D. Procaccia (Microsoft)

  2. Cake cutting • A cake must be divided between several children • The cake is heterogeneous • Each child has different value for same piece of cake • How can we divide the cake fairly? • What is “fairly”? • A metaphor for land disputes, divorce settlements, etc.

  3. The model • Cake is interval [0,1] • Set of agents {1,...,n} • Each agent has valuation vi over pieces of cake X  [0,1] • vi ([0,1]) = 1 • For disjoint X,Y: vi (XY) = vi (X) + vi (Y) • Find allocation X1,...,Xn • Not necessarily connected pieces

  4. Fairness criteria • Proportionality: i, vi(Xi)  1/n • Envy-Freeness: i,j, vi(Xi)  vi(Xj) • Envy-freeness  Proportionality • For n  3 envy-freeness is strictly stronger 1/3 1/2 1 1/6 1

  5. Cut-and-Choose • Agent 1 divides into two pieces X,Ys.t. v1(X)=1/2, v1(Y)=1/2 • Agent 2 chooses preferred piece • Protocol is proportional + envy free 1/2 1/3 1/2 2/3

  6. Dubins-Spanier • Referee continuously moves knife • Repeat: when piece left of knife is worth 1/n to agent, agent shouts “stop” and gets piece • Protocol is proportional 1/3 1/3

  7. Discrete Dubins-Spanier • Moving knife is not really needed • Repeat: each agent makes a mark at his 1/n point, leftmost agent gets piece up to its mark

  8. The Robertson-Webb model • A concrete complexity model • Two types of queries • Evali(x,y) = vi([x,y]) • Cuti(x,) = ys.t. vi([x,y]) =  • Can simulate all known discrete protocols

  9. Bounds in RW model • Proportional • Recursive protocol that requires O(nlogn) queries [Even and Paz, 1984] • Lower bound of (nlogn) [Edmonds and Pruhs, 2006] • Envy free (always exists) • n = 2: Cut and Choose • n= 3: “good” protocol [Selfridge and Conway, 1961] • n  4: known protocol requires unbounded number of queries • Lower bound for envy free cake cutting?

  10. Main result • Theorem: The query complexity (in the Robertson-Webb model) of achieving an envy free allocation is (n2) • Proof idea : • Consider a problem that has to be solved with respect to (n) agents separately in order to guarantee envy-freeness • Query complexity of this problem is (n)

  11. Discussion • Provides a separation between proportional and envy free • Future work: significantly improve the lower bound • Need completely different technique

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