# Ariel D. Procaccia (Microsoft)  A cake must be divided between several children  The cake is heterogeneous  Each child has different value for same.

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Ariel D. Procaccia (Microsoft)

 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. 2

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

 Proportionality:  i, v i (X i )  1/n  Envy-Freeness:  i,j, v i (X i )  v i (X j )  Envy-freeness  Proportionality  For n  3 envy-freeness is strictly stronger 4 1/3 1/2 1/6 1 1 1 1

 Agent 1 divides into two pieces X,Y s.t. v 1 (X)=1/2, v 1 (Y)=1/2  Agent 2 chooses preferred piece  Protocol is proportional + envy free 5 1/2 1/3 2/3

6 1/3  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

7  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

 A concrete complexity model  Two types of queries Eval i (x,y) = v i ([x,y]) Cut i (x,  ) = y s.t. v i ([x,y]) =   Can simulate all known discrete protocols 8

 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? 9

 Theorem: The query complexity (in the Robertson-Webb model) of achieving an envy free allocation is  (n 2 )  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) 10

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

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