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/
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