Truth, Justice, and Cake Cutting

1. Truth, Justice, and Cake Cutting Ariel Procaccia (Harvard SEAS)

2. Standing on the shoulders of giants Superman: “I’m here to fight for truth, justice, and the American Way.” Lois Lane: “You’re gonna wind up fighting every elected official in this country!” Superman (1978)

3. Truth, justice, and cake cutting • Division of a heterogeneous divisible good • The cake is the interval [0,1] • Set of agents N={1,...,n} • Piece of cake X  [0,1] = finite union of disjoint intervals • Each agent has a valuation function Vi over pieces of cake • Integral over a value density function vi • iN, Vi(0,1) = 1 • Find an allocation A1,...,An

4. 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

5. Some childhood nostalgia • Assume that n=2 • The cut and choose algorithm [Procaccia&Procaccia, circa 1987?]: • Player 1 cuts the cake into two pieces X1,X2s.t. V1(X1)=V1(X2) = ½ • Player 2 chooses the piece that he prefers • Player 1 gets the other piece • Not a bad algorithm! • Envy-free  proportional • (Contiguous pieces  one cut) 1/2 1/3 1/2 2/3

6. Cake cutting is not a piece of cake • Very cool envy-free algorithm for n=3[Selfridge&Conway, circa 1960] • Envy-free algorithm for n4 [Brams&Taylor, 1995] • May require an unbounded number of steps! • Recent lower bounds in a concrete complexity model • Envy-free unbounded assuming contiguous pieces [Stromquist, 2008] • (n2) lower bound for envy-free cake cutting [Procaccia, 2009]

7. 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 • Prior-free! • Truthful algorithm= truthfulness is a dominant strategy • Cut and choose is “strategyproof” but not truthful

8. An inconvenient truth • Goal: design truthful, fair (envy-free and proportional), and tractable cake cutting algorithms • Requires restricting the valuation functions • Valuation Vi is piecewise constant if its value density function vi is piecewise constant • Valuation is piecewise uniform if moreover vi is some uniform constant or zero • Agent is uniformly interested in piece of cake Ui • Representation: boundaries of these intervals • A natural (?) restriction and also proof of concept

9. Restricted valuations illustrated Piecewise constant valuation that is not piecewise uniform Piecewise uniform valuation 2 2 1 1 0 0 0 0.5 1 0 0.5 1 Vi([0,0.1][0.5,0.7]) = 0.4

10. The case of two agents: take 1 • We first assume n=2 (and piecewise uniform valuations) • A simple algorithm: • For each agent, make a mark at the beginning and end of each of the agent’s desired intervals • For each subinterval between consecutive marks, allocate left half to agent 1and right half to agent 2 • Each agent gets value ½  envy-free and proportional • ... but not truthful • If U1 = [0,0.5] and U2 = [0,1] then A1= [0,0.25][0.5,0.75]and A2 = [0.25,0.5][0.75,1] • Agent 1 can gain by reporting [0,1] A1= [0,0.5]

11. The case of two agents: take 2 Initialization phase: Discard [0,1]\U1U2 Make a mark at the beginning and end of each desired interval Allocate half of each subinterval between consecutive marks to agent 1 and half to agent 2 • Denote: • len(X) = the total length of intervals in X • X1 = U1\U2 , X2 = U2\U1, X12 = U1U2 • Assume len(U1)  len(U2) • Another simple algorithm (that works)

12. The case of two agents: take 2 Swapping phase: Swap pieces Y,Z of equal length where agent 1 owns Y, agent 2 owns Z, YX2, ZX1 Swap pieces Y,Z of equal length where agent 1 owns Y, agent 2 owns Z, YX2, ZX12 If there are still pieces of X2 owned by agent 1, give them to agent 2 Initialization phase: • Discard [0,1]\U1U2 • Make a mark at the beginning and end of each desired interval • Allocate half of each subinterval between consecutive marks to agent 1 and half to agent 2 U1 X1 X2 U1 X12 U2 X2

13. Properties of the algorithm (n=2) • Envy-free and proportional: obvious • There are two cases (given len(U1)  len(U2)): • len(U1)  len(U1U2)/2: the agents receive a desired piece of length len(U1U2)/2 (an exact allocation) • len(U1)  len(U1U2)/2: agent 1 gets U1and agent 2 gets X2 Swapping phase: • Swap pieces Y,Z of equal length where agent 1 owns Y, agent 2 owns Z, YX2, ZX1 • Swap pieces Y,Z of equal length where agent 1 owns Y, agent 2 owns Z, YX2, ZX12 • If there are still pieces of X2 owned by agent 1, give them to agent 2

14. The algorithm is truthful (n=2) • Assume agent 1 misreports U’1 we have X’1 , X’2 , X’12 • Can assume len(U1)  len(U1U2)/2 • Originally got len(U1U2)/2 = (len(X1)+len(U2))/2 • Now gets  len(U’1U2)/2 = (len(X’1)+len(U2))/2 • len(X’1) = len(X1)k  increases length of piece by k/2 but length of k is useless • Crucial: Agent 1 first trades for X1 • len(X’1) = len(X1)k  decreases length of A1 by k/2, before all of A1 was desired

15. The general algorithm: setup • Let S  N, X is a piece of cake • D(S,X) = (iSUi)X = portions of X desired by at least one agent in S • avg(S,X) = len(D(S,X))/|S| • A1,...,Anis exact wrt S,X if iS, len(Ai)=avg(S,X) and Ai is desired by agent i • For example, S={1,2} and X=[0,1] • U1=U2=[0,0.2]  A1=[0,0.1], A2=[0.1,0.2] is exact • U1=[0,0.2], U2=[0.3,0.7]  no exact allocation

16. The general algorithm • Initialization: • SN, X[0,1] • While S • Sminargminavg(S’,X) • Let E1,...,En be an exact allocation wrt Smin,X • iSmin , AiEi • SS\Smin • XX\D(Smin,X) 0.6 S’S 0.39 U3 U2 0.1 U1 0 0

17. The case of two agents revisited • Assume len(U1)  len(U2) • Sminis either {1} or {1,2} • len(U1)  len(U1U2)/2: Smin is {1,2},give exact allocation wrt {1,2},[0,1] • len(U1) < len(U1U2)/2: Smin is {1}, give 1 exact allocation wrt {1},[0,1] (U1), the rest to 2 in next iteration • Initialization: • SN, X[0,1] • While S • Sminargminavg(S,’X) • Let E1,...,En be an exact allocation w.r.t. Smin,X • iSmin , AiEi • SS\Smin • XX\D(Smin,X) S’S

18. Exact allocations and network flow • There are two problematic steps in while loop: • Step 1: computing Smin? • Step 2: existence and computation of exact allocation? • Solution: use network flow • Initialization: • SN, X[0,1] • While S • Sminargminavg(S,’X) • Let E1,...,En be an exact allocation w.r.t. Smin,X • iSmin , AiEi • SS\Smin • XX\D(Smin,X) S’S

19. Let it flow • Define a graph G(S,X) • Mark beginning and end of every interval in UiX • Nodes: consecutive markings, agents, s and t • For each I, edge (s,I) with capacity len(I) • Each iN connected to t with capacity avg(S,X) • Edge (I,i) with capacity  if agent i desires interval I 0.5,1  0.5  0,0.1 1 0.1  0.45 0.15 s 0.1,0.25 t 0.45  0.1 0.4,0.5 2 0.15  0.25,0.4 U1= [0,0.25][0.5,1] , U2 = [0.1,0.4]

20. A lemma • Lemma: Let SN, a piece of cake X. If for all S’S, avg(S’,X)  avg(S,X) then there is a network flow of size len(D(S,X)) in G(S,X) • Proof: • Max Flow = Min Cut • Disconnect subset TS from t at cost |T|avg(S,X) • Need to additionally disconnect len(D(S\T,X)) =|S\T|avg(S\T,X) |S\T|avg(S,X) 0.5,1  0.5  0,0.1 1 0.1  0.45 0.15 s 0.1,0.25 t 0.45  0.1 0.4,0.5 2 0.15  0.25,0.4

21. Properties of the algorithm • Lemma: Let SN, a piece of cake X. If there exists a network flow of size len(D(S,X)) in G(S,X) then there is an exact allocation wrt S,X • If Smin minimizes avg(S’,X) then there is an exact flow wrt Smin,X, can be computed using network flow algorithms • Computing Smin is similar but more involved • Theorem: assume that the agents have piecewise uniform valuations, then the algorithm is truthful, proportional, envy-free, and polynomial-time

22. 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? • Theorem: assume that the agents have piecewise linear valuations, then there is a randomized alg that is truthful in expectation, universally proportional, universally envy-free, and polynomial-time

23. Discussion • Conceptual contributions • Truthful cake cutting • Restricted valuations functions and tractable algorithms • Communication model • Many previous discrete algorithms can be simulated using eval and cut queries • Our algorithms are centralized • Future work • Generalize deterministic algorithm • Piecewise uniform valuations with minimum interval length

24. Bibliographic notes • Yiling Chen, John K. Lai, David C. Parkes and Ariel D. Procaccia. Truth, Justice, and Cake Cutting. In the proceedings of AAAI 2010 • Full version coming soon, will be posted online (rought draft available on request)

26. Properties of the algorithm • There are two problematic steps in while loop: • Step 1: computing Smin? • Step 2: existence and computation of exact allocation? • Solution: use network flow / max flow min cut • Theorem: assume that the agents have piecewise uniform valuations, then the algorithm is truthful, proportional, envy-free, and polynomial-time • Initialization: • SN, X[0,1] • While S • Sminargminavg(S,’X) • Let E1,...,En be an exact allocation w.r.t. Smin,X • iSmin , AiEi • SS\Smin • XX\D(Smin,X) S’S