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Better Ways to Cut a Cake

Better Ways to Cut a Cake. Steven Brams – NYU Mike Jones – Montclair State University Christian Klamler – Graz University Paris, October 2006. Fair Division 2. Various Procedures (Brams & Taylor 1996) Comparisons on the basis of: Complexity of the rules Properties satisfied

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Better Ways to Cut a Cake

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  1. Better Ways to Cut a Cake Steven Brams – NYU Mike Jones – Montclair State University Christian Klamler – Graz University Paris, October 2006

  2. Fair Division 2 • Various Procedures (Brams & Taylor 1996) • Comparisons on the basis of: • Complexity of the rules • Properties satisfied • Manipulability • Division of a heterogeneous divisible good among various players • land division, • service used over time by different players

  3. Desirable Properties3 • Efficiency • There is no other allocation that is better for one player and at least as good for all others. • Envy-freeness • Each player thinks it receives at least a tied-for-largest portion, so it does not envy another player. • Equitability • Each player’s valuation of the portion that it receives is the same as every other player’s valuation of the portion it receives.

  4. 0 1 Assumptions 4 • CAKE as the unit interval X=[0,1] • Cuts divide the cake into subintervals • Every player i has a continuous value function vion[0,1] with the following properties: • For all x  X, vi(x)  0 • vi() = 0 i.e. measure is non-atomic • For any disjoint x,y  X, vi(x+y) = vi(x) + vi(y), i.e. measure is finitely additive • vi(X) = 1 • Players are ignorant about other players’ value functions. • Goal of each player is to maximize the value of the minimum-size piece that it can guarantee for itself, regardless of what the other players do (maximin value), i.e. players are risk-averse; they never choose strategies that entail the possibility of giving them less than their maximin values.

  5. 0 1/2 1 Cut and Choose5 Satisfies Efficiency, Envy-Freeness but NOT Equitability

  6. 2/3 1/2 0 1 1/2 Does a “perfect” cut exist?6 Efficient, envy-free and equitable solution at x = 3/7 (see Jones (2002)) However, (even risk averse) players have no incentive to state their true value functions!

  7. The Surplus Procedure7 RULES: • Independently, A and B report their value functions fA(x) and fB(x) to a referee. • Referee determines the 50-50 points a and b. 0---------------------a-------------b---------1 • If a and b coincide, the cake is cut at that point and the pieces are randomly assigned. • Let a be to the left of b. Then A gets [0,a] and B gets [b,1].

  8. The Surplus Procedure8 • Let c be the point in [a,b] at which the players receive the same proportion p of the cake in this interval as each values it. 0---------------------a-----c--------b---------1 A receives portion [a,c] and B [c,b] for a total of [0,c] for A and (c,1] for B.

  9. The Surplus Procedure9 To solve for c we set: For the previous example we get: Which yields c = 7/16. This does not ensure “pure” equitability as they value the interval [a,b] differently – only proportional equitability!

  10. The Surplus Procedure10 For “pure” equitability we need to cut the cake at point e such that: for e = 3/7 (which is further to the left than c). There are conflicting arguments for cutting at c (proportional equitability) and e (equitability). Property: A procedure is strategy-vulnerable if a maximin player can, by misrepresenting its value function, assuredly do better, whatever the value function of the other players. A procedure that is not strategy-vulnerable is called strategy-proof.

  11. Theorem 111 Theorem 1: SP is strategy-proof, whereas any procedure that makes e the cut-point is strategy-vulnerable. Proof: • Misrepresenting a and/or b. 0-----------a-----b---a’------------1 • Misrepresenting their value functions over [a,b]. 0-----------a-----c----b----------1 Shift of c to the right for A possible if it either • decreases • increases But therefore A would have to know fB(x) which it does not!

  12. Theorem 112 If A knew the location of b manipulation was possible: • concentrate the value just to the left of b, what moves c rightward • Manipulation is possible when cake is cut at e! • submit fA(x) with the same 50-50 point • if a is to the left of b, then decrease • if a is to the right of b, then decrease

  13. Extensions to Three or More Players13 Consider the following value functions for 3 players:

  14. Extensions to Three or More Players14 It is not always possible to divide a cake among three players into envy-free and equitable portions using two cuts!

  15. Extensions to Three or More Players15 However, an envy-freeallocation that uses n-1 parallel, vertical cuts is always efficient. (Gale, 1993; Brams and Taylor, 1996) • There are 2 envy-free procedures for 3-person, 2-cut cake division: • Stromquist (1980): requires 4 simultaneously moving knifes • Barbanel & Brams (2004): requires 2 simultaneously moving knifes Beyond 4 players, no procedure is known that yields an envy-free division unless an unbounded number of cuts is allowed.

  16. Equitability Procedure (EP) 16 It isalways possible to find an equitable division of a cake among three or more players that is efficient. The rules of EP are: • Independently, A,B,C, … report their (possibly false) value functions fA(x), fB(x), fC(x), … over [0,1] to a referee. • The referee determines the cutpoints that equalize the common value that all players receive (for the n! possible assignments of pieces) • Choose the assignment that gives the players their maximum common value.

  17. Equitability Procedure (EP) 17 Using the above 3-player example, the cutpoints e1 and e2 have to be such that: giving e1≈ 0.269 and e2 ≈ 0.662 with a value of 0.393 for each player.

  18. Theorems 2 and 3 18 Theorem 2: EP is strategy-proof. In order to misrepresent, a player would have to know the borders of the pieces. As it does not do so it cannot ensure itself a more valuable piece. Theorem 3: If a player is truthful under EP, it will receive at least 1/n of the cake regardless of whether or not the other players are truthful; otherwise, it may not. We know that there is a division where each player receives at least 1/n (e.g. Dubins-Spanier moving knife procedure). As vi(X)=1, undervaluing the cake at one part will overvalue it at some other part, but an ignorant player might get the latter.

  19. Example 19 In the previous example assume C knows the value functions of A and B. Let c1 and c2 be the cutpoints. Then C should undervalue the middle portion between those points so that: It is maximal if B is indifferent between receiving the right portion and the middle portion, i.e. This leads to c1 ≈ 0.230 and c2 ≈ 0.707 where A and B receive a value of 0.354 and C receives a value of 0.477 (compared to the 0.393 before). However, a bit more undervaluation  C gets a value less than 0.393.

  20. Conclusion 20 • We have described a new 2-person, 1-cut cake cutting procedure (SP). • Like cut-and-choose it induces players to be truthful, but produces a proportionally equitable division. • SP is more information demanding. • For three persons, there may be no envy-free division that is also equitable. For four persons, there is no known minimal-cut envy-free procedure. However, EP ensures equitability and efficiency.

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