N O T J U S T A C H I L D ’ S P L A Y

How does one fairly divide goods among several people?

What is “fairness” ?
Envy-freeness
• Each participant prefers keeping his own allocation to swapping with any other participant.
Proportionality
• Each of the n participants receives at least 1/n of his value for getting everything.

Divisible goods and Indivisible goods
Divisible goods
• Such as land, time, or memory on a computer.
Indivisible goods
• Such as a house or the computer itself.

What’s cake-cutting ?

A l l o c a t i n g a h e t e r o g e n e o u s d i v i s i b l e g o o d a m o n g m u l t i p l e p l a y e r s w i t h d i f f e r e n t p r e f e r e n c e s

Outline
• 1. Cake cutting mechanisms
• 2. Complexity of cake cutting
• 3. A game-theoretic viewpoint
• 4. Optimizing welfare

C h a p t er 1

Cut and Choose
• The famous cut and choose algorithm:
• 1st step: the first player divides the cake into two pieces that he values equally.
• 2rd step: the second player then chooses the piece that he prefers.
• 3rd step: the first player receives the remaining piece.

Is the cut and choose algorithm fair?
Proportional?
• Yes! Note that --
• The first player values both pieces at exactly
1/2,
• While the second player receives his preferred piece, which must be worth at least
1/2.
Envy-free?
• Yes! In fact –
• For the case of two players, the concepts of envy-freeness and proportionality coincide!

What about n-player setting?
• As we move from the two player setting to the nplayer setting, fairness becomes harder to achieve.
• Nevertheless, several elegant algorithms guarantee
proportional allocations.

Dubins-Spanier 1961
• In each stage, a referee slowly moves a knife over the cake from left to right.
• When the knife reaches a point such that the piece of cake to the left of that point is worth 1/n to one of the players, this player shouts “stop”.
• The referee makes a cut, and the piece of cake to the left of the cut is given to the player.
• The satisfied player and allocated piece are then removed.
• The process is repeated with the remaining players and leftover cake, until there is only one player left. The last player receives the unclaimed piece.

Even-Paz 1984
• Assume for ease of exposition that the number of
players is a power of 2.
• Similarly to the discretized version of the latter algorithm, each time the procedure is executed, the players make marks where the cake to the left of the mark is valued at 1/2 (rather than 1/n, as before).

Even-Paz 1984
• Rather than removing a single player ----
• we separate the players into two subsets of equal size, such that all the marks made by the players of the first subset lie to the left of the marks made by the players of the second subset.
• The players in the first subset then receive the piece of cake that lies to the left of their rightmost mark, while the players in the second subset receive the remaining cake.

Even-Paz 1984
• Then, resursion !
• What about proportionality ?
------------------------------------------------------------------------------
• A single player participates in exactly log n calls to the procedure, and therefore each player receives a piece of cake worth at least (1/2) 𝑙𝑜𝑔𝑛 = 1/𝑛 .

How to guarantee envy-freeness?
• While proportionality is well understood, envyfreeness is a far more elusive property.
• Proportionality is always implied by envy-freeness, in the case of divide the whole cake.

How to guarantee envy-freeness?
• For three players,
----------- Selfridge-Conway 1960---------
• For an arbitrary number of players,
------------Brams-Taylor 1992-----------------

Selfridge-Conway 1960
Stage 0
• Player 1 divides the cake into three equal pieces according to his valuation.
• Player 2 trims the largest piece (that is, cuts off a slice) such that there is a tie between the two largest pieces in his eyes.
• We call the original cake without the trimmings Cake
1, and we call the trimmings Cake 2.

Selfridge-Conway 1960
Stage 1 (Division of cake 1)
• Order: 3 - 2 – 1
• Player 3 chooses one of the three pieces of Cake 1.
• Either player 2 or player 3 receives the trimmed piece; denote that player by U, and the other player by Ū.
• Player1 is allocated the remaining(untrimmed) piece.
Stage 2 (Division of cake 2)
• Ū divides cake 2 into three equal pieces according to his valuation.
• Players U, 1, and Ū choose the pieces of
Cake 2, in that order.

Brams-Taylor 1992
• The first envy-free cake cutting algorithm for an arbitrary number of players!
• However—
• Through the computational lens, the algorithm’s
running time is unbounded.

Brams-Taylor 1992
• A bounded envy-free algorithm for the five-player case was recently proposed by Saberi and Wang
2009.
• However, the n-player case remains open!
• Is envy-free cake cutting inherently complex?

C h a p t er 2

Robertson-Webb’s model 1998
Evaluation query
• Which asks a player i for his value for the
subinterval between two given points x and
y:
• 𝑒𝑣𝑎𝑙 𝑖 𝑥, 𝑦 = 𝑉 𝑖
[𝑥, 𝑦] .
Cut query
• which asks a player i to mark a subinterval worth a given value α starting at a given point
x:
•
• 𝑐𝑢𝑡 𝑖
• S.t.
𝑉 𝑖 𝑥, α = 𝑉
[𝑥, 𝑦] = α .
𝑖
[𝑥, 𝑦]

Robertson-Webb’s model
Dubins-Spanier
• The number of required cut queries:
• n+(n-1)+(n-2)+…+2 =
(𝑛+2)(𝑛−1) = O( 𝑛 2 )
2
Even-Paz
• The number of required cut queries:
• 1*n+2*(n/2)+4*(n/4)+…+(n/2)*2= 𝑛log𝑛

The complexity of proportional cake cutting
• Are there proportional cake cutting algorithms that require significantly fewer than 𝒏𝐥𝐨𝐠𝒏 queries?
• Woeginger-Sgall 2007
• allocates connected pieces requires at least c ∗ 𝑛log𝑛 queries , where c is a constant.
• Edmonds-Pruhs 2006
• allocates disconnected pieces, also requires at least c ∗ 𝑛log𝑛 queries.

The complexity of envy-free cake cutting
• We would like to be able to establish that bounded
envy-free cake cutting algorithms do not exist.
• Stromquist 2008
• established such a nonexistence result under the assumption that the algorithm must allocate
connected pieces (even for the 3-person case).

The complexity of envy-free cake cutting
• When connected pieces are not assumed—
• Procaccia 2009
• In the Robertson-Webb model, required to compute an envy-free allocation is at least on the order of 𝑛 2 .
• Envy-freeness is provably harder than proportionality.

Ways to circumvent the problem
• Approximately approach
----------- Lipton et al. 2004----------------
• Special valuation structure
------------Chen et al. 2010-----------------

C h a p t er 3

Strategyproof
• Strategyproof: Players must not be able to gain from manipulating the algorithm, regardless of the actions of others.

Cut and choose algorithm is not strategyproof

Strategyproof cake cutting algorithm
• Chen et al. 2010
• perfect partition: partitioning the cake into n pieces
𝑋
1
… 𝑋 𝑛 such that each player i has value exactly 1/n for each of these pieces (not just his own), that is,
𝑉 𝑖
( 𝑋 𝑗
) = 1/n for every j.
• Let’s suppose for a short while that we have a magical method for perfect partitioning the cake.

Strategyproof cake cutting algorithm
• Chen et al. 2010
• Chen’s algorithm first computes a perfect partition, and then gives each player a random piece.
• Clearly, the algorithm is strategyproof. Because for any partition, the expected value of a random piece is exactly 1/n, that’s
1 𝑛
𝑉 𝑖
( 𝑋
1
)+…+ 1 𝑛
𝑉 𝑖
( 𝑋 𝑛
)= 1 𝑛
(𝑉 𝑖
( 𝑋
1
)+…+ 𝑉 𝑖
( 𝑋 𝑛
))= 1 𝑛
∗ 1 =
1 𝑛

How to compute a perfect partition
• Chen et al. 2010 showed that perfect partitions can be computed efficiently when valuations have a
piecewise constant structure.
• piecewise constant: each player only desires certain pieces of cake, and values each of these pieces uniformly.
• Additionally, if each player has a single desired piece of cake that he values uniformly, fairness and truthfulness can be guaranteed without resorting to randomization.

Strategyproof cake cutting algorithm
• The design of strategyproof cake cutting algorithms is still largely an open problem .
• First, because the above algorithms (especially the deterministic one) can only handle restricted valuations.
• Second, because these algorithms cannot be simulated in the Robertson-Webb model.

C h a p t er 4

Social welfare
• Utilitarian social welfare and egalitarian social welfare
• We assume that all players have the same value for the whole cake, say $1.
• Price of fairness: the worst-case ratio between the social welfare of the optimal allocation, and the social welfare of the best fair allocation.

Price of fairness—an example
• Consider the following scenario. Suppose the cake
[0, 1] has 𝑛 disjoint subintervals, each of length
1/ 𝑛 .
• Each of the first “large” 𝑛 players only desires one of these subintervals; no two “large” players desire the same subinterval, and each large player values his subinterval uniformly.
• The remaining n − 𝑛 “small” players value the whole cake uniformly.

Price of fairness—an example
Any proportional allocation
• The social welfare is smaller than 2.
The welfare-max allocation
• Just divide the entire cake between the larger players.
• Secure social welfare of 𝑛
.

Price of fairness—an example
• So in this example, the price of proportionality is at least 𝑛/2 .
• The price of envy-freeness is at least as high because envy-freeness implies proportionality.

Dumping paradox
• Aumann –Dombb 2010 studied the price of fairness under the assumption that connected pieces must be allocated.
• A interesting insight in this context is
dumping paradox: by throwing away pieces of cake, one can increase the social welfare of the best proportional (or envyfree) allocation!

Dumping paradox

Optimal fair cake divisions
• For piecewise constant valuations, welfare-maximizing proportional or envy-free allocations can be computed in polynomial time.
----------------------Cohler et al. 2011-------------------------------
• Computing optimal fair cake divisions with connected pieces is significantly harder.
-----------------------X. Bei et al. 2012--------------------------------
• Even if we abandon fairness completely and just focus on optimizing welfare.
---------------------Aumann et al. 2012------------------------------

How about Pareto-efficient?
• Pareto-efficient: in the sense that no other allocation is valued at least as highly by all players, and is strictly better for at least one player.
• there are examples where no welfare-maximizing envy-free allocation is Pareto-efficient, even when there are only three players with piecewise constant valuations.
----------------------Brams et al. 2012-----------------------------

How about Pareto-efficient?
• Should we sacrifice social welfare to obtain Paretoefficiency? How much must be sacrificed?
• More generally, what would constitute an ideal
cake division?
• ??……….
• These conceptual questions may lead to significant technical insights on the role of optimization in cake cutting!

Z H E N G B O