Envy-free cake cutting: A polynomial number of queries with high probability
aa r X i v : . [ c s . CC ] M a y ENVY-FREE CAKE CUTTING:A POLYNOMIAL NUMBER OF QUERIESWITH HIGH PROBABILITY
GUILLAUME CH`EZE
Abstract.
In this article we propose a probabilistic framework in order tostudy the fair division of a divisible good, e.g. a cake, between n players. Ourframework follows the same idea than the “Full independence model” used inthe study of fair division of indivisible goods. We show that, in this framework,there exists an envy-free division algorithm satisfying the following probabilityestimate: P (cid:0) C ( µ , . . . , µ n ) ≥ n b (cid:1) = O (cid:16) n − b − +1+ o (1) (cid:17) , where µ , . . . , µ n correspond to the preferences of the n players, C ( µ , . . . , µ n )is the number of queries used by the algorithm and b >
4. In particular, thisgives lim n → + ∞ P (cid:0) C ( µ , . . . , µ n ) ≥ n (cid:1) = 0 . It must be noticed that nowadays few things are known about the complexityof envy-free division algorithms. Indeed, Procaccia has given a lower bound inΩ( n ) and Aziz and Mackenzie have given an upper bound in n n nnnn . As ourestimate means that we have C ( µ , . . . , µ n ) < n with a high probability, thisgives a new insight on the complexity of envy-free cake cutting algorithms.Our result follows from a study of Webb’s algorithm and a theorem of Tao andVu about the smallest singular value of a random matrix. Introduction
In this article we study the problem of fair resource allocation. It consists toshare an heterogeneous good between different players or agents. This good canbe for example: a cake, land, time or computer memory. This problem is old. Forexample, the Rhind mathematical papyrus contains problems about the division ofloaves of bread and about partition of plots of land. In the Bible we find the famous“Cut and Choose” algorithm between Abraham and Lot, and in the greek mythol-ogy we find the trick at Mecone. More recently, the “Cut and Choose” protocolhas been used in the United Nations Convention on the Law of the Sea (December1982, Annex III, article 8).The problem of fair division has been formulated in a scientific way by Steinhausin 1948, see [33]. Nowadays, there exist several papers, see e.g. [1, 5, 6, 20, 21, 22,25, 27, 30, 38], and books about this topic, see e.g. [3, 7, 28, 31]. These results
Guillaume Ch`eze: Institut de Math´ematiques de Toulouse, Universit´e Paul Sabatier,118 route de Narbonne, 31 062 TOULOUSE cedex 9, France
E-mail address : [email protected] . Date : May 6, 2020.
Key words and phrases.
Computational Fair division, Cake cutting. appear in the mathematics, economics, political science, artificial intelligence andcomputer science literature. Recently, the cake cutting problem has been studiedintensively by computer scientists for solving resource allocation problems in multiagents systems, see e.g. [8, 12, 13, 24].Throughout this article, the cake will be an heterogeneous good represented bythe interval C = [0 , n players and we associate to each player a non-atomic probabilitymeasure µ i on the interval C = [0 , µ i are absolutely continuous with respect to the Lebesgue measure. These measuresrepresent the preferences, the utility functions of the players. We have µ i ( C ) = 1for all i .The problem in this situation is to get a fair division of C = C ⊔ . . . ⊔ C n , wherethe i -th player get C i .When we study fair divisions, we have to precise what is the meaning of “fair”.Indeed, several notions exist. • We say that a division is proportional when for all i , we have µ i ( C i ) ≥ /n. • We say that a division is equitable when for all i = j , we have µ i ( C i ) = µ j ( C j ) . • We say that a division is exact in the ratios ( α , α , . . . , α n ), where α i ≥ α + α + · · · + α n = 1, when for all i and j we have µ i ( C j ) = α j . • We say that a division is envy-free when for all i = j , we have µ i ( C i ) ≥ µ i ( C j ) . A practical problem is the computation of fair divisions. In order to describealgorithms we thus need a model of computation. There exist two main classesof cake cutting algorithms: discrete and continuous protocols (also called movingknife methods). Here, we study discrete algorithms. These kinds of algorithmscan be described thanks to the classical model introduced by Robertson and Webband formalized by Woeginger and Sgall in [40]. In this model we suppose that amediator interacts with the agents. The mediator asks two type of queries: eithercutting a piece with a given value, or evaluating a given piece. More precisely, thetwo type of queries allowed are:(1) eval i ( x, y ): Ask agent i to evaluate the interval [ x, y ]. This means return µ i ([ x, y ]).(2) cut i ( x, a ): Ask agent i to cut a piece of cake [ x, y ] such that µ i ([ x, y ]) = a .This means: for given x and a , return y such that µ i ([ x, y ]) = a .In the Robertson-Webb model the mediator can adapt the queries from the previ-ous answers given by the players. In this model, the complexity counts the numberof queries necessary to get a fair division. For a rigourous description of this modelwe can consult: [9, 40].We can remark that this model of computation does not take into account the na-ture and the number of operations performed by the mediator. The BSSRW model NVY-FREE CAKE CUTTING 3 of computation introduced in [16] allows to avoid these drawbacks.The first studied notion of fair division has been proportional division, [33]. Pro-portional division is a simple and well understood notion. In [33] Steinhaus explainsthe Banach-Knaster algorithm which gives a proportional division. There also existsan optimal algorithm to compute a proportional division in the Robertson-Webbmodel, see [21, 22]. The complexity of this algorithm is in O (cid:0) n log( n ) (cid:1) . Further-more, the portion C i given to the i -th player in this algorithm is an interval.Exact divisions in the ratios ( α , . . . , α n ) exist for all ratios ( α , . . . , α n ). Theexistence of this kind of fair division follows from a convexity theorem given byLyapounov, see e.g. [20]. When we have α i = 1 /n , for all i , we just say that thedivision is exact. Unfortunately, there exist no algorithm to compute exact divi-sions, see [31].Equitable fair division is of the same kind. Indeed, there exist equitable fair di-visions where each C i is an interval, see [10, 14, 32]. However, there do not existalgorithms computing an equitable fair division, see [11, 15, 29].Envy-free fair division is difficult to obtain in practice. Indeed, whereas envy-freefair divisions where each C i is an interval exist, there does not exist an algorithmin the Robertson-Webb model computing such divisions. These results have beenproved by Stromquist in [34, 35].The first envy-free algorithm for n players has been given by Brams and Taylor in[6]. This algorithm has been given approximatively 50 years after the first algo-rithm computing a proportional fair division. The Brams-Taylor algorithm has anunbounded complexity in the Robertson-Webb model. This means that we cannotbound the complexity of this algorithm in terms of the number of players only. Itis only recently that a finite and bounded algorithm has been given to solve thisproblem, see [1]. The complexity of this algorithm is in O (cid:16) n n nnnn (cid:17) . When n = 2, n n nnnn is bigger than the number of atoms in the universe. . . A lower bound forenvy-free division algorithm has been given by Proccacia in [26]. This lower boundis in Ω( n ).We can remark that there is a huge difference between the complexity in theworst case O (cid:16) n n nnnn (cid:17) and the lower bound Ω( n ). Therefore a natural questionarises: Can we design an envy-free algorithm such that in practice the number of queriesis smaller than n d , where d is a given degree, with a high probability ? In order to answer to this question we have to define a probabilistc framework.When we consider indivisible goods there exist two probabilistic models, see e.g. [4].The first model is the
Full correlation model . In this model we suppose thatall agents have the same preference and all preferences are equiprobable. Whenwe want to share several indivisible goods, this case corresponds to the worst case.
CH`EZE, G.
However, in the cake cutting situation, if we suppose that µ = . . . = µ n then wecan easily obtain an envy-free division. Indeed, we ask to the first player to cutthe cake in n equals portions. Thus the full correlation model is not an interestingmodel in the cake cutting situation.The second model in the indivisible goods setting is the Full independence model .In this model we suppose that all preferences are equiprobable and that all agentshave independent preferences. In the cake cutting setting, in order to obtain asimilar situation we consider the following construction:First, we divide the interval [0 ,
1] into n witness intervals W j = [( j − /n, j/n ],where j = 1 , . . . , n .Second, we remark that for all probabilistic measures µ i on [0 , (cid:0) µ i ( W ) , . . . , µ i ( W n ) (cid:1) belongs to the standard ( n − µ i is aprobabilistic measure we have for all j = 1 , . . . , n , µ i ( W j ) ≥
0, and µ i ( W ) + · · · + µ i ( W n ) = µ i ( W ⊔ . . . ⊔ W n ) = µ i ([0 , . When we consider a random measure µ i , it is natural to suppose that all witnessintervals play the same role. For example, there is no reason to suppose that theplayers usually prefer the first part W of the cake.Our probabilistic framework is thus the following: We suppose that the distribution of (cid:0) µ i ( W ) , . . . , µ i ( W n ) (cid:1) follows a uniform dis-tribution over the standard ( n − -simplex. A classical way to obtain a uniform distribution on the standard ( n − n independent random variables X i with probability density function f i ( x ) = e − x . Set S = P ni =1 X i and Y i = X i /S ,then ( Y , . . . , Y n ) follows the uniform distribution on the standard ( n − agents have independent preferences .This means for example that µ ( W j ) is independent of µ ( W j ).Thus, in the cake cutting situation the Full independence model means that wesuppose that the following hypothesis holds:( H ) : Taking randomly a matrix of the following kind M = µ ( W ) . . . µ ( W n ) ... . . . ... µ n ( W ) . . . µ n ( W n ) means that we consider a random matrix M = ( m ij ) where m ij = X ij P nj =1 X ij and X ij are independent exponential random variables, i.e. with a probability den-sity function f ij ( x ) = e − x . We remark that we have define what is a random matrix M = (cid:0) µ i ( W j ) (cid:1) andthat we do not have define what is a random measure. Indeed, instead of takingrandom measures and then construct the matrix M , we have directly define the NVY-FREE CAKE CUTTING 5 probability distribution for the matrix M . This approach allows to obtain a simpleand explicit probabilistic framework.In the following, we are going to study an envy-free fair division algorithm. Wewill denote by C ( µ , . . . , µ n ) the number of queries used by this algorithm whenthe inputs are µ , µ , . . . , µ n . Theorem 1.
If we suppose that the hypothesis ( H ) is satisfied, then we have thefollowing result:There exists a protocol in the Robertson-Webb model of computation giving an envy-free fair division and such that for all b > we have the following probabilityestimate P (cid:0) C ( µ , . . . , µ n ) ≥ n b (cid:1) = O (cid:16) n − b − +1+ o (1) (cid:17) . This theorem says: the bigger the number of queries, the smaller the probability.We recall that f ( n ) = O (cid:0) g ( n ) (cid:1) means that there exists a constant C and aninteger n such that for all n ≥ n , we have | f ( n ) | ≤ Cg ( n ).The notation o (1) refer to a function f ( n ) such that lim n → + ∞ f ( n ) = 0. Examples: • If we choose b = 5 then − b −
13 + 1 + o (1) = −
13 + o (1) . When n is big enough we can suppose o (1) < / O (cid:16) n − b − +1+ o (1) (cid:17) = O (cid:0) n − / (cid:1) . This gives lim n → + ∞ P (cid:0) C ( µ , . . . , µ n ) ≥ n (cid:1) = 0 . • If we choose b = 11 then − b −
13 + 1 + o (1) = −
73 + o (1) . When n is big enough we can suppose o (1) < / O (cid:16) n − b − +1+ o (1) (cid:17) = O (cid:0) n − (cid:1) . Thus Theorem 1 gives P (cid:0) C ( µ , . . . , µ n ) ≥ n (cid:1) = O (cid:16) n (cid:17) . These bounds are not very sharp but they give a precise statement of the followingidea: when n is big the probability that the algorithm uses more than n (or n )queries is very small. Strategy of the algorithm and structure of the paper
The algorithm proposed in this article is just a slight modification of Webb’s superenvy-free division algorithm. Webb’s algorithm constructs an envy-free divisionfrom the matrix M = (cid:0) µ i ( W j ) (cid:1) when det( M ) = 0. The algorithm that we propose CH`EZE, G. works as follows: if det( M ) = 0 then use Webb’s algorithm else use another envy-free algorithm.When the hypothesis ( H ) is satisfied the probability that det( M ) = 0 is equal tozero. Thus, in practice our algorithm almost always corresponds to Webb’s algo-rithm. As the number of queries needed in Webb’s algorithm can be written interms of the smallest singular value of M , the strategy to prove Theorem 1 relieson a probabilistic study of the smallest singular value of M .The structure of this article in thus the following:In the first section, we recall what is a super envy-free fair division and we also recallWebb’s super envy-free algorithm. Then, we give our algorithm. In Section 2, westudy the number of queries used by this algorithm. This leads us to recall somestandard results on singular values of a matrix and to write the complexity of thealgorithm in terms of the smallest singular value of the matrix M . In Section 3, weuse a theorem of Tao and Vu, see [36], about the probability that M have a smallsingular value. This theorem will be the key point in the proof of Theorem 1.1. The algorithm
Super envy-free algorithm.
Super envy-free fair division is a strong kindof envy-free division. This notion has been introduced and studied by Barbanel,see [2, 3].
Definition 2.
We say that a division is super envy-free when for all i = j , we have µ i ( C i ) > n > µ i ( C j ) . This definition says that this division is proportional and all players think tohave stricly more than other players. Of course, this kind of fair division is notalways possible. For example, if µ = µ = · · · = µ n , then the previous inequalityis not possible. Indeed, we cannot have µ ( C ) > /n > µ ( C ) = µ ( C ).However, a super envy-free fair division exists when the measures µ i are linearlyindependent. Definition 3.
Let µ , . . . , µ n be n measures on a measurable set ( C , B ), where B is the Borel σ -algebra. We say that these measures are linearly independent whenthey are linearly independent as functions from B to [0 , Theorem 4 (Barbanel’s theorem) . A super envy-free division exists if and only ifthe measures µ , . . . , µ n are linearly independent. In the following we are going to use a witness matrix in order to know if themeasures are linearly independent.
Definition 5.
The witness matrix associated to the measures µ , . . . , µ n is thematrix M = (cid:0) µ i ( W j ) (cid:1) where W j is the interval [( j − /n, j/n ] and j = 1 , . . . , n . Remark . If det( M ) = 0 then the measures µ , . . . , µ n are linearly independent.In [39], Webb gives a strategy to compute super envy-free fair division. In orderto recall this strategy, we recall what is a ε near-exact fair division . NVY-FREE CAKE CUTTING 7
Definition 7.
Let A be a measurable subset of C .We say that a division of A = A ⊔ . . . ⊔ A n is ε near-exact in the ratios ( α , α , . . . , α n ),where α i ≥ α + α + · · · + α n = 1, when for all i and j we have (cid:12)(cid:12)(cid:12) µ i ( A j ) µ i ( A ) − α j (cid:12)(cid:12)(cid:12) < ε. Now, we can describe Webb’s algorithm.
Super Envy-free fair division algorithmInputs:
A partition C = A ⊔ . . . ⊔ A n , a matrix M = ( m ij ) where m ij = µ i ( A j ), M is non-singular. Ouputs:
A super envy free division C = C ⊔ . . . ⊔ C n .(1) Compute M − = ( ˜ m ij ).(2) Set δ := n − n (1 − tn ) where t = min i,j ( ˜ m ij ).(3) Set N := ( n ij ), where n ii := 1 /n + δ and n ij = 1 /n − δ/ ( n − R = ( r ij ) := M − N .(5) For j = 1 , . . . , n doCompute a ε = δ/n near-exact fair division of A j in the ratios ( r j , . . . , r jn ),this gives A j = A j ⊔ . . . ⊔ A jn .(6) For all i = 1 , . . . , n do C i := A i ⊔ A i ⊔ . . . ⊔ A ni . We remark that in Step 2, we have t ≤
0. Indeed, if t > M M − = I is impossible, because the coefficients m ij are non-negative. There-fore in Step 2, we have δ >
0. The formula used to define δ implies that thecoefficients r ij of R are non-negative. It is a straightforward computation to checkthat the coefficients r ij also satisfy P nj =1 r ij = 1.In order to explain this algorithm, suppose that in Step 5 we compute an exactfair division in the ratios ( r j , . . . , r jn ) instead of an ε near- exact fair division withthese ratios. Then, by construction the partition C = C ⊔ . . . ⊔ C n as the followingproperty µ i ( C j ) = n ij . This gives µ i ( C i ) = 1 /n + δ and µ i ( C j ) = 1 /n − δ/ ( n − r j , . . . , r jn )is impossible, since it has been proved that such algorithms cannot exist, see [31].That is the reason why a ε near-exact algorithm is used. Indeed, a ε near-exactalgorithm in the Robertson-Webb model exists, see [31, Theorem 10.2]. Thereforethe idea is to choose a small enough ε in order to obtain a result very close to thetheoretical result where ε = 0. Thus, we obtain in practice a partition where µ i ( C j )are very close to n ij and then the division is super envy-free.The number of queries used by the ε near-exact division algorithm is at most n × (2 + 2 n / ) /ε , see [31, Theorem 10.2]. As already remarked in [31], this boundcan be improved . However, we just want to get a bound on C ( µ , . . . , µ n ) in termsof a polynomial in n , thus this estimate is sufficient.Therefore, the number of queries used by Webb’s super envy-free algorithm is at CH`EZE, G. most n × (2 + 2 n / ) /δ , since ε = δ/n and in Step 5 we compute n ε -near-exactfair divisions. Thanks to the definition of δ we get the following lemma. Lemma 8.
The number of queries used by the super envy-free division algorithmin the Robertson-Wenn model is bounded by n × (2 + 2 n / ) × (1 − tn ) n − ∈ O (cid:0) | t | n . (cid:1) . An envy-free algorithm.
Envy-free fair division algorithmInputs:
A cake C = [0 , n measures µ , . . . , µ n . Ouputs:
An envy free division C = C ⊔ . . . ⊔ C n .(1) % Construct the matrix M = ( m ij ) where m ij = µ i ( W j ) . %For all i = 1 , . . . , n , doFor all j = 1 , . . . , n do m ij := eval i ( W j ).(2) If det( M ) = 0 then compute an envy-free fair division algorithm thanks toAziz-Mackenzie’s algorithm,Else compute a super envy-free fair division algorithm thanks to Webb’salgorithm. Remark . When det( M ) = 0, we have to use an algorithm different from Webb’salgorithm . Indeed, in this case Webb’s algorithm is not defined (we cannot compute M − ). Furthermore, it is not necessary to use Aziz-Mackenzie’s algorithm. Thebound given in Theorem 1 will not change if we use another envy-free algorithmwhen det( M ) = 0. 2. Complexity analysis
The number of queries used by our envy-free division algorithm is O (cid:0) | t | × n , (cid:1) when det( M ) = 0. In the next subsection, we are going to bound | t | by σ − n where σ n is the smallest singular value of M . Then, in the second subsection, we use anestimate on the probability P ( σ n ≤ n − b ) in order to prove our theorem.2.1. The smallest singular value.
We recall here the definition and a simpleresult about the singular values of a matrix.
Definition 10.
The singular values of a matrix M are the square roots of theeigenvalues of M T M . They are denoted by σ ( M ) ≥ · · · ≥ σ n ( M ). Remark . We have det( M ) = 0 ⇐⇒ σ n ( M ) = 0.The smallest singular value allows to bound the coefficients of the inverse of amatrix. Proposition 12.
Let M be a non-singular matrix, such that M − = ( ˜ m ij ) .Let σ n ( M ) be the smallest singular value of M . We have max ij (cid:0) | ˜ m ij | (cid:1) ≤ kM − k = σ − n ( M ) . Proof.
This is a classical result, see [23, Formula 2.3.8 page 56] and [18, Theo-rem 3.3]. (cid:3)
NVY-FREE CAKE CUTTING 9
The previous proposition allows us to obtain an upper bound on the complexityof the super envy-free algorithm in terms of σ n ( M ). Corollary 13.
When n ≥ , the number of queries used by the super envy-freedivision algorithm in the Robertson-Webb model is bounded by n × max (cid:0) , σ − n ( M ) (cid:1) . Proof.
We have already remarked that in Step 2 of the super envy-free algorithmwe have t = min i,j ( ˜ m ij ) ≤
0. Then Proposition 12 gives − t = | t | ≤ max i,j | ˜ m ij | ≤ σ − n ( M ) . Then Lemma 8 implies that the number of queries used by the super envy-freedivision algorithm in the Robertson-Webb model is bounded by n × (2 + 2 n / ) × (1 + nσ − n ( M )) n − . As we have supposed that n ≥
19, we get2 + 2 n / n − ≤ n . Then, we have n × (2 + 2 n / ) × (1 + nσ − n ( M )) n − ≤ n × (1 + nσ − n ( M )) ≤ n (cid:0) σ − n ( M ) (cid:1) ≤ n × max (cid:0) , σ − n ( M ) (cid:1) . (cid:3) The singular value σ n ( M ) measures how far M is from a singular matrix. There-fore, if σ n ( M ) is small then the measures µ i are nearly linearly dependent and theprevious corollary shows that the number of queries is big. This result satisfies thegeneral result: if the agents have “very different” preferences it will be easier to getan envy-free fair division. A precise statement of this result with an explicit boundhas been given in [17, Corollary 17].2.2. Proof of Theorem 1.
We consider the event C ( µ , . . . , µ n ) ≥ n b .During the envy-free algorithm two situations appear:First, det( M ) = 0, then in this situation the algorithm used the super envy-freealgorithm. Thanks to Corollary 13, the number of queries used in this situationsatisfies n b ≤ C ( µ , . . . , µ n ) ≤ n × max (cid:0) , σ − n ( M ) (cid:1) . As n >
1, it follows n b ≤ σ − n ( M ) . This means that we have the following inclusion( ⋆ ) { det( M ) = 0 } ∩ { C ( µ , . . . , µ n ) ≥ n b } ⊂ { σ n ( M ) ≤ n − b } . The second situation corresponds to det( M ) = 0. Thus the second situationcorresponds to σ n ( M ) = 0 and obviously σ n ( M ) ≤ n − b . This gives the followinginclusion ( ⋆⋆ ) { det( M ) = 0 } ⊂ { σ n ( M ) ≤ n − b } . Thanks to ( ⋆ ) and ( ⋆⋆ ) we get { C ( µ , . . . , µ n ) ≥ n b } ⊂ { σ n ( M ) ≤ n − b } . We deduce then the following inequality between probabilities P (cid:0) C ( µ , . . . , µ n ) ≥ n b (cid:1) ≤ P (cid:0) σ n ( M ) ≤ n − b (cid:1) . In the next section we are going to prove the following proposition.
Proposition 14.
If we suppose that hypothesis ( H ) is satisfied then the followingholds:Let b > be a constant, then there exists a constant c > depending on b such that P (cid:0) σ n ( M ) ≤ n − b (cid:1) ≤ c (cid:16) n − b − +1+ o (1) + ne −√ n + n e −√ n − (cid:17) . In order to finish the proof of Theorem 1, we remark that for all b > ne −√ n + n e −√ n − = O ( n − b − +1 ). Therefore, we get P (cid:0) C ( µ , . . . , µ n ) ≥ n b (cid:1) ≤ P (cid:0) σ n ( M ) ≤ n − b (cid:1) = O (cid:16) n − b − +1+ o (1) (cid:17) , which gives the desired estimate.3. An estimate for P ( σ n ( M ) ≤ n − b )In this section we prove Proposition 14.In order to bound P (cid:0) σ n ( M ) ≤ n − b (cid:1) , we are going to introduce some notations.As we suppose that hypothesis H is satisfied we have M = DX , where D is the diagonal matrix with coefficients in the i -th row equal to 1 / ( P nj =1 X ij ), X is the matrix with coefficients X ij and X ij are indepent exponential random vari-ables with probability density function f ij = e − x .We also consider the two following events: A = { σ n ( M ) ≤ n − b } ∩ { σ n ( D ) ≤ n − / } .B = { σ n ( M ) ≤ n − b } ∩ { σ n ( D ) ≥ n − / } . Obviously, we have ( ♯ ) { σ n ( M ) ≤ n − b } = A ∪ B. Now, we are going to bound P ( A ) and P ( B ). Lemma 15.
We have P ( A ) ≤ ne −√ n . NVY-FREE CAKE CUTTING 11
Proof.
We have A ⊂ { σ n ( D ) ≤ n − / } . Thus P ( A ) ≤ P (cid:0) σ n ( D ) ≤ n − / (cid:1) . As D is a diagonal matrix with coefficients 1 / ( P nj =1 X ij ), we have σ n ( D ) ≤ n − / ⇒ min i (cid:16) X i + · · · + X in (cid:17) ≤ n − / ⇒ X + · · · + X n ≥ n / ⇒ ∃ i, X i ≥ √ n. We set A i = { X i ≥ √ n } , we have A ⊂ ∪ ni =1 A i . Furthermore, P ( A i ) = Z + ∞√ n e − x dx = e −√ n . Therefore, we get P ( A ) ≤ n X i =1 P ( A i ) = ne −√ n . (cid:3) In order to to give a bound on P ( B ) we are going to use the following theoremdue to Tao and Vu, see [36], and see also the erratum in [37]. Theorem 16.
Let Y be a random variable with mean zero and bounded secondmoment, and let γ ≥ / , a ≥ be constants. Then there is a constant c dependingon Y , γ , and a such that the following holds. Let Y be the random matrix of size n whose entries are independent and identically distributed copies of Y , let M be adeterministic matrix satisfying k M k ≤ n γ . Then P (cid:0) σ n ( M + Y ) ≤ n − (2 a +2) γ +1 / (cid:1) ≤ c (cid:16) n − a + o (1) + P (cid:0) kYk ≥ n γ (cid:1)(cid:17) . In the following we will need the following lemma.
Lemma 17.
Let A and B be two n × n matrices, we have σ n ( A ) × σ n ( B ) ≤ σ n ( AB ) . Proof.
When M is an n × n matrice we have kM − k − = σ n ( M ), see [18, Theorem3.3]. Thus 1 σ n ( AB ) = k ( AB ) − k = kB − A − k ≤ kB − k × kA − k . Therefore σ n ( AB ) ≥ kB − k − × kA − k − , which gives the desired result. (cid:3) Proposition 18.
There exists a constant c such that the following holds P ( B ) ≤ c (cid:16) n − b − +1+ o (1) + n e −√ n − (cid:17) . Proof.
With our notations we have M = DX and by Lemma 17 we have( ⋆ ) σ n ( D ) × σ n ( X ) ≤ σ n ( M ) . If the event B is realized then by definition we have σ n ( M ) ≤ n − b and σ n ( D ) ≥ n − / . The inequality ( ⋆ ) implies σ n ( X ) ≤ n − b +3 / . We set C = { σ n ( X ) ≤ n − b +3 / } . We have thus shown B ⊂ C .Now, we are going to applied Theorem 16 to σ n ( X ).We set Y = X −
1, where X is the exponential distribution with the probabilitydensity function f ( x ) = e − x .Furthermore, we denote by M the n × n matrix with all its entries equal to 1. Wedenote by Y = ( Y ij ) the n × n matrix where its coeffcients Y ij are independent andidenticaly distributed copies of Y . Therefore, the matrix M + Y corresponds to ourmatrix X .Furthermore, we remark easily that we have k M k = n . Then we can set γ = 3 / P (cid:0) kYk ≥ n / (cid:1) . We recall the classical bound, see[23], kYk ≤ n max i,j | Y ij | , where Y = ( Y ij ). Therefore, we have kYk ≥ n / ⇒ max i,j | Y ij | ≥ √ n. We denote by C i,j the following set C i,j = {| Y ij | ≥ √ n } . We deduce then the following inclusion {kYk ≥ n / } ⊂ ∪ ni,j =1 C i,j . By definition of Y , we have Y ij = X ij − X ij follow the exponential distri-bution. Therefore X ij ≥ | Y ij | ≥ √ n ⇐⇒ | X ij − | ≥ √ n ⇐⇒ X ij ≥ √ n + 1 . This gives P ( C i,j ) = P (cid:0) | Y ij | ≥ √ n (cid:1) = P ( X ij ≥ √ n + 1) = Z + ∞√ n +1 e − x dx = e −√ n − , it follows P (cid:0) kYk ≥ n / (cid:1) ≤ n X i,j =1 P ( C i,j ) ≤ n e −√ n − . Then Theorem 16 gives P (cid:16) σ n ( M + Y ) ≤ n − a +2) / / (cid:17) ≤ c (cid:16) n − a + o (1) + n e −√ n − (cid:17) . By construction, we have M + Y = X , then if we set − b + 32 = − (2 a + 2) ×
32 + 12
NVY-FREE CAKE CUTTING 13 then we have a = b − − P (cid:0) σ n ( X ) ≤ n − b +3 / (cid:1) ≤ c (cid:16) n − b − +1+ o (1) + n e −√ n − (cid:17) . (cid:3) Now, we can prove Proposition 14.Thanks to ( ♯ ), we have P ( σ n ( M ) ≤ n − b ) ≤ P ( A ) + P ( B ).Furthermore, by Lemma 15 and Proposition 18, there exists a constant c such that P ( σ n ( M ) ≤ n − b ) ≤ P ( A ) + P ( B ) ≤ c (cid:16) n − b − +1+ o (1) + ne −√ n + n e −√ n − (cid:17) , which gives the desired result. Conclusion
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