fair allocation of goods or resources among various agents is a central task in multiagent systems and other fields .the specific setting where just one divisible resource is to be divided fairly is commonly referred to as cake - cutting , and agents are called players in this setting .research in the area of cake - cutting started off in the 1940s with the pioneering work of steinhaus  who , to the best of our knowledge , was the first to introduce the problem of fair division . dividing a good ( or a resource ) fairly among several players such that each of them is satisfied withthe portion received is of central importance in many fields . in the last 60 yearsthis research area has developed vividly , spreading out into various directions and with applications in areas as diverse as economics , mathematics , computer science , and psychology .while some lines of this research seek to find reasonable interpretations of what `` fairness '' really stands for and how to measure it  , others are concerned more with proofs of existence or impossibility theorems regarding fair division , or with the development of new cake - cutting procedures  and , relatedly , with the analysis of their complexity relative to both upper and lower bounds .since cake - cutting procedures involve several parties ( namely , the players ) , they are also referred to as `` protocols . ''cake - cutting protocols aim at achieving a _ fair _ division of an infinitely divisible resource among players , who each may have different preferences for ( and thus different valuations of ) different parts of the resource . in this paper, we focus on ( a notion of ) fairness in finite bounded cake - cutting protocols .many cake - cutting protocols are known today , both finite and continuous ones . while a _ finite _ protocol always provides a solution after only a finite number of decisionshave been made , a _protocol could potentially run forever . among finite protocols, one can further distinguish between bounded and unbounded ones .a finite _ bounded _ cake - cutting protocol is present if we know in advance that a certain number of steps ( that may depend on the number of players ) will suffice to divide the resource fairly independently of how the players may value distinct parts of the resource in a particular case and independently of the strategies chosen by the players .in contrast , in finite _ unbounded _ cake - cutting protocols , we can not predict an upper bound on how many steps will be required to achieve the same goal . aiming to apply cake - cutting procedures to real - world scenarios , it is important to develop fair _ finite bounded _ cake - cutting protocols . in this context ,`` fairness '' is often interpreted as meaning `` envy - freeness . ''a division is _ envy - free _ if no player has an incentive to switch his or her portion with the portion any other player received .steinhaus  proved that for any number of players an envy - free division of a single divisible good always exists .however , the current state of the art after six decades of intense research is that for arbitrary , and even for , the development of _ finite bounded _ envy - free cake - cutting protocols still appears to be out of reach , and a big challenge for future research . for players , hardly any envy - free cake - cutting protocol is known , and the ones that are known are either finite unbounded or continuous ( see , e.g. , ) .though , from an implementation perspective , finite bounded protocols are the ones that are most desirable .recently , stromquist  has shown that for more than two players there is no finite cake - cutting protocol that provides an envy - free division when all portions are required to consist of contiguous pieces .our goal in this paper is to look for compromises that can be made with respect to envy - freeness while keeping the protocol finite bounded : we propose an approach to evaluate finite bounded ( yet possibly non - envy - free ) cake - cutting protocols with respect to their `` degree of guaranteed envy - freeness '' ( dgef ) .informally put , this notion provides a measure of how good such a protocol can approximate the ( possibly for this particular protocol unreachable ) ideal of envy - freeness in terms of the number of envy - free - relations that are guaranteed to exist even in the worst case .this paper is organized as follows . after defining some basic notions in section [ sec : preliminaries ] ,we introduce the notion of degree of guaranteed envy - freeness , and specify the dgef for some well - known finite bounded proportional cake - cutting protocols in section [ sec : guaranteed - envy - freeness ] . in section [ sec : improved - algorithm - guaranteed - envy - freeness ] we present a new finite bounded proportional cake - cutting protocol with an enhanced degree of guaranteed envy - freeness , compared with the proportional protocols mentioned in section [ sec : guaranteed - envy - freeness ] .this new cake - cutting protocol makes use of parallelization in order to include as many matching valuations ( in terms of not raising envy ) as possible .section [ sec : survey ] briefly describes those protocols mentioned in section [ sec : guaranteed - envy - freeness ] , and determines their degree of guaranteed envy - freeness via a detailed analysis . in section [ sec : discussion ] , we compare the dgef approach with related work , and show that even small steps toward the development of cake - cutting protocols with an enhanced dgef are of significance .finally , we conclude in section [ sec : conclusion ] that our approach extends the scope for the development of new finite bounded cake - cutting protocols by `` approximating '' envy - freeness instead of insisting on it .cake - cutting is about dividing a cake into portions that are assigned to the players such that each of them feels , according to his or her individual valuation of the portions , to have received a fair amount of the cake .the cake is assumed to be infinitely divisible , and can be divided into arbitrary pieces without losing any of its value .moreover , we assume the cake to be heterogeneous . this assumption can be made without loss of generality , as any cake - cutting protocol providing a `` fair '' division of a heterogeneous resource can be applied in the same way to a homogeneous one .given players , cake is to be divided into portions that are to be distributed among the players so as to satisfy each of them .a portion is not necessarily a single piece of cake but rather can be a collection of disjoint , possibly noncontiguous pieces of  .furthermore , all players may have different individual valuations of the single pieces of the cake .for example , one player may prefer the pieces with the chocolate topping , whereas another player may prefer the pieces with the cherry topping .more formally , cake is represented by the unit interval  ] and s valuation function maps subintervals of  ] .we require each valuation function to satisfy the following properties :  1 .  _normalization : _ .positivity:_the literature is a bit ambiguous regarding this assumption .some papers require the players values for nonempty pieces of cake to be _ nonnegative _( i.e. , ) instead of positive .for example , robertson and webb  and woeginger and sgall  require nonnegative values for nonempty pieces of cake , whereas positive values for such pieces are required by brams and taylor  , brams , jones , and klamler  , and weller  .] for all , , we have ._ additivity : _ for all , , we have .divisibility:_ ] for intervals \subseteq [ 0,1] ] , define \| = y - x ] having equal size can be valued differently by the same player .moreover , distinct players may value one and the same piece of the cake differently , i.e. , their individual valuation functions will in general be distinct .every player knows only the value of ( arbitrary ) pieces of corresponding to his or her own valuation function .players do not have any knowledge about the valuation functions of other players .a _ division of _ is an assignment of disjoint and nonempty portions  , where , to the players such that each player receives a portion consisting of at least one nonempty piece .the goal of a cake - cutting division is to assign the portions to the players in as fair a way as possible .there are different interpretations , though , of what `` fair '' might mean .to distinguish between different degrees of fairness , the following notions have been introduced in the literature ( see , e.g. , robertson and webb  ) :  [ def : proportional - strongfair - envyfree - division ] let be the valuation functions of the players . a division of cake , where is the player s portion , is said to be :  1 ._ simple fair __ proportional _ ) if and only if for each  , , we have ; 2 ._ strong fair _ if and only if for each  , , we have ; 3 .  _ envy - free _ if and only if for each and  , , we have .a cake - cutting protocol describes an interactive procedure for obtaining a division of a given cake , without having any information on the valuation functions of the players involved .each protocol is characterized by a set of rules and a set of strategies ( see , e.g. , brams and taylor  ) .the rules just determine the course of action , such as a request to cut the cake , whereas the strategies define how to achieve a certain degree of fairness , e.g. , by advising the players where to cut the cake .if all players obey the protocol , it is guaranteed that every player receives a `` fair '' portion of the cake .cake - cutting protocols are characterized according to the degree of fairness of the divisions obtained :  [ def : proportional - strongfair - envyfree - protocol ] a cake - cutting protocol is said to be _ simple fair _ ( or _ proportional _ ) , _ strong fair _ , and _ envy - free _ , respectively , if every division obtained ( i.e. , regardless of which valuation functions the players have ) is simple fair ( or proportional ) , strong fair , and envy - free , respectively , provided that all players follow the rules and strategies of the protocol .  apparently , every division obtained by either a strong fair cake - cutting protocol or by an envy - free cake - cutting protocol is simple fair as well ( i.e. , every strong fair or envy - free cake - cutting protocol can be classified as being simple fair , too ) .moreover , every simple fair cake - cutting protocol can easily be applied to the case when there are unequal shares to be assigned , though with respect to rational ratios only . informally speaking ,this can be done by cloning players and their valuation functions so as to have in total as many players as the smallest common denominator specifies ( see , e.g. , ) .as mentioned in the introduction , the design of envy - free cake - cutting protocols for any number of players seems to be quite a challenge . for  players ,a number of protocols that always provide envy - free divisions have been published , both finite ( bounded and unbounded ) and continuous ones  . however , to the best of our knowledge , up to date no finite bounded cake - cutting protocol for  players is known to always provide an envy - free division . for practical purposes , it would be most desirable to have _ finite bounded _ cake - cutting protocols that always provide divisions as fair as possible . in this regard , it is questionable whether the advantage of always having an envy - free rather than just a proportional division would be big enough to justify the lack of finite boundedness. it may be worthwhile to be content with a certain lower degree of envy - freeness , rather than insisting on complete envy - freeness , for the benefit of having a finite bounded protocol in exchange .  in this paper, we propose an approach that weakens the concept of envy - freeness for the purpose of keeping protocols finite bounded . on the one hand , in section[ sec : survey - guaranteed - envy - freeness ] we are concerned with known simple fair ( i.e. , proportional ) cake - cutting protocols that are finite bounded , and determine their degree of guaranteed envy - freeness , a notion to be introduced in section [ sec : def - guaranteed - envy - freeness ] ( see definition [ def : degree - guaranteed - envy - freeness ] ) . on the other hand , in section [ sec : improved - algorithm - guaranteed - envy - freeness ]we propose a new finite bounded proportional cake - cutting protocol that compared with the known protocols has an enhanced degree of guaranteed envy - freeness .   when investigating the degree of envy - freeness of a cake - cutting protocol for  players , for each player , , the value of his or her portion needs to be compared to the values of the other portions ( according to the measure of player  ) .thus , pairwise relations need to be investigated in order to determine the degree of envy - freeness of a cake - cutting protocol for players .a player envies another player , , , when prefers player s portion to his or her own . if envies  , we call the relation from to an _ envy - relation _ ; otherwise , we call it an _ envy - free - relation_.  [ def : envy - relation - envy - free - relation ] consider a division of cake for a set of players , where is s valuation function and is s portion .  1 .an _ envy - relation for this division _ ( denoted by ) is a binary relation on  .player envies player , , , if and only if .we write .2 .  an _ envy - free - relation for this division _ ( denoted by ) is a binary relation on  .player does not envy player , , , if and only if .we write .the following properties of envy - relations and envy - free - relations are worth mentioning .no player can envy him- or herself , i.e. , envy - relations are irreflexive : the inequality never holds .thus , we trivially have that always holds .however , when counting envy - free - relations for a given division , we will disregard these trivial envy - free - relations , , throughout the paper .  furthermore , neither envy - relations nor envy - free - relations need to be transitive .this is due to the fact that each player values every piece of the cake according to his or her own valuation function .the valuation functions of different players will be distinct in general .for example , given three distinct players , , and with valuations and , we have that and .however , these valuations do not provide any information about player s valuation of portion , so we can not conclude that .an analogous argument applies to envy - free - relations .the above observations imply that envy - relations and envy - free - relations are either one - way or two - way , i.e. , it is possible that :  1 .  two players envy each other ( and ) , which we refer to as `` two - way envy , '' 2 .neither of two players envies the other ( and ) , which we refer to as `` two - way envy - freeness , '' and 3 .one player envies another player but is not envied by this other player ( and ) , which we refer to as both `` one - way envy '' ( from to  ) and `` one - way envy - freeness '' ( from to  ) .  assuming that all players are following the rules and strategies, some cake - cutting protocols always guarantee an envy - free division ( i.e. , they always find an envy - free division of the cake ) , whereas others do not .only protocols that _ guarantee _ an envy - free division in _ every _ case , even in the worst case ( in terms of the players valuation functions ) , are considered to be envy - free .note that an envy - free division may be obtained by coincidence , just because the players have matching valuation functions that avoid envy , and not because envy - freeness is enforced by the rules and strategies of the cake - cutting protocol used .in the worst case , however , when the players have totally nonconforming valuation functions , an envy - free division would not just happen by coincidence , but needs to be enforced by the rules and strategies of the protocol .an envy - free - relation is said to be _ guaranteed _ if it exists even in the worst case .we now define the degree of guaranteed envy - freeness in relation to the problem of cake - cutting .[ def : degree - guaranteed - envy - freeness ] for players , the _ degree of guaranteed envy - freeness _( dgef , for short ) of a given proportional cake - cutting protocol is defined to be the maximum number of envy - free - relations that exist in every division obtained by this protocol ( provided that all players follow the rules and strategies of the protocol ) , i.e. , the dgef ( which is expressed as a function of  ) is the number of envy - free - relations that can be guaranteed even in the worst case .  by a slight abuse of notation, we will sometimes speak of the number of guaranteed envy - free - relations ( rather than of the guaranteed number of envy - free - relations ) .when we do so , let us remind the reader that what matters is the _ total number _ of envy - free - relations that exist in the worst case , and not the _ identification _ of specific envy - free - relations .moreover , for technical reasons ( see the proof of lemma [ lem : divide - conquer ] ) , we also consider the case that there is only one player ( i.e. , ) . note , however , that in this case the dgef of any cake - cutting protocol is trivially zero , since we disregard the trivial envy - free - relation .definition [ def : degree - guaranteed - envy - freeness ] is based on the idea of weakening the notion of fairness in terms of envy - freeness in order to obtain cake - cutting protocols that are fair ( though perhaps non - envy - free ) _ and _ finite bounded , where the fairness level of a protocol is given by its degree of guaranteed envy - freeness .the higher the degree of guaranteed envy - freeness the fairer the protocol .we now give an upper and a lower bound on the degree of guaranteed envy - freeness for proportional cake - cutting protocols . for comparison ,note that feldman and kirman  observed that , for _ any _ division , the number of envy - relations is always between zero and ; zero if everyone is happy with his or her share of the cake , and if everyone is envious of everyone else .[ prop : dgef - minimum - maximum ] let be the degree of guaranteed envy - freeness of a proportional cake - cutting protocol for players .it holds that .  if then we obviously have , and this will be the case exactly if both players value their share of the cake as being at least  . note that this case , , reflects the fact that every proportional protocol for two players is envy - free .so , we now assume that .as stated above , we disregard the trivial envy - free - relation for each  , .consequently , each player can have at most envy - free - relations , one to each of the other players , which gives a total of at most guaranteed envy - free - relations .this proves the upper bound : .  to prove the lower bound , note that , by definition , in a proportional division every player  , , regards his or her portion being of value at least , i.e. , .thus , for each  .we will now prove that this implies that none of the players can envy each of the other players at the same time .we show this for player  ; the argument is analogous for the other players .so , assume that envies some other player , say  .thus .it follows that values the remaining cake as being less than , i.e. , .consequently , there is no way to divide the remaining cake into portions such that for each , , we have .hence , there must be at least one player  , , such that , so .considering all players , this gives at least guaranteed envy - free - relations for a proportional protocol , so .the degree of fairness of a division obtained by applying a proportional cake - cutting protocol highly depends on the rules of this protocol .specifying and committing to appropriate rules often increases the degree of guaranteed envy - freeness , whereas the lack of such rules jeopardizes it in the sense that the number of guaranteed envy - free - relations may be limited to the worst - case minimum of as stated in proposition [ prop : dgef - minimum - maximum ] . in this context ,`` appropriate rules '' are those that involve the players evaluations of other players portions and of pieces that still are to be assigned .concerning a particular piece of cake , involving the evaluations of as many players as possible in the allocation process helps to keep the number of envy - relations to be created low , since this allows to determine early on whether a planned allocation may later turn out to be disadvantageous and thus allows to take adequate countermeasures .in contrast , omitting mutual evaluations means to forego additional knowledge that could turn out to be most valuable later on .for example , say player  is going to get assigned piece  .if the protocol asks all other players to evaluate piece  according to their measures , all envy - relations to be created by the assignment of piece  to player  can be identified before the actual assignment and thus countermeasures ( such as trimming piece  ) can be undertaken . however , if the protocol requires no evaluations on behalf of the other players , such envy - relations can not be identified early enough to prevent them from happening .[ lem : dgef - no - evaluations ] let a proportional cake - cutting protocol for players be given . if the rules of the protocol require none of the players to value any of the other players portions , then the degree of guaranteed envy - freeness is  ( i.e. , each player is guaranteed only one envy - free - relation ) .having a certain number of guaranteed envy - free - relations means to have at least this number in any case , even in the worst case . for , proportionality implies envy - freeness , so the worst case is the best case . for players ,consider the following scenario . given a division of cake , without any restrictions other than aiming at a proportional division ( i.e., the rules of the protocol require none of the players to value any of the other players portions ) , we set the valuation functions of the players as follows . for each , , player values portion to be worth exactly , values exactly one portion  , , to be worth exactly , and values each of the remaining portions  , , to be worth exactly .these valuations make this division proportional , as every player values his or her portion to be worth exactly .moreover , each player has envy - relations and just one guaranteed envy - free - relation , the latter of which is due to proposition [ prop : dgef - minimum - maximum ] .hence , if the rules of the protocol require none of the players to value any of the other players portions , then no more than envy - free - relations can be guaranteed by the given proportional cake - cutting protocol in the worst case .   the argument in the proof of lemma [ lem : dgef - no - evaluations ]will be used to prove the upper bounds on the dgef of the proportional cake - cutting protocols considered in theorem [ thm : survey - dgef ] ( see also lemmas [ lem : last - diminisher ] through [ lem : divide - and - choose ] ) and theorems [ thm : n4-dgef ] and [ thm : n - dgef ] .  an envy - free cake - cutting protocol for playersguarantees that no player envies any other player  , i.e. , for all with , in each division obtained by this protocol .this is the case exactly if the protocol meets the upper bound on the degree of guaranteed envy - freeness given in proposition [ prop : dgef - minimum - maximum ] .that is , a cake - cutting protocol for players is envy - free ( or completely fair ) exactly if the degree of guaranteed envy - freeness equals .our next result shows the dgef for a number of well - known finite bounded _proportional _ cake - cutting protocols . for the sake of self - containment, these protocols will be described in section [ sec : survey ] .[ thm : survey - dgef ] for players , the trivial cases ( where one player receives all the cake ) and ( where each proportional division is always envy - free ) are ignored .specifically , an envy - free ( and thus proportional ) division for players can always be obtained by applying the cut - and - choose protocol : one player cuts the cake into two pieces both of which he or she considers to be worth exactly one half of the cake , and the other player chooses the piece that he or she considers to be worth at least half of the cake . ]the proportional cake - cutting protocols listed in table [ tab : dgef - survey ] have a degree of guaranteed envy - freeness as shown in the same table .the proof of theorem [ thm : survey - dgef ] can be found in section [ sec : survey ] .in particular , the proofs of lemmas [ lem : last - diminisher ] through [ lem : divide - and - choose ] provide the details of the analysis yielding the values in the dgef column of table [ tab : dgef - survey ] ..dgef of selected finite bounded cake - cutting protocols . [ cols="<,^,^",options="header " , ]in this section , we determine the degrees of guaranteed envy - freeness of the proportional cake - cutting protocols listed in table [ tab : dgef - survey ] .thus , we prove theorem [ thm : survey - dgef ] via lemmas [ lem : last - diminisher ] through [ lem : divide - and - choose ] .we investigate proportional protocols only , because proportional cake - cutting protocols have a dgef of at least according to proposition [ prop : dgef - minimum - maximum ] and , thus , show some degree of fairness already . over the years , several cake - cutting protocolshave been proven to be proportional and finite bounded for any number of players .a detailed description of various finite bounded proportional protocols can be found in the books by brams and taylor  and robertson and webb  .our analysis of the protocols in table [ tab : dgef - survey ] provides a basis for further algorithmic improvements in terms of the degree of guaranteed envy - freeness , and the protocol in figure [ algo : n ] is a first step in this direction .  in the following subsections ,we give a brief description of the protocols listed in table [ tab : dgef - survey ] and provide a detailed analysis of their dgef .note that the value of cake  to be divided is normalized such that for all players  with .the protocol works as follows : the first player cuts a piece he or she considers being worth exactly in his or her measure .this piece is given to the other players , one after the other .now , each player has the choice to either pass the piece on as it is , or to trim it before passing it on .if a player considers the piece to be worth more than , he or she trims it to exactly according to his or her measure .when the last player has evaluated this piece , it is given to the player who was the last trimming it , or to the player who cut it in the first place if no trimmings have been made .the trimmings are reassembled with the remainder of the cake , and the procedure is applied in the same way for the remaining players and the reassembled remainder of the cake .this process is repeated until only two players remain . in the final round ( of rounds in total ) , these last two players apply the simple cut - and - choose protocol to the remainder of the cake . for guaranteeing each player a proportional share of the cake ,the order of the players is of no significance .[ lem : last - diminisher ] the last diminisher protocol has a degree of guaranteed envy - freeness of .  concerning the analysis of the degree of guaranteed envy - freeness , it is quite evident that in the first round envy - free - relations are guaranteed , since each of the players not receiving the first piece consider this piece to be of value at most  .analogously , in the round , , additional envy - free - relations are guaranteed .the number of guaranteed envy - free - relations is consecutively decreasing by one per round , as the players who already received a piece are not involved in the evaluation process of subsequent rounds ( see the proof of lemma [ lem : dgef - no - evaluations ] ) .this sums up to guaranteed envy - free - relations .in addition , in the final round one more guaranteed envy - free - relation is created , as the simple cut - and - choose protocol guarantees that both of the players will not envy each other . finally , note that the player receiving the first portion is not involved in the evaluations of any other portions , butsince the last diminisher protocol is proportional , this player too can not envy each of the other players according to the argument in the proof of proposition [ prop : dgef - minimum - maximum ] .thus , one more guaranteed envy - free - relation must be added .consequently , the last diminisher protocol guarantees envy - free - relations .the lone chooser protocol was first proposed by fink  ( as cited in  ) .it can be described as follows . for two players ,the protocol is just the simple cut - and - choose protocol .for players , the protocol has rounds .the first round simply describes the cut - and - choose protocol executed by players and  , which results in two pieces , and  , with . assuming player received piece and player received piece , in the second round player has to divide piece with player , and player has to divide piece with player . to this end , cuts into three pieces each of which he or she considers to be worth at least , and so does with  .player then chooses one of the pieces of player and one of the pieces of player  , both being most valuable according to s measure .this guarantees each of the players , , and a portion of at least in their measures . carrying on in this way ,when the final round has been entered , each of the players is in possession of a portion that he or she considers to be worth at least in his or her measure .let us refer to those players as the `` cutters '' of round  .finally , each of the cutters cuts his or her portion into pieces each of value  , and player  , the `` chooser '' of round , chooses one piece of highest value ( according to his or her measure ) from each plate of the cutters of this round .  [lem : lone - chooser ] the lone chooser protocol has a degree of guaranteed envy - freeness of  .  in the course of the lone chooser protocol , none of the players evaluate the portion of any of the other players , which determines the dgef of the lone chooser protocol to be  as a result of lemma [ lem : dgef - no - evaluations ].    a complete algorithmic description of this protocol would be rather comprehensive .that is why we will give just a rough sketch of the procedure . for a more detailed description ,the reader is referred to kuhn  , see also , e.g. , .the lone divider protocol works as follows : some player , , is chosen to be the first - round `` divider .'' the divider cuts cake  into portions that he or she considers each to be worth , i.e. , with .subsequently , all other players ( the first - round `` choosers '' ) are asked to identify any of the portions they find acceptable , that is , each player identifies all portions that are worth at least in this player s measure . obviously , every chooser  , , needs to accept at least one portion  .depending on the players choices , there are different ways for how the protocol continues . in the simplest case ,the players choices allow for a _ fully decidable division _ ,i.e. , for a division such that each chooser receives one of the portions he or she previously identified as being acceptable .divider then receives the portion that has not been assigned to any of the choosers .all players drop out and the division is complete .however , if the players choices do not allow a fully decidable division , there is either a _ partially decidable division _ ( which means that only some not all of the choosers are assigned a portion and drop out ) , or a _ fully undecidable division _( which means that there are at least two portions that have not been identified as being acceptable by any of the choosers and that none of the choosers receive a portion ) . in the case of a partially decidable division ,the choosers accomplish only a partial allocation of the cake , i.e. , those choosers that identified acceptable portions in a nonconflicting way are assigned a portion they have marked as being acceptable and drop out .in addition , this round s divider is assigned any one of the other portions and drops out , the remaining portions ( that could not be assigned to players ) are reassembled , and a new round with the remaining players , among which a new divider is to be chosen , is started . in the case of a fully undecidable division ,none of the choosers are assigned a portion ; only this round s divider receives one of the two portions that have not been identified as being acceptable by any of the choosers and drops out , and a new round with the remaining players , among which a new divider is to be chosen , is started in which the remaining cake is divided .this procedure is repeated until the whole cake has been allocated .note that in each round at least this round s divider is assigned a portion and drops out .it is easy to see that the lone divider protocol is finite bounded and proportional .[ lem : lone - divider ] the lone divider protocol has a degree of guaranteed envy - freeness of .the analysis of the degree of guaranteed envy - freeness is done by a worst - case scenario in terms of the number of existing envy - free - relations .we claim that the maximum number of guaranteed envy - free - relations exists in the case that every chooser marks portions as being acceptable in the first round . in the following ,let us refer to this situation as the `` worst - case scenario . '' in this scenario ,the rules of the protocol imply that a proportional division will be achieved in the very first round ( i.e. , in this worst case scenario the first round results in a fully decidable division ) , and the following envy - free - relations are guaranteed to exist in this case .the divider will not envy any of the choosers , as he or she considers each of the portions to be , resulting in guaranteed envy - free - relations .furthermore , none of the choosers will envy the player ( be it the divider or any of the other choosers ) that received the portion he or she considers to be not acceptable , leading to additional guaranteed envy - free - relations . since the dgef is the maximum number of envy - free - relations that are guaranteed to exist in _ every _ case , the dgef of the lone divider protocol is at most . to argue that the scenario given above indeed represents the worst case for players ( and so the dgef is equal to ) , we will consider all possible cases different from the worst - case scenario .we will show that in each of these cases the number of existing envy - free - relations is , in fact , higher than .this implies that none of these other cases considered represent a worst - case scenario .  for notational convenience, we will use the term `` case - enforced envy - free - relation '' to refer to the number of those envy - free - relations that necessarily must exist in any of these cases , regardless of which particular valuation functions the players have ( other than what was causing the respective case to occur ) .recall that the term `` guaranteed envy - free - relation '' is reserved for the number of envy - free - relations that necessarily exist in the worst case ( and as we will see none of the cases below will describe the worst case ) .thus , the term `` case - enforced envy - free - relation '' is more general than and includes the term `` guaranteed envy - free - relation '' : the number of case - enforced envy - free - relations in the worst case ( i.e. , the minimum number of case - enforced envy - free - relations , where the minimum is taken over all possible cases ) is exactly the number of guaranteed envy - free - relations .[ case1 ] the first round results in a fully decidable division .consider the following two subcases .[ case1 - 1 ] the total number of _ inacceptable _ portions is greater than in the worst - case scenario .specifically , let us look at the situation that exactly one of the choosers considers just one more portion as being inacceptable than in the worst - case scenario .this change decrements the total number of acceptable portions by one and thus creates one additional case - enforced envy - free - relation , since this chooser will not envy the player receiving this particular portion .however , since the number of guaranteed envy - free - relations of the worst - case scenario persists also in this case , increasing the total number of inacceptable portions also increases the total number of case - enforced envy - free - relations .thus , the present case does not describe a worst - case scenario .[ case1 - 2 ] the total number of _ acceptable _ portions is greater than in the worst - case scenario .specifically , let us look at the situation that exactly one of the choosers considers just one more portion as being acceptable than in the worst - case scenario .this chooser then accepts all portions , and thus necessarily considers each of the portions to be worth exactly .accordingly , this chooser does not envy any of the other players , resulting in additional case - enforced envy - free - relations .in particular , since this chooser still does not envy the player he or she did not envy in the worst - case scenario , incrementing the total number of acceptable portions by just one increases the total number of case - enforced envy - free - relations by one ( if ) or even more ( if ) .thus , the present case does not describe a worst - case scenario .[ case2 ] the first round does not result in a fully decidable division .thus , the protocol runs over more than one round . for players, an additional round would be caused only by a fully undecidable division .for players , additional rounds are caused by either a fully undecidable division or a partially decidable division . in every round ,at least the divider of this round is assigned a portion and drops out . on the part of the choosers , entering a new round creates additional case - enforced envy - free - relations the number of which depends on the present circumstances .simply put , every additional round will increase the total number of case - enforced envy - free - relations compared with the worst - case minimum of .this justifies why a division obtained by the execution of more than one round does not present a worst - case scenario .  to see why running more than one round will lead to more than case - enforced envy - free - relations ,let us first , in cases [ case2].[case2 - 1 ] and [ case2].[case2 - 2 ] , have a closer look at the case - enforced envy - free - relations created in each nonfinal round if more than one round is executed in total .the final round will be handled separately in case [ case2].[case2 - 3 ] .in particular , if there are exactly two rounds , the total number of case - enforced envy - free - relations created is the sum of those explained in the first paragraph of either case [ case2].[case2 - 1 ] or case [ case2].[case2 - 2 ] ( which describe the number of case - enforced envy - free - relations created in the first round ) and those explained in case [ case2].[case2 - 3 ] ( which describes the number of case - enforced envy - free - relations created in the final round ) .[ case2 - 1 ] in the case of a _ fully undecidable division _ of the cake as the result of the first round , the first - round divider receives one of the portions that have not been marked as being acceptable by any of the choosers and drops out .all choosers enter the second round and will not envy the divider of the first round , resulting in case - enforced envy - free - relations .moreover , the first - round divider will not envy at least one of the other players , since there must be at least one player he or she does not envy , which follows from the proof of proposition [ prop : dgef - minimum - maximum ] .thus , case - enforced envy - free - relations result from the first round in this case .note that proposition [ prop : dgef - minimum - maximum ] can be applied only to the first - round divider .thus , analogously to the above argument , for every additional round with players that is caused by a fully undecidable division and that is not the final round , case - enforced envy - free - relations need to be added .an analysis of the final round follows in case [ case2].[case2 - 3 ] .[ case2 - 2 ] in the case of a _ partially decidable division _ of the cake as the result of the first round , let denote the set of those players , , that are in conflict with each other concerning the portions they identified as being acceptable in this first round .let be the set of the remaining players , where and .the players in can divide a part of the cake without any conflict , i.e. , each of the players in is assigned one of the portions he or she identified as being acceptable . note that would represent case [ case2].[case2 - 1 ] , and that the divider always is one of the players in . following the protocol ,each of the players in receives a portion and drops out while each of the players in enters the second round .since none of the players in would accept any of the portions the players in have received , each of the players in does not envy any of the players in  , resulting in case - enforced envy - free - relations .the players in ( except for the first - round divider ) that are assigned a portion and drop out in the first round are each guaranteed one envy - free - relation due to the argument in the proof of proposition [ prop : dgef - minimum - maximum ] , summing up to additional case - enforced envy - free - relations .moreover , the first - round divider will not envy any of the other players in  , resulting in more case - enforced envy - free - relations .consequently , case - enforced envy - free - relations result from the first round in this case .  now consider any additional round with players that is caused by a partially decidable division of the cake and that is not the final round , where players are in conflict with each other and players accomplish a partial allocation of the cake .analogously to the above argument ( except that proposition [ prop : dgef - minimum - maximum ] is no longer applicable ) , case - enforced envy - free - relations are to be added for the players being in conflict , and case - enforced envy - free - relations are to be added for this round s divider .an analysis of the final round follows in case [ case2].[case2 - 3 ] .[ case2 - 3 ] aside from the case - enforced envy - free - relations that are created by executing a nonfinal round , additional case - enforced envy - free - relations are created in the _ final round _ , in which even the last player receives a portion . in each case , the final round is characterized by providing a fully decidable division . consequently , considering a final round with  players at least case - enforced envy - free - relations are created according to cases [ case1].[case1 - 1 ] and [ case1].[case1 - 2 ] .  as a result , in the example of the protocol running a second round , either with ( see case [ case2].[case2 - 1 ] ) , or with ( see case [ case2].[case2 - 2 ] ) envy - free - relations are case - enforcedthus , since , a second round yields at least case - enforced envy - free - relations in total .as indicated above , every additional round increases the number of case - enforced envy - free - relations even more .cases [ case1 ] and [ case2 ] ( and their subcases ) completely characterize all situations different from the worst case scenario , since their number of case - enforced envy - free - relations is always greater than , the number of envy - free - relations guaranteed to exist in the scenario given in the first paragraph of this proof .this justifies that this scenario indeed represents the worst case .note that the protocol does not require any of the choosers to value any of the portions they marked as being acceptable ( i.e. , the only information provided on these portions is that they are considered to be worth at least ) , and thus , according to the proof of lemma [ lem : dgef - no - evaluations ] , the dgef of the lone divider protocol is .the protocol works as follows : every player marks adjacent pieces each valued in his or her measure , resulting in marks in total .afterwards , a cut is made at between and of the existing marks , resulting in at least  adjacent pieces .each player then is assigned a portion that contains at least one of the pieces he or she marked beforehand plus some optional supplement .there are different strategies for how to make cuts such that the resulting division is fair .the number of cuts to be made depends on the strategy chosen .note that there always is at least one strategy that guarantees each player a portion valued at least according to his or her measure , i.e. , for all  .if  cuts are made ( at of the existing marks ) , each player s portion consists of exactly one piece which is worth at least according to his or her measure .  [ lem : cut - your - own - piece ] the cut your own piece protocol has a degree of guaranteed envy - freeness of if no strategy is specified , and a degree of guaranteed envy - freeness of for the `` left - right strategy '' ( which , for convenience , will be explained in the proof ) .moreover , for this protocol no strategy can give a better dgef than .  as mentioned above, there are different strategies for how to make cuts such that the resulting division is fair .if no particular strategy is given , this protocol has a dgef of only  ( according to lemma [ lem : dgef - no - evaluations ] ) , since all players made their marks independently , and none of the players were asked to give an evaluation of the marks of any of the other players .hence , only the minimum of envy - free - relations can be guaranteed .steinhaus  did not mention a strategy for how to achieve a simple fair division ; he just mentioned that there always exists at least one .however , when we consider a strategy that always assigns the leftmost ( with respect to the interval  ] ) , which could be used for subsequent comparisons of the guaranteed sizes and thus values of the pieces marked , are known .the divide and conquer protocol , which was first presented by even and paz  ( see also , e.g. , ) , is based on the idea of dividing cake by simultaneously partitioning disjoint parts of  .the procedure slightly differs depending on whether the number of players is even or odd .if there is an even number of players , say for some integer  , all players but one divide cake in the ratio by a single cut , yielding two pieces of equal value for each of these players .the noncutter identifies either the piece to the left of the middle cut ( with respect to the interval  ] ) , and then to use this additional cut for splitting the particular piece of the cake into two subpieces for the following round if there is any ( or to be assigned if this has been the final round ) .[ lem : divide - conquer ] the divide and conquer protocol and the minimal - envy divide and conquer protocol both have a degree of guaranteed envy - freeness of .the divide and conquer protocol is recursively defined .put simply , in each subprocedure the given subpart of the cake is divided into two pieces and so are the players into two groups , the procedure then is applied recursively again and again to the resulting pieces and related players until in each subprocedure just one player remains . in each round , every player participating in any of the subprocedures of this round will not envy at least one of the players continuing with the corresponding other piece , as this other piece is of no more value ( according to his or her measure ) than the one he or she is continuing with .thus , in each round , for each player involved in this round , one envy - free - relation is guaranteed to exist .note that , for each subprocedure in any round except the final one , the numbers of players to be continued with in the resulting two subprocedures of the following round depend on whether the total number of players involved in the given subprocedure is even or odd .  from these remarksit follows that the divide and conquer protocol s degree of guaranteed envy - freeness , call it for players , can be described by the following recurrence : apparently , this recurrence relation can be simplified to :   the recurrence in ( [ eq : rec - solution - divide - conquer ] ) and similar versions are well known to occur also in other contexts .it is a matter of routine ( see , e.g. , ) to solve it ( i.e. , to bring it into closed form ) :   how does equation ( [ eq : solution - divide - conquer ] ) reflect the guaranteed number of envy - free - relations of the divide and conquer protocol ? as mentioned before , this protocol can be considered as a collection of several subprocedures that altogether yield a proportional division of the given cake .this collection of subprocedures can be represented as a balanced binary tree ( called the `` recursion tree '' ) , because in every subprocedure the given subpart of the cake is cut into two pieces for which the procedure is applied recursively again and again until just one player remains in each resulting subprocedure .since the depth of a balanced binary tree is logarithmic in the number of leaves , rounds are performed in total . if the number  of players is not a power of two , every round except for the last one ( i.e. , rounds ) is represented by a completely filled level of the binary tree in terms of the number of players , since all  players are participating in these rounds in different subprocedures though .note that is a power of two if and only if it holds that ( i.e. , the final round is numbered ) , and only in this case , all  players are involved in each of the rounds , even in the final round .recall that , in each round , every participating player will not envy at least one of the players continuing with the particular other piece , since he or she considers this piece to be of no more value than the one he or she is continuing with , i.e. , in each round one guaranteed envy - free - relation is created on behalf of each of the participating players .for this reason ,  envy - free - relations are guaranteed to be created in each of the first rounds .this is because in subsequent rounds the particular subparts to be divided will never get bigger again , and once two players have ended up in different subprocedures , they will never meet again in the same subprocedure of any of the following rounds .moreover , once a player has ended up in some group , he or she will not make future evaluations of pieces of the cake to be divided among the players in the other group .thus , those envy - free - relations that result from any of the first rounds are guaranteed to persist until all the cake has been allocated .however , it can not be determined which of the players continuing with the other piece of the particular subpart is not envied , since no evaluations of the pieces created in other subprocedures are made . in accordance with the proof of lemma [ lem : dgef - no - evaluations ] ,the latter also justifies why no more envy - free - relations can be guaranteed .hence , it can just be guaranteed that each player does not envy at least one of the players continuing with the other piece .summing up , as exactly guaranteed envy - free - relations are created in each of the first rounds , guaranteed envy - free - relations are created over all rounds , except for the final round if is not a power of two . note that if is a power of two , the round is the final round and equation ( [ eq : solution - divide - conquer ] ) simplifies to , so in this case we are done .in contrast , if is not a power of two then less than  players will be involved in the final round ( i.e. , in the round numbered ) , since in that case there is at least one subprocedure that involves an odd number of players .more specifically , in this case the number of players involved in the final round can be expressed by the term , where specifies the number of players that would be involved in the final round if the binary recursion tree would be a full binary tree , i.e. , if all  players would be involved in the final round . in order to analyze the final round for not being a power of two in detail , let -subprocedure denote a subprocedure involving exactly players . a -subprocedure can occur only in the second - to - last round , and if it occurs then one of its three players can not be participating in the final round , i.e. , only two out of three players are proceeding to the final round . since in a balanced binary treethe depth of all leaves differs by at most one , the second - to - last round can have only either -subprocedures and/or -subprocedures , or -subprocedures and/or -subprocedures . regarding a second - to -last round with at least one -subprocedure and any number of -subprocedures ( which can happen only if is not a power of two ) , the number of players involved in the final round will be twice the number of -subprocedures occurring in the second - to - last round . regarding a second - to -last round with at least one -subprocedure and any number of -subprocedures ( which again can happen only if is not a power of two ) , the number of players involved in the final round will be twice the number of -subprocedures plus four times the number of -subprocedures . consequently ,the number of -subprocedures and the number of -subprocedures in the second - to - last round determine how many players are participating in the final round , and thus also determine the number of guaranteed envy - free - relations to be created in the final round .if is not a power of two , analogously to the argument for the first rounds , also in the final round one guaranteed envy - free - relation is created with respect to each participating player .thus , for any number of players , at least guaranteed envy - free - relations are created in the round .is a power of two if and only if , and that in this case there is no round . ]according to the proof of lemma [ lem : dgef - no - evaluations ] , no more than that many envy - free - relations can be guaranteed .altogether , this sums up to guaranteed envy - free - relations in total , which is the number stated in equation ( [ eq : solution - divide - conquer ] ) .the degree of guaranteed envy - freeness of the minimal - envy divide and conquer protocol can be shown just as for the original divide and conquer protocol .the difference in the way of splitting the particular piece of the cake into two subpieces does not affect the number of guaranteed envy - free - relations .consequently , although the minimal - envy divide and conquer protocol does decrease envy according to the definition of brams , jones , and klamler  ( see section [ sec : discussion ] for more discussion of this point ) , its dgef is , just as for the original divide and conquer protocol .this protocol has been presented by tasndi  and describes a recursive procedure for how to always achieve a proportional division .it works as follows : in the case of , this is just the simple cut - and - choose protocol . in the case of , one of the players , the `` divider, '' divides the cake into three equal pieces according to his or her measure , and each of the two other players , the `` choosers , '' marks two pieces he or she considers to be worth the most , where ties may be broken arbitrarily .if both choosers marked the same two pieces , they divide these by applying the simple cut - and - choose protocol , and the divider receives the remaining piece .if the choosers marked different pieces , they divide the piece they both have marked via the simple cut - and - choose protocol , and each of the choosers divides the piece marked by just him- or herself with the divider , again via applying the simple cut - and - choose protocol .  in the case of players , this procedureis repeated recursively until all comes down to the simple cut - and - choose protocol involving two players only . in more detail , in the first round the divider cuts the cake into equal pieces according to his or her measure , and each of the choosers marks  pieces he or she considers to be worth the most , where ties may be broken arbitrarily .afterwards , a new round is started and each of the  pieces is divided among those choosers that identified this piece as being acceptable .concerning pieces that have been marked by less than  choosers , the divider fills out these empty slots by an appropriate number of clones . in other words ,each of the  pieces enters the next round of the protocol and induces a new subprocedure in the scope of which the particular piece is being divided among players .all  subprocedures are executed simultaneously but independently of each other .note that if in the very first round all  choosers marked the same  pieces as being acceptable than there is exactly one piece that has not been marked by any of the choosers . in this case , this piece is directly assigned to the divider and the divider drops out , whereas all choosers enter the next round for dividing the  remaining pieces among them .analogously , in the round , , there are  subprocedures , i.e. ,  pieces are to be divided simultaneously but independently among  players each . in every subprocedureany one player is determined to be the divider and cuts the particular piece of this subprocedure into  equal subpieces according to his or her measure .afterwards , each of the  choosers of this particular subprocedure marks  pieces he or she considers to be worth the most , where ties may be broken arbitrarily .each of the  pieces will induce a new subprocedure in the next round and will be divided among those  players that marked this piece as being acceptable where the divider fills out all empty slots regarding pieces that have been marked by less than  choosers .again , if in some round , , in any of the subprocedures , all  choosers agree on the same  pieces , then there will be exactly one piece that has not been marked by any of the choosers . in this case, the unmarked piece is directly assigned to the divider of this particular subprocedure , and in the following rounds , the divider will not be involved in any of the subprocedures that results from this one , i.e. , only  pieces enter the next round , in which these are to be divided among the  choosers of the previous round .nevertheless , this divider will enter the next round and will participate in all those subprocedures that result from procedures he has not dropped out from yet .  applying this procedure recursively until only two players remain in each subprocedure ( i.e. , running  rounds ) , and dividing the corresponding piece of the cake between these two via the simple cut - and - choose protocol , the recursive divide and choose protocol provides a proportional division of the cake .[ lem : divide - and - choose ] the recursive divide and choose protocol has a degree of guaranteed envy - freeness of  .  for  players , no evaluations of entire portions are made except for the one special case when the first - round divider drops out in the very first round ( which , for the sake of self - containment , will be considered separately below)and thus the scenario from the proof of lemma [ lem : dgef - no - evaluations ] is applicable .simply put , only  envy - free - relations can be guaranteed in every case due to the argument in the proof of proposition [ prop : dgef - minimum - maximum ] , as the missing evaluations of entire portions allow for any valuation functions , for example , those described in the proof of lemma [ lem : dgef - no - evaluations ] .if in the very first round all  choosers agree on the very same  pieces then the first - round divider will drop out with a portion that he or she values exactly and that all other players value at most .this results in guaranteed envy - free - relations , since none of the  choosers will envy the first - round divider and the first - round divider will not envy at least one of the other players by the argument in the proof of proposition [ prop : dgef - minimum - maximum ] . however , in this case no more envy - free - relations can be guaranteed in the following rounds due to the argument given above . consequently , the dgef of the recursive divide and choose protocol is  .the analysis of envy - relations dates back at least to feldman and kirman  .in contrast to our approach , they consider the number of envy - pairs in already existing divisions with the intention of maximizing fairness afterwards via trading .in particular , they do not consider the _ design _ of cake - cutting protocols that maximize fairness . in the majority of cases ,research in the area of cake - cutting from an economic perspective is concerned more with the existence of certain divisions and their properties than with how to achieve these divisions .a different approach measures the intensity of envy in terms of the distance between envied portions  .more recently , brams , jones , and klamler  proposed to minimize envy in terms of the maximum number of players that a player may envy .their notion of measuring envy differs from our notion of dgef in various ways , the most fundamental of which is that their notion takes an `` egalitarian '' approach to reducing the number of envy - relations ( namely , via minimizing the most - envious player s envy , in terms of decreasing the number of this single player s envy - relations ) .in contrast , the dgef aims at a `` utilitarian '' approach ( namely , via minimizing overall envy , in terms of increasing the total number of guaranteed envy - free - relations among all players ) . that is to say that , although these notions may seem to be very similar at first glance , the approach presented in  is not sensitive to a reduction in the number of envy - relations on the part of any other than the most - envious player , whereas the dgef does take each single improvement into account and adapts accordingly .the dgef , thus , is a more specific , more fine - tuned measure .note also that brams , jones , and klamler  focus primarily on presenting a new protocol and less so on introducing a new notion for measuring envy .another approach is due to chevaleyre et al . , who define various metrics for the evaluation of envy in order to classify `` the degree of envy in a society , '' and they use the term `` degree of envy '' in the quite different setting of multiagent allocation of _ indivisible _ resources .  besides , we stress that our approach of approximating envy - freeness differs from other lines of research that also deal with approximating fairness . for example , lipton et al . propose to seek for minimum - envy allocations of _ indivisible _ goods in terms of the value difference of the utility functions of envied players , and edmonds and pruhs  approximate fairness in cake - cutting protocols by allowing merely approximately fair pieces ( in terms of their value to the players ) and by using only approximate cut queries ( in terms of exactness ) .it may be tempting to seek to decrease envy ( and thus to increase the dgef ) via trading , aiming to get rid of potential circular envy - relations .although we do not consider trading to be an integral part of a cake - cutting protocol , let us for a moment digress to briefly discuss how trading may potentially affect the number of guaranteed envy - free - relations .indeed , if the dgef is _ lower than  _ , the number of guaranteed envy - free - relations can be improved to this lower bound , or to an even higher number , by resolving circular envy - relations ( of which two - way envy - relations are a special case ) by means of circular trades after the execution of the protocol .thus , in this case , involving subsequent trading actions adds on the number of guaranteed envy - free - relations .furthermore , having exactly guaranteed envy - free - relations after all circular envy - relations have been resolved , three more guaranteed envy - free - relations can be gained by applying an envy - free protocol ( e.g. , the selfridge conway protocol ) to the three most envied players , which yields to an overall lower bound of  guaranteed envy - free - relations . to give an example for an even higher impact of trading , when circular trades indeed are involved after executing either the divide and conquer protocol or the minimal - envy divide and conquer protocol ,their numbers of guaranteed envy - free relations can be improved to  , which follows from  .on the other hand , if the dgef of a proportional cake - cutting protocol is _ or higher _( such as the dgef of the protocol presented in figure [ algo : n ] ) then depending on the protocol circular envy - relations may not be _ guaranteed _ to exist , and if such cycles are not guaranteed to exist , trading has no impact on the number of guaranteed envy - free - relations .however , as mentioned above , we consider trading not to be part of a cake - cutting protocol , though it might be useful in certain cases ( for example , brams and taylor  mention that trading might be used `` to obtain better allocations ; however , this is not a procedure but an informal adjustment mechanism '' ) . in particular, the notion of dgef refers to ( proportional ) cake - cutting protocols without additional trading , i.e. , the dgef is defined to make a statement on the performance of a particular protocol and not about all sorts of actions to be undertaken afterwards .although the well - known protocols listed in table [ tab : dgef - survey ] have not been developed with a focus on maximizing the dgef , mentioned above . ] linking their degrees of guaranteed envy - freeness to the lower bound provided by involving , e.g. , the selfridge conway protocol and guaranteed trading opportunities indicates that the development of cake - cutting protocols with a considerably higher dgef or even with a dgef close to the maximum of  poses a true challenge .that is why we feel that the enhanced dgef of the protocol presented in figure [ algo : n ] constitutes a significant improvement .although different disciplines are engaged in the development of fair cake - cutting protocols for decades now , finite bounded protocols that guarantee an envy - free division for players are still a mystery . however , finite bounded protocols are the ones we are looking for in terms of practical implementations .that is why we in this paper have proposed to weaken the requirement of envy - freeness ( as much as needed and as little as possible ) , while insisting on finite boundedness . to this end, we introduced the notion of degree of guaranteed envy - freeness for proportional cake - cutting protocols .based on this definition , we gave a survey of the dgef in existing finite bounded proportional cake - cutting protocols , which shows that when one is trying to approximate the ideal of envy - freeness via the dgef , there is quite a bit room for improvements .in particular , we expect that the concept of dgef is suitable to extend the scope for the development of new finite bounded cake - cutting protocols by allowing to approximate envy - freeness step by step . in this context , we proposed a new finite bounded proportional cake - cutting protocol , explicitly demonstrated for and for arbitrary  , which provides a significantly enhanced degree of guaranteed envy - freeness , compared with the status quo given by the survey in table [ tab : dgef - survey ] ( see also section [ sec : survey ] ) .in particular , our protocol has more guaranteed envy - free - relations than the last diminisher protocol , which previously was the best finite bounded proportional cake - cutting protocol with respect to the dgef . to achieve this significantly enhanced dgef , our protocol makes use of parallelization with respect to the leftmost and the rightmost pieces . in this regard , adjusting the values of the pieces to be marked from to ( with players still in the game ) and applying an appropriate inner - loop procedure is crucial to make the parallelization work .in addition to an enhanced dgef , our protocol still has the other useful properties the last diminisher protocol is known to possess , such as strategy - proofness .  in general , we suggest to target improvements on the `` level of envy - freeness '' already in the design of cake - cutting protocols rather than trying to improve on the number of envy - free - relations afterwards for the division obtained ( e.g. , via trading  ) . in terms of future research , this approach encourages to develop new protocols with even higher degrees of guaranteed envy - freeness one may even think of modifications that focus on the development of protocols with `` balanced envy - freeness '' while keeping the dgef high .we are grateful to ariel procaccia for many interesting discussions and pointers to the literature .in particular , the second author thanks him for drawing his attention to the fascinating area of cake - cutting during a visit to hebrew university of jerusalem , and he thanks jeff rosenschein for hosting this visit . we also thank magnus roos for helpful comments .s. brams , m. jones , and c. klamler .divide - and - conquer : a proportional , minimal - envy cake - cutting procedure . in s. brams , k. pruhs , and g. woeginger , editors ,_ dagstuhl seminar 07261 : `` fair division''_. dagstuhl seminar proceedings , november 2007 .available on - line at http://drops.dagstuhl.de/opus/volltexte/2007/1221/pdf/07261.bramssteven.paper.1221.pdf .y. chevaleyre , u. endriss , s. estivie , and n. maudet .reaching envy - free states in distributed negotiation settings . in _ proceedings of the 20th international joint conference on artificial intelligence _, pages 12391244 .ijcai , 2007 .r. lipton , e. markakis , e. mossel , and a. saberi . on approximately fair allocations of indivisible goods . in _ proceedings of the 5th acm conference on electronic commerce _ , pages 125131 .acm press , 2004 .m. magdon - ismail , c. busch , and m. krishnamoorthy .cake - cutting is not a piece of cake . in _ proceedings of the 20th annual symposium on theoretical aspects of computer science _ , pages 596607 .springer - verlag _ lecture notes in computer science # 2607 _ , 2003 .w. stromquist .envy - free cake divisions can not be found by finite protocols . in s.brams , k. pruhs , and g. woeginger , editors , _ dagstuhl seminar 07261 : `` fair division''_. dagstuhl seminar proceedings , november 2007 .available on - line at http://drops.dagstuhl.de/opus/volltexte/2007/1220/pdf/07261.stromquistwalter.paper.1220.pdf .

Abstract: cake - cutting protocols aim at dividing a `` cake '' ( i.e. , a divisible resource ) and assigning the resulting portions to several players in a way that each of the players feels to have received a `` fair '' amount of the cake . an important notion of fairness is envy - freeness : no player wishes to switch the portion of the cake received with another player s portion . despite intense efforts in the past , it is still an open question whether there is a _ finite bounded _ envy - free cake - cutting protocol for an arbitrary number of players , and even for four players .   we introduce the notion of degree of guaranteed envy - freeness ( dgef ) as a measure of how good a cake - cutting protocol can approximate the ideal of envy - freeness while keeping the protocol finite bounded ( trading being disregarded ) . we propose a new finite bounded proportional protocol for any number of players , and show that this protocol has a dgef of . this is the currently best dgef among known finite bounded cake - cutting protocols for an arbitrary number of players . we will make the case that improving the dgef even further is a tough challenge , and determine , for comparison , the dgef of selected known finite bounded cake - cutting protocols .