Guarantees in Fair Division: general or monotone preferences
aa r X i v : . [ ec on . T H ] S e p Guarantees in Fair Division: general ormonotone preferences
Anna Bogomolnaia ♠ and Herv´e Moulin ♠ September 2020 ♠ University of Glasgow, UK, and Higher School ofEconomics, St Petersburg, Russia
Abstract
To divide a ”manna” Ω of private items (commodities, workloads,land, time intervals) between n agents, the worst case measure of fair-ness is the welfare guaranteed to each agent, irrespective of others’preferences. If the manna is non atomic and utilities are continu-ous (not necessarily monotone or convex), we can guarantee the min-Max utility: that of our agent’s best share in her worst partition ofthe manna; and implement it by Kuhn’s generalisation of Divide andChoose. The larger Maxmin utility – of her worst share in her bestpartition – cannot be guaranteed, even for two agents.If for all agents more manna is better than less (or less is betterthan more), our Bid & Choose rules implement guarantees betweenminMax and Maxmin by letting agents bid for the smallest (or largest)size of a share they find acceptable. Acknowledgement 1
We are grateful for the critical comments ofHaris Aziz, Steve Brams, Eric Budish, Ariel Procaccia, Richard Stong,and participants in seminars and workshops at LUISS University,UNSW Sydney, Universities of Gothenburg, St Andrews, the LondonSchool of Economics, the 12th International Symposium on Algorith-mic Game Theory, Athens, and webinars at at ETH Zurich and Cal-tech. Special thanks to Erel Segal-Halevi and Ron Hozman for theirdetailed comments on the first version of this paper.Bogomolnaia and Moulin acknowledge the support from the BasicResearch Program of the National Research University Higher Schoolof Economics. Introduction and the punchlines
The fair division of a common property manna – resources privately con-sumed – is a complicated problem if its joint owners have heterogenouspreferences over the manna. A coarse yet important benchmark is the wel-fare
Guarantee a division rule offers to each participant: this is the highestwelfare that a given agent can secure in this rule, irrespective of the pref-erences of other agents, even if our agent is clueless about the latter andassumes the worst. The more an agent is risk averse and the less she knowsabout others’ preferences, the more this worst case benchmark matters toher.Our goal is to throw some light on the feasible Guarantees in the verygeneral class of non atomic fair division problems: small changes in the sizeof a share result in small utility changes (a continuity property explainedbelow). Our model places no other restrictions on the structure of prefer-ences and corresponding utilities, or their direction: the manna may containsome desirable parts (money, tasty cake, valuable commodities), some not(unpleasant tasks, financial liabilities, burnt parts of the cake that must stillbe eaten [33]); agents may disagree over which parts are good or bad; utili-ties can be single-peaked over some parts (teaching loads, volunteering time,shares of a risky project), single-dipped on others, etc..Assume that the manna Ω and the domain D of potential preferences arecommon knowledge, and define a Fair Guarantee as a mapping ( u i , n ) → Γ( u i ; n ) selecting for each preference in D , described for clarity as a utilityfunction u i , and each number n of joint owners, a utility level. The mappingis fair because it ignores agent i ’s identity, and it must be feasible: for anyprofile ( u i ) ni =1 of utilities in D n there exists a partition ( S i ) ni =1 of Ω such that u i ( S i ) ≥ Γ( u i ; n ) for all i .Given the division problem (Ω , D ) we ask what are the best (highest) FairGuarantees? and what mechanism implements them?Observe first that any Fair Guarantee Γ( u ; n ) is bounded above by theutility, denoted M axmin ( u ; n ), of the worst share for u in the best n -partitionof the manna. For all u ∈ D and n we haveΓ( u ; n ) ≤ M axmin ( u ; n ) = max Π=( S i ) ni =1 min ≤ i ≤ n u ( S i ) (1)where the maximum (that may not be achieved exactly) bears on all n - In the simple sense of implementation described in the last paragraph of this section. S i ) ni =1 of Ω. This follows by feasibility of Γ( u ; n ): at theunanimous profile where u i = u for all i there is a partition Π such that u ( S i ) ≥ Γ( u ; n ) for all i , hence Γ( u ; n ) ≤ min ≤ i ≤ n u ( S i ) ≤ M axmin ( u ; n ).Therefore if ( u, n ) → M axmin ( u ; n ) is itself a Fair Guarantee (it is fair,but the issue is feasibility), it is the best possible one and answers the first ofthe two general questions above. This happens in two well known and muchdiscussed families of fair division problems.In the cake-cutting model due to Steinhaus ([35]) the manna Ω is a mea-surable space endowed with a non atomic measure, and utilities are additivemeasures, absolutely continuous with respect to the base measure. Additiv-ity of u implies M axmin ( u ; n ) ≤ n u (Ω); this is in fact an equality becausethe cake can be partitioned in n shares of equal utility. Agent i ’s share S i is Proportionally Fair if u i ( S i ) ≥ n u i (Ω): this is feasible for all agents atany preference profile ( u i ) ni =1 , therefore Proportional Fairness offers the bestpossible Guarantee in this model, and is the weakest and least controversialtest of fairness throughout the cake-cutting literature ([14] and [32]).In the microeconomic model of fair division the manna is a bundle ω ∈ R K + of K divisible and non disposable items, and D is the set of convex andcontinuous preferences over [0 , ω ] (not necessarily monotonic). It is feasibleto give an equal share n ω to every agent, so that Γ es ( u ; n ) = u ( n ω ) is afeasible Guarantee. In D the inequality M axmin ( u ; n ) ≤ u ( n ω ) is also true. Therefore the optimal Guarantee is Γ es , aka the Equal Split lower bound u i ( z i ) ≥ u ( n ω ) (where z i is i ’s share of ω ). Here too it is the starting pointof the discussion of fairness (see e. g., [38] and [28]).As soon as we drop either additivity in the former model, or convexityin the latter one, the Maxmin benchmark is not a Fair Guarantee any more:already in some two person problems no division of the manna yields at least M axmin ( u i ; 2) for both i = 1 ,
2. In a simple example Ann and Bob share10 units of a single non disposable divisible item (e.g., time spent in a givenactivity). Ann’s preferences are single-peaked (hence convex), while Bob’sare single-dipped ( see Figure 1 ): u A ( x ) = x (12 − x ) ; u B ( x ) = x ( x −
6) for 0 ≤ x ≤ Pick a hyperplane H supporting the upper contour of u at n ω ; the lower contour of u at n ω contains one closed half-space cut by H , and every division of ω as ω = P n z i includes at least one z j in that half-space. M axmin ( u A ) = 35 at Π = { , } ; M axmin ( u B ) = 0 at Π = { , } If Bob’s share is worth at least
M axmin ( u B ) then Ann gets either the wholemanna or at most 4 units: so her utility is at most 32 and we see that( M axmin ( u A ) , M axmin ( u B )) is not feasible.A second critical benchmark utility is minM ax ( u ; n ), the utility of thebest share for u in the worst possible n -partition of Ω: minM ax ( u ; n ) = min Π=( S i ) ni =1 max ≤ i ≤ n u ( S i )where as before the minimum bears on all n -partitions of Ω.Our first main result, Theorem 1 in Section 4, says that in any non atomicproblem, the mapping u → minM ax ( u ; n ) is a Fair Guarantee; in particular minM ax ( u ; n ) ≤ M axmin ( u ; n ) for all u ∈ D and n (by (1)). Moreoverthe minM ax Guarantee is implemented by Kuhn’s little known n -persongeneralisation of Divide and Choose ([20]), denoted here D&C n .The result is clear in two person problems, where ordinary Divide andChoose clearly guarantees her M axmin to the Divider and his
M inmax to the Chooser. For instance in the example above Ann would Divide asΠ = { , } and Bob would get utility −
5, exactly his minM ax ( u B ; 2); whileBob would Divide as Π = { , } , and Ann would Choose 10, thus achieving minM ax ( u A ; 2) = 20.In three persons problems D&C works as follows. The Divider Ann offersa 3-partition Π = { S , S , S } where all shares are of equal value to her; Bob accepts all shares worth at least minM ax ( u B ; 3), and Charles all those worthleast minM ax ( u C ; 3). If Bob and Charles can each be assigned a share theyaccept, we do so and Ann gets the last piece. If both accept a single share inΠ, the same one, we give one of the remaining shares S k to Ann (it does notmatter which one) and then run D&C between Bob and Charles for Ω (cid:31) S k (it does not matter who Divides or Chooses).The n -person division rule D&C n proceeds similarly in at most n − Maybe more than one such assignment is feasible; any choice implements the targetGuarantee, which is all we need. n the current Divider can find an equipartition : apartition of the remaining manna where all shares are equally valuable to thisDivider. Because we only assume that he manna is measurable and endowedwith a non atomic measure, and that utilities are continuous in that measure,the proof of Lemma 1 requires advanced tools in algebraic geometry: this theobject of the companion paper [4], see the discussion in Subsection 3.2.Our second main result, Theorem 2 in Subsection 5.2, focuses on nonatomic problems where preferences are also co-monotone : that is, increasing if enlarging a share cannot make it worse and we speak of a good manna ; or decreasing if the opposite holds and we have a bad manna . Either restrictionon preferences opens the door to a new family of division rules significantlysimpler than D&C n and implementing a higher Guarantee than the minM ax .These rules are inspired by the well known family of Moving Knife (MK n )rules (Dubins and Spanier [19]) that we recall first.Assume the manna is good: a knife cuts continuously an increasing shareof the cake; agents can stop the knife at any time; the first agent who doesgets the share cut so far. Repeat between the remaining agents and manna.For a bad manna, agents can drop at any time and the last one to drop getsthe share cut so far.A Moving Knife (MK) rule chooses a single arbitrary path for the knife,which tightly restricts the range of individual shares and partitions, hencecan result in a very inefficient allocation. We introduce a large family ofrules in the same spirit as MK but with all partitions in their range, thatwe call the Bid & Choose (B&C n ) rules. Each rule is defined by fixing abenchmark additive measure of the shares, diversely interpreted as their size,their market price, etc.. If the manna is good a bid b i by agent i is thesmallest measure of a share that i finds acceptable: the smallest bidder i ∗ chooses freely a share of measure at most b i ∗ , then we repeat between theremaining agents and manna. For a bad manna the bid b i is the largest sizeof a share that i finds acceptable, and the largest bidder i ∗ picks any shareof size at least b i ∗ .Theorem 2 in Section 5 shows that all B&C n rules, as well as all MK n rules implement a Guarantee between the minM ax and M axmin level.A handful of examples in Subsection 5.3 show that the B&C n Guaranteeimproves substantially the minM ax
Guarantee in the microeconomic modelof fair division. There the Equal Split Guarantee is the
M axmin benchmark5the best possible) for agents with convex preferences, while for agents with“concave” preferences (convex lower contours) Equal Split is the minM ax
Guarantee, which the B&C n Guarantee improves significantly.Throughout the paper we speak of implementation in the very simplesense adopted by most of the cake cutting literature (e. g., [14]), and formal-ized in the general collective decision context as implementation in “protec-tive equilibrium” (Barbera and Dutta [7]). A rule implements (guarantees)a certain utility level γ means this: no matter what her preferences, eachagent has a strategy that depends also upon Ω , n and D , such that what-ever other agents do the utility of her share is no less than γ . Moreover the“guaranteeing strategy” is essentially unique. The two welfare levels
M axmin and minM ax are key to our results. Inthe atomic model where the manna is a set of indivisible items, they areintroduced by Budish ([16]) and Bouveret and Lemaitre ([12]) respectively .If utilities are additive in that model, the basic inequality of our non atomic model is reversed:
M axmin ( u ; n ) ≤ n u (Ω) ≤ minM ax ( u ; n )and minM ax ( u ; n ) is obviously not a feasible Guarantee. It took a couple ofyears and many brain cells to check that the M axmin lower bound may notbe feasible either for three or more agents ([31]), though this happens in rareinstances of the model ([22]). Our paper is the first systematic discussionof these two bounds in the non atomic model of cake division.Kuhn’s 1967 n person generalisation of Divide and Choose ([20]) promptlyimplements the minM ax guarantee in our model: Theorem 1. Except fora recent discussion in [1] for additive utilities, D&C n has not received muchattention, a situation which our paper may help to correct. In particular,unlike the Diminishing Share ([35]) Moving Knife ([19]), and Bid and Chooserules, it is very well suited to divide mixed manna, i. e., containing subjec-tively good and bad parts, as when we divide the assets and liabilities of a If the manna is atomic and utilities are not necessarily additive, it is easy to constructexamples showing that all six orderings of
M axmin , minM ax , and n u (Ω) are possible. n rule stands out for its informational parsimony:each Divider only reports a partition with the understanding that she isindifferent between the two shares she just cut, and Choosers only only accepta subset of these shares. If the manna is mixed, no one is asked to explainwhich parts they view as good or bad: for instance if we divide tasks, I maynot want others to know which tasks I am actually happy to perform, orwhich ones are very painful to me.The “cuts” selected by Dividers and “queries” answered by Choosers re-quire only a modest cognitive effort: no one needs to form complete prefer-ence relations over all shares of the cake. Taking this feature to heart, a largeliterature in the cake cutting model evaluates the informational complexityof various mechanisms by the number of “cuts” and “queries” they involve:see [14] or [32], and more recently [17] and [18]. This line of research goesbeyond the test of Proportional Guarantee, and find cuts and queries divi-sion rules more complex than D&C n reaching an Envy-free division of thecake. The algorithms in Brams and Taylor ([13]), and more recently Aziz andMcKenzie ([6]), do exactly this when utilities are additive and non atomic;but because they involve an astronomical number of cuts and queries theyare of no practical interest and squarely contradict informational parsimony.See ([15], [21]) for some fine tuning of these general facts.The “equipartition” Lemma (Subsection 3.2) is critical to the proof ofTheorem 1, and proved in [4] by algebraic geometry techniques. These, orsubtle variants of Sperner’s Lemma, demonstrate the existence of an Envy-free division under very general preferences, where which share I like bestin a given partition can depend upon the partition itself, not just uponmy own share: Stromquist’s ([36]) and Woodall’s ([39]) seminal insights areconsiderably strenghtened by the recent results in [37], [33], [25] and [3].However all these results assume that, either all agents (weakly) prefer anynon empty share to the empty share, or all weakly prefer the empty share toany non empty one: this rules out a mixed manna.We noted earlier that the concept of unanimity utility (the common effi-cient utility level in the economy where everyone has the same preferences)leads to the Equal Split Guarantee when we divide private goods and pref-erences are convex (see Footnote 2). When applied to fair division problems7nvolving production, it defines some compelling Fair Guarantees as well assome meaningful upper bounds on individual welfare: [27], [26]. The manna Ω is a bounded measurable set in an euclidian space, endowedwith the Lebesgue measure | · | , and such that | Ω | >
0. A share S is a possiblyempty measurable subset of Ω, and B is the set of all shares. A n -partitionof Ω is a n -tuple of shares Π = ( S i ) ni =1 such that ∪ ni =1 S i = Ω and | S i ∩ S j | = 0for all i = j ; and P n (Ω) is the set of all partitions of Ω. We define similarlyan n -partition of S for any share S ∈ B , and write their set as P n ( S ).If S ⊗ T = ( S ∪ T ) (cid:31) ( S ∩ T ) is the symmetric difference of shares,recall that δ ( S, T ) = | S ⊗ T | is a pseudo-metric on B (a metric except that δ ( S, T ) = 0 iff S and T differ by a set of measure zero).A utility function u is a mapping from B into R such that u ( ∅ ) = 0and u is continuous for the pseudo-metric δ and bounded. So u does notdistinguish between two shares at pseudo-distance zero (equal up to a set ofmeasure zero): for instance u ( S ) = 0 if | S | = 0. Also if the sequence | S t | converges to zero in t , so does u ( S t ). We write D (Ω) for this domain of utilityfunctions.So a non atomic division problem consists of (Ω , B , ( u i ) ni =1 ∈ D (Ω) n ).Several subdomains of D (Ω) play a role below: • additive utilities: u ∈ A dd (Ω) iff u ( S ) = R S f ( x ) dx for all S , where f is bounded and measurable in Ω; • monotone increasing: u ∈ M + (Ω) iff S ⊂ T = ⇒ u ( S ) ≤ u ( T ) for all S, T ; • monotone decreasing: u ∈ M − (Ω) iff S ⊂ T = ⇒ u ( S ) ≥ u ( T ) for all S, T ; • separable: u ∈ S (Ω) iff there is a finite set A , a partition ( C a ) a ∈ A ∈P | A | (Ω) of Ω, and a continuous function v from R A + into R , such that u ( S ) = v (( | S ∩ C a | ) a ∈ A ) for all S ∈ B .8he separable domain S (Ω) captures the standard microeconomic fairdivision model: A is a set of divisible commodities, the manna is the bundle ω ∈ R A + such that ω a = | C a | for all a , a share S i gives to agent i the amount z ia = | S i ∩ C a | of commodity a , and the partition Π = ( S i ) ni =1 corresponds tothe division of the manna as ω = P n z i .In the general non atomic division problem, the set of shares B is not com-pact for the pseudo-metric δ . It follows that when we maximize or minimizeutilities over shares, or look for a partition achieving a benchmark utility minM ax or M axmin , we cannot claim the existence of an exact solution tothe program: the minM ax is not a true minimum, only an infimum, and
M axmin is only a supremum, not a true maximum. As this will cause noconfusion, we stick to the min and
M ax notation throughout.However in the microeconomic model, the set of shares and of partitionsare both compact so for this important set of problems (where all our exam-ples live) the min and
M ax notation are strictly justified.One can also specialise the general model by imposing constraints on theset of feasible shares. The most important instance is the familiar intervalmodel , where the manna is Ω = [0 ,
1] and a share must be an interval, soan n -partition is made of n adjacent intervals. Other instances assume Ωis a polytope, and shares are polytopes of a certain type: e.g. triangles ortetrahedrons ([34]). And sometimes shares must be connected subsets of Ω([8], [2]) . The Divide and Choose n rules, as well as our Bid and Choose n rules, donot work in these models , so our Theorems 1 and 2 do not apply. But theinterval model is still useful here in a technical sense: the proof of the criticalLemma 1 in Subsection 3.2 starts by projecting the general problem onto aninterval model and proving existence of an equipartition there. Definition 1 An n - equipartition of the share T ∈ B for utility u ∈ D ( T ) is a partition Π e = ( S i ) ni =1 ∈ P n ( T ) such that u ( S i ) = u ( S j ) for all i, j ∈{ , · · · , n } ; we write u (Π e ) for this common value, and E P n ( T ; u ) for the setof these n -equipartitions. For instance in the interval model, the first divider can find an equipartition made ofadjacent intervals (by our Lermma 1), but the next agent called to divide is typically leftwith disconnected intervals.
9t is clear that
E P n ( S ; u ) is non empty if u is additive: if B [ S ] is the subsetof shares included in S , Lyapunov Theorem implies that the range u ( B [ S ])is convex, so it contains n u ( S ); then we replace n by n − u is monotone ( u ∈ M ± (Ω)), and the proof, outlinedin Remark 1 below, is fairly simple. That of our next statement is muchharder. Lemma 1 ([4])
Fix a share S ∈ B and a utility u ∈ D (Ω) . The set E P n ( S ; u ) of n -equipartitions of S at u is non empty. Proof.
The Theorem in [4] proves Lemma 1 for the interval model (which,as mentioned above, is not a special case of our model). Fix a real valuedfunction f on the set of intervals [ a, b ] ⊂ [0 , f ( a, a ) = 0 for all a ∈ [0 , n subintervals [0 = x , x ] , [ x , x ] , · · · , [ x n − , x n = 1] of [0 ,
1] forming anequipartition of f : f ( x i − , x i ) is constant for i = 1 , · · · , n .Start now from a share S in the statement of Lemma 1 and pick a movingknife through S , i. e., a path κ : [0 , ∋ t → K ( t ) ∈ B from K (0) = ∅ to K (1) = S , continuous for the pseudo-metric δ on B and weakly inclusionincreasing: 0 ≤ t < t ′ ≤ ⇒ K ( t ) ⊆ K ( t ′ )(in Subsection 5.1 moving knifes must be strictly inclusion increasing). Thenthe function f ( a, b ) = u ( K ( b ) (cid:31) K ( a ))is as in the previous paragraph, and an f -equipartition ([ x i − , x i ]) ni =1 of [0 , u -equipartition ( K ( x i ) (cid:31) K ( x i − )) ni =1 of S . (cid:4) Remark 1 It is easy to prove Lemma 1 if we assume that the sign of u is constant: all shares are weakly preferred to the empty share, or all areweakly worse. Assume the former and use as above a moving knife to project S onto [0 , , where a n - partition is identified with a point in the simplex ofdimension n − Then apply the Knaster–Kuratowski–Mazurkiewicz Lemmato the sets Q i of partitions of the interval where the i -th interval gives thelowest utility: each Q i is closed, contains the i -th face of the simplex, andtheir union covers it entirely. Thus these sets intersect.One can also invoke the stronger results in [36] and [37] showing theexistence of an Envy-free partition under this assumption. But recall that a ey feature in the division of a mixed manna is that the sign of u is not constant across shares. Definition 2
Fix n , the manna (Ω , B ) and u ∈ D (Ω) : minM ax ( u ; n ) = min Π ∈P n (Ω) max ≤ i ≤ n u ( S i ) ; M axmin ( u ; n ) = max Π ∈P n (Ω) min ≤ i ≤ n u ( S i )(2)Recall that minM ax is the utility agent u can achieve by having first pickin the worst possible n -partition of Ω, and M axmin by having last pick inthe best possible n -partition of Ω. Proposition 1 i ) If u ∈ A dd (Ω) then minM ax ( u ; n ) = M axmin ( u ; n ) = n u (Ω) ii ) If u ∈ M ± (Ω) minM ax ( u ; n ) = min Π e ∈EP n (Ω; u ) u (Π e ) ; M axmin ( u ; n ) = max Π e ∈EP n (Ω; u ) u (Π e ) (3) iii ) If u ∈ D (Ω) minM ax ( u ; n ) ≤ min Π e ∈EP n (Ω; u ) u (Π e ) ≤ max Π e ∈EP n (Ω; u ) u (Π e ) ≤ M axmin ( u ; n )(4) Proof
Statement iii ) If Π e is an n -equipartition, u (Π e ) is the utility of its best share,hence minM ax ( u ; n ) ≤ u (Π e ); proving the other inequality in (4) is just aseasy. Statement i ) By additivity of u , for any n -partition Π we have max i u ( P i ) ≥ n u (Ω) implying minM ax ( u ; n ) ≥ n u (Ω); we check symmetrically n u (Ω) ≥ M axmin ( u ; n ), and the conclusion follows by comparing these inequalitiesto those in (4). Statement ii ) Assume u ∈ M + (Ω); the proof for M − (Ω) is identical. Thecontinuity and monotonicity of u imply: if S, T are two disjoints sharessuch that u ( S ) > u ( T ), we can trim part of S and add it to T to get twodisjoint shares with equal utility in between u ( S ) and u ( T ). Expanding thisargument, if S , · · · , S k and T are disjoint shares such that u ( S ) = u ( S ) = · · · = u ( S k ) > u ( T )11e can simultaneously trim S , · · · , S k keeping them of equal utility and addthe trimming to T , so that the resulting k + 1 shares are all equally goodand their common utility is between the two utilities above. Iterating thisprocess, we see that if Π = ( S i ) ni =1 ∈ P n (Ω) is such that max ≤ i ≤ n u ( S i ) > min ≤ i ≤ n u ( S j ), we can construct an equipartition Π e ∈ E P n (Ω; u ) such thatmax ≤ i ≤ n u ( S i ) > u (Π e ) > min ≤ j ≤ n u ( S j )Now fix ε >
0, arbitrarily small, pick Π = ( S i ) ni =1 ∈ P n (Ω) such thatmin ≤ j ≤ n u ( S j ) ≥ M axmin ( u ; n ) − ε , and assume that Π is not an equiparti-tion. By the argument above we can find Π e ∈ E P n (Ω; u ) such that u (Π e ) > min ≤ j ≤ n u ( S j ), therefore Π e too is an ε -approximation of M axmin ( u ; n ), andthe right-hand inequality in (3) follows. The proof of the left-hand inequalityis similar. (cid:4) In the general domain D (Ω), the partitions achieving the M axmin and minM ax utilities are not necessarily equipartitions. In the microeconomicexample of Section 1, Ann has single-peaked preferences and her minM ax is achieved by the all-or-nothing partition { ∅ , Ω } ; Bob has single-dippedpreferences and the same partition delivers his M axmin ; but { ∅ , Ω } is notan equipartition for either utility. Remark 2: In the interval model with a monotone utility u , it is easy tocheck that any two n - equipartitions have the same utility and in turn this im-plies minM ax ( u ; n ) = M axmin ( u ; n ) : hence this is the best Fair Guarantee.The numerical example above can be viewed as an instance of the intervalmodel where the two agents are indifferent between [0 , x ] and [1 − x, forall x : so only the inequality (4) holds true in the general (non monotone)interval model. Definition 3
Fix the manna (Ω , B ) and a subdomain D ∗ , D ∗ ⊆ D (Ω). AFair Guarantee in D ∗ is a mapping Γ : u → Γ( u ; n ) such that for any profile ( u i ) ni =1 ∈ ( D ∗ ) n there exists Π = ( S i ) ni =1 ∈ P n (Ω) such that u i ( S i ) ≥ Γ( u i ; n ) for all i . In Section 1 we observed, by looking at unanimity profiles, that
M axmin ( · ; n )is an upper bound for any Fair Guarantee: inequality (1). We also men-tioned two subdomains where
M axmin ( · ; n ) itself is a (hence the optimal)12air Guarantee: the additive domain A dd (Ω) and the subdomain of the sep-arable one S (Ω) where preferences are also convex. Finally we used theAnn and Bob microeconomic example with a single commodity to show that M axmin ( · ; n ) is not a Fair Guarantee in D (Ω), even for n = 2 and a onedimensional manna.Before proving in the next Section that minM ax ( · ; n ) is a Fair Guaranteein the whole domain D (Ω) we construct a microeconomic problem with twodivisible items and two agents u and u where minM ax ( u i ; 2) = 0 < M axmin ( u i ; 2) for i = 1 , minM ax ( u ) , minM ax ( u )) is weakly Pareto optimalThis implies that for any Fair Guarantee Γ, at least one of Γ( u ; 2) = 0 andΓ( u ; 2) = 0 must hold. In words, for some problems, no Fair Guarantee canreduce the gap from minM ax to M axmin for both agents. The manna is ω = (1 ,
1) and we a share as z = ( x, y ). Both utilities aresymmetric in x, y : u i ( x, y ) = u i ( y, x ) so it is enough to define them for x ≤ y : u ( z ) = 0 if x ≤ ≤ yu ( z ) = 1 − y if x ≤ y ≤ u ( z ) = 2 x − ≤ x ≤ yu ( z ) = 0 if x ≤ y ≤ or ≤ x ≤ yu ( z ) = 2 y − ≤ y ≤ − xu ( z ) = 1 − x if ≤ − x ≤ y The range of both utilities is [0 , u ( z ) is null in the NWand SE quadrant of the box [0 , with center at ( , ); it is strictly positivein the SW and NE quadrants except on the lines x = and y = . Agent 2’sutility u ( z ) is symetrically null in the SW and NE quadrants, and strictlypositive in the NW and SE quadrants except on the same two lines. Thereforefor any division (1 ,
1) = z + z of the manna we have u ( z ) · u ( z ) = 0:there is no feasible division s. t. u i ( z i ) > i = 1 , { (0 , , (1 , } achieves M axmin ( u ) = 1 and minM ax ( u ) =0; the partition { (0 , , (1 , } achieves M axmin ( u ) = 1 and minM ax ( u ) =0. Divide and Choose implements the utility profile ( minM ax ( u i ; 2) , M axmin ( u j ; 2)):this gap can be closed for one agent. The Divide & Choose n rule Start by a combinatorial observation. Let G be a bilateral graph betweenthe sets M of agents and R of shares: interpret ( m, r ) ∈ G as agent m likes share r . We say that the subset f M of agents are properly matched to thesubset e R of shares if | f M | = | e R | , agents in f M are each matched (one-to-one)to a share they like in e R , and no one outside f M likes any share in e R . Lemma 2 . Assume | M | = | R | , each agent in M likes at least one objectin R and some agent i ∗ likes all objects in R . Then there is a (non empty)largest set M ∗ of properly matchable agents containing i ∗ : if f M is properlymatched to e R , then f M ⊆ M ∗ . Proof . We apply the Gallai-Edmonds decomposition of a bipartite graph:see e.g. [23] Chap 3 (or Lemma 1 in [9]). If M can be matched with R thisis a proper match and the statement holds true. If M and R cannot bematched, then we can uniquely partition M as ( M + , M ∗ ) and R as ( R + , R ∗ )such that:1. | M + | > | R + | , the agents in M + do not like any object in R ∗ , and theycompete for the over-demanded objects in R + : every subset of R + is likedby a strictly larger subset of agents in M + ;2. | M ∗ | < | R ∗ | and the agents in M ∗ can be matched with some subset of R ∗ .By the general Gallai-Edmonds result, M + and R ∗ are non empty. Here M ∗ is non empty as well because it contains the special agent i ∗ . Everymatch of M ∗ to a subset of R ∗ is proper. Finally suppose f M is properlymatched to e R and c M = f M ∩ M + is non empty. Then c M is matched to somesubset b R of R + but b R is liked by more agents in M + than there are in c M ,therefore the match is not proper: contradiction. So f M does not intersect M + as was to be proved. (cid:4) Definition 4: the D&C n rule . Fix the manna (Ω , B ) and the ordered set of agents N = { , · · · , n } , eachwith a utility in D (Ω) .Step 1. Agent proposes a partition Π ∈ P n (Ω) ; all other agents reportwhich shares in Π they like (at least one). In the resulting bipartite graphbetween N and the shares in Π , where agent likes all the shares, we useLemma 2 to match properly the largest possible set of agents N (it containsagent ) with some set of shares R ; if N = N we are done, otherwise we go to tep 2. Repeat with the remaining manna Ω and agents in N (cid:31) N . Ask thefirst agent in the exogenous ordering to propose a partition Π ∈ P n −| N | (Ω ) ,while others report which of these new shares they like. And so on. At least one agent, the Divider, is served in each step, thus the algorithmjust described takes at most n − R of shares to assign in each step, and multiple ways toassign these to the agents.Our first main result is that minM ax is a Fair Guarantee, implementedby the D&C n rule in the full domain D (Ω). Theorem 1
Fix the manna (Ω , B ) and n . i ) In the D&C n rule, an agent with utility u ∈ D (Ω) guarantees the minM ax ( u ; n ) utility level by 1) when called to divide, proposing an equipartition Π e ∈E P m ( S ; u ) of the remaining share S of manna among the m remainingagents, and 2) when reporting shares he likes, accepting all shares, and onlythose, not worse than minM ax ( u ; n ) (the minM ax level in the initial prob-lem ). ii ) Moreover the first Divider (and no one else) guarantees her
M axmin utility by proposing her
M axmin partition in Step 1. Other agents cannotguarantee more than their minM ax utility . Proof.
Statement i ). Consider agent u using the strategy in the state-ment. At a step where he must report which shares he likes among thoseoffered at that step, he can for sure find one worth at least minM ax ( u ; n ): allshares previously assigned are worth to him strictly less than minM ax ( u ; n ),and together with the freshly cut shares they form a partition in P n (Ω); inany partition at least one share is worth minM ax ( u ; n ) or more.At a step where our agent is called to cut, he proposes to the remainingagents an m -equipartition Π e ∈ E P m ( S ; u ) of the remaining manna S . Tocheck the inequality u (Π e ) ≥ minM ax ( u ; n ) note that Π e together with thepreviously assigned shares is a partition of Ω in which the old shares areworth strictly less than minM ax ( u ; n ). After Step 1 an agent can secure his
M axmin utility for the smaller manna S among m agents, but this may be below the M axmin utility in the initial problem. tatement ii ). This is clear for the first Divider. Fix now an agent i withutility u and check that if he is not the first Divider, for certain moves ofthe other agents, agent u gets exactly his minM ax utility. Pick a partitionΠ ∈ P n (Ω) achieving minM ax ( u ; n ) (as usual, the existence assumption iswithout loss). Suppose that the first Divider, who is not agent i , offers Π,and all agents other than i (including the Divider) find all shares acceptable:then a full match is feasible ( i must accept at least one share) so i ’s sharecannot be worth more than minM ax ( u ; n ). (cid:4) We now assume that the manna is unanimously good, u ∈ M + (Ω), or unan-imously bad, u ∈ M − (Ω). Because u ( ∅ ) = 0, for all S we have u ( S ) ≥ u ( S ) ≤ minM ax (resp. M axmin ) utility is the smallest (resp. largest)equipartition utility: property (3) in Proposition 1.We check first that the profile of
M axmin utility levels still may notbe feasible, even in the simple microeconomic model (corresponding to theseparable domain S (Ω) in Subsection 3.1). We have one unit each of twodivisible goods, ω = (1 , u ( z ) = min { x, y } so his worst case partition isΠ = { (1 , , (0 , } and his best one is the equal split partition Π ′ = { ω, ω } : minM ax ( u ; 2) = 0 < = M axmin ( u ; 2). Agent 2 has anti-Leontief pref-erences: u ( z ) = max { x, y } . For her the equal split partition Π ′ is the worstand the best one is Π: minM ax ( u ; 2) = < M axmin ( u ; 2). Clearlythe profile of M axmin utilities ( ,
1) is not feasible, while D&C implements( , ) and (0 , minM ax guarantee is always improved, at least weakly,by the large family of Bid and Choose (B&C n ) rules, inspired by the familiarMoving Knives (MK n ) rules ([19]). κn and B&C θn rules A moving knife through the manna (Ω , B , | · | ) is a path κ : [0 , ∋ t → K ( t ) ∈ B from K (0) = ∅ to K (1) = Ω, continuous for the pseudo-metric δ B and strictly inclusion increasing:0 ≤ t < t ′ ≤ ⇒ K ( t ) ⊂ K ( t ′ ) and | K ( t ′ ) (cid:31) K ( t ) | > κ arranges shares of increasing value to all participantsalong the specific path of the knife. An example is K ( t ) = B ( t ) ∩ Ω, where t → B ( t ) is a path of balls with a fixed center and radius growing from 0 to1, so that B (1) contains Ω. Moving knifes can take many other shapes, forinstance hyperplanes.Our Bid and Choose rules offer more choices than Moving Knives to theagents, with the help of a benchmark measure θ of the shares, chosen by therule designer: θ is a positive σ -additive measure on (Ω , B ), normalised to θ (Ω) = 1. It is absolutely continuous w.r.t. the Lebesgue measure | · | andvice versa: the density of θ w.r.t. | · | is strictly positive. In particular θ isstrictly inclusion increasing: ∀ S, T ∈ B : S ⊂ T and | T (cid:31) S | > ⇒ θ ( S ) < θ ( T )In applications θ can evaluate for instance the market value, physical size, orweight of a share.Fixing a moving knife κ and a measure θ , we define in parallel the MovingKnife (MK κn ) and the Bid and Choose (B&C θn ) rules. In both cases a clock t runs from t = 0 to t = 1. Definition 5 the MK κn and B&C θn rules with increasing utilitiesStep 1. The first agent i to stop the clock, at t , gets the share K ( t ) inMK κn , or in B&C θn chooses any share in Ω s.t. θ ( S ) = t , say S i , and leaves;Step k: Whoever stops the clock first at t k gets the share K ( t k ) (cid:31) K ( t k − ) inMK κn , or in B&C θn chooses any share in Ω (cid:31) ∪ k − S i ℓ s.t. θ ( S ) = t k − t k − ,say S i k , and leaves;In Step n − the single remaining agent who did not stop the clock takes theremaining share Ω (cid:31) K ( t n − ) or Ω (cid:31) ∪ n − S i ℓ . Definition 5 ∗ with decreasing utilitiesIn each step all agents must choose a time to “drop”, and the last agent i whodrops, at t , gets K ( t ) in MK κn , or in B&C θn chooses S i s.t. θ ( S i ) = t .The other steps are similarly adjusted. Breaking ties between agents stopping the clock (or dropping) at thesame time is the only indeterminacy in these rules, much less severe than inD&C n , where we serve at each step an unambiguous set of agents, but thereare typically several ways to match them properly.17p to tie-breaking, B&C θn and MK κn are anonymous (do not discriminatesbetween agents) but not neutral (do discriminate between shares), whileD&C n is neutral but not anonymous.In MK κn the share of an agent takes the form K ( t ) (cid:31) K ( t ′ ) so it covers aset of dimension 2 (and feasible partitions move in a set of dimension n − P m (Ω) is feasible under the B&C θn rule.To check this fix Π = ( S i ) ni =1 and assume first | S i | > i . Consider n agents deciding (cooperatively) to achieve Π. By the strict monotonicityof θ the sequence t i = θ ( ∪ ij =1 S j ) increases strictly therefore they can stopthe clock (or drop) at these successive times and choose the correspondingshares in Π. If there are shares of measure zero they can all be distributedat time 0.On the other hand in B&C θn all but one agent must pick a share underconstraints, thus revealing more information than in MK κn . Loosely speaking,B&C θn is informationally comparable to D&C n . Remark 3. We can also implement the Guarantees described in the nextSubsection by alternative static versions of MK κn and B&C θn where agentsbid all at once for potential stopping times; we do not discuss these rules forthe sake of brevity. θ and MK κ Guarantees
We fix an increasing utility u ∈ M + (Ω). The results are identical, andidentically phrased, for a bad manna u ∈ M − (Ω). See also Remark 4 at theend of this Subsection.Define the triangle T = { ( t , t ) | ≤ t ≤ t ≤ } in R and the set Υ( n )of increasing sequences τ = ( t k ) ≤ k ≤ n in [0 ,
1] s.t. t = 0 ≤ t ≤ · · · ≤ t n − ≤ t n For a moving knife κ , utilities of the shares in MK κn are described by thefunction u κ on T : u κ ( t , t ) = u ( K ( t ) (cid:31) K ( t )) for all ( t , t ) ∈ T For a measure θ , the corresponding definition in B&C θ is the indirect utility u θ : u θ ( t , t ) = min T : θ ( T )= t max S : S ∩ T = ∅ ; θ ( S )= t − t u ( S ) for all ( t , t ) ∈ T (5)18oth u κ and u θ decrease (weakly) in t and increase (weakly) in t .We show below that the Guarantees Γ k and Γ θ implemented by MK κn andB&C θn respectively are computed asΓ α ( u ; n ) = max τ ∈ Υ( n ) min ≤ k ≤ n − u α ( t k ; t k +1 ) where α is κ or θ (6)For instance in MK κ with two agents, write τ κ for the (not necessarilyunique) position of the knife making our agent indifferent between the share K ( τ κ ) and its complement. ThenΓ κ ( u ; 2) = max ≤ t ≤ min { u ( K ( t )) , u (Ω (cid:31) K ( t )) = u ( K ( τ κ )) = u (Ω (cid:31) K ( τ κ )In B&C θ the bid τ θ makes the best share of size τ θ as good as the worstshare of size 1 − τ θ :Γ θ ( u ; 2) = max ≤ t ≤ min { max θ ( S )= t u ( S ) , min θ ( S )= t u (Ω (cid:31) S ) } = max θ ( S )= τ θ u ( S ) = min θ ( S )= τ θ u (Ω (cid:31) S )(7) Lemma 4 i ) The utility u κ and the indirect utility u θ are continuous. Both the minimumand maximum in (5) are achieved. ii ) The maximum of problem (6) (for both rules) is achieved at some τ ∈ Υ( n ) where the sequence t k increases in k , all the u α ( t k ; t k +1 ) are equal, and thiscommon utility is the optimal value of (6). Proof in the Appendix.
Theorem 2
Fix the manna (Ω , B ) , the number of agents n , and a utility u ∈ M + (Ω). i ) With the MK κn rule, an agent guarantees the utility Γ κ ( u ; n ) by committingto stop the knife at t kκ if exactly k − other agents have been served before; ii ) With the B&C θn rule, she guarantees Γ θ ( u ; n ) by stopping the clock at t kθ if exactly k − other agents have been served before; and choosing then thebest available share of size t k − t k − . iii ) minM ax ( u ; n ) ≤ Γ α ( u ; n ) ≤ M axmin ( u ; n ) where α is κ or θ . Proof . Statement i ) and iii ) for MK κn . Recall the equipartition Π = ( K ( t kκ ) (cid:31) K ( t k − κ )) n has u (Π) = Γ κ ( u ; n ). Thus (3) in Proposition 1 implies the inequalities iii ).Next if the knife has been stopped k − t k − κ therefore if she does stop the knife at t kκ K ( t kκ ) (cid:31) K ( t k − κ ). If shenever gets to stop the knife, the last stop is at or before t n − κ and she gets atleast Ω (cid:31) K ( t n − κ ). Statement ii ). If she is the first to stop the clock (perhaps also winning thetie break) at step k , in step k − t k − ≤ t k − θ and theshare T already distributed at that time has θ ( T ) = t k − : therefore she canchoose a share with utility u θ ( t k − ; t kθ ) ≥ u θ ( t k − θ ; t kθ ) = Γ θ ( u ; n ). If she is thelast to be served, having never stopped the clock (or lost some tie breaks)the share assigned to all other agents has θ ( T ) = t n − ≤ t n − θ therefore hershare is worth u θ ( t n − ; 1) ≥ u θ ( t n − θ ; 1) = Γ θ ( u ; n ). Statement iii ) for B&C θn . Right hand inequality . It is enough to construct a partition Π = ( S k ) n inwhich the utility of every share S k , ≤ k ≤ n − u θ ( t k − θ , t kθ ),implying min k u ( S k ) ≥ Γ θ ( u ; n ). We proceed by induction on the steps ofB&C θn . First S maximizes u ( S ) s.t. θ ( S ) = t θ so u ( S ) = u θ (0; t θ ) = Γ θ ( u ; n )and θ ( S ) = t θ . Assume the sets S ℓ are constructed for 1 ≤ ℓ ≤ k , mutuallydisjoint, s.t. θ ( S ℓ ) = t ℓθ − t ℓ − θ and u ( S ℓ ) ≥ u θ ( t ℓθ , t ℓ − θ ): then the set T = ∪ k S ℓ is of size t kθ and we pick S k +1 maximizing u ( S ) s.t. S ∩ T = ∅ and θ ( S ) = t k +1 θ − t kθ . By definition (5) we have u ( S k ) ≥ u θ ( t kθ ; t k +1 θ ) and the inductionproceeds. Note that in fact min k u ( S k ) = Γ θ ( u ; n ). Left hand inequality . We need now construct a partition Π = ( R k ) n s. t. u ( R k ) ≤ u θ ( t k − θ ; t kθ ) for 1 ≤ k ≤ n . We do this by a decreasing induction in n .In (the first) step n of the induction we define the 2-partition Π n = ( T n − , R n )of Ω where T n − is any solution of the program min T : θ ( T )= t n − θ u (Ω (cid:31) T ), and R n = Ω (cid:31) T n − . Thus u ( R n ) = u θ ( t n − θ ; 1) and θ ( T n − ) = t n − θ .Assume that in step k we constructed the ( n − k + 2)-partition Π k =( T k − , R k , R k +1 , · · · , R n ) s.t. θ ( T k − ) = t k − θ and u ( R ℓ ) ≤ u θ ( t ℓ − θ ; t ℓθ ) for k ≤ ℓ ≤ n . Pick e T a solution ofmin T : θ ( T )= t k − θ max S : S ∩ T = ∅ ; θ ( S )= t k − θ − t k − θ u ( S ) = u θ ( t k − θ ; t k − θ )As θ ( e T ∩ T k − ) ≤ t k − θ and θ ( T k − ) = t k − θ we can choose T k − s.t. e T ∩ T k − ⊆ T k − ⊆ T k − and θ ( T k − ) = t k − θ . Then we set R k − = T k − (cid:31) T k − so that u ( R k − ) ≤ u θ ( t k − θ ; t k − θ ) follows from R k − ∩ e T = ∅ and the definition of e T . This completes the induction step. We note finally that each set R k thusconstructed is of θ -size t kθ − t k − θ , and that max k u ( S k ) = Γ θ ( u ; n ). (cid:4)
20t is easy to check that no agent can secure more utility than Γ κn in MK κn or Γ θn in B&C θn . Remark 4. The minM ax
Guarantee and
M axmin upper bound for u ∈M ε (Ω) and − u ∈ M − ε (Ω) , where ε = ± , are related: minM ax ( − u ; n ) = − M axmin ( u ; n ) . With two agents the Guarantees Γ κ ( u ; 2) and Γ θ ( u ; 2) aresimilarly antisymmetric: Γ α ( − u ; 2) = − Γ α ( u ; 2) where α is κ or θ (8) This is clear for Γ κ and we check it for Γ θ by means of the change of variable S → S ′ = Ω (cid:31) S : Γ θ ( − u ; 2) = − min ≤ t ≤ max { min θ ( S )= t u ( S ) , max θ ( S )= t u (Ω (cid:31) S ) } = − min ≤ t ≤ max { min θ ( S ′ )=1 − t u (Ω (cid:31) S ′ ) , max θ ( S ′ )=1 − t u ( S ′ ) } = − min ≤ t ′ ≤ max { max θ ( S ′ )= t ′ u ( S ′ ) , min θ ( S ′ )= t ′ u (Ω (cid:31) S ′ ) } and the claim follows because if two continuous functions t → f ( t ) and t → g ( t ) intersect in [0 , and one increases while the other decreases, then min ≤ t ≤ max { f ( t ) , g ( t ) } = max ≤ t ≤ min { f ( t ) , g ( t } .The identity (8) generalises to n ≥ for the MK κ Guarantee, but not forthe B&C θ one. We must divide a good manna ω ∈ R K + in n shares z i ∈ R K + . Utilities u ∈ M + ( ω ) are continuous and weakly increasing on [0 , ω ].A Moving Knife is a continuous increasing path t → K ( t ) from 0 to ω :a natural choice is K ( t ) = tω, ≤ t ≤
1: the corresponding GuaranteeΓ κ ( u ; n ) = u ( n ω ) is the Equal Split utility Γ es ( u ; n ) = u ( n ω ). A positive,additive measure θ defining B&C θ is a “price” θ ( z ) = p · z , p ∈ R K + (cid:31) { } , sowe write the corresponding Guarantee as Γ p . Recall from Section 1 that if an agent’s preferences are convex her EqualSplit utility equals her
M axmin utility, the upper bound on all Fair Guar-antees ((1)), in particular it is weakly larger than the B&C p guarantee forany p . The converse inequality holds for “concave preferences”. Lemma 5 ) If the upper contours of the utility u ∈ M + ( ω ) are convex, then Γ p ( u ; n ) ≤ u ( n ω ) = M axmin ( u ; n ). ii ) If the lower contours of the utility u ∈ M + ( ω ) are convex, then minM ax ( u ; n ) = u ( n ω ) ≤ Γ p ( u ; n ).The equality in statement i ) was proven in Section 1. A symmetricalargument gives statement ii ).We turn to a handful of numerical examples where K = 2, ω = (1 , p · z = ( x + y ). Shares are z = ( x, y ), utilities are 1-homogenous andnormalised so that u ( ω ) = 10. We compute our three Guarantees: Bid andChoose Γ p , Equal Split, and minM ax , and compare them to the M axmin upper bound.The first three utilities (Leontief, Cobb Douglas and CES) define con-vex preferences, the last two define “concave preferences” (represented byquadratic and “anti-Leontief” utilities).Our first table assumes two agents, n = 2, and illustrates Lemma 5. Anagent with convex (resp. concave) preferences gets a better Guarantee underEqual Split (resp. Bid and Choose): u ( x, y ) minM ax ( u ; 2) Γ p ( u ; 2) u ( ω ) M axmin ( u ; 2)10 min { x, y } . √ x · y . ( √ x + √ y ) . . x + y ) 5 5 5 55 p x + y ) 5 5 . .
110 max { x, y } . M axmin utility for the first fourpreferences, and the minM ax utilities for the last three. The equipartitionΠ = { (1 , , (0 , } gives similarly the minM ax utilities of the first four, andthe M axmin ones for the last three.To compute Γ p ( u ; 2) we know from (7) that the optimal bid t (denoted t for simplicity) solvesmax ( x + y ) ≤ t u ( x, y ) = min ( x + y ) ≤ t u (1 − x, − y ) = min ( x + y ) ≥ − t u ( x, y )This equality implies 0 ≤ t ≤ . If u represents convex preferences symmetricin the two goods, u ( x, y ) is maximal under ( x + y ) ≤ t at x = y = t , and22inimal under x + y ≥ − t ) at x = 1 , y = 1 − t . So we must solve u ( t, t ) = u (1 , − t ): see Figure 2 .If u represents concave symmetric preferences its maximum under ( x + y ) ≤ t is at x = 0 , y = 2 t , and its minimum under x + y ≥ − t ) at x = y = 1 − t , so we solve u (0 , t ) = u (1 − t, − t ): see Figure 3 .We compute finally the same Guarantees with three agents: u ( x, y ) minM ax ( u ; 3) Γ p ( u ; 3) u ( ω ) M axmin ( u ; 3)10 min { x, y } . . √ x · y . . . ( √ x + √ y ) . . . x + y ) 3 . . . . p x + y ) 3 . . . .
110 max { x, y } . . minM ax equipartition for u = ( √ x + √ y ) and the M axmin equipartition for u ′ = 5 p x + y ) have the same form Π = { ( x, , (0 , x ) , (1 − x, − x ) } : in the former case we find x = and minM ax ( u ; 3) = 2, in thelatter we get x = 2 − √ M axmin ( u ′ ; 3) = 10( √ − ′ = { (1 , , (0 , ) , (0 , ) } fill the remaining values of minM ax and M axmin .To compute Γ p ( u ; 3) we know by Lemma 4 that the three terms in ( )are equal. They are u p (0 , t ) = max ( x + y ) ≤ t u ( x, y ) u p ( t , t ) = min ( x + y ) ≤ t max ( x + y ) ≤ t − t and ( x + x,y + y ) ≤ (1 , u ( x, y ) u p ( t ,
1) = min ( x + y ) ≤ t u (1 − x , − y )Clearly t ≤ (as t − t < < t and 1 − t < < t are bothimpossible). Therefore u p (0 , t ) = u p ( t , t ) is achieved by t = 2 t (theconstraint ( x + x, y + y ) ≤ (1 ,
1) does not bind). Writing t = t = t − t it remains to solvemax ( x + y ) ≤ t u ( x, y ) = min ( x + y ) ≤ t u (1 − x , − y ) = min ( x + y ) ≥ − t u ( x, y )When u represents convex preferences symmetric in the two goods, theminimum on the right-hand side is achieved by ( x, y ) = (1 − t,
1) so we solve u ( t, t ) = u (1 − t, See Figure 4. If u represents concave symmetric preferences, the minimum on the right-hand side is achieved by ( x, y ) = (1 − t, − t ) so we solve u (2 t,
0) = u (1 − t, − t ). See Figure 5 . 23
Concluding comments
Comparing B&C n versus D&C n rules The exogenous ordering of theagents greatly affects the outcome of D&C n , whereas B&C n treats the agentssymmetrically. On the other hand the choice of the benchmark measure inB&C n is exogenous, which allows much, perhaps too much flexibility to thedesigner.In D&C n the dividing agent may have many different strategies guaran-teeing her minM ax utility. By contrast in B&C n the solution to programs(7) and (6) is often unique. Multiple choices and the resulting indeterminacyof the outcome may be appealing for the sake of privacy preservation, less sofrom the implementation viewpoint. Two challenging open questions , B ) as in Theo-rem 1, and each of the n agents with his own utility in D (Ω). As mentionedin Section 2 and Subsection 3.2, Stromquist ([36]) showed that an Envy-freepartition of Ω exists if all utilities are non negative for all shares . Withoutthe sign assumption on utilities, Avvakumov and Karasev ([3]) prove exis-tence of an Envy-free partition if n is a power of a prime number. Whetherthis remains true for all n is still an open question.2) If the utilities vary in a domain U (Ω) where the M axmin utility is notfeasible, we would like to describe the family of undominated
Fair Guarantees u → Γ( u ; n ). For instance in the microeconomic domain M + ( ω ) of Subsec-tion 5.3, the Equal Split Guarantee is clearly undominated. We conjecturethat in the domains M ± (Ω) the B&C Guarantees Γ θ (Subsection 5.2) areundominated as well. First statement.
Recall that we can replace in definition ( ) the equalitieslike θ ( T ) = t with inequalities θ ( T ) ≤ t . We check first that the correspon-dence t → { S ∈ B| θ ( S ) ≤ t } is continuous. Upper hemi continuity follows bythe continuity of θ . For lower hemi continuity pick a sequence t n convergingto t and S ∈ B s.t. θ ( S ) ≤ t . If t n has a decreasing subsequence, we set S n = S so that θ ( S n ) ≤ t n and S n converges to S . If t n has an increasingsubsequence we construct an inclusion increasing sequence S m converging to S and s.t. | S m | < | S | for all m : because θ increases strictly, so does the24equence θ ( S m ) converging to θ ( S ), therefore we can pick subsequences S p of S m and t p of t n s.t. θ ( S p ) ≤ t p , as desired.Next we apply the Maximum Theorem twice. The first one to showthat the function ( T, t , t ) → C ( T, t , t ) = max { u ( S ) | S ⊂ Ω (cid:31) T ; θ ( T ∪ S ) ≤ t + t } is continuous because the correspondence ( T, t , t ) → { S | S ⊂ Ω (cid:31) T ; θ ( T ∪ S ) ≤ t + t } is continuous. The second one to deduce that thefunction min T : θ ( T ) ≤ t C ( T, t , t ) is continuous.2). Second statement.
For simplicity we assume n = 3, the general proofis entirely similar. Fixing u and t there is some t such that u θ ( t ; t ) = u θ ( t ; 1). This is because of the monotonicity properties of u θ and of theinequalities u θ ( t ; t ) = 0 ≤ u θ ( t ; 1) and u θ ( t ; 1) ≥ u θ (1; 1). Thiscommon value is unique (though t may not be) and defines a function g ( t ) = u θ ( t ; t ) = u θ ( t ; 1). It is easy to check from the continuity and monotonicityproperties of u θ that g is weakly decreasing and continuous. Then we find inthe same way t s.t. g ( t ) = u θ (0; t ).Check finally that if τ ∗ ∈ Υ( n ) is such that all terms u θ ( t k ∗ ; t k +1 ∗ ), 0 ≤ k ≤ n −
1, equal a common value λ , then τ ∗ solves program (6). If it doesnot there is a τ such that u θ ( t k ; t l +1 ) > λ for 0 ≤ k ≤ n −
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