Optimal Division of a Dollar under Ordinal Reports (Incomplete Version, do not distribute)

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1 Optimal Divisio of a Dollar uder Ordial Reports (Icomplete Versio, do ot distribute) Rube Juarez Chuog Mai Departmet of Ecoomics, Uiversity of Hawaii 44 Maile Way, Sauders Hall 54, Hoolulu, HI 968 ( rubej@hawaii.edu) Jue 3, 013 Abstract We study the problem of dividig a dollar whe agets report rakigs of the cotributios of other people. We fid optimal rules usig the maximum absolute loss from the true profile for ay umber of agets. If budget-balace is required, optimal rules exist oly for 3 ad 4 agets. Budget balace rules that are early optimal are provided for 5 or more agets. Keywords: Fair divisio, Judgmet aggregatio, Ordial reports, Maximum loss JEL classificatio: D70, D63, C78 1

2 1 Itroductio A dollar was eared by a group of agets ad eeds to be split amog them based o the work performed by each member toward geeratig this dollar. The problem is how to evaluate each aget s work. We study sharig rules where the full dollar is split based o the reports of the agets about the work performed by their peers. We focus o the case where agets report ordial rakigs. This problem has multiple applicatios. The caoical example is the divisio of the profit geerated by parterships of differet firms or agets. For istace, the divisio of the ed of year bous by a group of employees; the divisio of the collective profit geerated by a group of parters i a law firm; or the divisio of a uexpected amout of moey to professors i a departmet. We study the problem where each aget works ad observes the work of other agets. They evaluate ad report the ordial cotributio of the work performed by other agets but themselves. The available iformatio is the agets opiios that show their subjective evaluatio of his parters. It meas that each member has to report his rakig about the other colleagues. Based o the agets report, the divisio rule is costructed i the form of a fuctio such that the iput is the rakig of the agets ad the output is the amouts they will receive. This paper is the first to aalyze the divisio of the dollar uder ordial evaluatios. Cardial evaluatios have bee discussed i previous papers, see De Clippel et al. (008) or Juarez (008). Ordial evaluatios are more appealig tha cardial evaluatios i may scearios, sice agets might lack perfect exactess o the work performed by their peers, but could determie who worked more tha who (their ordial rakig). For istace, i views of agets i ad j, aget k might work more tha aget l but their relative rakig might differ (for istace i might thik k worked twice as hard tha l; but j might thik k worked oly 50% more tha l). Therefore, cosesuality, a property that respects the relative rakigs of the agets, is easier to be satisfied uder ordial reportig tha uder cardial reportig. The mai dowside of ordial evaluatios is that typically the fial paymet to the agets do ot coicide with the real ratio of true cotributios, therefore there might be a loss. For istace, assume that there are three agets whose relative work is (50%, 40%, 10%). The, uder commo kowledge, ideally aget 1 would report that worked more tha 3; aget would report that 1 worked more tha 3; ad aget 3 would report tha aget 1 worked more tha. However, if the oly available iformatio is the relative rakigs of the agets, the whe the relative work of the agets is (35%, 33%, 3%), the agets would report the same ordial rakig, ad thus obtai the same shares of the dollar. The loss of a rule is the absolute differece betwee the true profile ad the fial paymet allocated to the agets. For istace, if the true vector of cotributios is (50%, 40%, 10%) ad the rule allocates (r 1, r, r 3 ), the the loss at that profile is:.5 r r +.1 r 3. We look at the maximal

3 loss that the rule geerates for all potetial vectors of true cotributios. The we choose the rule that miimize the maximum loss. 1.1 Overview of the results I this paper, two types of rules are studied: uaimity rule, ad scorig method. The former, uaimity rule, is a rule i which if all members of the group agree o the mutual rakig the the distributio is give by a fixed sharig rule, otherwise they would get zero. The latter, a scorig method, is a rule where every aget gets a score based o his positio at the differet rakigs. The fial payoff of a aget is the ratio betwee his idividual score ad sum of the scores of all the agets. Amog the uaimity rules, for ay umber of agets the maximum loss of a divisio rule is ay value betwee ( 1)/ ad (Propositio 1). The the optimal profiles are the oes that have the maximum loss equal to ( 1)/. Moreover, the ecessary coditio for optimal profile is derived i Propositios. O the other had, for ay umber of the agets, there exists a optimal uaimity rule. Theorem 1 characterizes such rules. Oe of the dowplays of uaimity rules is that they do ot satisfy budget balace, e.g. whe the agets do ot agree o the rakig, it will ot allocate the full amout of moey. O the other had, scorig methods satisfy budget balace. The mai result of the paper, Theorem, shows that for 3 ad 4 agets there exist uique scorig rules that are optimal (that is, achieve the boud ( 1)/, where is the umber of agets). Theorem also shows that o optimal scorig rule exists for 5 or more agets; however, it provide rules that achieve a loss of /( + 1), which is a very good approximatio to ( 1)/. Fially, Theorem 3 shows that studyig uaimity ad scorig rules is by o meas a restrictio. I particular, it shows that if a rule always has a Nash equilibrium, the its loss will ot be smaller tha a uaimity rule. Moreover, if such rule is budget balace the the loss ca be bouded by a scorig rule. 1. Literature overview Accordig to Koblauch (008), may authors survey the literature pertaiig to the resolutio of coflictig claims over a resource. Thomso (003) ad Brams et al. (006) showed a literature o cake cuttig ad pie cuttig that focuses o fair divisio of a divisible good, parts of which are valued differetly by differet parties. Holzma (010) preseted the rule of votig to fid the wier to award of a prize. De Clippel et al. (008) raised a similar questio of dividig a dollar usig a rule that depeds oly o agets cardial evaluatios of their associates. This paper is the first oe usig the cocept of rakig cotributio to costruct a model that ca divide a dollar for a team of agets by their peer review. It also metios the potetial properties of divisio rule such as exactess, impartiality ad coset. The model requires at least three agets. With exactly three, there 3

4 is a uique impartial ad cosesual divisio rule. Their rule is aoymous ad feasible. However, that rule distributes exactly the dollar oly whe the three reports are cosistet; otherwise it distributes strictly less. Four or more agets, de Clippel et al (008) propose may aoymous, impartial ad cosesual rules that always distribute the dollar exactly. 1.3 Orgaizatio of the paper The structure of this paper is as follows. Sectio itroduces the model. Sectio 3 discusses uaimity rules ad gives the set of optimal uaimity rules. Sectio 4 studies budget balace rules ad presets the mai result of the paper i optimal scorig rules. Sectio 5 states the mai theorem o o-truthful rules. Sectio 6 cocludes. All proofs are left to the appedix. The Model Let N {1,,..., } be the set of agets where 3. For all i N, defie R i, the set of strict rakigs 1 over N\{i}. Let R i R i. Defie f, the sharig rule, to be f : R 1 R 1 {0} where 1 is the ( 1)-dimesioal simplex. Let v 1 be a truthful rakig of cotributios. Assume that there is commo kowledge about the cotributio of people toward the project. Let v be a vector of true cotributios. A rule will elicit the rakigs of the agets over other agets but himself. Give those rakigs, the rule decides o a distributio of moey. We focus i two types of rules: the uaimity rules that we discuss i sectio 3, ad the scorig methods, which is preseted i sectio 4. 3 Uaimity Rules Cosider a vector u 1 such that u 1 u u. The uaimity rule ξ u is such that if all members of N agree o the mutual rakig of N, say i 1 i i the ξi u k u k for all k. If there is o commo agreemet the ξi u k 0 for all k. For istace, let N {1,, 3} ad cosider the uaimity rule for the vector u such that u 1 u u 3 0. For all i, let R i R i. Cosider the rakig R 1 3, 1R 3 ad 1R 3. I this case, agets agree i the rakig 1 3. Therefore, aget 1 gets a payoff u 1, aget gets a payoff u ad aget 3 gets a payoff u 3. O the other had, cosider the rakig R 1 3, 1R 3 ad 1R 3. The, the agets do ot agree i the rakig, therefore all agets get zero payoff. 1 A strict rakig over a set is a complete, trasitive ad atisymmetric biary relatio over that set. 4

5 3.1 The total loss of a rule Cosider the uaimity rule ξ u. The loss of the uaimity rule ξ u at the vector of cotributios v 1, deoted L u (v), where 1, is: L u (v) u k k. k1 I a ideal world, the vector of cotributios ad the assiged amout would be the same. The loss of a rule measures the absolute differece betwee these two vectors, that is it aggregates over all the agets the errors i their assigemets. I practice, the vector of cotributios is ukow, therefore we evaluate the loss of a rule as the maximum loss over all potetial vector of cotributios. This type of worse-case measures have bee used i the literature before, see for istace Mouli ad Sheker[001], Juarez[008]. The maximum loss of a uaimity rule is L(ξ u ) max L u (v) max u 1 v u v v 1... v v 1 v A uaimity rule ξ u is optimal if it has the smallest maximal loss over all uaimity rules. That is, L(ξ u ) L(ξ u ) for ay other uaimity rule ξ u. The profile u is a optimal profile if the rule ξ u is optimal. 3. Results for uaimity rules 3..1 Bouds o the rule Propositio 1 For ay uaimity rule ξ u : 1 L(ξ u ) For a group of three agets, the miimum of the maximum total loss is /3. For a group of 4 agets, it is 3/4. As we ca see ext, this loss will be bidig. The goal of the paper is to fid rules that achieve the smallest loss 1. We say that a uaimity rule ξ u is optimal if L(ξ u ) Necessary ad sufficiet coditios for optimality We ext state the first mai theorem of the paper. It states the ecessary ad sufficiet coditios for optimality of a uaimity mechaism based o the utility profile. Theorem 1 A profile (u 1,..., u ) is optimal if ad oly if u 1 +1, u i 1 for i 1, 5

6 (u +... u k ) 1 + k k for k, 3,..., 1. Note that at the optimal profile the aget who is highest raked gets +1, which teds to 1 as icreases. O the other had, all the other agets gets o more tha 1. There are multiple profiles that geerate a optimal uaimity mechaism. Corollary The followig profiles are optimal for the umber of agets. i. If is odd: ii. If is eve: u i u i if i 1 if i otherwise if i 1 if i if i otherwise Fially, we describe below the class of optimal profiles for 3 ad 4 agets. Corollary 3 For 3, Φ 3 {( 3, u, u 3 ) u + u ad u u 3 } For 4, cosider u 1 { 5 8, 1 4, 1 8, 0}, u { 5 8, 5 10, 1 1, 1 1 }, u3 { 5 8, 1 4, 1 16, 1 16 }, u 4 { 5 8, 3 16, 3 16, 0} ad u5 { 5 8, 7 48, 7 48, 1 1 }. The, Φ 4 is the covex hull of u 1, u, u 3, u 4 ad u 5. That is, Φ {λ 1 u 1 + λ u + λ 3 u 3 + λ 4 u 4 + λ 5 u 5 (λ 1, λ, λ 3, λ 4, λ 5 ) 1 }. 4 Budget balace rules A huge dowplay of uaimity rule is that they do ot distribute the full amout of moey if agets do ot coicide o the rakig of their peers. I fact, whe there is o agreemet, uaimity rules distribute o moey at all. I particular, if oe aget overlooks the cotributio of aother aget, the everyoe gets puished. I this sectio we study rules that always distribute the full amout of moey eve whe the agets do ot coicide o the geeral rakig. Defiitio 3. A rule f is budget balace if f : R 1... R N 1. 6

7 4.1 The scorig rules Let N {1,..., } be a set of agets with 3. Each aget reports his rakig of the other parters. We deote by c k i the umber of agets who would put aget i as their k th highest rakig. We defie a (a 1,..., a k,..., a ) as the scorig vector where a k is the score assiged to a aget who has the positio k. It is further assumed that a 1... a. The total score of aget i is defied as s i ( ) c k i a k. The fial shares of k1 the agets are proportioal to their scores, e.g. f i si s i Remark 4 Scorig rules are budget balace. To see this, s i s i 1 f i s i s i Remark 5 Scorig rules are impartial, that is, the report of aget i does ot (c k i a k) (c k k1 affect his payoff. To see this, f i (c k i a k) i a k) k1. a k k1 Remark 6 Scorig rule are ot ecessarily optimal for some scorig vectors. Let a be a Borda scorig vector, e.g. a ( 1,,...1, 0). Let v (v 1,..., v ), ad let all the agets agree o the rakig give by v. The the share of aget 1 is give by k1 f 1 s 1 s i ( ) c k 1 a k k1 k1 ( 1)( 1) ( ) c k ( ) i a 1 k 4. Mai Theorem: Optimal scorig rule ( 1) + 1 u 1. Now we will fid scorig vectors that would give us optimal rules. Theorem 7 a) For 3, there exist a uique scorig rule with the optimal shares u ( 3, 1 3, 0) that achieves the optimal loss of 3 with the scorig profile a 1 1, a 0. b) For 4, there exist a uique scorig rule with the optimal shares u ( 5 8, 1 4, 1 16, 1 16 ) that achieves the optimal loss of 3 4 with the scorig profile a 1 5 6, a 1 1, a c) For > 4, there is o scorig rule that achieves the optimal loss of 1. However, there exist scorig rules that achieve the loss of We have Borda scorig method whe a ( 1,,..., 0)

8 5 More geeral rules I this sectio, we see that focusig o Uaimity ad Scorig rule is by o meas a restrictio, sice they will always geerate a loss ot larger tha ay other rule that always has a Nash equilibrium. Let v 1 be a truthful rakig of cotributios. Let r i (v) R i be the rakig over N\i geerated by v. Defiitio 1 A rule truthfully implemets the rakig of cotributio v reportig (r 1 (v ),..., r (v )) is a Nash equilibrium. if We simply refer to these rules as truthful. Theorem 8 For ay rule G that always has a Nash equilibrium i Pure strategies: a) L(G) L(U) for some uaimity rule U. b) If G is budget-balaced, the L(G) L(Sc) for some scorig rule Sc. 6 Coclusio We provide a ew framework for aggregatig miimal iformatio (ordial rather tha cardial) of the cotributios of agets towards a project, ad dividig the rewards i a optimal way. Whe we allow rules that are ot budgetbalaced, uaimity rules are optimal. For rule that are budget-balace, we show that scorig rules are optimal. We compute the uique optimal scorig for 3 ad 4 agets. No scorig rule would be optimal for more tha 4 agets. Multiple questios remai ope. For istace, are there rules that are less demadig o iformatio tha uaimity rules while still truthful? the partial aswer to this questio is Yes, sice ay rule such that ay pair of aget is reported by at least two agets would satisfy the properties above. However, less demadig scearios are easily coceivable. Fially, it is left ope the study of other settig where truthful implemetatio is a meaigful property. 8

9 7 Appedix Proof of Propositio 1. 1 max L. v 1... v a) We first prove the left had side of iequality: Cosider two truthful cotributio vectors {1, 0,..., 0} ad { 1,..., 1 }. We wat to show that max{ u u u 0, u 1 1,..., u 1 } 1 (1) Suppose (1) is ot true. So we have ad u u u 0, u 1 1 < 1 u u 1 < 1 () (3) for some k {1,..., }. We have 1 <u k 1 ad 0<u 1 1 u 1 + u j u 1 < 1 j u j k 1 j + k u j k u j j j u j (1 k u j ) + k k u j k < 1 From (1) >u 1 > + 1 From assumptio 1 <u k 1 k u 1 + k u j k i i u j > + k 1 of {L} must be greater tha 1 that has {L} 1 u 1 > + 1 > 1 the () u u j k 1 k 1 + k u j < + k 1 i u j k 1 The a cotradictio. At least oe of compoets or max {L} 1. The, If X* is a profile the X* is a optimal profile. ) We prove the right had side iequality:. L u 1 v 1 + u v u v L u i < u i + u i + The max v 1 v... v {L} < 9

10 Proof of Propositio. a) We show that if X* is a optimal profile tha u Assume that X* is a optimal the X* has to satisfies propositio 1. It meas that: max... v { u 1 v 1 + u v u v } 1 for every v 1 v v {v 1, v,..., v ). I the other word: { u 1 v 1 + u v u v } 1 for every v {v 1, v,...,v t ) Because it true for every v {v t 1, v,..., v )the it true for v{1, 0,..., 0). We have: u u u u 1 + (u + u u 1 ) 1 u u t 1 1 (1 u 1 ) 1 u because u t 1 + u + u u 1 Suppose u 1 > + 1, cosider profiles u {u 1> + 1, u i 1 u} ad v ( 1, 1,... 1 ). We have: u 1 + u + u u 1 u + u u ( 1) (u) 1 u 1 < u 1 > + 1 u< 1 ( 1) 1 < 1 >L u u u 1 u ( 1)(u 1 ) because L >( ) + ( 1)( 1 1 ) The X{u 1,> + 1, u i 1 u} is ot a optimal profile. At the optimal u b) We show that at the optimal u i 1 1 for every i... Cosider u {u 1 + 1, u, u 3... u k u... u }for k<. L { u 1 v 1 + u v u v } L k k 1 k L 3 4k + 1 i + k i j

11 4k + L + k 4k + vi> + k i This profile does ot satisfy propositio 1 the it is ot a optimal oe. At the optimal, u i 1 1 c) However, ot all X* {u 1 + 1, u i 1 1 } are optimal. For example, 4 cosider profile { 5 8, 1 8, 1 8, 1 } that satisfies the optimal coditio but is 8 ot optimal profile because the loss at v { 1 3, 1 3, 1 3, 0} L { } 5 6 > The the coditio for optimal profile X* {u 1 + 1, u i 1 ecessary but ot sufficiet. 1 } is Proof of Theorem 1. a) Case 1: v 1 > + 1 >v >v 3 >... >v k > 1 >v >v k+ >...>v L a v i + 1 k 1 + 4k + ( Suppose that L a > 1 ( k ( k i +3 i +3 However k <1 Cotradictio k k + 1 k + v 1 + )> 1 )> +1 i +3 ) 1 4k 4k +1 i i 4k + ( k 4 + 4k >1 because k> k +1 i i +3 + ( i k +3 i +3 )> i +3 ) Therefore L a + 1 v 1 + v v v

12 v Case : >v 1>v >v 3 >... >v k > >v >v k+ >... >v L a + 1 v 1 + i + 1 k 1 + k k v k k 4k + 4 v 1 + ( Suppose that L a > 1 ( k +3 )> k v k 4 4k O the other had, we have: v 1 + k k k i k i i Therefore: v l 1 +1 i i i ) 1 4 4k + ( k k 6 +3 v 1 <1 v 1 because v l <1 1 1 because v 1 > 1 + 4k 6 < k +3 i + 4k 6 k... L a + 1 v 1 + v v + 1 > 1 1 Or max{l a } 1 b) ( )> 1 ( k >1 ) v 1 v l < 1.It is impossible because + 4k 6> 4k 4>0 k>1 true for 1 + v v 0 + 4k 6 )> 1

13 Case 3: v 1 > + 1 k, 3,..., >v >v 3 >... >v k > 1 >v >v k+ >...v > 1 >... >v for L b v k + i + 1 k 1 + 4k + ( Suppose that L b > 1 ( k )> 1 ( k )> However k k ) v 1 + v i 4k 4k + ( k 4k 4 + 4k >1 because k>1 + 1 k 4 + 4k + + ( j1 ) )> 1 1 <Cotradictio Therefore: L b + v 1 + v v 1 + v 1 + v v 0 1 Case 4: v 1 > + 1 >v >v 3 >... >v k > 1 >v vv k+ >... > 1 >v >... >v for k, 3,.., L b v i + 1 k 1 + 4k + ( j1 k 1 + k ) 1 + k v 1 + 4k i + ( v + 1 j1 + + )

14 Suppose that L b > 1 4k ( k + )> 1 4k ( k + However Cotradictio. Therefore: L b )> 4 + 4k 1 + ( k because k> k k + v 1 + v v Case 5: + 1, 3,..., L b + 1 v v 1 + v + >v 1>v >v 3 >...>v k > 1 >v >v k+ >...v i + 1 k 1 + k 1 + k k + k 1 4k v 1 + ( Suppose that L b > 1 ( k i i O the other had,we have v 1 + k k i k i i k 1 v 1 + v 1 + i i )> 1 1 < v > 1 >... >v for k v 1 ) 1 4k + ( 4k + ( k i i )> 1 )> 1 + 4k + 4k ( k i 1 k i 1 + ) )> 1 v 1 v 1 <1 v 1 because 0 1 v 1 <1 v 1 <1 v 1 < k 4 because

15 v 1 > 1 + 4k Therefore: < k < 1.It is impossible because i + 4k > 1 + 4k > 4k>0 k>0 true for k.. L b + 1 v 1 + v v v 0 1 or max {L a } 1 Case 6:, 3,..., 1 + v 1 + v >v 1>v >v 3 >...>v k > 1 >v >v k+ >...> 1 >v... >v for k L b + 1 v 1 + i + 1 k 1 + k 1 + k k + k + 1 4k + v 1 + ( Suppose that L b > 1 ( k i ( k i + + j1 k + 1 v 1 + v k + )> 1 +4k )> 1 +4k O the other had,we have v 1 + k k i k i i i i v + 1 ) 1 4k + + ( k i + + 4k 4 ( k i + 4k 4 ( k + 1 k + i + + i ( + i )> ) + 1 v 1 1 v 1 <1 v 1 because 0 + 4k 4 )> + 4k 4 )> 1 v 1 <1 v 1 <1 v 1 <1 1 1 because 15

16 v 1 > 1 + 4k Therefore: < k + < 1.It is impossible because i + 4k > 1 + 4k 4> 4k> k> 1 true for k.. L b + 1 v 1 + v v v 0 1 or max {L a } 1 c) Case 7: Suppose that we have u i i [1, k] u i i [k + 1, 1] u i i [l + 1, ] L c ( i 1 1 i ) + j1 j1 ( ( l j1 j1 l+1 l v 1 + v + l 1 ( i 1 1 i ) + l+1 + (1 1 k ) + ( 1 k 1 l ) (1 l 1 ) ) ) + k l l+1 + l + l If L c > 1 the we have L b ( k ) + il il+1 k l 1 ( k )>3 il il+1 k + l 1 ( k )> 3 il il+1 1 k + 1 l 1 However we ca prove that 3 1 k + 1 l + 1 >1 because 3 1 k + 1 l 1 >1 3 1 k + 1 l 1 >0 (kl l) + (k kl)>0 l(k ) + k( l)>0 It is true because 1 k 3. The ( k ) 3 1 k + 1 l 1 >1 il il ( i 1 1 i ) l+1

17 O the otherh ad: k l v i + 1 k 1 il+1 il+1 l <1 because l >0 a Cotradictio The L c u 1 v 1 + u v + u 3 v u v 1 Case 8: Suppose that we have + 1 >v 1>v > 1 u i i [3, k] u i i [k + 1, l] u i i [l + 1, ] L c + 1 v 1 + v ( i 1 1 i ) il ( i 1 1 i ) + 3 l 1 ( i 1 1 i ) l + il+1 L c v 1 + v + 1 l ) (1 l 1 ) L c ( i il+1 3 l il+1 ) 1 + k l + 1 If L c > 1 the we have L c ( i ( )>1 k + l i il+1 ( )> k + l ( i i il+1 il+1 I additio v 1 + i il+1 )>1 1 k + 1 l 1 i l 1 v 1 il+1 l 1 i (1 1 k ) + ( 1 k il+1 il+1 ) 1 + k l v 1 because l >0 17

18 However we have :v 1 >v >v The i k + 1 l <( i il+1 il+1 1 v 1 < )< 5 6 ( ) For k 6( ) ca ot happe because k + 1 l 5 6 We also prove that L b u 1 v 1 + u v + u 3 v u v 1 for k<6 Propositio 4 Proof. Assume that X* ad Y* are two profile that satisfy the followig coditio with all true profile {v 1, v,..., v } L u u 1 v 1 + u v u v 1 L Y Y 1 v 1 + Y v Y v 1 Cosider a combiatio profile αu + (1 α)y, for ay 0<α<1, the we have L c αu 1 + (1 α)y v 1 + αu + (1 α)y v αu + (1 α)y v L c αu 1 + (1 α)y αv 1 (1 α)v 1 + αu + (1 α)y αv (1 α)v αu + (1 α)y αv (1 α)v L c [ α(u 1 v 1 ) + α(u v ) α(u v ) ]+ (1 α)(y 1 v 1 ) + (1 α)(y v ) (1 α)(y v ) L c [ α(u 1 v 1 ) + α(u v ) α(u v ) ]+[ (1 α)(y 1 v 1 ) + (1 α)(y v ) (1 α)(y v ) ] L c α[u 1 v 1 +u t v u v ] + (1 α)[y 1 v 1 + Y v Y v ] L c αl u + (1 α)l Y α 1 + (1 α) 1 ) 1 Propositio 5 Proof. I order to show that, assume there exist a optimal profile Z* {Z 1, Z, Z 3 }. It meas that for every{v 1, v, v 3 },the loss must be satisfied: L Z 1 v 1 + Z v + Z 3 v 3 ( ) The ( )must be true with v {1, 0, 0}, we have: L 1 Z 1 + Z + Z 3 3 (1 Z 1 ) 3 Z 1 3 Suppose that Z 1 > 3 Z + Z 3 therefore both Z ad Z 3 < 1 3. Because (***) also true with v { 1 3, 1 3, 1 3 } L Z Z Z (Z + Z 3 ) 4 3 (Z + Z 3 ) 3 18

19 (Z + Z 3 ) 1 3 a Cotradictio Therefore Z 1 3 ad Z + Z We also have two origial profiles as: u { 3, 1 3, 0} ad Y { 3, 1 6, 1 6 }the Φ {αu + (1 α)y, 0<α<1}. Φ { 3, (α1 3 + (1 α)1 6 ), (α 0 + (1 α) 1 6 )) Propositio 6 Proof. Step 1: Prove that U { 5 8, 5 10, 1 1, 1 1 }, u { 5 8, 1 4, 1 16, 1 16 }, Y { 5 8, 3 16, 3 16, 0}ad Z { 5 8, 7 48, 7 48, 1 } are optimal 1 a) We prove that U { 5 8, 5 10, 1 1, 1 1 } is a optimal profile Case 1: v 1> 5 8 >v > 5 10 >v 3> 1 1 >v 4 L v v v v 4 L 1 v v 4< 0 10 <3 4 because v 4 0 Case : v 1 > 5 8 >v > 5 10 >v 3> 1 1 >v 4 L v v v v 4 L 1 (v + v 4 ) (v + v 4 )< <3 4 because v v 4 0 Case 3: v 1 > 5 8 > 5 10 >v > 1 1 v 3>v 4 L v v v v 4 L 1 (v + v 3 + v 4 ) (v + v 3 + v 4 ) 90 because v + v 3 + v 4 0 Case 4: 5 8 >v 1>v > 5 10 >v 3> 1 1 >v 4 L v v v v 4 10 <3 4 L 1+(v +v 3 ) (v +v 3 ) ( ) <3 4 because v 1 4 (propositio ) ad v 3 < 5 10 (assumptio). Case 5: 5 8 >v 1> 5 10 >v >v 3 > 1 1 >v 4 L v v v v 4 L 1 + v v <3 4 because v 3 < 5 10 (assumptio). Case 6: 5 8 >v 1>v > 5 10 > 1 1 >v 3>v 4 L v v 5 10 v v 4 19

20 L 1 + v v <3 4 because v 1 (propositio ) 4 We ca see that i the proof above, maximum loss occurs at case 3 ad case 4. Therefore, for remaiig profiles we cosider case 3 ad case 4 oly. b) u { 5 8, 1 4, 1 16, 1 16 } Case 3b: v 1> 5 8 >1 4 >v > 1 16 >v 3>v 4 L v v v v 4 L 1 (v + v 3 + v 4 ) (v + v 3 + v 4 ) because v + v 3 + v 4 0 Case 4b: 8 >v 1>v > 1 4 >v 3> 1 16 >v 4 L v v v v 4 L 1 + (v + v 3 ) (v + v 3 ) 5 8 <( ) <3 4 because v 1 4 (propositio ) ad v 3< 1 4 (assumptio). Y { 5 8, 3 16, 3 16, 0} Case 3c: v 1 > 5 8 > 3 16 >v >v 3 > L v v v v 4 L 1 (v + v 3 ) (v + v 3 ) because v + v 3 0 Case 4c: 8 >v 1>v > 3 16 >v 3>v 4 L v v 3 16 v v 4 L 1 + (v + v 3 ) (v + v 3 ) 3 8 <( ) <3 4 because v 1 4 (propositio ) ad v 3< 3 16 (assumptio). d)z { 5 8, 7 48, 7 48, 1 1 } Case 3d: v 1 > 5 8 > 7 48 >v >v 3 > 1 1 >v 4 L v v v v 4 L 1 (v + v 3 + v 4 ) (v + v 3 + v 4 ) because v + v 3 + v 4 0 Case 4d: 8 >v 1> v > 7 48 >v 3> 1 1 >v 4 L v v 7 48 v v 4 L 1 + (v + v 4 ) (v + v 4 ) ( ) <3 4 because v 1 4 (propositio ) ad v 4< 7 (assumptio). Step : Prove that Φ 48 0

21 {αt*+βu*+χx*+δy*+γ Z*} for ay α, β, χ, δ,γ are o egative ad α+β +χ +δ +γ 1. a)accordig to propositio 4, we have Φ {αt*+βu*+χx*+δy*+γ Z*} the o the flat limited by u ad u 3 all poits belog to the area (X*Y*Z*U*T*) are optimal. b)we show that all poit out side that area are ot optimal. Because u 1 5/8, u + u 3 + u 4 3/8 u 4 3/8 (u + u 3 ) 0 3/8 (u + u 3 ) (1) Accordig to propositio, 1 4 u () Based o our assumptio the u u 3 (3) Based o our assumptio the u 3 u 4 u 3 3/8 (u + u 3 ) 0 u / + u 3 3/16 (4) Combie (1), (), (3), ad (4) we have Φ (X*Y*G*T*) However every poit belogs to (G*U*Z*) is ot optimal. Because, we ca take ay poit that has u u (Z ) adu 3 u 3(Z ) + ε. The we have a ew profile (5/8, 7/48,7/48+ε, 1/1 -ε), the total loss of this profile 1

22 L 5/8 v 1 + 7/48 v + 7/48 + ε v 3 + 1/1 ε v 4 L 5/8 v 1 + 7/48 v + 7/48 v 3 + 1/1 v 4 + ε > L (Z ) Because Z* is a optimal file, G is ot optimal profile. {αt*+βu*+χx*+δy*+γ Z*} Φ Combie (a) ad (b) we have Φ {αt*+βu*+χx*+δy*+γ Z*} Proof. Theorem a) For 3, the scorig profile is (a 1, a, a 3 )such that a 1 a a 3 0. Assume that every member kows the order 1 3. Usig the same rule costructed above, we have the scorig matrix as follow: a1 a a1 a 3 a1 a a 1 The, the vector of moey distributio is ( 3(a 1 + a ), a 1 + a 3(a 1 + a ), a 3(a 1 + a ) )or a 1 ( 3(a 1 + a ), 1 3), a 3(a 1 + a ) ). Because u 1 the there is oly the optimal 3 X*( 1 3, 1 3, 0)3 feasible with scorig method. The we have: a 3(a1 + a) 0 a1 1, a 0 a3 a 3(a1 + a) 3 Therefore, for group of 3 agets we should use scorig profile (1, 0, 0) for 1 st, d, 3 rd positio respectively. b) For 4, the scorig profile is (a 1, a, a 3, a 4 )such that a 1 a a 3 a 4 0 Assume that every member kows the order Usig the same rule costructed above, we have the scorig matrix as follow a1 a a3 a1 a a3 3 a1 a a3 4 a1 a a3 The, the vector of moey distributio is 3a 1 ( 4(a 1 + a + a 3 ), a 1 + a 4(a 1 + a + a 3 ), a + a 3 4(a 1 + a + a 3 ), 3a 3 4(a 1 + a + a 3 ) ). 3 Accordig to profile X*( 1 3, 1, 0) is optimal 3

23 We should write u i f(a i )to fid which optimal profile is feasible with scorig method. Without loss of geeral, we ca assume that a 1 +a +a3+a 4 1 u 1 3a 1 4 u a 1 + a 4 u 3 a + a3 4 u 4 3a 3 4 Accordig to propositio, u a ad u 1 4 a 1 1. Suppose that u < 1 4 the a < 1 leads to u 4 u 3 < ad 1 a cotradictio. The u 1 4 a. 1 1 Sice u 4 + u ad u 4 u 3 the u We also have a a 3 the a 3 < 1 1, that If u the a 3 < 1 1 that leads tou 3< 1 16 ad u 4 + u 3 < 1 8 a cotradictio. The u 3 u , a ad a 4 0 Therefore, oly the optimal profile X*( 5 8, 1 4, 1 16, 1 16 )4 is feasible with scorig method at scorig profile: ( 5 6, 1 1, 1 1, 0) 4 Step 1 of the of propositio 6 shows that X* is optimal. 3

24 Refereces [1] Brams, S. J., Joes, M. A., ad Klamler, C. (006). Better ways to cut a cake. Notices of the America Mathematical Associatio, 53, [] de Clippel, G., Mouli, H., ad Tidema, N. (008). Impartial divisio of a dollar: Joural of Ecoomic Theory, 139, [3] Thomso, W. (003). Axiomatic ad game-theoretic aalysis of bakruptcy: Mathematical Social Scieces, 45, [4] Koblauch, V., (008). Three-aget Peer Evaluatio. Mimeo Uiversity of Coecticut. 4