A Fair Division Based on Two Criteria

Transcription

1 A Fair Division Based on Two Criteria Ken Naabayashi * Biresh K. Sahoo and Kaoru Tone National Graduate Institute for Poliy Studies 7-- Roppongi Minato-uToyo Japan Amrita Shool of Business Amrita Vishwa Vidyapeetham Ettimadai Post Coimbatore India Abstrat This paper deals with the problem of sharing membership fees amongst member soieties of the International Federation of Operational Researh Soieties (IFORS. To our nowledge there has so far been little researh attempt made to deal with this problem. This paper however is an attempt to fill in this void by arguing that the membership fee of any ountry should be in proportion to both its average inome and qualified membership based on whih we present here an innovative DEA game whih provides a possible solution to this problem. In partiular we show in ase of the two-riterion that the Shapley and nuleolus alloations are nothing but the simple average of the observed values of two individual riteria: qualified membership and average inome with regards to their proportion. Keywords: Cost-sharing opinion aggregation DEA ooperative game theory Shapley value nuleolus. Introdution The problem of sharing ost/burden among members appears to be most ommon in various soieties ranging from loal ommunities to international organizations. Examples are numerous but we ite here only three speial examples: Maret arade problem Apportionment of the expenses of the United Nations (UN and Membership fee sharing problem in the International Federation of Operational Researh Soieties (IFORS to potentially exhibit the importane that the problem of ost/burden sharing in multi-riteria environment is inreasingly hallenging. [Example ] Maret arade problem A overed arade provides reasonably a omfortable ondition for shopping and waling. So a lot of shopping distrits need arades to attrat ustomers. It osts them a great deal of money to * en@stu.grips.a.jp s_biresh@ettimadai.amrita.edu tone@grips.a.jp: Corresponding author

2 onstrut and maintain them too. One shopping mall assoiation in Japan pratially made the following agreement on the arade maintenane fee: Every shop faing the arade street has to pay a monthly fee. The method for arriving at this fee for eah shop was disussed and approved at the general meeting. Share of ost was determined based on parameters suh as ategory of business the size of the shop the number of employees and so on. [Example ] Apportionment of the expenses of the UN The United Nations is an international organization made up of almost all the ountries in the world and its prime objetive is to maintain international seurity and attain peae. The member ountries have to pay an annual membership fee for this purpose. The sale of assessments for the apportionment of the expenses is based on a number of parameters suh as gross national produt (GNP external debt per apita inome and so on. In the UN General Assembly the representatives of the member ountries disussed based on the aforementioned parameters various ideas onerning how to share and estimate the expenses in order to arrive at a single omprehensive sheme. Colletively they are though framed within a set of potential parameters individually they depart by expressing their own favorable opinions onerning the importane of various parameters. Tadooro (996 summed up at reoniling the disussion in the meeting as follows: The deision-maing proess of the General Assembly on the methodology for estimating share of expense often appears to be a politial battle in that eah member ountry supports its preferable methods or riteria so as to minimize its own share. [Example 3] Membership fee sharing problem in IFORS: The IFORS is an umbrella organization omprising the National Operations Researh (OR Soieties of over 45 ountries from four geographial regions: Asia Paifi Europe North Ameria and South Ameria. The prime onern of this organization has not only been the development of OR as a unified siene but also its advanement in all nations around the globe. To mae this happen eah member ountry needs to pay an annual membership fee. Artile IV of IFORS states that All Member Soieties shall pay an annual membership fee. The amount of this fee will be in proportion to the qualified membership of eah Member Soiety. In Example two or more riteria were disussed to arrive at the arade maintenane fee for eah shop. The estimate of assessing and sharing membership fees in Example was too multi-dimensional in nature. However in Example 3 the IFORS taes into aount just one riterion i.e. qualified membership. A multi-riteria problem is interestingly more ompliated ompared to its single-riterion ounterpart. Why both the shopping mall assoiation and the UN should have taen the trouble in applying the multi-riteria priniple to arrive at a deision.

3 The possible reason for the underlying problem in Example may be that it is often diffiult to mae a fair judgment based on just one riterion. The reason for Example may be that the members have different opinions on the methods for estimating an equitable sharing and reoniling all the members opinions to arrive at a onsensus is muh more ompliated than most of us let on. One natural question over here is: an we always assert the IFORS s alloation rule based on a single riterion to be fair? The IFORS has dereed that all the 45 member ountries pay the same expense per apita irrespetive of the differene no matter whether small or big in their respetive per apita inome levels. For a better exposition let us now illustrate this problem onsidering two speifi ountries: India a developing ountry and Switzerland a developed one who are both members of IFORS. Aording to the World Ban data the gross national inomes (GNI per apita of India and Switzerland are respetively US$460 and US$38330 in 00. India and most other developing ountries are under the impression that differenes in per apita inomes among ountries might be taen into onsideration in the IFORS s proposed membership fee. Therefore we have a try at proposing another alloation rule based on the average inome whih is taen here as the produt of GNI per apita and qualified membership. Note that our main intention of this paper is to mae with our familiar example an aademi study of ost-sharing problems under multi-riteria environment but not to hallenge the views of IFORS. Now suppose one another rule whih is based on average inome is proposed and plaed on the IFORS s agenda. The IFORS would then be faed with the problem of nowing whih one is more appropriate: the present rule or the proposed one. Three logial possibilities that would ideally emerge to deal with the above problem are as follows:. If all the members of IFORS regard the present rule as more appropriate the proposed one would be rejeted.. If all the members onsider the present rule less appropriate Artile IV of IFORS would be reworded after the adoption of the proposed one. 3. There is another possibility that the members have different opinions regarding the importane of riteria. In this study we however onsider the third possibility to analyze this most problemati and ontroversial situation using the DEA-game sheme developed by Naabayashi and Tone (003. We assume that every member ountry desires to redue its own burden by favoring its most preferable riterion. Suh a selfish behavior is normally observed in all real-life problems. In this situation a division annot be determined as long as all the members sti to their own preferable riteria. That is to mae a olletive deision some members need to 3

4 ompromise. If so then how far should one member ompromise? How do we find an equitable ompromise between these two measures? This issue is essentially related to priority whih is a lass of oneptions of equity as mentioned by Young (994. More often priority based on a mixture of riteria is not one-dimensional it involves trade-offs among various priniples. Multi-riteria problems ould be solved in priniple by assigning various weights to the riteria in order of their importane. However partiipants in general have their own independent opinions onerning the various riteria and it is therefore often diffiult to attain a onsensus with a single riterion or a ommon set of weights. In this ase fairness redues to a proedural question of how to strie an equitable balane between diverse points of view. This brings us to the lassial problem in group deision maing onerning how to design a proess that fairly aggregates individual opinions into a olletive deision. This is the opinion aggregation or soial hoie problem and it is not a simple one. In this paper we introdue using ooperative game theory an approah to deal with this opinion aggregation problem. To our nowledge we have not ome aross any study showing this ind of appliation of ooperative game. In partiular we deal with a problem involving two riteria and 45 players and demonstrate that this bi-riteria ase is very speial where the Shapley and nuleolus solutions an easily be obtained simply by taing the average although it is in general diffiult to ompute them for large games. The remainder of this paper unfolds as follows. Setion begins with a real-life data and then desribes the membership-fee sharing problem of IFORS with two riteria. In Setion 3 we apply our DEA-game sheme to find out a reasonable solution to the IFORS problem. Finally Setion 4 onludes with some remars.. Problem: The egoist s dilemma The data on qualified membership its proportion and the average inome of member soieties of IFORS in 00 are all shown in Table. We treat all the 45 soieties as distint players and onsider qualified membership and average inome respetively as Criterion and Criterion. Let us represent the matrix 45 X ( xij R+ in Table where ij x is the player j s 45 proportion of riterion i. It holds that x (i. Heneforth we deal with this j ij normalized sore matrix. Suppose that player desires to minimize his/her proportion by seleting the favorable riterion in the following linear program: We all this the normalization of Criterion i (i. 4

5 where ( w w w d min w subjet to w ( w x + w x + w w w 0 is a vetor of weights assigned to the riteria by player. The optimal weight vetor w may differ from player to player. This arbitrary weight seletion proedure is the distinguishing feature underlying DEA models initiated by Charnes et al. (978. This game is regarded as the variable weight game as opposed to an a priori fixed one. Refer to Cooper et al. (999 pp.-3 for the detailed explanation on this variable weight issue. Computation of d is an easy tas. The optimal solution d is given by min { x x } as exhibited in the last olumn of Table. ( Table : Two proportions of membership and average inome in IFORS Data soure: IFORS Membership; from the internet: [ 03/4/004. The World Ban World Ban Atlas

6 The optimal 45 d yields d < whih means that d fails to satisfy the membership fee sharing problem. That is if eah member ountry stis to its preferable weight seletion behavior it is not guaranteed that IFORS an gather the full amount ( of the annual fee. Naabayashi and Tone (003 all this problem the egoist s dilemma. Perhaps the most famous model for the tension between individual interests and the ommon good is the non-ooperative game nown as Prisoner s Dilemma in the game theory literature. We present here the reasons why another model alled the egoist s dilemma should be needed with a disussion of the differene between these two dilemmas. In the ase of Prisoner s Dilemma prisoners have no ontraditions if they an ooperate. Most researhers inluding us do reognize that Prisoner s Dilemma is nowadays in fat a dilemma. The reason may be that the prisoners who are not supposed to tal to eah other ould not individually hoose ooperation over non-ooperation in theory. In the ases of real-life problems suh as ost-sharing partiipants unlie prisoners used to disuss their subjet in the soiety. During the disussion they may insist on their own favorable opinions on the methods for estimating share of burden/ost. They have absolutely no problem if their opinions are in aord. However more often they have in pratie different opinions onerning various riteria. There exist a lot of problemati soial situations where partiipants sti to their own favorable opinions and are not in a position to be able to easily attain a onsensus onerning the olletive deision. One may propose that no one should sti to his/her selfish behavior but this proposal does not yield a satisfatory solution to a pratial problem. We have therefore tried to find out how to sientifially strie an equitable balane on the assumption that eah partiipant stis to his/her most preferable riterion. We believe the ase of Prisoner s Dilemma is an exellent framewor for some of the problemati soial situations but this annot give us a lear explanation of situations in the onsensus-maing proess. In other words through the use of only Prisoner s Dilemma one annot reognize and analyze a different type from a single-riterion problem. Hene we need a different framewor for sientifi analyses in the ase of multi-riteria environment. 3. DEA game approah This setion applies our DEA-game sheme to the ost-sharing problem of IFORS under the bi-riteria environment. We demonstrate that game-theoreti solutions suh as the Shapley value and nuleolus an be easily obtained in this problem. 3.. Applying the DEA-game sheme to bi-riteria problem We onsider how to aggregate the various weights hosen by players into a ommon weight 6

7 vetor w to determine a division z ( z Λ z with z. Toward this aggregation problem we tae an approah to onstrut the harateristi funtion d(s as follows. Let a oalition S be a subset of the player set N { Λ n} oalition S is defined as j j. The reord for the xi ( S xij ( i. ( Similar to an individual player members of the oalition S are supposed to olletively hoose their favorable riterion in the problem S min{ x ( S x ( S } d (. The d(s with the ondition d ( φ 0 defines a harateristi funtion of the oalition S. We assume that the value of d(s is transferable. Thus we have a DEA min game in oalition form with transferable utility represented by (N d. The game (N d is super-additive in nature and further it holds that d(n. Using this game one an obtain game-theoreti solutions suh as Shapley value and nuleolus from this alloation problem. We an define another game (N in this problem with S max{ x ( S x ( S } ( whih is just the reverse of the egoist s game (N d. This DEA max game (N is sub-additive in nature and it further holds that (N. On omparison these (N and (N d it holds that d( S + ( N \ S S N. (3 The equation (3 reveals that (N and (N d are dual games (see e.g. Peleg and Sudhölter (003. In this two-riterion ase we have the following lemma. Lemma : In the two-riterion ase the game (N +d is additive i.e. it holds that Proof: We have {} j ( S + d( S ( { j} + d( { j} for every S N. ( d( { j} x + x ( j N + j j and ( S d( S x ( S + x ( S x + x ( S N + j j. Hene it holds that ( S + d( S ( { j} + d( { j}. Note that Lemma does not hold true in the ase involving three or more riteria. The Shaple value for the game (N d is defined by ( d S: N ( s! ( n s! [ d( S d( S \ { j} ] φ j n! where s is the number of members of oalition S. We have the following proposition in this 7

8 two-riterion ase. Proposition : In the two-riterion ase the Shapley value ( d game (N d is given by ({ } + d( { }. φ of player for the DEA min φ ( d (4 Proof: Sine the Shapley value satisfies the property of duality and additivity we have respetively φ ( φ ( d and φ ( + d φ ( + φ ( d. Hene φ ( φ ( d φ ( d. Lemma we have φ ( + d ( { } d( { } +. Therefore it holds that ({ } + d( { }. + From φ ( φ ( d Proposition reveals that the Shapley alloation in the two-riterion ase is nothing but the average solution of the two diametrially opposite values: ({ } and d ({}. That is the Shapley alloation with the ommon weight w ( w w ( ( φ ( d w x + wx an be expressed as φ and is obtained as the average sore of these two riteria. Allowing the number of player to inrease renders it hard to obtain Shapley alloation beause it involves omplex alulations where one needs to determine a large number of harateristi funtions for all the possible potential oalitions. For example in the ase of the IFORS problem involving 45 players the number of all possible potential oalitions is 45 > However from Proposition in the two-riterion ase one does not need to onsider all the potential oalitions. The Shapley alloation an be obtained as the average of two proportion sores made on individual riteria by the player. The data on proportion sores with respet to qualified membership and average inome as well as the Shapley alloation for all the member ountries in IFORS are shown in Table. We now demonstrate that the nuleolus an also be obtained by taing the average. For a payoff vetor z ( z with z the exess of eah oalition S of the game (N d Λ z n n j is defined by e ( S z d( S. Let ( z z j the exesses of all oalitions S N; ( S φ N j θ be the vetor (with n omponents of ordered with inreasing magnitude i.e. n θ ( z ( e( S z e( S z Λ n with e( S z e( S z Λ. 8

9 Table : An evaluation of the IFORS s membership fee with two riteria θ i.e. θ ( z > L θ ( y if n i i there exists { Λ } suh that e( S z e( S y ( i Λ and e( S z > e( S y. Let Z be an entire set of payoff vetors of the game (N d. The nuleolus of (N d is defined by We introdue a lexiographi ordering of the vetors ( z { z Z θ ( z ( y y Z} µ ( Z L θ. Proposition : In the bi-riteria ase for any player it holds that where ( d ({ } + d( { } µ ( d (5 µ is the nuleolus of the DEA min game (N d for player. Proof: We define a vetor z ( z Λ z n ( {} + d( { } suh that z { Λ n}. 9

10 n Then from Lemma it holds that z. For any oalition S we have and e e ( S z ( N \ S z d( S j j ({} j + d( { j} d( N \ S j N \ S d ( S ( S + d( S d( S ( S ({ j} + d( { j} d( N \ S ( N \ S. Sine the games (N and (N d are dual games It holds that d ( S ( S d( N \ S ( N \ S Hene we have e ( S z e( N \ S z. By applying Lemma of Kohlberg (97 it is lear that z. should be nuleolus. 3.. Inorporating the assurane region (AR method Now suppose that all players agree on the ondition that Criterion is more important than Criterion. Then we need to add the following onstraint to the original LP formulation (: w w w or. (6 w In general if all players agree on a preferene relation involving two riteria we an apply the following assurane region (AR method originally developed in DEA literature (Thompson et al. (986 and Dyson and Thanassoulis (988. Let the base riterion be w. We set bounds on this preferene relation as follows: w l u (7 w where l and u denote respetively the lower and upper bounds on the ratio w w. These bounds must be set in agreement among the players. In this AR ase the harateristi funtions (S and d(s of the oalition S an be omputed as follows: d ( S ( S x max x min ( S + lx ( S x ( S + ux ( S + l + u (8 ( S + lx ( S x( S + ux ( S ( S N + l + u. (9 We have now the following lemma. Lemma : In a bi-riteria DEA game Lemma holds no matter whether the AR method is applied or not. 0

11 Proof: ( S + d( S x ( S + x ( S x + x lu ( x ( S x ( S ( + l( + u lu j j ( + l( + u ({} j + d( { j}. + + x j x j Sine Lemma is ruial for the proofs of both Propositions and we have now from Lemma the following proposition. Proposition 3: In a bi-riteria DEA game both Proposition and Proposition hold no matter whether the AR method is applied or not. Let us now show the AR-based results on the Shapley/nuleolus alloation in Table 3. We set one ommonly agreed onstraint:.5 w w (say and obtain the Shapley and 0 nuleolus alloations as the average of two vetors: ( ({ } Λ ({ n} and ( d ({} d( { n} Λ. This AR ase assigns more weight to qualified membership and thereby favors the developed soieties. To illustrate our earlier ase for omparison our evaluation shows that Switzerland redues its burden by 7% (as 0.69 / whereas India inreases its outlay by 9% (as.45 / Conluding remars In this paper we dealt with a membership fee sharing problem among the member soieties of IFORS. While deiding on setting the membership fee of any soiety IFORS taes only a single riterion i.e. qualified membership. However doing so is potentially unfair for it fosters in the member ountries minds the belief that the differenes in per apita inomes among ountries are taen into onsideration. This is preisely due to the fat that both per apita inome and qualified membership are inextriably lined. Therefore to avoid fostering the wrong pereption and to reinfore the member ountries beliefs we propose one additional riterion i.e. average inome to be taen into onsideration while deiding on the membership fee. To aomplish this we applied our DEA-game sheme to propose game-theoreti solutions to this bi-riteria problem. We found in the two-riterion ase that the Shapley and nuleolus alloations were nothing but the simple average of the observed values of these two riteria with regard to their proportion. Furthermore we demonstrated the use of AR DEA game providing solution to this bi-riteria problem when most member ountries are in broad

12 agreement on ertain preferene relations. Table 3: Evaluation with inorporating the AR method:.5 w w 0 In ase of two riteria though the results obtained in this paper are analytially enrihing deriving them are mathematially too simple beause of whih one ould view our paper not only important but also interesting. In ase of more than two riteria however one needs to ompute the Shapley or nuleolus alloation just from their definitions. The two-riterion ase is very speial where the Shapley and nuleolus solutions an easily be obtained simply by taing the average although it is in general diffiult to ompute them in large games. This is what Littlehild and Owen (973 mentioned in their paper by saying: In the twenty years sine its proposal the Shapley value has reeived surprisingly few appliations. Perhaps one reason is the diffiulty of omputing it for large games. And if there was another bottlene in the appliation of ooperative game it would be the diffiulty of onstruting the harateristi funtion in real problems. They onstruted the harateristi funtion in the real-life problem whih yields a very simple but available expression of the Shapley value.

13 Their study greatly ontributes to the literature in terms of offering an exellent appliation of game theory. In our study we also showed in a speial ase a simple expression for the Shapley value of the real-life problem. This is so important that it deserves speial mention. In the two-riterion ase we too found out using two OR methodologies DEA and ooperative game theory a simple solution i.e. the way of taing the average. One an simply say here that this way is not the authors original one. That is adopting the golden mean between two proposals that had long been in dispute may be a matter of ommon pratie in our soiety. We ould however interpret this ommon pratie as a potential OR solution and therefore this interpretation we believe would be interesting from a wider perspetive. We hope this study adds a new dimension to the nowledge of the equitable sharing problem. Referenes [] Charnes A. Cooper W. W. and Rhodes E. (978 Measuring the effiieny of deision maing units European Journal of Operational Researh [] Cooper W. W. Seiford L. M. and Tone K. (999 Data Envelopment Analysis A Comprehensive Text with Models Appliations Referenes and DEA-Solver Software Boston: Kluwer Aademi Publishers. [3] Dyson R. G. and Thanassoulis E. (988 Reduing weight flexibility in data envelopment analysis Journal of the Operational Researh Soiety [4] Kohlberg E. (97 On the nuleolus of a harateristi funtion game SIAM Journal of Applied Mathematis [5] Littlehild S. C. and Owen G. (973 A simple expression for the Shapley value in a speial ase Management Siene [6] Peleg B. and Sudhölter P. (003 Introdution to the Theory of Cooperative Games Kluwer Aademi Publishers. [7] Naabayashi K. and Tone K. (003 Egoist s Dilemma: A DEA Game Omega: The International Journal of Management Siene (forthoming. [8] Tadooro M. (996 The U.N. Finane (in Japanese Yuuhiau. [9] Thompson R. G. Singleton Jr. F. D. Thrall R. M. and Smith B. A. (986 Comparative site evaluations for loation a high-energy physis lab in Texas Interfaes [0] Young H.P. (994 Equity: In Theory and Pratie Prineton: Prineton University Press. 3