A DUAL APPROACH TO COMPROMISE VALUES

Transcription

1 A DUAL APPROACH TO COMPROMISE VALUES J. M. BILBAO,E.LEBRÓN,A.JIMÉNEZ-LOSADA, AND S. H. TIJS Escuela Superior de Igeieros, Camio de los Descubrimietos, Sevilla, Spai CetER ad Departmet of Ecoometrics & OR, Tilburg Uiversity, P.O. Box 90153, 5000 LE Tilburg, The Netherlads. Abstract. The aim of this paper is to aalyse the Tijs value as a solutio cocept for cost allocatio problems. Give a cost game c, we defie its dual τ-value as τ (c ), if the dual game c is quasi-balaced. The we show that the dual τ-value coicides, for a wide class of cost games, with the alterate cost avoided (ACA) allocatio proposed i the 1930 s by the Teessee Valley Authority. It turs out that the ceter of the imputatio set, the egalitaria oseparable cotributio ad the ACA allocatio are colliear. 1. Itroductio A trasferable utility game is a pair (N,v), wheren is a fiite set ad v :2 N R, is a fuctio with v( ) =0. The elemets of N = {1,...,} are called players, the subsets S 2 N coalitios ad v(s) is the worth of S. The worth of a coalitio is iterpreted as the maximal profit ormiimal cost for the players i their ow coalitio. We will cosider profit gamesif v(s) measures the profit ofthecoalitios ad cost games c :2 N R if the fuctio measures the cost c(s) icurred by S. We will use a shorthad otatio ad write i for the set {i}, S i for S {i}, ads \ i for S \{i}. I a game (N,v), a vector x R is called efficiet if it distributes the worth v(n) amog the players, i.e., P x i = v(n). The set of all efficiet vectors is called the preimputatio set ad is deoted by I (v). The imputatio set of a profit game(n,v) is defied by I (v) ={x I (v) : x i v (i) for all i N}. Note that I (v) 6= if ad oly if v(n) P v(i). If (N,c) is a cost game, the the set of (efficiet) vectors x R with P x i = c(n) is deoted by I (c), ad its imputatio set is I (c) ={x I (c) : x i c (i) for all i N}. Therefore, I (c) 6= if ad oly if c(n) P c(i). Assumig that the coalitio N of all players will be formed, the followig solutio cocept Mathematics Subject Classificatio Primary 91A12. Key words ad phrases. ACA allocatio, τ-value, cost allocatio, dual game. 1

2 2 J. M. BILBAO,E.LEBRÓN,A.JIMÉNEZ-LOSADA, AND S. H. TIJS will prescribe a distributio of the cost c(n) or the profit v(n) amog the players. Defiitio 1.1. The core of a cost game c ad the core of a profit gamev are respectively defied by Core(c) = {x I (c) :x(s) c(s) for all S N}, Core(v) = {x I (v) :x(s) v(s) for all S N}, where x (S) = P i S x i ad x( ) =0. Games with a oempty core are called balaced games. Defiitio 1.2. The dual game (N,v ) of (N,v) is defied by the dual fuctio v :2 N R, where v (S) =v(n) v(n \ S), for all S N. Note that (v ) = v. Weremarkthatif(N,v) is a profit (cost)gamead (N,v ) is a cost (profit) game, the Core(v) =Core(v ). The Tijs value of a game is a feasible compromise betwee the upper ad the lower vectors for the game, itroduced by Tijs [4, 6, 7]. The upper vector of the game (N,v) is the vector M(v) R, where M i (v) =v(n) v(n \ i), for all i N. The compoet M i (v) is called the utopia payoff for player i i the grad coalitio. The remaider of i S if the coalitio S forms ad all other players i S obtai their utopia payoff is R v (S, i) =v(s) M j (v). The lower vector is the vector m(v) R,defied by m i (v) = max {S N : i S} Rv (S, i). Defiitio 1.3. Agame(N,v) is called quasi-balaced if it satisfies: 1. m(v) M(v), 2. P m i(v) v(n) P M i(v). The family of quasi-balaced games cotais the family of balaced games as a subset (see Tijs [4]). For a quasi-balaced game v the Tijs value, deoted by τ(v), is the uique preimputatio (efficiet vector) o the closed iterval [m(v),m(v)] i R. So we have τ(v) =m(v)+λ (M(v) m(v)), where λ R is determied by τ i (v) =v(n).

3 A DUAL APPROACH TO COMPROMISE VALUES 3 2. Cost allocatios problems The Teessee Valley Authority was a developmet project to costruct dams ad reservoirs alog the Teessee River. A method used by civil egieers to allocate the costs of the project is kow as the alterate cost avoided (ACA) allocatio. We ow describe the versio of this method give by Youg [8]. Let (N,c) be a cost game. For each project i N we defie its separable cost s i = c (N) c (N \ i). The alterate cost for i is the cost c (i) ad the differece r i = c (i) s i is the alterate cost avoided. Let s (N) = P s i ad r (N) = P r i. The the ACA allocatio assigs to each i N the cost allocatio give by the formula ( si + r i (c (N) s (N)), if r (N) 6= 0, ACA i (c) = r (N) (2.1) s i, otherwise. Notice that every i pays the sum of its separable cost ad a proportio of the oseparable cost c (N) s (N). Notice further that r i 0 for all i N if c is subadditive, i.e., for all S, T 2 N such that S T =, we have c (S T ) c (S)+c(T). Defiitio 2.1. Let (N,c) be a cost game. The associated savigs game (N,v) is defied by v (S) = P i S c (i) c (S), for all S N. The worth v (S) is the cost savigs obtaied from cooperatio betwee the players. Driesse [2, Chapter IV] obtaied the relatioship betwee two separable cost allocatio methods ad the Tijs value of the associated savigs game. If the cost game c is subadditive, the for every S N ad i N \ S, we have that c (S i) c (S)+c(i). It follows that v (S i) v (S) =c (i) (c (S i) c (S)) 0. We coclude that its savigs game is mootoic ad hece oegative. Let (N,c) be a cost game. The its savigs game (N,v) is mootoic if ad oly if for all T S N, c (i) c (T ) c (i) c (S) c (S) c (T )+ c (i). (2.2) i T i S i S\T The mootoicity of the savigs game implies, for all i N, s i = c (N) c (N \ i) c (i), or r i 0. Tijs ad Driesse [5] itroduced the reverse τ-value, τ r (c), of a cost game (N,c) defied as τ r (c) := τ ( c) if c is quasi-balaced. They proved that the reverse τ-value of a cost game c coicides with the ACA allocatio if the cost game is such that m r i (c) =c (i) for all i N, i.e., if c is semi-covex.

4 4 J. M. BILBAO,E.LEBRÓN,A.JIMÉNEZ-LOSADA, AND S. H. TIJS 3. The dual Tijs value Give a cost game c, we defie its dual τ-value as τ (c) :=τ (c ),ifthe dual game c is quasi-balaced. First, we study the properties of the upper ad the lower vectors for the game (N,c ). Theorem 3.1. Let (N,c) be a cost game such that its savigs game is mootoic. The the compoets of the upper ad the lower vectors of c satisfy 1. M i (c )=c(i) for all i N. 2. m i (c )=c(n) c (N \ i) for all i N. 3. m i (c ) M i (c ) for all i N, ad c (N) P M i (c ). Proof. 1. For every i N, M i (c )=c (N) c (N \ i) =c (N) (c (N) c (i)) = c (i). 2. For every i N, m i (c ) = max {S N : i S} c (S) M j (c ) = max c (N) c (N \ S) c (j) {S N : i S} = c (N) mi c (N \ S)+ c (j) {S N : i S} = c (N) mi c (T )+ c (j). T N\i j (N\i)\T From the mootoicity of the associated savigs game ad (2.2) we obtai c (N \ i) c (T )+ c (j), for all T N \ i. Therefore, mi c (T )+ T N\i j (N\i)\T j (N\i)\T c (j) = c (N \ i), ad we coclude that m i (c )=c (N) c (N \ i). 3. The iequality (2.2) implies that c (N) c (N \ i)+c (i). From 1 ad 2, it follows that m i (c )=c (N) c (N \ i) c (i) =M i (c ). Moreover, c (N) =c (N) c (i) = M i (c ).

5 A DUAL APPROACH TO COMPROMISE VALUES 5 Theorem 3.2. Let (N,c) be a cost game such that its savigs game is mootoic. The 1. The dual game c is quasi-balaced if ad oly if s (N) c (N). 2. If s (N) c (N), the the dual Tijs value τ (c ) coicides with the allocatio ACA (c). Proof. 1. It follows from Theorem that the dual game c is quasibalaced if ad oly if (c (N) c (N \ i)) c (N), or s (N) c (N). 2. For every i N, the compoet i of the dual Tijs value is τ i (c )=s i + λ (c (i) s i )=s i + λr i, where λ R is such that τ i (c )=c(n). Sice r i 0 we have r (N) =0 r i =0 for all i N τ (c )=M(c )=m(c ). Otherwise, implies that c(n) = (s i + λr i )=s(n)+λr(n) c (N) s (N) λ =. r (N) Thus, the compoets of the dual Tijs value satisfy ( si, if r (N) =0, τ i (c )= s i + r i (c (N) s (N)), if r (N) > 0, r (N) which coicides with the formula (2.1) of the ACA allocatio. 4. Colliearity of cost allocatio rules The ceter of the imputatio set (CIS) ad the egalitaria oseparable cotributio (ENSC) are well kow cost allocatio rules (cf. [1], [2], [3]). Defiitio 4.1. Let (N,c) be a cost game. For each i N, the CIS ad the ENSC allocatios are give by CIS i (c) = c (i)+ 1 c (N)! c (i), ENSC i (c) = s i + 1 (c (N) s (N)).

6 6 J. M. BILBAO,E.LEBRÓN,A.JIMÉNEZ-LOSADA, AND S. H. TIJS Obviously CIS (c) =ENSC (c ) ad ENSC (c) =CIS (c ). I the ext theorem we compute the reverse τ-value of the dual of a cost game (N,c), that is, the vector τ r (c )= τ ( c ). This vector is the uique efficiet compromise betwee the two vectors M r (c ) ad m r (c ). Theorem 4.1. Let (N,c) be a cost game such that its savigs game (N,v) is oegative. The we have 1. Mi r (c )=c(i) for all i N. 2. m r i (c )=c(i)+ ³c (N) P (j) j N c for all i N. 3. The reverse τ-value of the dual game c coicides with the allocatio CIS (c). Proof. 1. M r i (c )= M i ( c )=c (N) c (N \ i) =c (i), for all i N. 2. The compoets of the lower vector satisfy m r i (c ) = m i ( c ) = max {S N : i S} c (S) M j ( c ) = max c (N \ S) c (N)+ c (j) {S N : i S} = c (N) max c (T )+ c (j). T N\i For every T N \ i we have that c (T )+ c (j) = c (T )+ j (N\i)\T = j N\i j N\i j (N\i)\T j N\i c (j) j T c (j) v (T ) c (j), c (j) sice the savigs game (N,v) is oegative. Moreover, this upper boud is attaied i T = ad hece m r i (c )=c(n) c (j) =c (i)+ c (j), j N\i c (N) j N for all i N. 3. First, we show that the game c is quasi-balaced. By hypothesis v (N) 0 ad hece m r i (c )=c (i) v (N) c (i) =M r i (c ), for all i N.

7 A DUAL APPROACH TO COMPROMISE VALUES 7 Furthermore, c (i) c (N) 0 implies c (N) Mi r (c ). Fially, m r i (c ) = c (i)+c (N) c (i) = c (N)+( 1) c (N)! c (i) = c (N) ( 1) v (N) c (N). Therefore, the reverse τ-value is defied ad equals the uique efficiet vector betwee the upper ad lower vectors with compoets M r i (c )=c (i) ad m r i (c )=c (i) v (N). The τ r i (c )=c (i) v (N)+λv (N) =c (i)+(λ 1) v (N). The efficiecy implies that c (N) = τ r i (c )= c (i)+(λ 1) v (N). From this we coclude that for all i N, τ r i (c )=c(i)+ 1 c (N)! c (i), which coicides with the CIS allocatio. I order to obtai the colliearity of the three allocatios rules CIS, ENSC ad ACA, it is useful to recall some properties of orthogoal projectios of R oto a hyperplae. The ier product of vectors x ad y is the umber hx, yi = P i=1 x iy i. The a hyperplae H i R is a set of the form H = {x R : hm, xi = α}, where m is a ozero vector ormal to H ad α is a real umber. The orthogoal projectio P H : R H oto H is give by α hm, qi P H (q) =q + kmk 2 m, for each q R. Note that (P1) P H (q) =q ifadolyifq H, (P2) P H is a affie map, i.e., P H (λq + ηr) =λp H (q) +ηp H (r) for all q,r R ad λ, η R with λ + η =1. From (P2) it follows that the orthogoal projectio o H maps a lie i R otoalie(orapoit)ih. This implies:

8 8 J. M. BILBAO,E.LEBRÓN,A.JIMÉNEZ-LOSADA, AND S. H. TIJS (P3) If p, q, r R are colliear, i.e., there is a lie i R cotaiig p, q ad r, the also P H (p),p H (q) ad P H (r) are colliear. Furthermore, if p = λq + ηr with λ + η =1, the P H (p) =λp H (q)+ηp H (r). I the ext propositio we obtai the orthogoal projectios of the vectors k =(c (i)) ad s =(s i ) o the preimputatio set I (c). For a geometric illustratio see Figure 1. Propositio 4.2. Let H be the preimputatio set I (c) of a cost game (N,c) ad let k, s be the vectors defied above. The P H (k) =CIS (c) ad P H (s) =ENSC (c). Proof. Note that H = {x R : hm, xi = c(n)}, where m =(1,...,1). The! P H (k) =k + 1 c(n) c (i) (1,...,1), which implies (P H (k)) i = c (i)+ 1 c(n) for each i N. Further, P H (s) =s + 1 c(n) Hece, for each i N, we obtai i=1! c (i) = CIS i (c), i=1! s i (1,...,1). i=1 (P H (s)) i = s i + 1 (c(n) s (N)) = ENSC i (c). Theorem 4.3. Let (N,c) be a cost game with s (N) c (N) k (N). The the allocatios CIS (c), ENSC (c) ad ACA (c) are colliear. Moreover, if s (N) 6= k (N) the c (N) s (N) ACA (c) =ENSC (c)+ r (N) Otherwise, ACA (c) =ENSC (c). (CIS (c) ENSC (c)). Proof. The defiitio (2.1) of the ACA allocatio implies that ACA (c) =s + c (N) s (N) (k s) k (N) s (N) if s (N) 6= k (N), ad ACA (c) =s, otherwise. Thus the vector ACA (c) is a covex combiatio of the vectors k =(c (i)) ad s =(s i ). Further

9 A DUAL APPROACH TO COMPROMISE VALUES 9 ACA (c) I (c) =H. Ithecasethats (N) 6= k (N) the properties (P1), (P2) ad (P3) imply that ACA (c) =P H (ACA (c)) = P H (s)+ c (N) s (N) k (N) s (N) (P H (k) P H (s)). The, by Propositio 4.2, we obtai c (N) s (N) ACA (c) =ENSC (c)+ (CIS (c) ENSC (c)). r (N) If s (N) =k (N) the s (N) =c (N) ad so ACA (c) =ENSC (c). Remark 4.1. If s (N) =k (N) ad the savigs game is mootoic, the the allocatios ACA (c) =CIS (c) =ENSC (c). Theorem 4.3 suggests us to defie the followig cocept. Defiitio 4.2. The compromise set of a cost game (N,c) is the covex hull of the vectors CIS (c) ad ENSC (c). I future research, we expect to see ew applicatios of this set, cosistig of the lie segmet with ed poits CIS (c) ad ENSC (c). Refereces [1] Draga, I., Driesse, T. S. H., ad Fuaki Y. (1996) Colliearity betwee the Shapley value ad the egalitaria divisio rules for cooperative games. OR Spektrum 18, [2] Driesse, T. S. H. (1988) Cooperative Games, Solutios ad Applicatios. KluwerAcademic Publishers, Dordrecht. [3] Mouli, H. (1985) The separability axiom ad equal-sharig methods. Joural of Ecoomic Theory 36, [4] Tijs, S. H. (1981) Bouds for the core ad the τ-value. I: Game Theory ad Mathematical Ecoomics (O. Moeschli ad D. Pallaschke, eds.) North-Hollad, Amsterdam, [5] Tijs, S. H., ad Driesse, T. S. H. (1986) Game theory ad cost allocatio problems. Maagemet Sciece 32, [6] Tijs, S. H. (1987) A axiomatizatio of the τ-value. Mathematical Social Scieces 12, [7] Tijs, S. H., ad Otte, G.-J. (1993) Compromise values i cooperative game theory. Top 1, [8] Youg, H. P. (1994) Cost allocatio. I Hadbook of Game Theory, vol. II (R. J. Auma ad S. Hart, eds.) North-Hollad, Amsterdam,

10 10 J. M. BILBAO,E.LEBRÓN,A.JIMÉNEZ-LOSADA, AND S. H. TIJS k ENSC HcL ACA HcL CIS HcL s Figure 1. Orthogoal projectios o I (c)