- Email: [email protected]
An axiomatic characterization of the Baker-Thompson rule
Economics Letters 107 (2010) 85–87
Contents lists available at ScienceDirect
Economics Letters j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / e c o l e t
An axiomatic characterization of the Baker-Thompson rule Vito Fragnelli a,⁎, Maria Erminia Marina b a b
University of Eastern Piedmont, Department of Sciences and Advanced Technologies, Viale T.Michel 11 - 15121 Alessandria, Italy University of Genova, Italy
a r t i c l e
i n f o
Article history: Received 21 February 2009 Received in revised form 5 December 2009 Accepted 17 December 2009 Available online 24 December 2009
a b s t r a c t The airport problem is a widely studied allocation problem, with the aim of providing simple and fair sharing rule for the landing fees. In this note we introduce some fairness criteria and characterize the classical BakerThompson allocation rule. © 2009 Elsevier B.V. All rights reserved.
JEL Classification: D63 C71 Keywords: Airport problem Baker-Thompson rule Fairness
1. Introduction In this note we consider a cost allocation problem in which a set of agents N = {1, 2, ..., n} that uses a certain facility has to share the cost C of the facility itself. The cost may represent the building cost and/or the maintenance cost. We suppose that the agents require the facility at different levels of quality so that each agent i ∈ N is associated to a cost ci. We suppose that the agents are ordered in such a way that C = c1 ≥ c2 ≥ ... ≥ cn > 0. Moreover, we suppose that the agents with the equal cost are grouped; consequently, we consider an allocation problem with disjoint sets of agents N1, N2, ..., Nk, where N1 ∪ N2 ∪ ... ∪ Nk =N; let n1, n2, ..., nk be the number of agents of the groups with n1 + n2 + ... +nk =n and let c1 =C1 >C2 > ...>Ck =cn be the costs of the groups. A classical example of this kind of problems is the airport problem (see Littlechild and Thompson, 1977), where N = {1, 2, ..., n} is the set of planes landing in a given time period, C is the cost of the landing strip to be allocated and c1, c2, ..., cn are the costs of the landing strip required by the planes. The airport problem may be extended to a more general situation in which the agents may use k different facilities with the requirement that if an agent asks for using facility r he has to use (or at least he has to pay) also for the facilities r + 1, ..., k. We have in mind a set of independent services that cannot be purchased separately, but sequentially, as explained above. This approach is common in car
⁎ Corresponding author. Tel.: +39 0131 360224; fax: +39 0131 360391. E-mail address: [email protected] (V. Fragnelli). 0165-1765/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2009.12.033
market where cars are available with increasing packages (trim quality levels) plus separate optional extras. The economists Baker (1965) and Thompson (1971) proposed a nice allocation rule for sharing the cost C among the agents, now called the Baker-Thompson (BT) rule: BTi ðN1 ; :::; Nk ; C1 ; :::; Ck Þ =
∑
Cj −Cj
+ 1
j = r;:::;k ∑l = 1;:::;j nl
; i∈Nr ; r∈f1; :::; kg
where Ck + 1 = 0. In 1973 Littlechild and Owen showed that the BT rule coincides with the Shapley value (see Shapley, 1953) of the airport game (see Littlechild and Thompson, 1977). In Alparslan Gök, Branzei and Tijs (2009) the authors notice that no axiomatic characterization is available for the BT rule; so, our motivation in this note is to provide some fairness criteria for possible allocation rules and mainly to characterize the BT rule. 2. The fairness criteria Formally, an allocation rule for an allocation problem is a map F that associates with each allocation problem (N1, ..., Nk, C1, ..., Ck), a unique point F (N1, ..., Nk, C1, ..., Ck) ∈ Rn with ∑i∈N F i (N1, ..., Nk, C1, ..., Ck) =C1. The first criterion of satisfaction is the individual equal sharing (IES): as all the n agents have the right of usage, each agent i ∈ Nr may Cr not ask to pay less then ; an allocation rule F satisfies IES if for every n allocation problem (N1, ..., Nk, C1, ..., Ck): F i ðN1 ; :::; Nk ; C1 ; :::; Ck Þ≥
Cr ; i∈Nr ; r = 1; :::; k: n
86
V. Fragnelli, M.E. Marina / Economics Letters 107 (2010) 85–87
The second criterion of satisfaction is the collective usage right (CUR): each agent of the subset N1 ∪ N2 ∪ ... ∪ Nr needs a facility of cost at least Cr and may be required of contributing to the cost of the facility at the level required by agent i ∈ Nr; while the other agents in the subset Nr + 1 ∪ Nr + 2 ∪ ... ∪ Nk that need a facility cheaper than Cr could behave as free riders; so, agent i ∈ Nr may ask to pay at most Cr ; an allocation rule F satisfies CUR if for every allocation ∑l = 1;:::;r nl problem (N1, ..., Nk, C1, ..., Ck): F i ðN1 ; :::; Nk ; C1 ; :::; Ck Þ≤
Cr ; i∈Nr ; r = 1; :::; k: ∑l = 1;:::;r nl
l
pay in order to reach a satisfactory allocation. These two criteria do not imply that two agents in the same group pay the same amount. For instance, consider N = {N1, N2} where N1 = {1, 2}, N2 = {3} with C1 = 12, C2 = 6; in this case l1 = 4, l2 = 2, u1 = 6, u2 = 2 and a possible allocation is (6, 4, 2). Consequently, we consider the widely used criterion of equal treatment (ET): all the agents in a group have to pay the same amount. By definition, the BT rule satisfies ET. Now, we check that it satisfies also IES and CUR: BTi ðN1 ; :::; Nk ; C1 ; :::; Ck Þ =
∑
Cj −Cj +
1
j = r;:::;k ∑l = 1;:::;j nl
Cj −Cj n j = r;:::;k
≥ ∑
+ 1
=
Cr ; n
i∈Nr ; r∈f1; :::; kg BTi ðN1 ; :::; Nk ; C1 ; :::; Ck Þ =
∑
Cj −Cj
+ 1
j = r;:::;k ∑l = 1;:::;j nl
≤ ∑
Cj −Cj
+ 1
j = r;:::;k ∑l = 1;:::;r nl
=
Cr ; ∑l = 1;:::;r nl
i∈Nr ; r∈f1; :::; kg:
In order to satisfy the IES criterion, we may suppose that each Cr plus a fraction of the remaining player i ∈ Nr, r ∈ {1, ..., k} pays n Cl part (surplus) s = C1 − ∑ nl . An allocation (x1, ..., xn) satisfies n l = 1;:::;k also the criteria CUR and ET when for player i ∈ Nr, r ∈ {1, ..., k} we have xi =
Cr n
+ sr , with ∑l= 1,...,knlsl = s and 0≤sr ≤ ∑
Cr l = 1;:::;r nl
− Cnr :
Remark 1. The allocations corresponding to the equal sharing of the surplus and to the proportional sharing of the surplus w.r.t. the cost of the landing strip of each agent do not satisfy CUR as in both cases sk > 0. Moreover, the latter allocation coincides with the proportional allocation rule Pi ðN1 ; :::; Nk ; C1 ; :::; Ck Þ = C1 ∑ Cr n C ; i∈Nr ; r∈f1; :::; kg: l = 1;:::;k
l l
Remark 2. Sharing the surplus proportionally to the gap of each group gr = ∑ Cr n − Cnr ; r∈f1; :::; kg provides the allocation rule l = 1;:::;r
l
Zi ðN1 ; :::; Nk ; C1 ; :::; Ck Þ =
Cr gr +s ; i∈Nr ; r∈f1; :::; kg n ∑l = 1;:::;k nl gl
that satisfies the criteria IES, CUR and ET. Now, we introduce a further criterion of fairness, consistency on last group (CLAST). An allocation rule F satisfies CLAST if for every allocation problem (N1, ..., Nk, C1, ..., Ck) and for each h ∈ Nk: ˆ ; :::; N ˆ ; Cˆ ; :::; Cˆ Þ; i∈N∖fhg F i ðN1 ; :::; Nk ; C1 ; :::; Ck Þ = F i ð N 1 k 1 k where N̂l = Nl, l = 1, …, k − 1, Nk̂ = Nk\{h}1 and C ̂l = Cl − F h (N1, ..., Nk, C1, ..., Ck), l = 1, ..., k. 1
BTi ðN1 ; :::; Nk ; C1 ; :::; Ck Þ =
∑
Cj −Cj + 1
j = r;:::;k−1 ∑l = 1;:::;j nl
+
Ck ; i∈Nr ; r∈f1; :::; kg: n
Analogously, for the allocation problem with N 1̂ , N 2̂ , …, N k̂ (N ̂k ≠ Ø) and C 1̂ , C ̂2, ..., C k̂ we have:
The previous criteria, enable us to state a lower bound lr = Cnr and an upper bound ur = ∑ Cr n on the amount that agent i ∈ Nr has to l = 1;:::;r
This criterion states that if an agent h ∈ Nk withdraws after paying F h (N1, ..., Nk, C1, ..., Ck) and the remaining agents in N ∖ {h} reallocate the remaining amount C1 − Fh(N1, ..., Nk, C1, ..., Ck) redefining the costs C ̂l and the sets N ̂l, l = 1, ..., k the resulting new allocation coincides with the previous one for each agent i ∈ N∖{h}. We check that the BT rule satisfies also CLAST. For the allocation problem with N1, N2, ..., Nk and C1, C2, ..., Ck we have:
If Nk = {h} then N̂k = Ø so the definition of C ̂k is no longer needed.
ˆ ; :::; N ˆ ; Cˆ ; :::; Cˆ Þ = BTi ð N 1 k 1 k
∑
Cˆ j − Cˆ j
+ 1
j = r;:::;k−1 ∑l = 1;:::; j nl
+
Cˆ k ˆ ; r∈f1; :::; kg: ; i∈ N r n−1
Trivially, Cj −Cj + 1 =Cĵ −Cĵ + 1, j=1, ..., k−1; as BTh ðN1 ; :::; Nk ; C1 ; Cˆ k :::; Ck Þ = Cnk , then we have n−1 = Cnk . 3. Characterization Now we can show the main result of this note, that the three fairness criteria IES, CUR and CLAST characterize the BT rule. Theorem 1. The BT rule is the unique allocation rule that satisfies IES, CUR and CLAST for every airport problem. Proof. We already proved that the BT rule satisfies the three criteria. We have to check the uniqueness. Given an allocation rule F , let 1 ≤r ≤k − 1; withdrawing one by one the agents in the groups Nk, Nk− 1, ..., Nr + 1, if F satisfies CLAST we obtain F i ðN1 ; :::; Nk ; C1 ; :::; Ck Þ = F i ðN1 ; :::; Nr ; Cˆ 1 ; :::; Cˆ r Þ; i∈Nr where Cl̂ =Cl − ∑i ∈ Nr + 1 ∪…∪NkF i (N1, …, Nk, C1, …, Ck), l = 1, …, r. If F satisfies IES and CUR then F i ðN1 ; :::; Nk ; C1 ; :::; Ck Þ =
Ck ; i∈Nk n
and F i ðN1 ; :::; Nr ; Cˆ 1 ; :::; Cˆ r Þ =
Cˆ r ∑l = 1;:::r nl
; i∈Nr :
The previous equalities define a system of linear equations with a unique solution. □ Remark 3. The criteria IES, CUR and CLAST are logically independent. share the total The allocation rule in which theagents in N1 equally cost, i.e. F ðN1 ; :::; Nk ; C1 ; :::; Ck Þ = Cn11 ; :::; Cn11 ; 0; :::; 0 satisfies CUR and CLAST but not IES. The equal sharing allocation rule, i.e. F ðN1 ; :::; Nk ; C1 ; :::; Ck Þ = C1 C1 C1 ; ; :::; satisfies IES and CLAST but not CUR. n n n The allocation rule Z in Remark 2 satisfies IES and CUR but not CLAST. 4. Concluding remarks In this note we considered the airport problem. This was widely studied under a game theoretic setting, from the pivotal paper by Littlechild and Thompson (1977); this interest depends on the matter that some game theoretical solution concepts may be computed easily for this class of games. We already mentioned the paper by Littlechild and Owen (1973) on the Shapley value, and we may add also the papers by Littlechild and Owen (1976) and by Branzei et al. (2006) on the
V. Fragnelli, M.E. Marina / Economics Letters 107 (2010) 85–87
computation of the nucleolus (see Schmeidler, 1969). The Shapley value for a cooperative game with transferable utility was characterized by several properties; in particular Hart and Mas-Colell (1989) presented the internal consistency property: “eliminating some of the players, after paying them according to the solution, does not change the outcome for any of the remaining ones”, i.e. what the allocation rule assigns them in the original game is the same as what it would give them in a suitable reduced (smaller) game. The idea of consistency was introduced in the framework of bargaining theory by Harsanyi (1959) and applied in a wider setting of sharing problems by Moulin (1984). In our note we took into account only the airport problem, and proved that a different criterion of consistency that involves only a single agent of the last group, jointly with the criteria IES and CUR, characterize the BakerThompson rule. References Alparslan Gök, Z., Branzei, R., Tijs, S., 2009. Airport interval games and their Shapley value. Operations Research and Decision 2, 9–18.
87
Baker, J., 1965. Airport runway cost impact study. Report submitted to the Association of Local Transport Airlines, Jackson, Mississippi. Branzei, R., Inarra, E., Tijs, S., Zarzuelo, J.M., 2006. A simple algorithm for the nucleolus of airport profit games. International Journal of Game Theory 34, 259–272. Harsanyi, J.C., 1959. A bargaining model for the cooperative n-person games. Annals of Mathematics Studies 40, 325–355. Hart, S., Mas-Colell, A., 1989. Potential, value, and consistency. Econometrica 57, 589–614. Littlechild, S.C., Owen, G., 1973. A simple expression for the Shapley value in a special case. Management Science 20, 370–372. Littlechild, S.C., Owen, G., 1976. A further note on the nucleolus of the “airport game”. International Journal of Game Theory 5, 91–95. Littlechild, S.C., Thompson, G.F., 1977. Aircraft landing fees: a game theory approach. Bell Journal of Economics 8, 186–204. Moulin, H., 1984. The separability axiom and equal-sharing methods. Journal of Economic Theory 36, 120–148. Schmeidler, D., 1969. The nucleolus of a characteristic function game. SIAM Journal of Applied Mathematics 17, 1163–1170. Shapley, L.S., 1953. A value for n-person games. Annals of Mathematics Studies 28, 307–317. Thompson, G.F., 1971, Airport costs and pricing, PhD. Dissertation, University of Birmingham.
Contents lists available at ScienceDirect
Economics Letters j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / e c o l e t
An axiomatic characterization of the Baker-Thompson rule Vito Fragnelli a,⁎, Maria Erminia Marina b a b
University of Eastern Piedmont, Department of Sciences and Advanced Technologies, Viale T.Michel 11 - 15121 Alessandria, Italy University of Genova, Italy
a r t i c l e
i n f o
Article history: Received 21 February 2009 Received in revised form 5 December 2009 Accepted 17 December 2009 Available online 24 December 2009
a b s t r a c t The airport problem is a widely studied allocation problem, with the aim of providing simple and fair sharing rule for the landing fees. In this note we introduce some fairness criteria and characterize the classical BakerThompson allocation rule. © 2009 Elsevier B.V. All rights reserved.
JEL Classification: D63 C71 Keywords: Airport problem Baker-Thompson rule Fairness
1. Introduction In this note we consider a cost allocation problem in which a set of agents N = {1, 2, ..., n} that uses a certain facility has to share the cost C of the facility itself. The cost may represent the building cost and/or the maintenance cost. We suppose that the agents require the facility at different levels of quality so that each agent i ∈ N is associated to a cost ci. We suppose that the agents are ordered in such a way that C = c1 ≥ c2 ≥ ... ≥ cn > 0. Moreover, we suppose that the agents with the equal cost are grouped; consequently, we consider an allocation problem with disjoint sets of agents N1, N2, ..., Nk, where N1 ∪ N2 ∪ ... ∪ Nk =N; let n1, n2, ..., nk be the number of agents of the groups with n1 + n2 + ... +nk =n and let c1 =C1 >C2 > ...>Ck =cn be the costs of the groups. A classical example of this kind of problems is the airport problem (see Littlechild and Thompson, 1977), where N = {1, 2, ..., n} is the set of planes landing in a given time period, C is the cost of the landing strip to be allocated and c1, c2, ..., cn are the costs of the landing strip required by the planes. The airport problem may be extended to a more general situation in which the agents may use k different facilities with the requirement that if an agent asks for using facility r he has to use (or at least he has to pay) also for the facilities r + 1, ..., k. We have in mind a set of independent services that cannot be purchased separately, but sequentially, as explained above. This approach is common in car
⁎ Corresponding author. Tel.: +39 0131 360224; fax: +39 0131 360391. E-mail address: [email protected] (V. Fragnelli). 0165-1765/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2009.12.033
market where cars are available with increasing packages (trim quality levels) plus separate optional extras. The economists Baker (1965) and Thompson (1971) proposed a nice allocation rule for sharing the cost C among the agents, now called the Baker-Thompson (BT) rule: BTi ðN1 ; :::; Nk ; C1 ; :::; Ck Þ =
∑
Cj −Cj
+ 1
j = r;:::;k ∑l = 1;:::;j nl
; i∈Nr ; r∈f1; :::; kg
where Ck + 1 = 0. In 1973 Littlechild and Owen showed that the BT rule coincides with the Shapley value (see Shapley, 1953) of the airport game (see Littlechild and Thompson, 1977). In Alparslan Gök, Branzei and Tijs (2009) the authors notice that no axiomatic characterization is available for the BT rule; so, our motivation in this note is to provide some fairness criteria for possible allocation rules and mainly to characterize the BT rule. 2. The fairness criteria Formally, an allocation rule for an allocation problem is a map F that associates with each allocation problem (N1, ..., Nk, C1, ..., Ck), a unique point F (N1, ..., Nk, C1, ..., Ck) ∈ Rn with ∑i∈N F i (N1, ..., Nk, C1, ..., Ck) =C1. The first criterion of satisfaction is the individual equal sharing (IES): as all the n agents have the right of usage, each agent i ∈ Nr may Cr not ask to pay less then ; an allocation rule F satisfies IES if for every n allocation problem (N1, ..., Nk, C1, ..., Ck): F i ðN1 ; :::; Nk ; C1 ; :::; Ck Þ≥
Cr ; i∈Nr ; r = 1; :::; k: n
86
V. Fragnelli, M.E. Marina / Economics Letters 107 (2010) 85–87
The second criterion of satisfaction is the collective usage right (CUR): each agent of the subset N1 ∪ N2 ∪ ... ∪ Nr needs a facility of cost at least Cr and may be required of contributing to the cost of the facility at the level required by agent i ∈ Nr; while the other agents in the subset Nr + 1 ∪ Nr + 2 ∪ ... ∪ Nk that need a facility cheaper than Cr could behave as free riders; so, agent i ∈ Nr may ask to pay at most Cr ; an allocation rule F satisfies CUR if for every allocation ∑l = 1;:::;r nl problem (N1, ..., Nk, C1, ..., Ck): F i ðN1 ; :::; Nk ; C1 ; :::; Ck Þ≤
Cr ; i∈Nr ; r = 1; :::; k: ∑l = 1;:::;r nl
l
pay in order to reach a satisfactory allocation. These two criteria do not imply that two agents in the same group pay the same amount. For instance, consider N = {N1, N2} where N1 = {1, 2}, N2 = {3} with C1 = 12, C2 = 6; in this case l1 = 4, l2 = 2, u1 = 6, u2 = 2 and a possible allocation is (6, 4, 2). Consequently, we consider the widely used criterion of equal treatment (ET): all the agents in a group have to pay the same amount. By definition, the BT rule satisfies ET. Now, we check that it satisfies also IES and CUR: BTi ðN1 ; :::; Nk ; C1 ; :::; Ck Þ =
∑
Cj −Cj +
1
j = r;:::;k ∑l = 1;:::;j nl
Cj −Cj n j = r;:::;k
≥ ∑
+ 1
=
Cr ; n
i∈Nr ; r∈f1; :::; kg BTi ðN1 ; :::; Nk ; C1 ; :::; Ck Þ =
∑
Cj −Cj
+ 1
j = r;:::;k ∑l = 1;:::;j nl
≤ ∑
Cj −Cj
+ 1
j = r;:::;k ∑l = 1;:::;r nl
=
Cr ; ∑l = 1;:::;r nl
i∈Nr ; r∈f1; :::; kg:
In order to satisfy the IES criterion, we may suppose that each Cr plus a fraction of the remaining player i ∈ Nr, r ∈ {1, ..., k} pays n Cl part (surplus) s = C1 − ∑ nl . An allocation (x1, ..., xn) satisfies n l = 1;:::;k also the criteria CUR and ET when for player i ∈ Nr, r ∈ {1, ..., k} we have xi =
Cr n
+ sr , with ∑l= 1,...,knlsl = s and 0≤sr ≤ ∑
Cr l = 1;:::;r nl
− Cnr :
Remark 1. The allocations corresponding to the equal sharing of the surplus and to the proportional sharing of the surplus w.r.t. the cost of the landing strip of each agent do not satisfy CUR as in both cases sk > 0. Moreover, the latter allocation coincides with the proportional allocation rule Pi ðN1 ; :::; Nk ; C1 ; :::; Ck Þ = C1 ∑ Cr n C ; i∈Nr ; r∈f1; :::; kg: l = 1;:::;k
l l
Remark 2. Sharing the surplus proportionally to the gap of each group gr = ∑ Cr n − Cnr ; r∈f1; :::; kg provides the allocation rule l = 1;:::;r
l
Zi ðN1 ; :::; Nk ; C1 ; :::; Ck Þ =
Cr gr +s ; i∈Nr ; r∈f1; :::; kg n ∑l = 1;:::;k nl gl
that satisfies the criteria IES, CUR and ET. Now, we introduce a further criterion of fairness, consistency on last group (CLAST). An allocation rule F satisfies CLAST if for every allocation problem (N1, ..., Nk, C1, ..., Ck) and for each h ∈ Nk: ˆ ; :::; N ˆ ; Cˆ ; :::; Cˆ Þ; i∈N∖fhg F i ðN1 ; :::; Nk ; C1 ; :::; Ck Þ = F i ð N 1 k 1 k where N̂l = Nl, l = 1, …, k − 1, Nk̂ = Nk\{h}1 and C ̂l = Cl − F h (N1, ..., Nk, C1, ..., Ck), l = 1, ..., k. 1
BTi ðN1 ; :::; Nk ; C1 ; :::; Ck Þ =
∑
Cj −Cj + 1
j = r;:::;k−1 ∑l = 1;:::;j nl
+
Ck ; i∈Nr ; r∈f1; :::; kg: n
Analogously, for the allocation problem with N 1̂ , N 2̂ , …, N k̂ (N ̂k ≠ Ø) and C 1̂ , C ̂2, ..., C k̂ we have:
The previous criteria, enable us to state a lower bound lr = Cnr and an upper bound ur = ∑ Cr n on the amount that agent i ∈ Nr has to l = 1;:::;r
This criterion states that if an agent h ∈ Nk withdraws after paying F h (N1, ..., Nk, C1, ..., Ck) and the remaining agents in N ∖ {h} reallocate the remaining amount C1 − Fh(N1, ..., Nk, C1, ..., Ck) redefining the costs C ̂l and the sets N ̂l, l = 1, ..., k the resulting new allocation coincides with the previous one for each agent i ∈ N∖{h}. We check that the BT rule satisfies also CLAST. For the allocation problem with N1, N2, ..., Nk and C1, C2, ..., Ck we have:
If Nk = {h} then N̂k = Ø so the definition of C ̂k is no longer needed.
ˆ ; :::; N ˆ ; Cˆ ; :::; Cˆ Þ = BTi ð N 1 k 1 k
∑
Cˆ j − Cˆ j
+ 1
j = r;:::;k−1 ∑l = 1;:::; j nl
+
Cˆ k ˆ ; r∈f1; :::; kg: ; i∈ N r n−1
Trivially, Cj −Cj + 1 =Cĵ −Cĵ + 1, j=1, ..., k−1; as BTh ðN1 ; :::; Nk ; C1 ; Cˆ k :::; Ck Þ = Cnk , then we have n−1 = Cnk . 3. Characterization Now we can show the main result of this note, that the three fairness criteria IES, CUR and CLAST characterize the BT rule. Theorem 1. The BT rule is the unique allocation rule that satisfies IES, CUR and CLAST for every airport problem. Proof. We already proved that the BT rule satisfies the three criteria. We have to check the uniqueness. Given an allocation rule F , let 1 ≤r ≤k − 1; withdrawing one by one the agents in the groups Nk, Nk− 1, ..., Nr + 1, if F satisfies CLAST we obtain F i ðN1 ; :::; Nk ; C1 ; :::; Ck Þ = F i ðN1 ; :::; Nr ; Cˆ 1 ; :::; Cˆ r Þ; i∈Nr where Cl̂ =Cl − ∑i ∈ Nr + 1 ∪…∪NkF i (N1, …, Nk, C1, …, Ck), l = 1, …, r. If F satisfies IES and CUR then F i ðN1 ; :::; Nk ; C1 ; :::; Ck Þ =
Ck ; i∈Nk n
and F i ðN1 ; :::; Nr ; Cˆ 1 ; :::; Cˆ r Þ =
Cˆ r ∑l = 1;:::r nl
; i∈Nr :
The previous equalities define a system of linear equations with a unique solution. □ Remark 3. The criteria IES, CUR and CLAST are logically independent. share the total The allocation rule in which theagents in N1 equally cost, i.e. F ðN1 ; :::; Nk ; C1 ; :::; Ck Þ = Cn11 ; :::; Cn11 ; 0; :::; 0 satisfies CUR and CLAST but not IES. The equal sharing allocation rule, i.e. F ðN1 ; :::; Nk ; C1 ; :::; Ck Þ = C1 C1 C1 ; ; :::; satisfies IES and CLAST but not CUR. n n n The allocation rule Z in Remark 2 satisfies IES and CUR but not CLAST. 4. Concluding remarks In this note we considered the airport problem. This was widely studied under a game theoretic setting, from the pivotal paper by Littlechild and Thompson (1977); this interest depends on the matter that some game theoretical solution concepts may be computed easily for this class of games. We already mentioned the paper by Littlechild and Owen (1973) on the Shapley value, and we may add also the papers by Littlechild and Owen (1976) and by Branzei et al. (2006) on the
V. Fragnelli, M.E. Marina / Economics Letters 107 (2010) 85–87
computation of the nucleolus (see Schmeidler, 1969). The Shapley value for a cooperative game with transferable utility was characterized by several properties; in particular Hart and Mas-Colell (1989) presented the internal consistency property: “eliminating some of the players, after paying them according to the solution, does not change the outcome for any of the remaining ones”, i.e. what the allocation rule assigns them in the original game is the same as what it would give them in a suitable reduced (smaller) game. The idea of consistency was introduced in the framework of bargaining theory by Harsanyi (1959) and applied in a wider setting of sharing problems by Moulin (1984). In our note we took into account only the airport problem, and proved that a different criterion of consistency that involves only a single agent of the last group, jointly with the criteria IES and CUR, characterize the BakerThompson rule. References Alparslan Gök, Z., Branzei, R., Tijs, S., 2009. Airport interval games and their Shapley value. Operations Research and Decision 2, 9–18.
87
Baker, J., 1965. Airport runway cost impact study. Report submitted to the Association of Local Transport Airlines, Jackson, Mississippi. Branzei, R., Inarra, E., Tijs, S., Zarzuelo, J.M., 2006. A simple algorithm for the nucleolus of airport profit games. International Journal of Game Theory 34, 259–272. Harsanyi, J.C., 1959. A bargaining model for the cooperative n-person games. Annals of Mathematics Studies 40, 325–355. Hart, S., Mas-Colell, A., 1989. Potential, value, and consistency. Econometrica 57, 589–614. Littlechild, S.C., Owen, G., 1973. A simple expression for the Shapley value in a special case. Management Science 20, 370–372. Littlechild, S.C., Owen, G., 1976. A further note on the nucleolus of the “airport game”. International Journal of Game Theory 5, 91–95. Littlechild, S.C., Thompson, G.F., 1977. Aircraft landing fees: a game theory approach. Bell Journal of Economics 8, 186–204. Moulin, H., 1984. The separability axiom and equal-sharing methods. Journal of Economic Theory 36, 120–148. Schmeidler, D., 1969. The nucleolus of a characteristic function game. SIAM Journal of Applied Mathematics 17, 1163–1170. Shapley, L.S., 1953. A value for n-person games. Annals of Mathematics Studies 28, 307–317. Thompson, G.F., 1971, Airport costs and pricing, PhD. Dissertation, University of Birmingham.