A simple algorithm for the nucleolus of airport profit games

Int J Game Theory 2006 34:259 272 DOI 10.1007/s00182-006-0019-4 ORIGINAL ARTICLE A simple algorithm for the nucleolus of airport profit games Rodica Brânzei Elena Iñarra Stef Tijs José M. Zarzuelo Accepted: 15 December 2005 / Published online: 4 July 2006 Springer-Verlag 2006 Abstract In this paper we present a procedure for calculating the nucleolus for airport profit games which are a generalization of the airport cost games. Keywords Airport cost profit games Nucleolus R. Brânzei A.I Cuza University, Iasi, 700506, Romania e-mail: branzeir@infoiasi.ro http://www.thor.info.uaic.ro/ branzeir E. Iñarra Department of Foundations of Economics Analysis I, Faculty of Economics and Business Administration, Basque Country University, Av. Lehendakari Agirre, 83, 48015, Bilbao, Spain e-mail: jepingae@bs.ehu.es J. M. Zarzuelo B Department of Applied Economics IV, Faculty of Economics and Business Administration, Basque Country University, Av. Lehendakari Aguirre, 83, 48015, Bilbao, Spain e-mail: elpzazaj@bs.eu.es http://www.ehu.es/zarzuelo S. Tijs Department of Mathematics, University of Genoa, Genoa, Italy S. Tijs University of Tilburg, Tilburg, The Netherlands e-mail: S.H.Tijs@uvt.nl http://www.center.uvt.nl/staff/tijs

260 Rodica Brânzei et al. 1 Introduction When Littlechild and Thompson 1977 investigated the cost sharing problem arising from the construction of a landing strip for Birmingham airport they proposed a cooperative game-theoretic approach to solve this problem. In the characteristic function of the game only the airport runway cost function was taken into account. Taking advantage of the special structure of these airport cost games, Littlechild 1974 derived an extremely simple algorithm for the calculation of the nucleolus Schmeidler 1969. Formerly, Littlechild and Owen 1973 had obtained a remarkably simplified formula for the Shapley value Shapley 1953 see also Dubey 1982. More recently, Potters and Sudhölter 1999 have studied some consistency properties for these games. Later, Littlechild and Owen 1977 pointed out that: airport cost games are only a partial representation of the actual situation, for they take no account of the revenues or other benefits generated by aircraft movements. For instance, revenues are important in determining the optimal size of the runway, or the final payoffs to agents, since none of them would agree to pay a fee schedule higher than his revenue. Consequently, these authors introduced airport profit games defining the worth of a coalition as the maximum revenue minus cost attainable by the coalition, whenever that maximum is non-negative, otherwise setting it equal to 0. In their paper, these authors claim that the fee schedule corresponding to the nucleolus of an airport profit game is independent of the revenues and that the nucleolus for these games could be directly determined from the simple expression of the nucleolus for airport cost games derived by Littlechild 1974. This claim is not true 1 and in this paper we determine a procedure to calculate the nucleolus for airport profit games. As we shall see, the proposed algorithm, based on the maximal excesses of coalitions, considers a sequence of airport problems, starting from the original one, where each problem is a reduction of the preceding one. The outline of the paper is as follows. Section 2 contains the model and some preliminary results. In Sect. 3, we obtain the results concerning the coalitions with maximal excess. Section 4 describes the algorithm. 2 Airport problems The tuple N,, C, b is an airport problem if a N is a finite nonempty set. b is a total order relation on N. c C : N R + is non-decreasing i.e., i j implies Ci Cj. d b R N ++. 1 We acknowledge J. Arin by pointing out that simple examples of three-person games can be used to check that the two statements of Littlechild Owen s note are incorrect. As far as we know this mistake has not been corrected until now. This led us to amend a former version of this work.

A simple algorithm for the nucleolus of airport profit games 261 The set N represents the set of agents and every non-empty subset of N is called a coalition. Every agent i wishes to implement a project that generates a cost Ci 0. Moreover, i j means that the project of agent j is an extension of the project of agent i, and accordingly, we assume Ci Cj. In addition, if the project of agent i is carried out, then that player receives a revenue b i > 0. Remark 2.1 The ordering has been included in the description of the model for the sake of clarity of the proofs. Needless to say, this ordering could have been any one induced by C. If there is no confusion we simply write N, C, b. Let x R N, and S N, S =. We write x S for the restriction of x to R S, and xs = i S x i. Therefore, bs is the total revenue of the members of S. Since actually C R N +, with some abuse of notation we denote CS = max {Ci : i S}. Thus, CS represents the cost of attending to the needs of coalition S. By convention C = 0. With each problem N, C, b, we associate a TU game N, v N,C,b, called an airport profit game, where v N,C,b S = max { br CR : R S } for each S N, 1 represents the profits obtained by coalition S. Thus, the members of S N will carry out the most profitable project that is feasible, whenever that project generates a non-negative surplus. The proof of the following proposition is standard and it is omitted. Proposition 2.1 Let N, C, b be an airport problem, then the associated profit game N, v is convex. Let N,, C, b be an airport problem, S a proper coalition of N, and x R N +. The reduced airport problem of N,, C, b with respect to S and x, is the problem S, S, C S,x, b S, where C S,x i = { min C Q {i} bq xq } : Q N\S + for every i S a + denotes max{a,0}. We simply write C x, and S, C x, b if no confusion arises. Note that C S,x depends only on x N\S, so abusing notation, we also write C S,x when x R N\S +. To illustrate the meaning of the reduced problem, suppose that agents in N constructing a runway, agree that agent i should receive a surplus of x i, i.e. i s payment to finance the runway is b i x i. Then we imagine agent i leaving the remaining agents negotiating, and assume agent i specifies the exact part of the runway that he will pay for. It is reasonable to think that he will pay for a fragment between the origin and the last point of the runway that he may use.

262 Rodica Brânzei et al. If agent i defrays the part of the section of the runway closest to this last point, the remaining agents will face a reduced airport problem, whose cost function is exactly C x, and their revenues are the same as in the original situation. Let N, v be a TU game, S N a coalition, and x R N +.Thereduced game of N, v with respect to S and x Davis and Maschler 1965 is the game S, v S,x, given by vn xn \ S if T = S, v S,x T = max { vt Q xq : Q N \ S } if T =, S, 0 if T =. Note that v S,x depends only on x N\S, so abusing notation, we also write v S,x when x R N\S +. Proposition 2.2 Let N, C, b be an airport problem, S a proper coalition of N, and x a core element of v N,C,b. Then v N,C,b S,x = v S,C x,b. In words, the reduced airport surplus game is the game corresponding to the reduced airport problem. Proof First note that if T S N are two coalitions, and x is a core element of v N,C,b, then T, C S,x T,x S, b S = T, C T T,x, b T, and v S,x T,x S = v T,x. Hence it is enough to consider the case in which S = N \{i} for some i N. First of all, notice that { C x R = min CR, C R {i} } b i x i for every R N \{i}. To simplify, write v = v N,C,b, and w = v N\{i},Cx,b. Thus, we have to show that wt = v x T for every T N\{i}. We consider first the case T = N\{i}. Then v x T = max {vt, v T {i} } x i { = max max { br CR, b R {i} C R {i} } } x i R T = max R T {br min { CR, C R {i} b i x i } } = max { br C x R : R T } = wt. Now, let T = N\{i}. Let M N, and M N\{i} be such that vn = b M C M, and v N\{i} = bm CM respectively. From the definition of w and C x it follows

A simple algorithm for the nucleolus of airport profit games 263 w N\{i} b M\{i} C x M\{i} b M\{i} C M + b i x i = b M C M x i = vn x i. Thus, it is enough to show that br C x R vn x i for every R N\{i}. We consider two cases: a C x R = C R. Since x is a core element of v N,C,b then x M = b M C M and xm bm CM. Hence C M C M b M x M b M x M b M\M x i. Notice that if i / M, then x i = 0. Consequently br C x R = br C R b M C M = b M x i C M b M\M + x i + C M C M b M x i C M = vn x i. b C x R = C R {i} b i x i. In this case br C x R = br C R {i} + b i x i = b R {i} C R {i} x i vn x i. 3 The maximal excess at the nucleolus of an airport profit game For the rest of the paper N,, C, b is a fixed airport problem with N 2. We also assume that N ={1, 2,..., n}, and is the usual order. By v we denote the associated airport profit game, and ν its nucleolus. The following numbers are the basic tool for the algorithm that is presented in the next section. α i = b {i + 1,..., n} C N C i, i = 1,..., n 1, n i + 1 β i = Ci, i = 1,..., n 1, i + 1 γ i = b i, i = 1,..., n 1, 2 b N C N δ =. N Define also λ = min {α i : i = n} {β i : i = n} {γ i : i = n} {δ}.

264 Rodica Brânzei et al. Next we determine the nucleolus for some players. We consider separately two cases: λ>0 and λ 0. 2 3.1 Case λ>0. Remark 3.1 a Since 0 <λ α i then b {i,..., n} > Cn Ci 1 for every i N in particular vn = bn CN. b Since 0 <λ β i then Ci >0 for every i N. In what follows we examine the structure of the collection of coalitions with maximal excess at the nucleolus for the case λ>0. Let S N, and x R N +. Then the excess of S at x is We also denote es, x = vs xs. D 1 x ={S N : es, x et, x, for all T N, S = N, }. i.e., the set of proper coalitions of N with maximal excess at x. If {S 1,..., S k } is a partition of N. the family {N\S 1,..., N\S k } formed by the complements is called an antipartition. Some upcoming proofs use the following result. Theorem 1 Arin and Iñarra 1998 For every convex game on N, the family of proper coalitions with maximal excess at the nucleolus contains a partition or an antipartition of N. Our goal is to determine some partitions or antipartitions in D 1 ν. We need some lemmas. Let N, v be a TU game, x R N +, and B a family of coalitions of N. Define eb, x = S B es, x B i.e., the average excess of coalitions in B at x. Remark 3.2 Arin and Iñarra 1998 If B is a partition or an antipartition of N and xn = vn then eb, x is independent of x. Indeed, it is easy to check that a if P is a partition, ep, x = b if A is an antipartition, ea, x = S P vs vn P, S A vs A 1 vn A. 2 It can be shown that these cases correspond respectively to the situation when there are not dummy players in v case λ>0 and to the opposite situationcase λ 0.

A simple algorithm for the nucleolus of airport profit games 265 Lemma 3.1 If P ={S 1,..., S k } is a partition of N, then vn > vs i. S i P Proof If vs j = 0 for every j = 1,..., k, the result follows by Remark 3.1a. So we can assume that v S j0 = 0forsomej0. For every j = 1,..., k, letr j S j be such that vs j = br j CR j.also let T = R l :vr l =0 R l and l t = max{k : k T}. We consider two cases. a l t = n. Let us assume w. l. o. g. that n R k. Then vn = bn CN = bs 1 + +bs k 1 + bs k CN > vr 1 + +vr k 1 + br k CN = vs 1 + +vs k 1 +vs k, where the inequality follows since b S j v Rj for every j = 1,..., k, and b S j0 > v Rj0 by Remark 3.1b. b l t = n. Then let T 0 ={1,..., l t }, and assume w. l. o. g. that l t R k. Then vn = bn CN = bn\t 0 CN CT 0 + bt 0 CT 0 > bt 0 CT 0 br 1 + +br k 1 + br k CR k vr 1 + +vr k = vs 1 + +vs k, where the first equality and the strict inequality follow from Remark 3.1a. Lemma 3.2 If A = {S 1,..., S k } is an antipartition of N, then k 1vN > k vs j. j=1 Proof If vs j = 0forsomej, then the proof is immediate by the monotonicity of v. So we can assume that vs j = 0 for every j = 1,..., k. Let R j S j be such that vs j = br j CR j for every j = 1,..., k. Assume w. l. o. g. that CR 1 = min{cr j : j = 1,..., k}. Since vs 1 >0 then R 1 = ; hence by Lemma 3.1 bwe have CR 1 >0, and therefore vs 1 + vs 2 + +vs k <br 1 + br 2 CR 2 + +br k CR k. 2 Further, since A is an antipartition {N\S 2 R 1,..., N\S k R 1 } is a partition of R 1, and, moreover, since R j S j then R j N\S j R 1 = for j = 2,..., k. So br 1 + br 2 + +br k = b R 2 N\S 2 R 1 + +b R k N\S k R 1.

266 Rodica Brânzei et al. This, together with inequality 2, implies vs 1 + vs 2 + +vs k <b R 2 N\S 2 R 1 CR 2 + +b R k N\S k R 1 CR k. Furthermore, since CR 1 CR j, then C R j N\S j R 1 = CR j for every j = 2,..., k. Hence vs 1 + vs 2 +...+ vs k <b R 2 N\S 2 R 1 C R 2 N\S 2 R 1 + +b R k N\S k R 1 C R k N\S k R 1 k 1vN, where the last inequality follows from the monotonicity of v. Lemma 3.3 For all S D 1 ν, we have es, ν < 0. Proof It follows from Theorem 1, Remark 3.2, and Lemmas 3.1 and 3.2. Lemma 3.4 For all i N, 0 <ν i < b i. Proof Obviously ν i 0 for every i N. Ifν i = 0forsomei N, then e {i}, ν = v {i} 0. But this is in contradiction with Lemma 3.3. Since v S {i} vs b i for every S N, it follows that ν i b i for every i N. Ifν i = b i for some i N, then by Lemma 3.3 we have 0 > e N\{i}, ν = vn\{i} vn + b i. Hence vn vn\{i} >b i and this is impossible. Lemma 3.5 a If S D 1 ν and S > 1, then vs >0. b If S D 1 ν and S > 1, then vs = bs CS. c If {j} D 1 ν, and j = 1, then v {j} = 0. Proof a Assume vs = 0. Let R S, R =. Then vr = 0 and since νs\r >0 it follows er, ν = vr νr = vs νs + νs\r >es, ν. But this contradicts S D 1 ν. b Let R S be a coalition such that vs = vr = br CR. Then R = by part a of this lemma. By Lemma 3.4, er, ν = vr νr = vs νs + νs\r >es, ν. This contradicts S D 1 ν.

A simple algorithm for the nucleolus of airport profit games 267 c If v {j} > 0, then v {j} = b j Cj, and therefore v {1, j} = b 1 +b j Cj. Then e {1, j}, ν = b 1 + b j Cj ν 1 ν j e {j}, ν + b 1 ν 1 > e {j}, ν. This contradicts {j} D 1 ν. A coalition is said to be complete if a S =1, or b S =n 1, or c there exists i 0 = n such that S ={1, 2,..., i 0 }. Lemma 3.6 D 1 ν only contains complete coalitions. Proof Assume that S D 1 ν is not complete. Then n 1 > S > 1, and by Lemma 3.5b it follows that vs = bs CS.Leti N be a predecessor of some member of S. Then v S {i} = bs + b i CS. Hence e S {i}, ν = bs + b i CS = bs CS νs ν i = es, ν + b i ν i > es, ν, where the last inequality follows from Lemma 3.4. This contradicts S D 1 ν. The following proposition identifies some partitions or antipartitions contained in D 1 ν according to Theorem 1. Cases A, C, and D correspond to partitions, while cases B and C to antipartitions. Case C corresponds simultaneously to a partition and an antipartition. Proposition 3.1 At least one of the following statements is true. A there exists { i 0 = n such that i {1,..., i0 }, {i 0 + 1},...,{n} } D 1 ν and ii v {1,..., i 0 } = 0, and v {i 0 + 1} =...= v {n} = 0. B there exists { i 0 = n such that i {1,..., i0 }, N\{1},..., N\{i 0 } } D 1 ν and ii v {1,..., i 0 } = 0. C there exists { i 0 = n such that i {i0 }, N\ {i 0 } } D 1 ν and ii v {i 0 } = 0. D i {1}, {2},...,{n} } D 1 ν and ii v {1} =...= v {n} = 0. Proof We have two cases: a D 1 ν contains a partition P. LetS P of maximal cardinality. We consider three subcases:

268 Rodica Brânzei et al. i S = 1. If v {1} > 0, we are in case A with i 0 = 1. Otherwise v {1} = 0, and we are in case D. ii S =n 1. Here, S = N\{i 0 } for some i 0 N. Ifi 0 = 1 and v {1} > 0 we are in case B. Otherwise, by Lemma 3.5, we are in case C. iii 1 < S < n 1. Here, since S is complete, then S ={1,..., i 0 } for some i 0 N, and by Lemma 3.5 a, vs >0. Moreover, if T P, T = S, then by Lemma 3.6 coalition T is complete, hence it has to be a singleton. So by Lemma 3.5c, we are in case A. b D 1 ν contains an antipartition A. LetS A of minimal cardinality. We consider three subcases: i S =1. Here, since A is an antipartition, A ={S, N\S}. LetS ={i 0 }. Then, if i 0 = 1, and v {1} > 0, we are in case B. Otherwise, we are in case C. ii S =n 1. Here, since A is an antipartition, A = { N\{1},..., N\{n} }. So we are in case B with i 0 = n 1. iii 1 < S < n 1. Here, S is complete, so S = {1,..., i 0 } for some i 0 N, and by Lemma 3.5a, vs > 0. Let T A, T = S. Since S has minimal cardinality, then T > 1. If T < n 1, since T must be complete, N\T N\S =, and A is not an antipartition. So T =n 1 for every T A, T = S, and since A is an antipartition, we are in case B. The following proposition permits to calculate very easily the nucleolus for some players in an airport profit game. Proposition 3.2 At least one of the following statements is true. A Let P = { {1,..., i 0 }, {i 0 + 1},...,{n} },i 0 = n, such that v {1,..., i 0 } = 0, and v {i 0 + 1} =...= v {n} = 0. Then a ep, ν α i0 ; b If P D 1 ν, then ep, ν = α i0. B Let A ={1,..., i 0 }, N\{1},..., N\{i 0 } }, and v {1,..., i 0 } = 0. Then a ea, ν β i0 ; b If A D 1 ν, then ea, ν = β i0. C Let P = { {i 0 }, N\ {i 0 } }, and v {i 0 } = 0. Then a ep, ν γ i0 ; b If P D 1 ν, then ep, ν = γ i0. D Let P ={1}, {2},...,{n} }, and v {1} =...= v {n} = 0. Then i ep, ν δ; ii If P D 1 ν, then ep, ν = δ. Proof We only prove case A. The proofs of the remaining cases are similar.

A simple algorithm for the nucleolus of airport profit games 269 For case i ep, ν = v {1,..., i 0 } + k>i 0 v {k} vn n i 0 + 1 b {1,..., i 0 } C i 0 bn + CN n i 0 + 1 = b {i 0 + 1,..., n} C N Ci 0 n i 0 + 1 = α i0. where the first equality follows from Remark 3.2. For case ii, notice that the inequality above becomes an equality by Lemma 3.5 parts b and c. Proposition 3.3 A If λ = α i0, then ν i = λ for every i > i 0. B If λ = β i0, then ν i = b i λ, for every i i 0. C If λ = γ i0, then ν i0 = λ. D If λ = δ, then ν i = λ for every i N. Proof We only prove case A. By Propositions 3.1 and 3.2, it follows that es, ν = λfor every S D 1 ν. Now if λ = α i0 then by Proposition 3.2, for every i > i 0, we have {i} D 1 ν and v {i} = 0. Hence α i0 = e {i}, ν = ν i. 3.2 Case λ 0. Notice that if λ 0, then necessarily λ = γ i for all i = 1,..., n. Proposition 3.4 a If λ = α i0 0 for some i 0 N, then ν i = 0 for every i > i 0. b If λ = β i0 for some i 0 N then ν i = b i for every i i 0. c If λ = δ 0 then ν i = 0 for every i N. Proof a Let i N such that i 0 < i. We show that i is a null player. First note that since λ = α i0 α i, and α i0 < 0, then for every j i 0 Hence b {k N : i 0 < k} + C i 0 b {k N : j k} + Cj. b {k N : i 0 < k j} Cj C i 0 for all j i 0. 3 Let S N\{i}, and Q S {i} be such that v S {i} = bq CQ. If Q =, then by monotonicity of v, we have v S {i} = vs. So assume Q =, and denote l Q = max{k : k Q}. Then

270 Rodica Brânzei et al. v S {i} = bq CQ = b {k Q : k i 0 } C i 0 + b {k Q : i 0 < k} Cl Q C i 0 vs + b {k Q : i 0 < k} Cl Q C i 0 vs + b {k N : i 0 < k l Q } Cl Q C i 0 vs, where the first inequality follows since i 0 < i implies {k Q : k i 0 } S, and the last one from expression 3 since i 0 l Q. From the monotonicity of v, it follows v S {i} = vs, and we can conclude that i is a null player in v. Consequently, ν i = 0. b If λ = β i0, then Ci 0 = 0. Hence for every i i 0 and every S N\{i}, we have v S {i} = vs + b i. Consequently, ν i = b i. c First, note that since δ α i for every i = n, and δ<0, bn Ci + b {k N : i k} for all i = n. 4 Now let S N, and Q S be such that vs = bq CQ.IfQ =, then 0 vs = bq CQ b {k Q : k l Q } C l Q = bn b {k Q : l Q < k} C l Q 0, where the last inequality follows from expression 4. So we can conclude that vs = 0 for every S N, and the result immediately follows. 4 The algorithm From Propositions 3.3 and 3.4 the following result easily follows. Proposition 4.1 a If λ = α i0, then ν i0 = λ + for every i > i 0. b If λ = β i0, then ν i0 = b i λ + for every i i 0. c If λ = γ i0, then ν i0 = λ + for every i N. d If λ = δ, then ν i = λ + for every i N. Propositions 2.2 and 4.1 permit us the design of an algorithm for calculating the nucleolus of the airport profit game v as follows. We construct a finite sequence of airport problems N m, C m, b m,withm = 1,..., M, where N 1, C 1, b = N, C, b. In each step m we calculate the nucleolus, x, for a subgroup of players Z m N m according to Proposition 4.1. If Z m = N m the algorithm stops. Otherwise, let N m+1 = N m \Z m, and C m+1 = C x m, and consider the reduced problem N m+1, C m+1, b Nm+1.

A simple algorithm for the nucleolus of airport profit games 271 To be precise, at each step m we calculate the following numbers for every i N m \ {l m }, where l m = max{i : i N m }: α m i = {k b Nm : k > i } C m N m C m i { k N m : k > i }, + 1 βi m C m i = { k N m : k i }, + 1 γi m = b i 2, δ m = b N m C m N m. N m Further, we consider the minimum λ m of all these numbers, that is {δ λ m m = min } { αi m : i N m \ {l m } } { β m i : i N m \ {l m } } { γ m i : i N m \ {l m } }. Now define x m i according to: i If λ m = αk m then x m i = λ m + for every i N m, i > k m. ii If λ m = βk m then x m i = b i λ m +, for every i N m, i k m. iii If λ m = γi m then x m i = λ m +. iv If λ m = δ m then x m i = λ m + for every i N m. is define according to these condi- Let Z m be the set of players, for which x m i tions i iv Theorem 2 The nucleolus of the game v is given by ν i = x m i, for every i Z m, m = 1,..., M. Proof Since the nucleolus of a convex game satisfies the reduced game property, the result immediately follows from Propositions 2.2 and 4.1. Remark 4.1 The procedure described above has at most n steps. For each step m, given the special nature of the sets N\N m, the reduced cost function for every i N m is C N m,x i = { min C {i}, C N\N m {i} bn\n m xn\n M }. + Thus in each step, On calculations are made, implying that this algorithm can be performed in On 2 time.

272 Rodica Brânzei et al. References Arin J, Iñarra E 1998 On the nucleolus of convex games. Games Econ Behav 23:12 24 Davis M, Maschler M 1965 The kernel of a cooperative game. Naval Res Logist Q 12:223 259 Dubey P 1982 The Shapley value as aircraft landing fees-revisited. Manag Sci 28:869 874 Littlechild SC 1974 A simple expression for the nucleolus in a special case. Int J Game Theory 3:21 29 Littlechild SC, Owen G 1973 A simple expression for the Shapley value in a special case. Manag Sci 20:370 372 Littlechild SC, Owen G 1977 A further note on the nucleolus of the airport game. Int J Game Theory 5:91 95 Littlechild SC, Thompson GF 1977 Aircraft landing fees: a game theory approach. Bell J Econ 8:186 204 Potters J, Sudhölter P 1999 Airport problems and consistent allocation rules. Math Soc Sci 38:83 102 Schmeidler D 1969 The nucleolus of a characteristic function game. SIAM J Appl Math 17:1163 1170 Shapley LS 1953 A value for n-person game. In: Kuhn HW, Tucker AW eds Contributions to the theory of games II in Annals of Mathematic Studies, vol 28. Princeton University Press, Princeton, pp 307 317