Airport Problems and Consistent Allocation Rules

Mathematical Social Sciences 38 (1999) 83±102

Airport problems and consistent allocation rules

Jos Pottersa,* , Peter SudholterÈ b aMathematical Institute, University of Nijmegen, Postbox 9010, 6500 GL Nijmegen, The Netherlands bInstitute of Mathematical Economics, University of Bielefeld, Postfach 100131, D-33501 Bielefeld, Germany Received 18 November 1997; received in revised form 20 October 1998; accepted 4 November 1998

Abstract

A class of single valued rules for airport problems is considered. The common properties of these rules are ef®ciency, reasonableness and a weak form of consistency. These solutions are automatically members of the core for the associated airport game. Every weighted Shapley value, the nucleolus, and the modi®ed nucleolus turn out to belong to this class of rules. The t-value, however, does not to belong to this class. As a side result we prove that, for airport games, the modi®ed nucleolus and the prenucleolus of the dual game coincide. Furthermore, we investigate monotonicity properties of the rules and axiomatize the Shapley value, nucleolus, and modi®ed nucleolus on the class of airport games.  1999 Elsevier Science B.V. All rights reserved.

Keywords: Airport problem; Airport game; Weighted Shapley value; Nucleolus; Modi®ed nucleolus

JEL classi®cation: 90D30

1. Introduction

Airport games were introduced by Littlechild and Owen (1973). The problem they pose is how the cost of a landing strip should be distributed among users who need runways of different lengths. Further discussion of the model can be found e.g. in Littlechild and Thompson (1977) and Dubey (1982). These authors studied the Shapley value and the nucleolus for this kind of cost sharing problems. In an airport problem there is a ®nite population N and a nonnegative cost function → C: N R1. For technical reasons it is assumed that the population is taken from the set

*Corresponding author. E-mail address: [email protected] (J. Potters)

0165-4896/99/$ ± see front matter  1999 Elsevier Science B.V. All rights reserved. PII: S0165-4896(99)00004-9 84 J. Potters, P. SudholterÈ / Mathematical Social Sciences 38 (1999) 83 ±102 of the natural numbers: players are identi®ed with their `ranking number'. The cost function satis®es the inequality C(i) # C( j) whenever i # j. It is typical for airport problems that the cost C(i) is assumed to be a part of the cost C( j)ifi , j, i.e. a coalition S is confronted with costs c(S): 5 maxi[S C(i). In this way an airport problem generates an airport game (N, c). As the value of each one-person coalition (i) equals C(i), we can rediscover the airport problem from the airport game. In the paper we investigate rules for airport problems (games) and the properties they have. In this introduction we give a short verbal description of and a motivation for the properties we shall meet in the subsequent sections. Many of these properties have a wider scope; they can be de®ned for rules on the set of all TU-games as well. The easiest property is ef®ciency, saying that the contributions of the players cover exactly the total costs. This is such an obvious requirement that one could even be inclined to consider only ef®cient rules as `real rules': this is what we want to achieve. In the theory of TU-games a rule is called reasonable if the payment of each player is between his smallest and his largest marginal. As the cost games related to airport games turn out to be concave, the largest marginal is c(i), the stand-alone costs as Moulin and Shenker (1992) called it, and the smallest marginal is c(N) 2 c(N\i). Moreover, the latter marginal is often zero. Only if i is a player with the largest needs and there are no other players with the same needs, the smallest marginal is positive. The reasonableness property for airport problems says that each player is paying an amount between 0 and his stand-alone costs. He is not paying more than `what he uses' and his co-players are not subsidizing him. Covariance is also a property adopted from the theory of the TU-games. It says that the addition of an additive game changes the solution accordingly. For airport games the addition of an additive game brings us mostly outside the class of airport games. So, the scope of this axiom is very limited: only the addition of games generated by vectors of the form (0, 0, 0,..., x) with x $ 0 can be taken into consideration. It also says that multiplication of the cost function with a positive factor yields a solution that is multiplied with the same factor. An important role is played by consistency properties. If we have an airport problem and a player i has agreed to pay the contribution x ± the amount he has to pay according some rule ± the remaining players are left with the same airport problem but one player less and an amount x that can be used to cover a part of the total costs. The question which part of the costs is covered allows for different answers and each of them leads to a different kind of reduced airport problem. As it is, however, player i's intention to pay only for the part he will use, it seems natural that his payment should be subtracted from the costs of this part. So, every player with higher needs (than i) bene®ts completely from this reduction: C( j) → C( j) 2 x, whenever j . i. For the remaining players k (with k , j) it is important to know where the payment x of player i is used for. There are two simple (and many complicated) answers to that question. In the terminology of landing strips, one can start to cover the costs C(i) from the starting point or from the end point of the landing strip needed by i. The ®rst possibility leads to a new cost function

(C(k) 2 x)1 for k , i and the latter possibility to a new cost function min hC(k), C(i) 2 xj for k , i. We call these two kinds of reduced problems the c-reduced and the n-reduced airport problem. Of course, some technical conditions must be satis®ed to make sure J. Potters, P. SudholterÈ / Mathematical Social Sciences 38 (1999) 83 ±102 85 that the reduced problem is an airport problem again. However, by assuming reasonableness these conditions are satis®ed a great deal. If the leaving player i is the player with the lowest needs, both kinds of reduction agree and, in fact, every reduction that subtracts the amount x from the costs of the part used by the ®rst player gives the same reduced airport problem. In general, consistency says that the rule prescribes for every player in the reduced problem the same contribution as in the original problem. So we get here two consistency concepts called c-consistency and n-consistency. The letters c and n refer to the notation for the modi®ed nucleolus and the nucleolus that is often used. As will be proved in Section 5 c-consistency and n-consistency are the most characteristic properties for the modi®ed nucleolus (c) and the nucleolus (n), respectively. If we require the consistency property only for a player with the lowest needs, only one kind of reduction is possible and we call the consistency property ®rst-player consistency. It is a property shared by many rules. If a rule satis®es ®rst-player consistency, the rule is completely determined by the payment of the `®rst' player in each airport problem, since the payment of the `second' player is the payment of the `®rst' player in the reduced airport problem and so on. The function assigning to each airport problem the payment of the `®rst' player is called the generating function of the rule. To deal with this concept it must be possible to identify in each population the `®rst',the `second' up to the `last' player. For that reason we assumed that the ®nite population in an airport problem was always taken from (a subset of) the natural numbers. Players are identi®ed by their ranking number (like athletes at the Olympic Games). To summarize, many ways to de®ne reduced airport problems are possible. In this paper we investigate two of them leading to c- and n-consistency. If the leaving player is a player with lowest needs (the ®rst player in the population), all kinds of reduction gives the same reduced problem, when we do justice to the intention of the leaving player to pay for his own part only. If a rule is not completely ad-hoc, one may expect that it exhibits some kind of consistency. The problem, however, is to ®nd the kind of reduction that is needed. The answer to that question gives an idea about the `nature of the rule'. Therefore, it is not the consistency property that needs a motivation but the way in which the reduced airport problem is de®ned. The c- and the n-reduction are quite natural ways to come to such a reduction. A different class of desirable properties are the monotonicity properties. It stipulates that players with higher costs are also paying a (weakly) larger contribution. We follow Moulin and Shenker (1992) again in calling this property fair ranking. Next we investigate what happens with the solution if the cost of exactly one player increases or if the size of the population (player set) increases. It hardly needs saying that it would be nice if the player with the increased cost would be going to pay more (cost monotonicity) or if every member of the population was bene®tting from an increase of the population size ( population monotonicity), certainly if the total costs remain the same. In this paper we de®ne and investigate cost allocation rules for airport problems. In Section 2 we recall the de®nition of an airport problem and introduce the idea of ®rst-player consistency and the thereby de®ned generating function. We show that, under mild conditions, a rule satisfying ®rst-player consistency is a `core selector'. 86 J. Potters, P. SudholterÈ / Mathematical Social Sciences 38 (1999) 83 ±102

Section 3 presents known rules like the weighted Shapley value (Kalai and Samet, 1988), the nucleolus (Schmeidler, 1969), the modi®ed nucleolus (Sudholter,È 1996, 1997) and the prenucleolus of the dual game as ®rst-player consistent rules. In fact, the modi®ed nucleolus and the prenucleolus of the dual cost game coincide on the class of airport games. The t-value is not ®rst-player consistent on the class of airport games. In Section 4 we investigate monotonicity properties of these solution concepts. The properties we will discuss are strong monotonicity (Young, 1985), coalitional monotonicity (Young,1985), population monotonicity (Thomson, 1993, Sprumont, 1990) and fair ranking (Moulin and Shenker, 1992). Fair ranking and population monotonicity are common properties of the Shapley value, the nucleolus, and the modi®ed nucleolus. The weighted Shapley values are known to be strongly monotonic (Young, 1985). The nucleolus-like solution concepts turn out to be coalitionally monotonic but not strongly monotonic. In the last section we axiomatize the Shapley value, the nucleolus, and the modi®ed nucleolus on the class of airport games using common axioms like ef®ciency, the equal treatment property and covariance. We also use speci®c axioms like strong monotonicity, n-consistency and c-consistency, respectively.

2. De®nition of airport problems and airport games

Let us assume that in a community a list of public projects is under discussion and a a a a that the projects can be ordered p12p ... pqijwhere p p means that project p j is an extension of project pi. The members of the community have given their opinion about the different plans and niimembers voted for project p . The coalition of people voting for project piiis denoted by N . We assume that n i$ 1 for i 5 1,..., q. The realization of project pi generates cost equal to C(i) and we assume that C(i) , C( j)if a pijp and that C(i) $ 0 for every project p i. The community authorities decide to realize the largest project pq and to charge the community members with the cost C(q)to be made. The paper studies cost allocation rules for this type of problems, rules that take the voting behavior of the community members into consideration. We call such problems airport (cost sharing) problems, just as in the literature, where the projects pi are landing strips of different lengths. To be able to deal with non-symmetric solutions and to have an environment in which `reduced airport problems' make sense, we choose the following framework. Let U 7 IN be a universe of players. An airport problem consists of a ®nite nonvoid player set 7 → N U and a map C: N IR1 satisfying

C (i) # C ( j), if i , j and i, j [ N.

N Sometimes we will understand C as a nonnegative vector in IR1. For every coalition S 7 N we will use

l S 5 max S, and s 5 uSu J. Potters, P. SudholterÈ / Mathematical Social Sciences 38 (1999) 83 ±102 87 to denote the last player in S and the number of players in S, respectively. The members of N ordered after an increasing index will also be denoted by [1], ..., [n] where n 5 uNu. So, e.g., [n] 5 l N. To avoid an unwieldy notation we delete brackets if there is no danger for confusion. We write C[i], N\[i] and c[i] instead of C([i]), N\h[i]j and c(h[i]j). The set of airport problems on U will be denoted by !. An (allocation) rule s (on ! ) assigns a payment vector s(N, C) [ IRN to each airport problem (N, C)in!. Also s(N, C) will sometimes be understood as a vector in IRN. So we will deal with single valued solutions all the time. A rule s is called ef®cient if

N O sii(N, C) 5 max [N C (i) 5 C (l ) 5 C[n]. i[N De®ning ef®ciency in this way implies that we understand the cost C (i) as a `part' of the cost C ( j)ifC(i) # C( j) (cf. the story we started with). The next property of rules is the main topic of this paper. Suppose we have a rule s.

Then x 5 s[1](N, C) is the payment of the ®rst player of N, when the rule s is applied. If only this payment has been done, the remaining players j [ N\[1] are faced with a cost function CÅ( j) 5 C( j) 2 x. These considerations motivate the following de®nition.

De®nition 2.1.

1. Let (N, C) [ ! be an airport problem with n $ 2 and x [ IR. De®ne C [1],x: N\[1] → IR by

C [1],x( j) 5 C ( j) 2 x for j [ N\[1].

If C [1],x $ 0, then (N\[1], C [1],x)isthereduced airport problem of (N, C) with respect to [1] and x. 2. A rule s on ! is ®rst-player consistent, if for every (N, C) [ ! with n $ 2 the following conditions are satis®ed for x 5 s(N, C): 2.1. (NÅÅ, C ): 5 (N\[1], C [1],x [1]) [ ! and ÅÅ 2.2. s(N, C ) 5 xNÅ . 3. For a ®rst-player consistent rule s the associated generating function g s :! → IR is de®ned by

s g (N, C) 5 s[1](N, C) for (N, C) [ !.

4. A function g:! → IR satisfying

g(N, C) # C ( j) for j [ N\[1]

for (N, C) [ ! with n $ 2, is called a generating function. 5. A generating function g generates the rule s g recursively by g 5.1. s [1](N, C) 5 g(N, C) and gg[1],g(N,C ) 5.2. s N\[1](N, C) 5 s (N\[1], C ), if n $ 2. 88 J. Potters, P. SudholterÈ / Mathematical Social Sciences 38 (1999) 83 ±102

Remark 2.2.

1. Note that ®rst-player consistency implies that the rule s is completely determined by the payment s (N,C) made by the ®rst player in each problem. [1] s 2. If s is a ®rst-player consistent rule, then gs is a generating function and s g 5 s. g 3. If g is a generating function, then s g is a ®rst-player consistent rule and g s 5 g.

A further property of rules is needed. We call a rule s reasonable (on both sides) (cf.

Milnor, 1952 and Sudholter,È 1997) if, for all j [ N,0# sj(N, C) # C( j) for all airport problems (N, C) [ !.

Example 2.3. The condition sj(N, C) # C( j) is also `reasonable' in a non-technical meaning of the word. No player pays more than the total cost of the plan (s)he voted for.

The other condition sj(N, C) $ 0 is less `reasonable' than it seems to be. The following rather convincing rule does not satisfy the second condition: rj(N, C) 5 C( j) 2 l where l is chosen such that the rule is ef®cient. So every player j pays the total costs of his plan C( j) but gets some `discount' l to make the solution ef®cient. In fact l 5 oi[N\[n]C(i)/n. It is easy to check that this rule is ®rst-player consistent, ef®cient, and satis®es the ®rst reasonability condition. We use this rule as a counter example at the end of the paper. When we discuss the modi®ed nucleolus in the next section, we will meet a variant of this rule.

First-player consistent, ef®cient and reasonable solutions of airport problems will be the topic of this paper. To say it differently, we will investigate rules generated by functions g: ! → IR satisfying

g(N, C) 5 C[1], if uNu 5 1 and 0 # g(N, C) # C[1], otherwise.

If (N, C) is an airport problem, we de®ne the associated airport game as the cooperative cost game (N, c) with coalition values c(S) 5 maxi[SC(i). Notice that the values of C can be rediscovered from the game (N, c), because C(i) 5 c(i). Therefore the airport problem and the associated cost game are frequently identi®ed. Every airport game is clearly a concave game and has, therefore, a nonempty core. The ®rst theorem states that rules satisfying ®rst-player consistency, ef®ciency, and reasonableness are core selectors.

Theorem 2.4. If s is a rule on ! that satis®es ®rst-player consistency, ef®ciency and reasonableness, then s(N, C) [ Core (N, c) for every airport problem (N, C) [ ! and the associated cost game (N, c).

Proof. The proof is by induction on the number n of players. If there is one player, the theorem follows from ef®ciency. Suppose that the theorem has been proved for problems with less than k players and suppose that (N, C) is an airport problem with k players. Let ÅÅ Å x 5 s(N, C). By ®rst-player consistency we have s (N, C ) 5 xNÅ , where N 5 N\[1], J. Potters, P. SudholterÈ / Mathematical Social Sciences 38 (1999) 83 ±102 89

CÅ 5 C [1],x [1]. By the inductive hypothesis this is a core allocation of the game (NÅ, cÅ ) associated with the reduced airport problem (NÅÅ, C ). If S 7 N Å, then Å cÅ(S) 5 maxi[SiC (i) 5 max [S C(i) 2 x[1]5 c(S) 2 x [1]. In view of x $ 0, reasonableness, and x(S) # cÅ(S) we have x(S) # c(S). If S is a coalition containing player [1], we have x(S\[1]) # cÅ (S\[1]) 5 c(S\[1]) 2 x[1] and therefore, also in this case, x(S) # c(S). h

3. First-player consistency of solution concepts for airport games

In this section we start with the airport game associated with an airport problem and investigate which single valued solution concepts for TU-games are ®rst-player consistent. The solution concepts we will consider are the nucleolus, the prenucleolus of the dual game (also called the antinucleolus of the game), the weighted Shapley value, the modi®ed nucleolus and the t-value. All these rules will turn out to be ®rst-player consistent except the t-value. For each of the ®rst-player consistent rules we will give the generating function.

3.1. The weighted Shapley value

We start with the weighted Shapley value. Let us brie¯y repeat the de®nition (see Kalai and Samet, 1988 for more details). Like the classical Shapley value, the weighted Shapley value is a linear function of the game. So it is enough to de®ne the solution for the elements of a basis of the vector space G N of all TU-games with player set N. For this basis the set of representation games h(N, uSS*)j 7N, the set of duals of the unanimity 7 games (N, uS ), (S N) is chosen. The dual game (N, c*) of a cost game (N, c) is the cost game de®ned by c*(S) 5 c(N) 2 c(N\S). So the dual game assigns to a coalition S the 7 marginal cost of coalition S,ifS joins the coalition N\S.IfS N, the game (N, uS*)is ± 5 the simple game with uS*(T ) 5 1 if and only if S > T . To de®ne a weighted Shapley → value we need a positive weight function on the universe U, i.e. w:U IR11, and a collection 6 5 (Sjj)[I of coalitions such that

1. I 7 IN, and 7 5 2. SjjU, < [IjS 5U, S j> S k5 for j, k [ I.

5 ± 0 ± 5 5 For any coalition T ,U we de®ne T : 5 T > Siiwhere T > S and T > S j5 for j , i. The (w, 6)-weighted Shapley value f w,6 is de®ned to be the linear solution concept given by w(i)/w(S 00), if i [ S f w,6(N, u*) 5 iSH0, otherwise The weighted Shapley value incorporates the idea of a `hierarchical system' (like the 90 J. Potters, P. SudholterÈ / Mathematical Social Sciences 38 (1999) 83 ±102 caste system in traditional Hinduism) as well as the idea of `measures of importance'. The coalition T 0 are the representatives of the `highest caste' among the members of T. The (symmetric) Shapley value f (see Shapley, 1953) is obtained as a weighted Shapley value if all weights coincide and the index set I is a singleton. The ®rst theorem of this section states the ®rst-player consistency of the weighted Shapley value.

Theorem 3.1. The (w, 6 )-weighted Shapley value f w,6 is ®rst-player consistent on the set of airport games and its generating function g 5 gw,6 is de®ned by w([1]) ]]C ([1]), if [1] [ N 0 g(N, C) 5 w(N 0) . 50, otherwise Proof. Let (N, c) be the game with (N, C) [ !. Then

(N, c) 5 C[1](N, uN**) 1 O sdC[i 1 1] 2 C[i](N, u([i11],...,[n])). 1#i,n Therefore

w [1] 0 w,6 w,6 C[1]]]0 , if [1] [ N x 5 f [1](N, c) 5 C[1] f [1] (N, uN*) 5 w (N ) 50, otherwise holds true by the de®nition of f w,6. Hence, if the weighted Shapley value has a generating function, it is the function mentioned in the theorem. The game (NÅ, cÅ ) associated with the reduced problem (NÅÅ, C ), i.e. N Å5 N\[1] and C Å5 C [1],x,is

ÅÅÅ Å (N, c ) 5 (C[2] 2 x)(N, u([2],...,[**n])) 1 O sdC[i 1 1] 2 C[i](N,u ([i11],...,[n])). 2#i,n It is clear from the de®nition of the weighted Shapley value that w,6 ÅÅw,6 7 f (N, u**SSN) 5 f (N, u ) Å if S N. So we are left to prove that w,6 Å w,6 sdC[2] 2 x f (N, u**NNNÅÅ) 5 C[1] f (N, u ) w,6 1sdC[2] 2 C[1] f (N, u([2],...,[* n])),NÅ or equivalently w,6 Å w,6 sdC[1] 2 x f (N, u**NNNÅÅ) 5 C[1] f (N, u ). There are three cases to be considered:

Case (a). [1] [¤ N 0.

Then x 5 0 and N 005 NÅ . The equality follows from the de®nition of f w,6.

Case (b). N 0 consists of [1] only. J. Potters, P. SudholterÈ / Mathematical Social Sciences 38 (1999) 83 ±102 91

w,6 Then x 5 C[1] and we have to prove that f (N, uNN*) Å 5 0. This is clear from the de®nition of f w,6.

Case (c). [1] [ N 00and N contains also other players.

Then NÅÅ005 N \[1]. For i [ N 0we have to prove that w [i] w [i] sdC[1] 2 x ]] 5 C[1]]] . w (NÅ 00) w (N ) This can be done by substituting the expression for x. h

3.2. The nucleolus

The next result is about the ®rst-player consistency of the nucleolus of airport games. For a cost game (N, c), x [ X(N, c) 5 hx [ IRNux(N) 5 c(N)j and a coalition S 7 N let e(S, x, c) 5 x(S) 2 c(S) denote the excess of S at x (with respect to c). The prenucleolus of (N, c) is the unique member n(N, c) of the set

hx [ X(N, c)uu(e(S, x, c))S7Nlex# u(e(S, y, c)) S7N for y [ X(N, c)j.

Nn Here u : IR22→ IR orders the coordinates of a vector in a weakly decreasing way. Note that in the case of an airport game the prenucleolus is automatically individually rational (i.e. ni(N, c) # c(i) for i [ N) by concavity of (N, c), thus it is the nucleolus (see Schmeidler, 1969). It is well-known (see Sobolev, 1975) that the prenucleolus satis®es the reduced game property, i.e., if x 5 n(N, c) and (S, cÅ ) is the reduced game of (N, c) in the sense of Davis and Maschler (1965) de®ned by c(N) 2 x(N\S), if T 5 S cÅ (T ) 5 0, if T 5 5 , 5min fgc(T < Q) 2 x(Q) , otherwise Q,N\S 5 ± 7 then n (S, cÅ ) 5 n (N, c)S for all coalitions S N. Note that the associated game of a reduced airport problem is the (Davis±Maschler)- reduced game of the game associated with the original airport problem. In Littlechild and Thompson (1977) the nucleolus of airport games has been determined. From the results of this paper one can deduce that C[i] n (N, C) 5 min ]] [1] i,n i 1 1 in case uNu . 1. These observations imply directly the following result.

Theorem 3.2. The nucleolus is ®rst-player consistent on the class of airport problems and the generating function g 5 gn is de®ned by 92 J. Potters, P. SudholterÈ / Mathematical Social Sciences 38 (1999) 83 ±102

C [i] min]] , if n $ 2 g (N, C) 5 i,n i 1 1 5C[1], if n 5 1 for (N, C) [ !.

3.3. The modi®ed nucleolus

For the modi®ed nucleolus we proceed differently. We give a formula for a rule and prove that it is a ®rst-player consistent rule. The next step will be that we prove that this rule coincides with the modi®ed nucleolus and the prenucleolus of the dual game. If (N, C) is an airport problem, we can do the following: every player pays the total cost of the plan he is supporting and later on he obtains a discount l in order to make the solution ef®cient. If the discount makes a player's payment negative, he pays nothing. m So more formally, s (N, C)i 5 (C(i) 2 l)1 5 max (0, C(i) 2 l) and l is chosen in such a way that s m is ef®cient . If C is not identically zero, then the number l is uniquely determined by this condition. If C 5 0, then we de®ne l 5 0.

± n21 Remark 3.3. Let (N, C) [ ! satisfy C 0 and n $ 2. Then the ®nite sequence (lii),51 n21 de®ned by lij5 o 5i C[ j]/(n 1 1 2 i) is unimodal, i.e. there is k [ h1,..., n 2 1j such that lii, l 11 for i 5 1,..., k 2 1 and l jj$ l 11 for j 5 k,..., n 2 2. Moreover, with mm l 5 lkjthe rule s is determined by s (N, C) 5 (C( j) 2 l)1 for j [ N. The straightforward proof is left to the reader.

Theorem 3.4. The rule s m on airport problems is ®rst-player consistent.

Proof. Let (N, C) be an airport problem with n $ 2 and C ± 0. Let x 5 s m(N,

C)[1] 5 (C[1] 2 l)1 be the payment by the ®rst player [1] and consider the reduced ÅÅ Å problem (N, C ). Let l be the solution of the equation oj$2 (C[ j] 2 x 2 t)1 5 C[n] 2 x. → Å If we substitute t 5 l 2 x in the function qÅ : t oj$2 (C[ j] 2 t)1 , we obtain oj$2 (C[ j] 2 l)115 C[n] 2 (C[1] 2 l) 5 C[n] 2 x. The ®rst equality follows from the de®nition of l.Sot 5 l 2 x is the unique solution of qÅ (t) 5 CÅÅ[n], i.e. l. Then the ®rst-player consistency of s m follows. h

The next step is to prove that the ®rst-player consistent rule s m assigns to each airport problem the modi®ed nucleolus of the associated airport game. First we repeat the de®nition of the modi®ed nucleolus (SudholterÈ 1996, 1997). When the nucleolus is an attempt to make the excesses of cost games lexicographically as small as possible, the modi®ed nucleolus tries to make the differences between the excesses lexicographically as small as possible. To de®ne the modi®ed nucleolus we need the bi-excess of a pair of coalitions, de®ned by be(S, T, x, c) 5 e(S, x, c) 2 e(T, x, c) for a game (N, c), coalitions S, T 7 N, and x [ X(N, c). The second ingredient to de®ne J. Potters, P. SudholterÈ / Mathematical Social Sciences 38 (1999) 83 ±102 93 the modi®ed nucleolus is the coordinate ordering map Q that orders the coordinates of a NN vector from R223 R in a weakly decreasing way. The modi®ed nucleolus c(N, c) of a game (N, c) is the unique member of the set

hx [ X(N, c)uQ((be(S, T, x, c))S,T7Nlex# Q((be(S, T, y, c)) S,T7N for y [ X(N, c)j. The modi®ed nucleolus is an example of a general nucleolus studied in Maschler et al. (1992). It shares several properties with the (pre)nucleolus (cf. Sudholter,È 1997). But it has also two properties in common with the Shapley value. The ®rst property is self-duality, i.e. the modi®ed nucleolus of the dual cost game is equal to the modi®ed nucleolus of the original game. Another related property that the modi®ed nucleolus shares with the Shapley value is the independence of the interpretation of the game as pro®t game or cost game. The next theorem states that the modi®ed nucleolus of an airport game is the solution s m we introduced before. This means that the modi®ed nucleolus is ®rst-player consistent. As a side result we will see that the modi®ed nucleolus coincides with the prenucleolus of the dual cost game (also called the antinucleolus of the original game).

Theorem 3.5. c(N, C) 5 s m (N, C) for every airport problem (N, C).

Proof. Let (N, C) be an airport problem, x 5 s m (N, C), and (N, c) the associated airport game. In order to prove that c(N, c) 5 x it can be assumed without loss of generality that ± n $ 2 and C 0. Let K 5 h j [ Nuxj 5 0j and G 5 N\K, thus K 5 h[1], ...,[k]j for some k 5 1,...,n 2 1. Let l be the number de®ned in Remark 3.3. If uGu # 1, then C[1] 5?? ?5C[n 2 1] 5 0. The associated airport game assigns the value C[n] to every coalition containing [n]. Then it is clear that the modi®ed nucleolus equals c(N, C) 5 C[n]e[n] because all excesses and therefore all bi-excesses vanish with respect to this point. The other equality C[n]e[n] 5 smm(N, C) follows trivially from the de®nition of s . Therefore we assume uGu $ 2. For the excesses e(S, x, c) the following properties are obvious: e(S, x, c) 5 0, if G 7 S or S 5 5 (3.1)

e(S, x, c) 52l,ifuS > Gu 5 1 (3.2)

2 l # e(S, x, c) # 0 for S 7 N (3.3)

Therefore maxS,T7Nbe(S, T, x, c) 5 l and it suf®ces to show for every y [ X(N, c) satisfying minS7Ne(S, y, c) $2l that y 5 x holds true. Let

} 5 h( j)u j [ Gj < h(i, j)ui [ K, j [ Gj < hK < j u j [ Gj.

Then e(S, x, c) 52l for all S [ } and } is balanced, i.e. there are positive coef®cients dS . 0 such that

O dSS1 5 1, N S[} 94 J. Potters, P. SudholterÈ / Mathematical Social Sciences 38 (1999) 83 ±102

where 1S denotes the indicator function of S considered as element of the Euclidean space IRN. Taking the inner product with y and x gives

y(N) 5 O dSSy(S) $ O d (c(S) 2 l) 5 O d Sx(S) 5 x(N). S[} S[} S[} We conclude e(S, y, c) 52l for S [ } by ef®ciency of y. The proof is ®nished by the N h observation that h1SuS [ } j contains a basis of IR .

Note that, in fact, in the proof of Theorem 3.5 the smallest excess is maximized. Therefore, we also have the following result.

Corollary 3.6. The modi®ed nucleolus of an airport game coincides with the nucleolus of the dual of the airport game.

3.4. The t-value

Example 3.7. (The t-value is not ®rst-player consistent for airport games.)

First we recall the de®nition of the t-value (Tijs, 1981). If (N, c) is a cost game, we de®ne the marginal vector M(N, c)byM(N, c):i 5 c (N) 2 c (N\i), (i [ N). If the game (N, c) has a nonempty core, oi[NiM(N, c) # c (N). The remaining costs are measured by the vector m(N, c) with coordinates

m(N, c):iS5 min :i[Sjc (S) 2 O M(N, c). FGj[S, j±i

For cost games with a nonempty core oi[Nim(N, c) $ c (N). The t-value of a cost game is the unique ef®cient point on the line segment between M(N, c) and m(N, c). For concave cost games we have m(N, c)i 5 c (i) for all players i [ N. The next example shows that the t-value is not ®rst-player consistent. Let N 5 h1,...,4j and C(1)5 C(2)5 1 and C(3)5 C(4)5 2. The t-value can be computed as t (N, C) 5 (1, 1, 2, 2)/3, whereas t(N\1, C 1,1/3) 5 (10, 25, 25)/36.

4. Monotonicity properties

A rule s on the set ! of airport problems satis®es

1. fair ranking: if, for (N, C) [ ! and i, j [ N satisfying C(i) # C( j),

sij(N, C) # s (N, C);

2. monotonicity in costs: if, for (N, C), (N, C9) [ ! and i [ N satisfying C9(i) $ C(i) and C9( j) 5 C( j) for j [ N\i

sii(N, C9) $ s (N, C); J. Potters, P. SudholterÈ / Mathematical Social Sciences 38 (1999) 83 ±102 95

3. population monotonicity: if, for (N, C) [ ! and 5 ± S 7 N

s(N, C)SS# s(S, C );

4. strong monotonicity: if, for (N, C),(N, C9) [ ! and i [ N satisfying C9(i) 2 C9( j) $ C(i) 2 C( j) for j , i and C9(i) $ C(i)

sii(N, C9) $ s (N, C).

These properties have generalizations in the game theoretical context. Let sÄ be a U U 7 single valued solution concept on the set 5 U of cost games (N, c) with N U. Moreover, let s denote its restriction to !.IfsÄ preserves desirability in the sense of Maschler and Peleg (1966), then s satis®es fair ranking. If sÄ satis®es coalitional monotonicity (i.e., if (N, c),(N, c9) [ Ca and i [ N satisfy c(S) 5 c9(S) for all S 7 N\i 7 and c9(S) $ c(S) for all S N with i [ S, then sÄÄii(N, c9) $ s (N, c)), then s satis®es monotonicity in costs. Population monotonicity of sÄ is the straightforward generalization of population monotonicity of s.IfsÄ satis®es strong monotonicity (i.e., if (N, c), (N, c9) [ Ca satisfy c9(S < i) 2 c9(S) $ c(S < i) 2 c(S) for all S 7 N and some i [ N, then sÄÄii(N, c9) $ s (N, c)), then s satis®es strong monotonicity. Some well-known results are the following assertions.

1. The Shapley value, the nucleolus, and the modi®ed nucleolus preserve desirability on U, thus satisfy fair ranking on ! (see Maschler and Peleg, 1966 and Sudholter,È 1997). 2. Every weighted Shapley value satis®es strong monotonicity on U, thus on ! (see Young, 1985). 3. The nucleolus on ! is population monotonic (see Sonmez,È 1993).

Lemma 4.1.

1. The nucleolus on ! satis®es monotonicity in costs. 2. The modi®ed nucleolus on ! satis®es monotonicity in costs.

Proof. Let (N, C), (N, C9) [ !,[i] [ N such that C9[i] $ C[i], C9[ j] 5 C[ j] for [ j] [ N\[i]. If n 5 1, both assertions follows from ef®ciency. Therefore we assume that n $ 2 and that both assertions are valid for airport problems with strictly less players.

n C[ j] ] 1. Let x 5 n(N, C) and x95n(N, C9). Then g (N, C) 5 j 1 1 for some j 5 1,..., n 2 1 by Theorem 3.2. If [ j] , [i], then g nn(N, C) 5 g (N, C9) (by the same theorem), hence the proof follows from ®rst-player consistency and the induction hypothesis. If

[ j] $ [i], one has x[99i]$ x [1]by fair ranking, x [1]9$ x [1]by Theorem 3.2 and x [1] 5?? ?5x[i] follows from Littlechild and Thompson (1977). 2. Let y 5 c(N, C) and y95c(N, C9). By Theorem 3.5 yj 5 (C( j) 2 l)1 and yj9 5 (C9( j) 2 l9)1 for j [ N and some l,l9$0 satisfying l # C[n],l9#C9[n]. There- fore, by ef®ciency, 96 J. Potters, P. SudholterÈ / Mathematical Social Sciences 38 (1999) 83 ±102

O (C ( j)-l)1 5 C[n] j[N and

O (C9( j) 2 l9)1 5 C9[n]. j[N

For i 5 n,we®ndl9#l.Ifi , n,(C9[i] 2 l9)11, (C[i] 2 l) would imply that

O (C9( j) 2 l9)11,O (C ( j) 2 l), j[Nj[N a contradiction. h

Lemma 4.2. The modi®ed nucleolus on ! satis®es population monotonicity.

Proof. Let (N, C) [ ! satisfy n $ 2. Moreover, let [i] [ N, x 5 c(N, C), and xÅ 5 c(NÅ, Å N CNNÅÅ), where N 5 N\[i]. It suf®ces to show that x # xÅ. Choose 0 # l # C[n] 5 C(l ) and ÅÅÅNÅ 0 # l # C(l ) such that xj 5 (C( j) 2 l)1 for j [ N and xÅj 5 (C( j) 2 l )1 for j [ N.We distinguish two cases.

1. i ± n or i 5 n and C[n 2 1] 5 C[n]. Then lÅ # l by ef®ciency. 2. i 5 n and C[n] . C[n 2 1]. As x belongs to the core by Theorem 2.4, we have x(NÅÅÅÅ) # C[n 2 1]. From ef®ciency we have xÅÅ(N ) 5 C[n 2 1]. Therefore, x(N ) # x(N ) and lÅ # l. h

Three-person airport problems show that the nucleolus and the modi®ed nucleolus do not satisfy strong monotonicity on !.

5. Axiomatizations

This section is devoted to axiomatizations of the nucleolus, the modi®ed nucleolus, and the Shapley rule on !. In the following de®nition we give two ways to extend the concepts of reduced airport problem and consistency in case a player [i] ± [1] leaves the game after paying an amount x. The player's payment is subtracted from the cost C[i]. In case of a n-reduced airport problem the last part of C[i] is diminished. For a c-reduced airport problem the ®rst part of C[i] is diminished.

De®nition 5.1. A rule s on ! satis®es

1. the equal treatment property,ifsij(N, C) 5 s (N, C) for (N, C) [ ! and i, j [ N satisfying C(i) 5 C( j). 2. covariance,if 2.1. s(N, aC) 5 as(N, C) for (N, C) [ !,a . 0, and 2.2. whenever (N, C), (N, C9) [ ! satisfy C[ j] 5 C9[ j] for [ j] [ N\[n], and C9[n] 5 C[n] 1 y with y $ 0, then J. Potters, P. SudholterÈ / Mathematical Social Sciences 38 (1999) 83 ±102 97

s(N, C9) 5 s(N, C) 1 ye[n],

where e[n] denotes the [n]-th unit vector; 3. n-consistency, if for (N, C) [ ! with n $ 2 and [i] [ N the following conditions hold: 3.1. (NÅÅ, C ) 5 (N\[i],nC [i],x), de®ned by minhC[ j], C[i] 2 x), if j , i x 5 s (N, C), CÅ [ j] 5 , [i] HC[ j] 2 x, if j . i belongs to !. In this case (NÅÅ, C ) is called the n-reduced airport problem of (N, C) with respect to [i] and x. ÅÅ 3.2. s(N, C ) 5 xNÅ . 4. c-consistency, if for (N, C) [ ! with n $ 2 and [i] [ N\[n] the following conditions hold: 4.1. (NÅÅ,C ) 5 (N\[i],cC [i],x), de®ned by (C[ j] 2 x), if j , i x 5 s (N, C), CÅ [ j] 5 1 , [i] HC[ j] 2 x, if j . i belongs to !. In this case (NÅÅ, C ) is called the c-reduced airport problem of (N, C) with respect to [i] and x. ÅÅ 4.2. s(N, C ) 5 xNÅ .

Remark 5.2.

1. Every n-consistent and every c-consistent rule is ®rst-player consistent, because the corresponding de®nitions coincide for [i] 5 [1]. U 2. If a single valued solution concept sÄÄon U satis®es covariance (i.e., s(N, U N ac 1 b ) 5 as(N, c) 1 b for (N, c) [ U, a . 0, b [ IR ), then its restriction to ! satis®es covariance. U 3. If a single valued solution concept sÄ on U satis®es reasonableness (i.e., 7 7 minh c(S < i) 2 c(S)uS N\ij # sÄ i(N, c) # maxh c(S < i) 2 c(S)uS N\ij U for i [ N and (N, c) [ U ) and the reduced game property, then its restriction to ! satis®es automatically n-consistency, since the associated game of a n-reduced airport problem is the (Davis±Maschler)-reduced game of the game associated with the original airport problem. It is well-known that the prenucleolus has these properties.

This remark directly implies the following result.

Corollary 5.3. The nucleolus rule on ! satis®es n-consistency.

Theorem 5.4. The modi®ed nucleolus rule on ! satis®es c-consistency.

Proof. Let (N, C) [ ! satisfy n $ 2 and let x denote c(N, C), i.e. xj 5 (C( j) 2 l)1 for 98 J. Potters, P. SudholterÈ / Mathematical Social Sciences 38 (1999) 83 ±102 some l $ 0. Let [i] [ N\[n] and de®ne CÅÅ5 cC [i],x [i]. Then (N\[i],C ) is an airport problem by reasonableness and fair ranking. We have to ®nd a number lÅ satisfying the equality

ÅÅÅ O (C[ j] 2 l )1 5 C[n] (5.1) [ j][N\[i] ÅÅ and to prove that (C[ j] 2 l )115 (C[ j] 2 l) for j [ N\[i]. We can rewrite equality 5.1 ÅÅÅ as:o[ j][N\[i][(C[ j] 2 x i] 2 l )1 5 C[n] 2 x[i][.Ifl is de®ned by l: 5 l 2 x i], then we ± h have a solution of 5.1, because x[ j] 5 (C[ j] 2 l)1 for all j i.

Theorem 5.5.

1. The nucleolus is the unique rule on ! that satis®es the equal treatment property, covariance, and n-consistency. 2. The modi®ed nucleolus is the unique rule on ! that satis®es the equal treatment property, covariance, and c-consistency.

Proof.

U 1. It is well-known that the prenucleolus on U satis®es the `equal treatment property' for games, which is a generalization of the property de®ned above, covariance, and consistency. Therefore, the nucleolus rule satis®es the desired properties. Let s be a rule on ! that satis®es the desired properties. Then s satis®es ef®ciency by covariance and n-consistency. The proof of Lemma 6.2.11 of Peleg (1988/89) can be copied literally. Note that ®rst-player consistency and covariance guarantee this property. Moreover, it is a member of the prekernel (cf. Maschler et al. (1979). For a proof of this assertion in a more general framework see Peleg (1988/89). The prekernel is a singleton, hence coincides with the nucleolus, in this case because the corresponding airport games under consideration are concave (see Maschler et al. (1972). 2. The modi®ed nucleolus satis®es the desired properties by Remark 3.3, Theorem 3.5, and Theorem 5.4. In order to show uniqueness let s be a rule that satis®es the desired properties and, thus, ef®ciency (see 1). Then s and c are the same for airport problems with less than three players by covariance and equal treatment. Assume that s coincides with c for every airport problem with less than n players for some n $ 3. Let (N, C) [ !. By covariance we can assume without loss of

generality that C[n 2 1] 5 C[n]. Let x 5 c[n21](N, C) and y 5 s [n21](N, C). By c-consistency it suf®ces to show that x 5 y. Assume, on the contrary, that x ± y. Let ÅÅ Å (N, C1 )bethec-reduced airport problem with respect to [n 2 1] and x and let (N, Å C2 )bethec-reduced airport problem with respect to [n 2 1] and y. By the induction hypothesis and c-consistency of s and c we ®nd: ÅÅ ÅÅ ÅÅ ÅÅ s(N, C11) 5 c(N, C ) 5 cNÅÅ(N, C) and c(N, C22) 5 s(N, C ) 5 sN(N, C). Å Å ÅÅ ÅÅ Let l12and l be the discount in (N, C 1) and (N, C 2), respectively. Then J. Potters, P. SudholterÈ / Mathematical Social Sciences 38 (1999) 83 ±102 99

ÅÅ Å ÅÅ Å c[ j]1(N, C ) 5 (C[ j] 2 x)1 2 l1 )1 and c[ j]2(N,C ) 5 (C[ j] 2 y)1 2 l2 )1 for j ± n 2 1. The equal treatment property and c-consistency of c give ÅÅ Å x 5 c[n][(N, C) 5 c n]1(N, C ) 5 (C[n] 2 x 2 l 1). Equal treatment and c-consistency of s give, together with the induction hypothesis: ÅÅ ÅÅ Å y 5 s[n][(N, C) 5 s n]2[(N, C ) 5 c n]2(N, C ) 5 (C[n] 2 y 2 l 2). Therefore, we ®nd ÅÅ l125 C[n] 2 2x and l 5 C[n] 2 2y. If x . y, we ®nd, for j , n 2 1 ÅÅ c[ j]1(N, C ) 5 ((C[ j] 2 x)11111 2x 2 C[n]) $ ((C[ j] 2 y) 1 2y 2 C[n]) ÅÅ 5 c[ j]2(N, C ). Then ÅÅ ÅÅ O c[ j]1(N, C ) $ O c [j]2(N, C ). [ j][NÅÅ[ j][N By ef®ciency C[n] 2 x $ C[n] 2 y and, therefore, x # y. If we start with y . x we ®nd by the same reasoning y # x. Only x 5 y is possible. h

Remark 5.6. The symmetric Shapley value is the unique rule on the set ! of airport problems satisfying ef®ciency, the equal treatment property, and strong monotonicity.

U For a proof of this assertion, where ! is replaced by the larger set U, refer to, e.g., Neyman (1989) or Peleg (1986). For the sake of completeness we present an analogous proof for the restricted set of airport problems.

Proof. For an airport problem (N, C) we introduce h (N, C): 5 uh[i] u C [i] , C [n]ju. The proof is by induction on h (N, C). If h (N, C) 5 0, we have a completely symmetric C [n] ] situation and s[i] 5 n for all players [i] [ N by ef®ciency and the equal treatment property. The same is true for the Shapley value. If the theorem holds for h (N, C) , k and (N, C) is an airport problem with h (N, C) 5 k, we have C [1] # C [2] #???# C [k] , C [k 1 1] 5???5C [n]. If we de®ne CÅÅby C [i]: 5 minhC[i], C[k]j, we have by the induction hypothesis s (N, CÅÅ) 5 f (C ). Notice that the marginals of the players Å [i] with i # k do not change if we go from C to C and therefore, s[i][(N, C) 5 s i] (N, ÅÅ C ) 5 f[i][(N, C ) 5 f i] (N, C) for i # k by strong monotonicity of s and f. The equal treatment property gives s[ j][(N, C) 5 s n][(N, C) and f j][(N, C) 5 f n] (N, C) for j . k h and ef®ciency gives s[n][(N, C) 5 f n] (N, C).

Some results are summarized in the following tables. The logical independence of the 100 J. Potters, P. SudholterÈ / Mathematical Social Sciences 38 (1999) 83 ±102 properties used in the characterizations of the nucleolus, modi®ed nucleolus, and Shapley value is shown. For a weighted Shapley value f w,6 we assume that it is not symmetric, i.e. that it does not coincide with f. The cardinality of the universe U is assumed to be at least 3. A ` 1 ' is the abbreviation for ìs satis®ed', whereas `2' indicates that this property is not satis®ed. We use the abbreviations ÈFF' for ef®ciency, ÈTP' for the equal treatment property, `COV' for covariance, `SM' for strong monotonicity, `FR' for fair ranking, `MC' for monotonicity in costs, `PM' for population monotonicity, `REAS' for reasonableness, `FPC' for ®rst-player consistency, `n-CONS' for n-consistency, and `c-CONS' for c-consistency. Many of the properties summarized in Table 1 are explicitly checked in the present paper. The remaining properties can easily be shown. Theorem 5.5 and Remark 5.6 present axiomatizations of both nucleolus rules and the Shapley rule each by three properties, respectively. To show this we need rules that satisfy all characterizing properties except one. The rules s 16,...,s are de®ned as follows. Let (N, C) [ !.

1. 0, if C[i] , C[n] s 1 (N, C) 5 C[n] , where p 5 uhi u C[i] 5 C [n]ju. [i] ]],ifC[i] 5 C[n] 5 p 2. 0, if i ± n 2 s [i](N, C) 5 C[1], if i 5 n 5 1. 5C[n] 2 C[n 2 1], otherwise 3. s 3 is the ®rst-player consistent rule de®ned by the generating function g 3:! → IRN 3 that satis®es g (N, C) 5 min[k][N (C[k]/k). 4. 0, if i ± n s 4 (N, C) 5 . [i] HC[n], if i 5 n

5. s 5 is de®ned with the help of a distinguished potential player w [U.Ifw [¤ N, then 55w w ± w s (N, C) 5 n(N, C). If [ N, then de®ne s iiN(N, C) 5 n (N\ ,C \w ) for i and

Table 1 Properties EFF ETP COV SM FR MC PM REAS FPC n-CONS c-CONS f 111 11111 12 2 n 111 21111 11 2 c 111 21111 12 1 t 111 21121 22 2 f w,6 121 12111 12 2 r 111 21122 12 2 J. Potters, P. SudholterÈ / Mathematical Social Sciences 38 (1999) 83 ±102 101

Table 2 Independence EFF ETP COV SM n-CONS c-CONS FPC f 111 12 2 1 n 111 21 2 1 c 111 22 1 1 s 1 112 22 1 1 s 2 211 12 2 2 s 3 112 21 2 1 s 4 122 11 1 1 s 5 121 21 2 1 s 6 121 22 1 1

55 s w(N, C) to be the unique real number such that s satis®es ef®ciency. It should be remarked that the n-orc-reduced airport problem with respect to the exceptional player coincides with the restriction of the airport problem to all other players. 6. s 65is de®ned analogously to s by interchanging the roles of n and c.

It is straightforward to verify the properties of the rules on ! summarized in the Table 2. Therefore n, s 24, s show the logical independence of ef®ciency, the equal treatment property, and strong monotonicity, whereas f, s 35, s or f, s 16, s show that the equal treatment property, covariance, and n-consistency or c-consistency, respectively, are logically independent.

References

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