A characterization of the nucleolus without ... - Semantic Scholar
Soc Choice Welf (2012) 38:355–364 DOI 10.1007/s00355-010-0524-z ORGINAL PAPER
A characterization of the nucleolus without homogeneity in airport problems Yan-An Hwang · Chun-Hsien Yeh
Received: 22 July 2009 / Accepted: 5 December 2010 / Published online: 6 January 2011 © Springer-Verlag 2011
Abstract We consider the problem of sharing the cost of a public facility among agents who have different needs for the facility. We show that the nucleolus is the only rule satisfying equal treatment of equals, last-agent cost additivity, and consistency. 1 Introduction We consider the problem of sharing the cost of a landing strip among airlines who need airstrips of different lengths.1 A “rule” is a function that associates with each such an “airport problem”, an allocation of the cost of the landing strip, which is called a “contribution vector.” Any solution defined on the class of TU games can be used to provide recommendations for airport problems. A well-known example is the nucleolus (Schmeidler 1969), which lexicographically maximizes the “welfare” of the worst-off coalitions. It is known that when applied to airport problems, the nucleolus satisfies the following properties. Equal treatment of equals says that airlines with equal cost parameters (the cost parameter of an airline is the cost of fulfilling the airline’s need) should contribute equal amounts. Homogeneity says that if all cost parameters are multiplied by the same positive number, so should the contribution vector. Last-agent cost additivity says that if the cost parameter of an airline with the largest cost parameter increases by δ, then its contribution should increase by δ, and all other airlines should contribute the same 1 For a survey of this literature, initiated by Littlechild and Owen (1973), see Thomson (2004).
Y.-A. Hwang Department of Applied Mathematics, National Dong Hwa University, Hua-Lien 974, Taiwan e-mail: [email protected] C.-H. Yeh (B) Institute of Economics, Academia Sinica, Taipei 115, Taiwan e-mail: [email protected]
123
356
Y.-A. Hwang, C.-H. Yeh
amounts as they did initially.2 However, the nucleolus is not the only rule satisfying the above properties. The “Shapley value” (Shapley 1953) and the “modified nucleolus” (Sudhölter 1997) are other examples. One may wonder whether the nucleolus can be singled out from the family of rules satisfying the properties defined above. Potters and Sudhölter (1999) provide an interesting answer to this question. They propose the following variable population property, which is an application for airport problems of “Davis-Maschler consistency”(Davis and Maschler 1965) introduced in the theory of TU games.3 When we apply the formula of Davis-Maschler consistency to airport problems, it amounts to the following. Let ci be airline i’s cost parameter. Consider a problem and a contribution vector x chosen by a rule for it. Imagine that airline i pays its contribution xi and “leaves”, and reassess the situation from the viewpoint of the remaining airlines. To take into account airline i’s contribution, we revise the remaining airlines’ cost parameter as follows: (i) for each airline j with c j ≥ ci , its revised cost parameter is ci − xi , and (ii) for each airline j with c j < ci , its revised cost parameter is the maximum of c j and ci − xi . Then, “consistency” of the rule says that for the reduced problem just defined, the components of x pertaining to the remaining airlines should still be chosen by the rule. Potters and Sudhölter (1999) show that the nucleolus is the only rule satisfying homogeneity, equal treatment of equals, last-agent cost additivity, and consistency. We show that in this characterization, homogeneity is redundant.
2 Notation and definitions Let U N be a universe of agents with at least two elements, where N is the set of natural numbers. An airport problem, or simply a problem, is a pair (N , c) where N is the profile of their cost N U is a finite nonvoid agent set and c = (ci )i∈N ∈ R+ parameters satisfying ci ≤ c j , if i < j and i, j ∈ N . For each coalition S N , we use s = |S| to denote the number of agents in S. Agents in N are numbered in the order of increasing cost parameters, and are denoted by [1], . . . , [n] where n = |N |. Let A be the class of all problems on U . A rule is a function defined on A that associates with each problem (N , c) ∈ A a vector x ∈ R N . Our generic notation for rules is S. For each coalition N ⊂ N , (ci )i∈N by c N , (Si (N , c))i∈N by S N (N , c), and so on are denoted. We now introduce the nucleolus. Since for general games, the payoff vector the rule chooses is obtained by solving a sequence of linear programs, it is in general not easy to compute. However, for airport problems, the nucleolus can be calculated by an explicit formula (Littlechild 1974; Sönmez 1994). For our purpose, Sönmez’s formula is the most convenient.
2 Potters and Sudhölter (1999) combine last-agent cost additivity and homogeneity as a property, referred to as covariance. See also Sudhölter (1998). 3 Consistency often plays a key role in axiomatizations, for instance, Peleg et al. (1996); Chun (2005); Hwang (2006), and so on. For a survey of the literature on a general principle of consistency and its converse, see Thomson (2000).
123
Characterization of the nucleolus
357
Nucleolus, N u: For each (N , c) ∈ A, N u [1] (N , c) =
c[k] min1≤k≤n−1 k+1 i−1
N u [i] (N , c) = mini≤k≤n−1 N u [n] (N , c) =
c[n] −
c[k] −
n−1 p=1
N u [ p] (N ,c) k−i+2
p=1
where 2 ≤ i ≤ n − 1
N u [ p] (N , c) .
We are interested in the following properties. Efficiency: For each (N , c) ∈ A,
i∈N
Si (N , c) = max j∈N c j .
Equal treatment of equals: For each (N , c) ∈ A and each pair {i, j} N , if ci = c j , then Si (N , c) = S j (N , c).
Last-agent cost additivity: For each pair (N , c) , N , c of elements of A and = c[n] + δ and for each [ j] ∈ N \{[n]}, c[ j] = c[ j] , then each δ ∈ R+ , if c[n] S[n] N , c = S[n] (N , c) + δ and for each [ j] ∈ N \{[n]}, S[ j] N , c = S[ j] (N , c). In order to formulate consistency, an additional piece of notation is needed. Let N (N , c) ∈ A with n ≥ 2, [i] ∈ N , and x x∈ R . The reduced problem of (N, c) with respect to N = N\{[i]} and x, N , c N , is defined by
(i) for each [ j] ∈ N such that j < i, (c xN )[ j] = min c[ j] , c[i] − x[i] , and (ii)
for each [ j] ∈ N such that j > i, (c xN )[ j] = c[ j] − x[i] .
Consistency: For each (N , c) ∈ A with n ≥ 2 and each [i] ∈ N , if x = S (N , c), then N \ {[i]} , c xN \{[i]} ∈ A and x N \{[i]} = S N \ {[i]} , c xN \{[i]} .4 In contrast to other models of fair allocation, for which a unique definition of the reduced problem usually stands out as most natural, there are several ways of defining the reduced problem. Informally, this is because what has to be divided is not a homogeneous whole (such as a social endowment), but it is composed of segments used differently by different airlines. Our paper adopts one simple and natural way of defining the reduced problem. The following figure helps illustrate the justification of the reduced problem.5 Consider the three-airline problem in Fig. 1 and a contribution vector x chosen by a rule for it. Imagine that one airline, say airline 2, pays its contribution x2 and 4 Since some of the coordinates of c x may be negative, we have to add the requirement that the object N N , c xN should be a well-defined airport problem. Similar requirements have often been imposed in the
theory of TU games. In addition, the reduction operator will preserve the property that agents are numbered in the order of increasing cost parameters. 5 The justification is borrowed from Thomson (2004).
123
358
Y.-A. Hwang, C.-H. Yeh
Fig. 1 Three-airline problem: there are three airlines. For each airline, say airline i, the cost of satisfying its need is represented by the cost parameter ci . For simplicity, assume that c1 < c2 < c3 . Thus, airline 1 uses segment 1. Airline 2 uses segments 1 and 2. Airline 3 uses segments 1,2, and 3. The cost of segment 1 is c1 . The cost of segment 2 is c2 − c1 . The cost of segment 3 is c3 − c2 . The total cost of the airstrip is c3 , which is the sum of c1 (the cost of segment 1), c2 − c1 (the cost of segment 2), and c3 − cs2 (the cost of segment 3)
“leaves”, and reassess the situation from the viewpoint of the remaining airlines.6 The remaining airlines are now left with the amount x2 that can be used to cover part of the total cost. Instead of thinking of the contribution x2 as abstractly covering a part of the total cost, it is appealing to impute x2 to these various segments. But how should these imputations be defined? First, it seems very natural to impute x2 to whatever segments airline 2 uses, namely, segments 1 and 2. In other words, because airline 2’s contribution is meant to pay only for the segments it uses, it makes sense to subtract x2 from the costs of segments 1 and 2. This implies that for airline 3 whose cost parameter is greater than c2 , airline 3 benefits completely from the contribution of airline 2 since contributing to the segments airline 2 uses implies contributing to the segments airlines 2 and 3 use jointly. The cost parameter of airline 3 is then revised down by the amount x2 , and its revised cost parameter is c3 − x2 . For airline 1 whose cost parameter is smaller than c2 , it is important to know where x2 is used. Potters and Sudhölter (1999) propose two simple and natural answers to this question: (i) one can first cover the cost of segment 1, and if there is money left, then the remainder is used to help cover the cost of segment 2; (ii) alternatively, one can first cover the cost of segment 2, and if there is money left, then the remainder is used to help cover the cost of segment 1. We adopt the latter formula.7 6 The consistency property allows the possibility of the departure of an arbitrary airline. For purpose of illustration, we focus on the departure of airline 2. 7 To elaborate on the idea of the former formula, we think of x as a contribution to the part of the 2
airstrip that airlines 1 and 2 use jointly. Namely, x2 is used to cover the cost of segment 1. For airline 1 whose cost parameter is smaller than c2 , if x2 ≤ c1 (or equivalently, c1 − x2 ≥ 0), then the cost of segment 1 is either not completely covered or just covered. In either case, there is no money left to help cover the cost of segment 2. Since airline 1 benefits completely from the contribution of airline 2, then the cost parameter of airline 1 is revised down by the amount x2 , and its revised cost parameter is c1 − x2 . If x2 > c1 (or equivalently, c1 − x2 < 0), then the cost of segment 1 is completely covered, and there is money left (namely, x2 − c1 ) to help cover the cost of segment 2. Since airline 1 can’t
123
Characterization of the nucleolus
359
To elaborate on the idea, we think of x2 as a contribution first to the part of the airstrip that airline 2 uses but airline 1 does not. Thus, x2 is used to cover the cost of segment 2. The following two cases are considered: •
•
If x2 ≤ c2 − c1 (or equivalently, c1 ≤ c2 − x2 ), then the cost of segment 2 is either not completely covered or just covered. In either case, there is no money left to help cover the cost of segment 1. Thus, airline 1 does not benefit from the contribution of airline 2, and its cost parameter remains unchanged. If x2 > c2 − c1 (or equivalently, c1 > c2 − x2 ), then the cost of segment 2 is completely covered, and there is money left (namely, x2 − (c2 − c1 )) to help cover the cost of segment 1. In this case, airline 1 benefits by the amount x2 − (c2 − c1 ) from the contribution of airline 2. The cost parameter of airline 1 is revised down by the amount x2 − (c2 − c1 ). Its revised cost parameter is c2 − x2 .
In sum, if c1 ≤ c2 − x2 , then the revised cost parameter of airline 1 is c1 , and if c1 > c2 − x2 , then its revised cost parameter is c2 − x2 . Thus, the definition of the revised cost parameter of airline 1 is the minimum of c1 and c2 − x2 . The consistency property being adopted is based on the above simple and natural way of defining the reduced problem. The nucleolus is consistent. As Potters and Sudhölter (1999) point out, however, it is not the only consistent rule. Another wellknown example is the so-called “constrained equal contributions” rule (Aadland and Kolpin 1998).8 3 Result To prove the new characterization of the nucleolus, we use the following two lemmas. The first lemma says that the nucleolus satisfies a cost monotonicity property. −c Lemma 1 Let (N , c) , N , c ∈ A. If c[1] [1] ≤ · · · ≤ c[n] −c[n] and c[n] −c[n] ≥ 0, then N u [n] (N , c) ≤ N u [n] N , c . Proof Let x = N u (N , c) and y = N u N , c . The proof that x[n] ≤ y[n] is by induction on n. For n = 1, by efficiency of the nucleolus, x[n] ≤ y[n] . The induction hypothesis is that x[n] ≤ y[n] for 1 ≤ n ≤ k. We now show that x[n] ≤ y[n] for n = k + 1. Let n = k + 1 and N = N \ {[1]}. Consider the two reduced problems, N , c xN y y and N , c N . By consistency of the nucleolus, N , c xN ∈ A and N , c N ∈ A. y
− y , and that Note that for each [i] ∈ N , (c xN )[i] = c[i] − x[1] and (c N )[i] = c[i] [1]
Footnote 7 continued benefit more than its cost parameter c1 , its revised cost parameter is defined to be 0. In sum, if c1 − x2 ≥ 0, then the revised cost parameter of airline 1 is c1 − x2 , and if c1 − x2 < 0, then its revised cost parameter is 0. Thus, the definition of the revised cost parameter of airline 1 is the maximum of 0 and c1 − x2 . 8 Aadland and Kolpin (1998) refer to it as the “restricted average cost share rule”. The terminology being adopted here is borrowed from Thomson (2004). Given an airport problem, we can transform the problem into a TU game. When the worth of each coalition is defined to be the largest cost parameter of any member of that coalition, the contribution vector recommended by the constrained equal contributions rule coincides with that prescribed by a well-known solution in TU games, the so-called “Dutta-Ray solution” (Dutta and Ray 1989).
123
360
Y.-A. Hwang, C.-H. Yeh y
y
x x −c c[2] [2] ≤ · · · ≤ c[n] − c[n] . Thus, (c N )[2] − (c N )[2] ≤ · · · ≤ (c N )[n] − (c N )[n] . c[k] c[k] , It can be shown that since x[1] = min1≤k≤n−1 k+1 and y[1] = min1≤k≤n−1 k+1 y
−c . Thus, (c ) −(c x ) then y[1] − x[1] ≤ c[n] ≥ 0. Since |N | = n −1 < n, by [n] N [n] N [n] x y the induction hypothesis, N u [n] N , c N ≤ N u [n] N , c N . Then by consistency of y the nucleolus, x[n] = N u [n] N , c xN and y[n] = N u [n] N , c N . Thus, x[n] ≤ y[n] .
The next lemma says that efficiency is implied by the properties considered. Lemma 2 If a rule satisfies equal treatment of equals, last-agent cost additivity, and consistency, then it satisfies efficiency. Proof Let S be a rule satisfying the three properties. We first show that S satisfies “one-agent efficiency,” which is efficiency for one-agent problems. We then use the induction argument to show that S is efficient in general. To verify one-agent efficiency, we first show that for each p ∈ U, S({ p}, 0) = 0. By last-agent cost additivity, we then conclude that S({ p}, c) = c. Let p ∈ U . Choose l ∈ U and l = p. Consider the two-agent problem ({l, p}, c) ∈ A with 0 = cl = c p . Let x = S({l, p}, c). By equal treatment of equals, xl = x p . Suppose that x ), ({ p}, c x ) ∈ A the order of {l, p} is l = [1] and p = [2]. By consistency, ({l}, c{l} { p} x x x and xl = S({l}, c{l} ), x p = S({ p}, c{ p} ). Thus, c{l} = min{cl , c p − x p } ≥ 0 and x = 0. Hence c{xp} = c p − xl ≥ 0. It follows that xl = x p ≤ c p = 0 and c{l} x x xl = S({l}, c{l} ) = S({l}, 0) and x p = S({ p}, c{ p} ) = S({ p}, c p −xl ) = S({ p}, −x p ). We now suppose that the order of {l, p} is l = [2] and p = [1]. By the same argument, x ) = S({l}, c −x ) = we derive that x p = S({ p}, c{xp} ) = S({ p}, 0) and xl = S({l}, c{l} l p S({l}, −xl ). Thus, x p = S({ p}, −x p ) and x p = S({ p}, 0). By last-agent cost additivity, x p = S({ p}, −x p ) = S({ p}, 0) − x p = x p − x p = 0. Hence S({ p}, 0) = 0. We next show that S is efficient in general. Let (N , c) ∈ A be a n-agent problem. The induction hypothesis is that S is efficient for 1 ≤ n ≤ k. We show that S is efficient for n = k + 1. Let n = k + 1, x = S(N , c), and [1] ∈ N . By consistency, N \ {[1]} , c xN \{[1]} ∈ A and x N \{[1]} = S N \ {[i]} , c xN \{[1]} . By the definition of c xN \{[1]} , (c xN \{[1]} )[n] = c[n] − x[1] . By the induction hypothesis, i∈N \{[1]} xi = c[n] − x[1] . Hence i∈N xi = c[n] . Thus, S is efficient. With the help of Lemmas 1 and 2, we are ready to prove our announced characterization. Theorem 1 The nucleolus is the only rule satisfying equal treatment of equals, lastagent cost additivity, and consistency. Proof Clearly, the nucleolus satisfies equal treatment of equals, last-agent cost additivity, and consistency. Conversely, let S be a rule satisfying the three properties. Let (N , c) ∈ A, x = S (N , c), and y = N u (N , c). The proof that x = y is by induction on n. For n = 1, the three properties altogether imply efficiency (Lemma
123
Characterization of the nucleolus
361
2). Thus, x = y. Suppose that the statement holds for 1 ≤ n ≤ k. We show that ∗ x = y for n = k + 1. The induction for each N , c ∈ A with ∗ hypothesis is that ∗ N ⊂ N and n ≤ n − 1, S N , c = N u N , c . We first show that x[n] = y[n] . By consistency and the induction hypothesis, we then conclude that x = y. Let N = N \ {[n]}. We consider two cases. Case: 1 c[n] = c[n−1] . By equal treatment of equals, x[n−1] = x[n] and y[n−1] = y[n] . Suppose, by contradiction, that x[n] = y[n] . Thus, either x[n] > y[n] or x[n] < y[n] . If x[n] > y[n] , then x[n−1] > y[n−1] . We now show that the two reduced y problems, N , c xN and N , c N , satisfy the hypotheses of Lemma 1. By con y x sistency, N , c N ∈ A and N , c N ∈ A. Since x[n] > y[n] , it follows that
y (c xN )[n−1] = min c[n−1] , c[n] − x[n] ≤ min c[n−1] , c[n] − y[n] = (c N )[n−1] . Since y y c[1] ≤ · · · ≤ c[n] and x[n] > y[n] , then (c N )[1] − (c xN )[1] ≤ · · · ≤ (c N )[n−1] − y (c xN )[n−1] . By Lemma 1, N u [n−1] N , c N ≥ N u [n−1] N , c xN . By consistency , x y x[n−1] = S[n−1] N , c N and y[n−1] = N u [n−1] N , c N . Note that |N | = n − 1 < n. x By the induction hypothesis, S[n−1] N , c N = N u [n−1] N , c xN . It follows that x[n−1] ≤ y[n−1] , in violation of y[n−1] < x[n−1] . If x[n] < y[n] , then we derive the desired contradiction by a similar argument. = c Case: 2 c[n−1] < c[n] . Let c[n] [n−1] , and let c[ j] = c[ j] for each [ j] ∈ N \{[n]}. By Case 1, S[n] N , c = N u [n] N , c . Let δ ∈ R+ with δ = c[n] − c[n−1] , by lastagent cost x[n] = S[n] (N , c) = S[n] N , c + δ and y[n] = N u [n] (N , c) = additivity, N u [n] N , c + δ. Since S[n] N , c = N u [n] N , c , we conclude that x[n] = y[n] .
The independence of properties listed in Theorem 1 can be established by adopting the rules φ, σ 3 , and σ 5 in Table 2 of Potters and Sudhölter (1999).
4 Discussion In addition to our Theorem 1 strengthening Potters and Sudhölter (1999)’s characterization of the nucleolus, our method of proof has the advantage of being self-contained and direct. Indeed, Potters and Sudhölter (1999) use Peleg (1988/89)’s characterization of the “prekernel”9 (Davis and Maschler 1965), and coincidence between the prekernel and the nucleolus on the class of convex games to characterize the nucleolus. Potters and Sudhölter (1999) find that efficiency is implied by homogeneity, lastagent cost additivity, and “first-agent consistency,” which is a weaker version of consistency, obtained by focusing on the departure of an agent with the smallest cost parameter. Strengthening first-agent consistency to consistency in the above finding, our Lemma 2 says that homogeneity can be replaced by equal treatment of equals. One may wonder whether consistency can be weakened to first-agent consistency in Lemma 2. The following modified Shapley value, Sh , shows the answer to be negative. 9 See also Peleg (1986).
123
362
Y.-A. Hwang, C.-H. Yeh
For each (N , c) ∈ A and for each [i] ∈ N , let a c[ j] − c[ j−1] + , n n− j +1 i
Sh [i] (N , c) =
j=1
where c[0] = 0 and a < 0. It can be shown that the modified Shapley value satisfies equal treatment of equals, last-agent cost additivity, and first-agent consistency, but violates efficiency. Acknowledgements We would like to thank William Thomson for helpful suggestions and discussions. We are grateful to Takashi Hayashi, Fan-Chin Kung, Xiao Luo, Man-Chung Ng, and Çaˇgatay Kayı for detailed comments. We are also grateful to an anonymous referee and the Associate Editor in charge for their suggestions. Yeh greatly acknowledges financial support from National Science Council of Taiwan under grant NSC96-2628-H-001-060-MY2, and 2010 Career Development Award of Academia Sinica, Taiwan. We are responsible for any remaining deficiency.
Appendix In the discussion, we claim that the modified Shapley value satisfies equal treatment of equals, last-agent cost additivity, and first-agent consistency, but violates efficiency. Here is a proof of this assertion. Note that the Shapley value, denoted by Sh, is defined as follows: for each (N , c) ∈ A and for each [i] ∈ N , Sh [i] (N , c) =
i
c[ j] − c[ j−1] j=1
n− j +1
,
where c[0] = 0.
Thus, the modified Shapley value, denoted by Sh , is defined by adding a constant term to the definition of the Shapley value. Formally, for each (N , c) ∈ A and for each [i] ∈ N ,
Sh [i] (N , c) =
a + Sh [i] (N , c), n
where a < 0. claim 1 The modified Shapley value satisfies equal treatment of equals, last-agent cost additivity, and first-agent consistency, but violates efficiency. Proof It is well known that the Shapley value satisfies equal treatment of equals and last-agent cost additivity. Thus, it is clear that the modified Shapley value satisfies these properties as well. Since a < 0, by efficiency of the Shapley value, the modified Shapley value violates efficiency. To verify first-agent consistency, let (N , c) ∈ A, x = Sh (N , c) and y = Sh (N , c). The Shapley value satisfies first-agent consistency (Theorem 3.1 of Potters and Sudhölter (1999)). Hence by first-agent consistency of the Shapley value,
123
Characterization of the nucleolus
363
(N \[1], c xN \{[1]} ) ∈ A and x N \{[1]} = Sh N \ {[1]} , c xN \{[1]} . It follows that for each y
[ j] ∈ N \{[1]}, (c xN \{[1]} )[ j] ≥ 0. It is clear that for each [ j] ∈ N \{[1]}, (c N \{[1]} )[ j] = (c xN \{[1]} )[ j] − an . By combining this fact, (c xN \{[1]} )[ j] ≥ 0, and a < 0, we derive y y (c N \{[1]} )[ j] ≥ 0. Thus, (N \{[1]}, c N \{[1]} ) ∈ A. Moreover, by the formula of the Shapley value, we have for each [i] ∈ N \{[1]}, i (c
N \{[1]} )[ j] − (c N \{[1]} )[ j−1] x
Sh [i] (N \{[1]}, c xN \{[1]} )
=
x
n−1− j +2
j=2
y
y Sh [i] (N \{[1]}, c N \{[1]} )
=
Since
y
[1]
= (c N \{[1]} )
y (c N \{[1]} )[ j]
=
n−1− j +2
− an , we have obtained y
=
,
= 0.
[1] (c xN \{[1]} )[ j]
y Sh [i] (N \{[1]}, c N \{[1]} )
y
i (c
N \{[1]} )[ j] − (c N \{[1]} )[ j−1] j=2
where (c xN \{[1]} )
, and
y
i (c
N \{[1]} )[ j] − (c N \{[1]} )[ j−1] j=2
n−1− j +2
x i (c x
N \{[1]} )[ j] − (c N \{[1]} )[ j−1] −a + = n(n − 1) n−1− j +2 j=2
−a = + Sh [i] (N \{[1]}, c xN \{[1]} ). n(n − 1) Thus, by first-agent consistency of the Shapley value and the definition of the modified Shapley value, we have for each [i] ∈ N \{[1]},
y
Sh [i] (N \{[1]}, c N \{[1]} ) = = = = =
a y + Sh [i] (N \{[1]}, c N \{[1]} ) n−1 −a a + + Sh [i] (N \{[1]}, c xN \{[1]} ) n − 1 n(n − 1) a + Sh [i] (N \{[1]}, c xN \{[1]} ) n a + Sh [i] (N , c) n Sh [i] (N , c).
References Aadland D, Kolpin V (1998) Shared irrigation costs: an empirical and axiomatic analysis. Math Soc Sci 35:203–218 Chun Y (2005) The separability principle in bargaining. Econ Theory 26:227–235 Davis M, Maschler M (1965) The kernel of a cooperative game. Nav Res Logist Q 12:223–259
123
364
Y.-A. Hwang, C.-H. Yeh
Dutta B, Ray D (1989) A concept of egalitarianism under participation constraints. Econometrica 57: 615–635 Hwang Y-A (2006) Associated consistency and equal allocation of nonseparable costs. Econ Theory 28:709–719 Littlechild SC, Owen G (1973) A simple expression for the Shapley value in a special case. Manag Sci 3:370–372 Littlechild SC (1974) A simple expression for the nucleolus in a special case. Int J Game Theory 3:21–29 Peleg B (1986) On the reduced game property and its converse. Int J Game Theory 15:187–200 Peleg B (1988/1989) Introduction to the theory of cooperative games. Hebrew University of Jerusalem, Israel Peleg B, Potters J, Tijs S (1996) Minimality of consistent solutions for strategic games, in particular for potential games. Econ Theory 7:81–93 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 games. SIAM J Appl Math 17:1163–1170 Shapley LS (1953) A value for n-person games. In: Kuhn H, Tucker AW (eds) Contributions to the theory of games. Princeton University Press, Princeton pp 307–317 Sönmez T (1994) Population monotonicity of the nucleolus on a class of public good problems. University of Rochester, mimeo Sudhölter P (1997) The modified nucleolus: properties and axiomatizations. Int J Game Theory 26:147–182 Sudhölter P (1998) Axiomatizations of game theoretical solutions for one-output cost sharing problems. Games Econ Behav 24:142–171 Thomson W (2000) Consistent allocation rules. University of Rochester, mimeo Thomson W (2004) Cost allocation and airport problems. University of Rochester, mimeo
123
A characterization of the nucleolus without homogeneity in airport problems Yan-An Hwang · Chun-Hsien Yeh
Received: 22 July 2009 / Accepted: 5 December 2010 / Published online: 6 January 2011 © Springer-Verlag 2011
Abstract We consider the problem of sharing the cost of a public facility among agents who have different needs for the facility. We show that the nucleolus is the only rule satisfying equal treatment of equals, last-agent cost additivity, and consistency. 1 Introduction We consider the problem of sharing the cost of a landing strip among airlines who need airstrips of different lengths.1 A “rule” is a function that associates with each such an “airport problem”, an allocation of the cost of the landing strip, which is called a “contribution vector.” Any solution defined on the class of TU games can be used to provide recommendations for airport problems. A well-known example is the nucleolus (Schmeidler 1969), which lexicographically maximizes the “welfare” of the worst-off coalitions. It is known that when applied to airport problems, the nucleolus satisfies the following properties. Equal treatment of equals says that airlines with equal cost parameters (the cost parameter of an airline is the cost of fulfilling the airline’s need) should contribute equal amounts. Homogeneity says that if all cost parameters are multiplied by the same positive number, so should the contribution vector. Last-agent cost additivity says that if the cost parameter of an airline with the largest cost parameter increases by δ, then its contribution should increase by δ, and all other airlines should contribute the same 1 For a survey of this literature, initiated by Littlechild and Owen (1973), see Thomson (2004).
Y.-A. Hwang Department of Applied Mathematics, National Dong Hwa University, Hua-Lien 974, Taiwan e-mail: [email protected] C.-H. Yeh (B) Institute of Economics, Academia Sinica, Taipei 115, Taiwan e-mail: [email protected]
123
356
Y.-A. Hwang, C.-H. Yeh
amounts as they did initially.2 However, the nucleolus is not the only rule satisfying the above properties. The “Shapley value” (Shapley 1953) and the “modified nucleolus” (Sudhölter 1997) are other examples. One may wonder whether the nucleolus can be singled out from the family of rules satisfying the properties defined above. Potters and Sudhölter (1999) provide an interesting answer to this question. They propose the following variable population property, which is an application for airport problems of “Davis-Maschler consistency”(Davis and Maschler 1965) introduced in the theory of TU games.3 When we apply the formula of Davis-Maschler consistency to airport problems, it amounts to the following. Let ci be airline i’s cost parameter. Consider a problem and a contribution vector x chosen by a rule for it. Imagine that airline i pays its contribution xi and “leaves”, and reassess the situation from the viewpoint of the remaining airlines. To take into account airline i’s contribution, we revise the remaining airlines’ cost parameter as follows: (i) for each airline j with c j ≥ ci , its revised cost parameter is ci − xi , and (ii) for each airline j with c j < ci , its revised cost parameter is the maximum of c j and ci − xi . Then, “consistency” of the rule says that for the reduced problem just defined, the components of x pertaining to the remaining airlines should still be chosen by the rule. Potters and Sudhölter (1999) show that the nucleolus is the only rule satisfying homogeneity, equal treatment of equals, last-agent cost additivity, and consistency. We show that in this characterization, homogeneity is redundant.
2 Notation and definitions Let U N be a universe of agents with at least two elements, where N is the set of natural numbers. An airport problem, or simply a problem, is a pair (N , c) where N is the profile of their cost N U is a finite nonvoid agent set and c = (ci )i∈N ∈ R+ parameters satisfying ci ≤ c j , if i < j and i, j ∈ N . For each coalition S N , we use s = |S| to denote the number of agents in S. Agents in N are numbered in the order of increasing cost parameters, and are denoted by [1], . . . , [n] where n = |N |. Let A be the class of all problems on U . A rule is a function defined on A that associates with each problem (N , c) ∈ A a vector x ∈ R N . Our generic notation for rules is S. For each coalition N ⊂ N , (ci )i∈N by c N , (Si (N , c))i∈N by S N (N , c), and so on are denoted. We now introduce the nucleolus. Since for general games, the payoff vector the rule chooses is obtained by solving a sequence of linear programs, it is in general not easy to compute. However, for airport problems, the nucleolus can be calculated by an explicit formula (Littlechild 1974; Sönmez 1994). For our purpose, Sönmez’s formula is the most convenient.
2 Potters and Sudhölter (1999) combine last-agent cost additivity and homogeneity as a property, referred to as covariance. See also Sudhölter (1998). 3 Consistency often plays a key role in axiomatizations, for instance, Peleg et al. (1996); Chun (2005); Hwang (2006), and so on. For a survey of the literature on a general principle of consistency and its converse, see Thomson (2000).
123
Characterization of the nucleolus
357
Nucleolus, N u: For each (N , c) ∈ A, N u [1] (N , c) =
c[k] min1≤k≤n−1 k+1 i−1
N u [i] (N , c) = mini≤k≤n−1 N u [n] (N , c) =
c[n] −
c[k] −
n−1 p=1
N u [ p] (N ,c) k−i+2
p=1
where 2 ≤ i ≤ n − 1
N u [ p] (N , c) .
We are interested in the following properties. Efficiency: For each (N , c) ∈ A,
i∈N
Si (N , c) = max j∈N c j .
Equal treatment of equals: For each (N , c) ∈ A and each pair {i, j} N , if ci = c j , then Si (N , c) = S j (N , c).
Last-agent cost additivity: For each pair (N , c) , N , c of elements of A and = c[n] + δ and for each [ j] ∈ N \{[n]}, c[ j] = c[ j] , then each δ ∈ R+ , if c[n] S[n] N , c = S[n] (N , c) + δ and for each [ j] ∈ N \{[n]}, S[ j] N , c = S[ j] (N , c). In order to formulate consistency, an additional piece of notation is needed. Let N (N , c) ∈ A with n ≥ 2, [i] ∈ N , and x x∈ R . The reduced problem of (N, c) with respect to N = N\{[i]} and x, N , c N , is defined by
(i) for each [ j] ∈ N such that j < i, (c xN )[ j] = min c[ j] , c[i] − x[i] , and (ii)
for each [ j] ∈ N such that j > i, (c xN )[ j] = c[ j] − x[i] .
Consistency: For each (N , c) ∈ A with n ≥ 2 and each [i] ∈ N , if x = S (N , c), then N \ {[i]} , c xN \{[i]} ∈ A and x N \{[i]} = S N \ {[i]} , c xN \{[i]} .4 In contrast to other models of fair allocation, for which a unique definition of the reduced problem usually stands out as most natural, there are several ways of defining the reduced problem. Informally, this is because what has to be divided is not a homogeneous whole (such as a social endowment), but it is composed of segments used differently by different airlines. Our paper adopts one simple and natural way of defining the reduced problem. The following figure helps illustrate the justification of the reduced problem.5 Consider the three-airline problem in Fig. 1 and a contribution vector x chosen by a rule for it. Imagine that one airline, say airline 2, pays its contribution x2 and 4 Since some of the coordinates of c x may be negative, we have to add the requirement that the object N N , c xN should be a well-defined airport problem. Similar requirements have often been imposed in the
theory of TU games. In addition, the reduction operator will preserve the property that agents are numbered in the order of increasing cost parameters. 5 The justification is borrowed from Thomson (2004).
123
358
Y.-A. Hwang, C.-H. Yeh
Fig. 1 Three-airline problem: there are three airlines. For each airline, say airline i, the cost of satisfying its need is represented by the cost parameter ci . For simplicity, assume that c1 < c2 < c3 . Thus, airline 1 uses segment 1. Airline 2 uses segments 1 and 2. Airline 3 uses segments 1,2, and 3. The cost of segment 1 is c1 . The cost of segment 2 is c2 − c1 . The cost of segment 3 is c3 − c2 . The total cost of the airstrip is c3 , which is the sum of c1 (the cost of segment 1), c2 − c1 (the cost of segment 2), and c3 − cs2 (the cost of segment 3)
“leaves”, and reassess the situation from the viewpoint of the remaining airlines.6 The remaining airlines are now left with the amount x2 that can be used to cover part of the total cost. Instead of thinking of the contribution x2 as abstractly covering a part of the total cost, it is appealing to impute x2 to these various segments. But how should these imputations be defined? First, it seems very natural to impute x2 to whatever segments airline 2 uses, namely, segments 1 and 2. In other words, because airline 2’s contribution is meant to pay only for the segments it uses, it makes sense to subtract x2 from the costs of segments 1 and 2. This implies that for airline 3 whose cost parameter is greater than c2 , airline 3 benefits completely from the contribution of airline 2 since contributing to the segments airline 2 uses implies contributing to the segments airlines 2 and 3 use jointly. The cost parameter of airline 3 is then revised down by the amount x2 , and its revised cost parameter is c3 − x2 . For airline 1 whose cost parameter is smaller than c2 , it is important to know where x2 is used. Potters and Sudhölter (1999) propose two simple and natural answers to this question: (i) one can first cover the cost of segment 1, and if there is money left, then the remainder is used to help cover the cost of segment 2; (ii) alternatively, one can first cover the cost of segment 2, and if there is money left, then the remainder is used to help cover the cost of segment 1. We adopt the latter formula.7 6 The consistency property allows the possibility of the departure of an arbitrary airline. For purpose of illustration, we focus on the departure of airline 2. 7 To elaborate on the idea of the former formula, we think of x as a contribution to the part of the 2
airstrip that airlines 1 and 2 use jointly. Namely, x2 is used to cover the cost of segment 1. For airline 1 whose cost parameter is smaller than c2 , if x2 ≤ c1 (or equivalently, c1 − x2 ≥ 0), then the cost of segment 1 is either not completely covered or just covered. In either case, there is no money left to help cover the cost of segment 2. Since airline 1 benefits completely from the contribution of airline 2, then the cost parameter of airline 1 is revised down by the amount x2 , and its revised cost parameter is c1 − x2 . If x2 > c1 (or equivalently, c1 − x2 < 0), then the cost of segment 1 is completely covered, and there is money left (namely, x2 − c1 ) to help cover the cost of segment 2. Since airline 1 can’t
123
Characterization of the nucleolus
359
To elaborate on the idea, we think of x2 as a contribution first to the part of the airstrip that airline 2 uses but airline 1 does not. Thus, x2 is used to cover the cost of segment 2. The following two cases are considered: •
•
If x2 ≤ c2 − c1 (or equivalently, c1 ≤ c2 − x2 ), then the cost of segment 2 is either not completely covered or just covered. In either case, there is no money left to help cover the cost of segment 1. Thus, airline 1 does not benefit from the contribution of airline 2, and its cost parameter remains unchanged. If x2 > c2 − c1 (or equivalently, c1 > c2 − x2 ), then the cost of segment 2 is completely covered, and there is money left (namely, x2 − (c2 − c1 )) to help cover the cost of segment 1. In this case, airline 1 benefits by the amount x2 − (c2 − c1 ) from the contribution of airline 2. The cost parameter of airline 1 is revised down by the amount x2 − (c2 − c1 ). Its revised cost parameter is c2 − x2 .
In sum, if c1 ≤ c2 − x2 , then the revised cost parameter of airline 1 is c1 , and if c1 > c2 − x2 , then its revised cost parameter is c2 − x2 . Thus, the definition of the revised cost parameter of airline 1 is the minimum of c1 and c2 − x2 . The consistency property being adopted is based on the above simple and natural way of defining the reduced problem. The nucleolus is consistent. As Potters and Sudhölter (1999) point out, however, it is not the only consistent rule. Another wellknown example is the so-called “constrained equal contributions” rule (Aadland and Kolpin 1998).8 3 Result To prove the new characterization of the nucleolus, we use the following two lemmas. The first lemma says that the nucleolus satisfies a cost monotonicity property. −c Lemma 1 Let (N , c) , N , c ∈ A. If c[1] [1] ≤ · · · ≤ c[n] −c[n] and c[n] −c[n] ≥ 0, then N u [n] (N , c) ≤ N u [n] N , c . Proof Let x = N u (N , c) and y = N u N , c . The proof that x[n] ≤ y[n] is by induction on n. For n = 1, by efficiency of the nucleolus, x[n] ≤ y[n] . The induction hypothesis is that x[n] ≤ y[n] for 1 ≤ n ≤ k. We now show that x[n] ≤ y[n] for n = k + 1. Let n = k + 1 and N = N \ {[1]}. Consider the two reduced problems, N , c xN y y and N , c N . By consistency of the nucleolus, N , c xN ∈ A and N , c N ∈ A. y
− y , and that Note that for each [i] ∈ N , (c xN )[i] = c[i] − x[1] and (c N )[i] = c[i] [1]
Footnote 7 continued benefit more than its cost parameter c1 , its revised cost parameter is defined to be 0. In sum, if c1 − x2 ≥ 0, then the revised cost parameter of airline 1 is c1 − x2 , and if c1 − x2 < 0, then its revised cost parameter is 0. Thus, the definition of the revised cost parameter of airline 1 is the maximum of 0 and c1 − x2 . 8 Aadland and Kolpin (1998) refer to it as the “restricted average cost share rule”. The terminology being adopted here is borrowed from Thomson (2004). Given an airport problem, we can transform the problem into a TU game. When the worth of each coalition is defined to be the largest cost parameter of any member of that coalition, the contribution vector recommended by the constrained equal contributions rule coincides with that prescribed by a well-known solution in TU games, the so-called “Dutta-Ray solution” (Dutta and Ray 1989).
123
360
Y.-A. Hwang, C.-H. Yeh y
y
x x −c c[2] [2] ≤ · · · ≤ c[n] − c[n] . Thus, (c N )[2] − (c N )[2] ≤ · · · ≤ (c N )[n] − (c N )[n] . c[k] c[k] , It can be shown that since x[1] = min1≤k≤n−1 k+1 and y[1] = min1≤k≤n−1 k+1 y
−c . Thus, (c ) −(c x ) then y[1] − x[1] ≤ c[n] ≥ 0. Since |N | = n −1 < n, by [n] N [n] N [n] x y the induction hypothesis, N u [n] N , c N ≤ N u [n] N , c N . Then by consistency of y the nucleolus, x[n] = N u [n] N , c xN and y[n] = N u [n] N , c N . Thus, x[n] ≤ y[n] .
The next lemma says that efficiency is implied by the properties considered. Lemma 2 If a rule satisfies equal treatment of equals, last-agent cost additivity, and consistency, then it satisfies efficiency. Proof Let S be a rule satisfying the three properties. We first show that S satisfies “one-agent efficiency,” which is efficiency for one-agent problems. We then use the induction argument to show that S is efficient in general. To verify one-agent efficiency, we first show that for each p ∈ U, S({ p}, 0) = 0. By last-agent cost additivity, we then conclude that S({ p}, c) = c. Let p ∈ U . Choose l ∈ U and l = p. Consider the two-agent problem ({l, p}, c) ∈ A with 0 = cl = c p . Let x = S({l, p}, c). By equal treatment of equals, xl = x p . Suppose that x ), ({ p}, c x ) ∈ A the order of {l, p} is l = [1] and p = [2]. By consistency, ({l}, c{l} { p} x x x and xl = S({l}, c{l} ), x p = S({ p}, c{ p} ). Thus, c{l} = min{cl , c p − x p } ≥ 0 and x = 0. Hence c{xp} = c p − xl ≥ 0. It follows that xl = x p ≤ c p = 0 and c{l} x x xl = S({l}, c{l} ) = S({l}, 0) and x p = S({ p}, c{ p} ) = S({ p}, c p −xl ) = S({ p}, −x p ). We now suppose that the order of {l, p} is l = [2] and p = [1]. By the same argument, x ) = S({l}, c −x ) = we derive that x p = S({ p}, c{xp} ) = S({ p}, 0) and xl = S({l}, c{l} l p S({l}, −xl ). Thus, x p = S({ p}, −x p ) and x p = S({ p}, 0). By last-agent cost additivity, x p = S({ p}, −x p ) = S({ p}, 0) − x p = x p − x p = 0. Hence S({ p}, 0) = 0. We next show that S is efficient in general. Let (N , c) ∈ A be a n-agent problem. The induction hypothesis is that S is efficient for 1 ≤ n ≤ k. We show that S is efficient for n = k + 1. Let n = k + 1, x = S(N , c), and [1] ∈ N . By consistency, N \ {[1]} , c xN \{[1]} ∈ A and x N \{[1]} = S N \ {[i]} , c xN \{[1]} . By the definition of c xN \{[1]} , (c xN \{[1]} )[n] = c[n] − x[1] . By the induction hypothesis, i∈N \{[1]} xi = c[n] − x[1] . Hence i∈N xi = c[n] . Thus, S is efficient. With the help of Lemmas 1 and 2, we are ready to prove our announced characterization. Theorem 1 The nucleolus is the only rule satisfying equal treatment of equals, lastagent cost additivity, and consistency. Proof Clearly, the nucleolus satisfies equal treatment of equals, last-agent cost additivity, and consistency. Conversely, let S be a rule satisfying the three properties. Let (N , c) ∈ A, x = S (N , c), and y = N u (N , c). The proof that x = y is by induction on n. For n = 1, the three properties altogether imply efficiency (Lemma
123
Characterization of the nucleolus
361
2). Thus, x = y. Suppose that the statement holds for 1 ≤ n ≤ k. We show that ∗ x = y for n = k + 1. The induction for each N , c ∈ A with ∗ hypothesis is that ∗ N ⊂ N and n ≤ n − 1, S N , c = N u N , c . We first show that x[n] = y[n] . By consistency and the induction hypothesis, we then conclude that x = y. Let N = N \ {[n]}. We consider two cases. Case: 1 c[n] = c[n−1] . By equal treatment of equals, x[n−1] = x[n] and y[n−1] = y[n] . Suppose, by contradiction, that x[n] = y[n] . Thus, either x[n] > y[n] or x[n] < y[n] . If x[n] > y[n] , then x[n−1] > y[n−1] . We now show that the two reduced y problems, N , c xN and N , c N , satisfy the hypotheses of Lemma 1. By con y x sistency, N , c N ∈ A and N , c N ∈ A. Since x[n] > y[n] , it follows that
y (c xN )[n−1] = min c[n−1] , c[n] − x[n] ≤ min c[n−1] , c[n] − y[n] = (c N )[n−1] . Since y y c[1] ≤ · · · ≤ c[n] and x[n] > y[n] , then (c N )[1] − (c xN )[1] ≤ · · · ≤ (c N )[n−1] − y (c xN )[n−1] . By Lemma 1, N u [n−1] N , c N ≥ N u [n−1] N , c xN . By consistency , x y x[n−1] = S[n−1] N , c N and y[n−1] = N u [n−1] N , c N . Note that |N | = n − 1 < n. x By the induction hypothesis, S[n−1] N , c N = N u [n−1] N , c xN . It follows that x[n−1] ≤ y[n−1] , in violation of y[n−1] < x[n−1] . If x[n] < y[n] , then we derive the desired contradiction by a similar argument. = c Case: 2 c[n−1] < c[n] . Let c[n] [n−1] , and let c[ j] = c[ j] for each [ j] ∈ N \{[n]}. By Case 1, S[n] N , c = N u [n] N , c . Let δ ∈ R+ with δ = c[n] − c[n−1] , by lastagent cost x[n] = S[n] (N , c) = S[n] N , c + δ and y[n] = N u [n] (N , c) = additivity, N u [n] N , c + δ. Since S[n] N , c = N u [n] N , c , we conclude that x[n] = y[n] .
The independence of properties listed in Theorem 1 can be established by adopting the rules φ, σ 3 , and σ 5 in Table 2 of Potters and Sudhölter (1999).
4 Discussion In addition to our Theorem 1 strengthening Potters and Sudhölter (1999)’s characterization of the nucleolus, our method of proof has the advantage of being self-contained and direct. Indeed, Potters and Sudhölter (1999) use Peleg (1988/89)’s characterization of the “prekernel”9 (Davis and Maschler 1965), and coincidence between the prekernel and the nucleolus on the class of convex games to characterize the nucleolus. Potters and Sudhölter (1999) find that efficiency is implied by homogeneity, lastagent cost additivity, and “first-agent consistency,” which is a weaker version of consistency, obtained by focusing on the departure of an agent with the smallest cost parameter. Strengthening first-agent consistency to consistency in the above finding, our Lemma 2 says that homogeneity can be replaced by equal treatment of equals. One may wonder whether consistency can be weakened to first-agent consistency in Lemma 2. The following modified Shapley value, Sh , shows the answer to be negative. 9 See also Peleg (1986).
123
362
Y.-A. Hwang, C.-H. Yeh
For each (N , c) ∈ A and for each [i] ∈ N , let a c[ j] − c[ j−1] + , n n− j +1 i
Sh [i] (N , c) =
j=1
where c[0] = 0 and a < 0. It can be shown that the modified Shapley value satisfies equal treatment of equals, last-agent cost additivity, and first-agent consistency, but violates efficiency. Acknowledgements We would like to thank William Thomson for helpful suggestions and discussions. We are grateful to Takashi Hayashi, Fan-Chin Kung, Xiao Luo, Man-Chung Ng, and Çaˇgatay Kayı for detailed comments. We are also grateful to an anonymous referee and the Associate Editor in charge for their suggestions. Yeh greatly acknowledges financial support from National Science Council of Taiwan under grant NSC96-2628-H-001-060-MY2, and 2010 Career Development Award of Academia Sinica, Taiwan. We are responsible for any remaining deficiency.
Appendix In the discussion, we claim that the modified Shapley value satisfies equal treatment of equals, last-agent cost additivity, and first-agent consistency, but violates efficiency. Here is a proof of this assertion. Note that the Shapley value, denoted by Sh, is defined as follows: for each (N , c) ∈ A and for each [i] ∈ N , Sh [i] (N , c) =
i
c[ j] − c[ j−1] j=1
n− j +1
,
where c[0] = 0.
Thus, the modified Shapley value, denoted by Sh , is defined by adding a constant term to the definition of the Shapley value. Formally, for each (N , c) ∈ A and for each [i] ∈ N ,
Sh [i] (N , c) =
a + Sh [i] (N , c), n
where a < 0. claim 1 The modified Shapley value satisfies equal treatment of equals, last-agent cost additivity, and first-agent consistency, but violates efficiency. Proof It is well known that the Shapley value satisfies equal treatment of equals and last-agent cost additivity. Thus, it is clear that the modified Shapley value satisfies these properties as well. Since a < 0, by efficiency of the Shapley value, the modified Shapley value violates efficiency. To verify first-agent consistency, let (N , c) ∈ A, x = Sh (N , c) and y = Sh (N , c). The Shapley value satisfies first-agent consistency (Theorem 3.1 of Potters and Sudhölter (1999)). Hence by first-agent consistency of the Shapley value,
123
Characterization of the nucleolus
363
(N \[1], c xN \{[1]} ) ∈ A and x N \{[1]} = Sh N \ {[1]} , c xN \{[1]} . It follows that for each y
[ j] ∈ N \{[1]}, (c xN \{[1]} )[ j] ≥ 0. It is clear that for each [ j] ∈ N \{[1]}, (c N \{[1]} )[ j] = (c xN \{[1]} )[ j] − an . By combining this fact, (c xN \{[1]} )[ j] ≥ 0, and a < 0, we derive y y (c N \{[1]} )[ j] ≥ 0. Thus, (N \{[1]}, c N \{[1]} ) ∈ A. Moreover, by the formula of the Shapley value, we have for each [i] ∈ N \{[1]}, i (c
N \{[1]} )[ j] − (c N \{[1]} )[ j−1] x
Sh [i] (N \{[1]}, c xN \{[1]} )
=
x
n−1− j +2
j=2
y
y Sh [i] (N \{[1]}, c N \{[1]} )
=
Since
y
[1]
= (c N \{[1]} )
y (c N \{[1]} )[ j]
=
n−1− j +2
− an , we have obtained y
=
,
= 0.
[1] (c xN \{[1]} )[ j]
y Sh [i] (N \{[1]}, c N \{[1]} )
y
i (c
N \{[1]} )[ j] − (c N \{[1]} )[ j−1] j=2
where (c xN \{[1]} )
, and
y
i (c
N \{[1]} )[ j] − (c N \{[1]} )[ j−1] j=2
n−1− j +2
x i (c x
N \{[1]} )[ j] − (c N \{[1]} )[ j−1] −a + = n(n − 1) n−1− j +2 j=2
−a = + Sh [i] (N \{[1]}, c xN \{[1]} ). n(n − 1) Thus, by first-agent consistency of the Shapley value and the definition of the modified Shapley value, we have for each [i] ∈ N \{[1]},
y
Sh [i] (N \{[1]}, c N \{[1]} ) = = = = =
a y + Sh [i] (N \{[1]}, c N \{[1]} ) n−1 −a a + + Sh [i] (N \{[1]}, c xN \{[1]} ) n − 1 n(n − 1) a + Sh [i] (N \{[1]}, c xN \{[1]} ) n a + Sh [i] (N , c) n Sh [i] (N , c).
References Aadland D, Kolpin V (1998) Shared irrigation costs: an empirical and axiomatic analysis. Math Soc Sci 35:203–218 Chun Y (2005) The separability principle in bargaining. Econ Theory 26:227–235 Davis M, Maschler M (1965) The kernel of a cooperative game. Nav Res Logist Q 12:223–259
123
364
Y.-A. Hwang, C.-H. Yeh
Dutta B, Ray D (1989) A concept of egalitarianism under participation constraints. Econometrica 57: 615–635 Hwang Y-A (2006) Associated consistency and equal allocation of nonseparable costs. Econ Theory 28:709–719 Littlechild SC, Owen G (1973) A simple expression for the Shapley value in a special case. Manag Sci 3:370–372 Littlechild SC (1974) A simple expression for the nucleolus in a special case. Int J Game Theory 3:21–29 Peleg B (1986) On the reduced game property and its converse. Int J Game Theory 15:187–200 Peleg B (1988/1989) Introduction to the theory of cooperative games. Hebrew University of Jerusalem, Israel Peleg B, Potters J, Tijs S (1996) Minimality of consistent solutions for strategic games, in particular for potential games. Econ Theory 7:81–93 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 games. SIAM J Appl Math 17:1163–1170 Shapley LS (1953) A value for n-person games. In: Kuhn H, Tucker AW (eds) Contributions to the theory of games. Princeton University Press, Princeton pp 307–317 Sönmez T (1994) Population monotonicity of the nucleolus on a class of public good problems. University of Rochester, mimeo Sudhölter P (1997) The modified nucleolus: properties and axiomatizations. Int J Game Theory 26:147–182 Sudhölter P (1998) Axiomatizations of game theoretical solutions for one-output cost sharing problems. Games Econ Behav 24:142–171 Thomson W (2000) Consistent allocation rules. University of Rochester, mimeo Thomson W (2004) Cost allocation and airport problems. University of Rochester, mimeo
123
