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Coalitional games: Monotonicity and core
European Journal of Operational Research 216 (2012) 208–213
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European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor
Decision Support
Coalitional games: Monotonicity and core q J. Arin a,⇑,1, V. Feltkamp b a b
Dpto. Ftos. A. Económico I, University of the Basque Country, L. Agirre 83, 48015 Bilbao, Spain Maastricht School of Management, P.O. Box 1203, 6201 BE Maastricht, The Netherlands
a r t i c l e
i n f o
Article history: Received 3 January 2011 Accepted 13 July 2011 Available online 23 July 2011 Keywords: Monotonicity Core TU games Per capita nucleolus
a b s t r a c t We characterize a monotonic core solution defined on the class of veto balanced games. We also discuss what restricted versions of monotonicity are possible when selecting core allocations. We introduce a family of monotonic core solutions for veto balanced games and we show that, in general, the per capita nucleolus is not monotonic. Ó 2011 Elsevier B.V. All rights reserved.
1. Introduction Young (1985) formulates an impossibility result for the problem of finding core solutions satisfying monotonicity on the domain of balanced TU games. A positive result holds on the domain of convex games where the Shapley value (Shapley, 1953) is a core solution satisfying several monotonicity requirements. On the domain of veto balanced games, Arin and Feltkamp (2005) introduce a monotonic core solution that does not entirely accomplish the goals we would expect from a fair solution concept. The solution is very extreme and divides the worth of the grand coalition equally among the veto players (we call this the all for veto players solution). We also argue that the per capita nucleolus is not monotonic on the class of veto balanced games even if it does satisfy strong N-monotonicity. Therefore, the existence of monotonic core solutions in the class of veto balanced games is still an open question. This paper deals with this problem. We discuss the possibilities of combining monotonicity and core selection. We introduce restricted monotonicity requirements and given these restricted versions of monotonicity properties, we characterize a core solution on the class of veto balanced games. We call this core solution
serial rule and we also offer an alternative characterization where the requirement of being a core solution is not needed. The core solution characterized proves to be a member of a family of core solutions that also includes the all for veto players solution. The class of veto balanced games2 have been used to model social situations in which a group of agents has the power of veto. This and the class of convex games may be the most significant classes of TU games. Various economic problems have been studied using this class of games, and in this way, interesting subclasses of games have been generated. Some examples are Information market games (Muto et al., 1988), Clan games (Muto et al., 1989) and Landowner economies (Shapley and Shubik, 1967). The class is also useful for modeling hierarchical situations where a top player is necessary for a project to be developed (see, for example, Brink, 1997). A different approach opened up by the result of Young is to define new properties of monotonicity (weaker than the one formulated by Young) and to deal with the class of all TU balanced games. Following this approach we also discuss whether the serial rule may be a candidate for a monotonic core solution in the class of all balanced games. 2. Preliminaries
q We thank R. Velez-Cardona and two anonymous referees for their helpful comments. ⇑ Corresponding author. E-mail address: [email protected] (J. Arin). 1 This research is supported by the Spanish Ministerio de Ciencia e Inno-vación under projects SEJ2006-05455 and ECO2009-11213, co-funded by ERDF, and by Basque Government funding for Grupo Consolidado GIC07/146-IT-377-07. This author also thanks the Department of Economics of the University of Rochester for its hospitality and the ‘‘Salvador de Madariaga’’ Program of the Spanish Ministerio de Educación y Ciencia for the financial support provided.
0377-2217/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2011.07.025
2.1. TU Games A cooperative n-person game in characteristic function form is a pair (N, v), where N is a finite set of n elements and v : 2N ! R is a real-valued function on the family 2Nof all subsets of N with 2 This paper does not set out to analyze the solution concept characterized here in different settings. We consider that for some interesting subclasses of veto balanced games further research may be of interest.
J. Arin, V. Feltkamp / European Journal of Operational Research 216 (2012) 208–213
v(;) = 0. Elements of N are called players and the real valued function v the characteristic function of the game. Any subset S of N is called a coalition. For notational convenience we sometimes denote the singleton {i} by i. The number of players in S is denoted by jSj. Given S N we denote by NnS the set of players of N that are not in S. We only consider games where the worth of all coalitions is nonnegative. A distribution of v(N) among the players is a real-valued vector x 2 RN where xi is the payoff assigned by x to player i. A distribution satisfying xi P v(i) for all i 2 N is called an imputation and P the set of imputations is denoted by I(v). We denote i2S xi by x(S). The core of a game is the set of imputations that cannot be blocked by any coalition, i.e.
Cðv Þ ¼ fx 2 Iðv Þ : xðSÞ P v ðSÞ for all S Ng: It has been shown that a game with a non-empty core is balanced3 and therefore games with non-empty core are called balanced games. Player i is a veto player if v(S) = 0 for all S where player i is not present. A balanced game with at least one veto player is called a veto balanced game. We denote by CB the class of balanced games and by CVB the class of veto balanced games. A solution / on a class of games C0 is a correspondence that associates with every game (N, v) in C0 a set /(N, v) in RN such that x(N) 6 v(N) for all x 2 /(N, v). This solution is efficient if this inequality holds with equality. The solution is single-valued if the set contains a unique element for every game in the class. For notational convenience we write a single-valued solution as /ðv Þ 2 Rn : Also for notational convenience and if there is no confusion we write /i instead of /i(v). We say that a solution satisfies core selection if it selects core allocations whenever the core is non-empty. Schmeidler (1969) introduced the nucleolus of a game v, denoted by g(v). For any game v with a non-empty imputation set, the nucleolus is a single-valued solution, is contained in the kernel and lies in the core provided that the core is non-empty. The per capita nucleolus (Grotte, 1970) is related with the nucleolus and selects core allocations whenever the game is balanced. In this paper, we study solution concepts that select precisely one core allocation for each balanced game. We call such concepts core solutions. Let / be a single-valued solution concept defined on a class of games C0. We say that / satisfies the equal treatment property (ETP) if for each (N, v) 2 C0 interchangeable players i, j are treated equally, i.e., /i(N, v) = /j(N, v). Players i and j are interchangeable if v(S [ i) = v(S [ j) for all S # Nn{i, j}.
2.2. On monotonicity properties We present some monotonicity properties for single-valued solution concepts. Let / be a single-valued solution on a class of games C0. We say that solution / satisfies: Monotonicity: if for all v, w 2 C0, such that for all T containing player i, v(T) 6 w(T), and for all S # Nn{i}v(S) = w(S), then, /i(w) P /i(v). Coalitional monotonicity: if for all v, w 2 C0, if for all S – T, v(S) = w(S) and v(T) < w(T), then for all i 2 T, /i(v) 6 /i(w). Strong coalitional monotonicity: if for all v, w 2 C0, if for all S – T, v(S) = w(S) and v(T) < w(T), then for all i, j 2 T, /i(w) /i(v) = /j(w) /j(v) P 0. N-monotonicity (Meggido, 1974): if for all v, w 2 C0, if for all S – N, v(S) = w(S) and v(N) < w(N), then for all i 2 N, /i(v) 6 /i(w). 3
See Peleg and Sudholter (2007).
209
Strong N-monotonicity: if for all v, w 2 C0, if for all S – N, and v(N) < w(N), then for all i, j 2 N, /i(w) /i(v) = /j(w) /j(v) P 0. The strong versions of the properties capture the idea of equal treatment of players equally affected (by the monotonic change). Monotonicity and coalitional monotonicity are equivalent whenever each coalitional change generates games belonging to the domain in which the solution has been defined. Coalitional monotonicity implies N-monotonicity. It is also immediately apparent that strong coalitional monotonicity implies strong Nmonotonicity. In the class of veto balanced games there exist core solutions satisfying monotonicity. We call the following core solution the all for veto players solution (denoted by AVS). Let (N, v) be a veto balanced game with T as the set of veto players and let AVS be the solution for veto balanced games defined as follows:
v(S) = w(S)
( AVSi ðN; v Þ ¼
v ðNÞ jTj
for all i 2 T
0
for all i 2 N n T:
This solution does not satisfy the strong versions of the monotonicity properties. Young (1985) proves that in the class of balanced games there is no solution satisfying coalitional monotonicity and core selection. The next section discusses the impossibility of finding a core solution (on the class of veto balanced games) that satisfies the strong monotonicity requirements. But still is possible to find more core solutions satisfying different properties of monotonicity. It is also interesting to note that the per capita nucleolus satisfies strong N-monotonicity on the class of all TU games. The per capita nucleolus selects a core allocation whenever the game is balanced. 3. On monotonic core selections 3.1. Examples and properties We use two examples to illustrate what kind of restricted monotonicity properties are allowed for core solutions on the domain of veto balanced games. Example 1. Let N = {1, 2, 3} a set of players and consider the following 3-person veto balanced games:
8 8 > > < 6 if S ¼ f1; 2g < 6 if S 2 ff1; 2g; f1; 3gg and wðSÞ ¼ 6 v ðSÞ ¼ > 6 if S ¼ N if S ¼ N > : : 0 otherwise: 0 otherwise A core solution satisfying ETP should choose (6, 0, 0) and (3, 3, 0) in games (N, v) and (N, w). Player 3 (unlike player 1) does not receive any benefit from the fact that he is a member of the only coalition changing its worth by increasing it. Therefore, core solutions do not satisfy strong coalitional monotonicity. However there are core solutions that satisfy strong N-monotonicity. Example 2. Let N = {1, 2, 3, 4} a set of players and consider the following 4-person veto balanced games:
8 8 6 if 1 2 S and jSj ¼ 3 > > > > < 6 if1 2 S and jSj ¼ 3 < or S ¼ f1; 2g v ðSÞ ¼ > 6 and wðSÞ ¼ if S ¼ N > 6 if S ¼ N : > > 0 otherwise : 0 otherwise: A core selection should choose (6, 0, 0, 0) in the games (N, v) and (N, w).
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Players 1 and 2 do not receive any benefit from the fact that they are the members of the only coalition changing its worth by increasing it. Therefore, core selections impose some restrictions on the monotonicity properties we can ask for. The fact that, given a game, only one coalition increases its worth does not imply necessarily that all members of the coalition should benefit from this increase. The aim of a monotonic core solution, given these restrictions, should be to be as strongly monotonic as possible. This notion is formalized below. The first problem is to detect whether or not increasing the worth of a coalition is strictly profitable for some (maybe all) members of the coalition or not. Example 1 shows that player 3 does not receive any benefit from the fact that the only coalition that has increased its worth contains him. This is so because player 3 is not necessary for players 1 and 2 in order to obtain a worth of six. Player 3 has no power to claim that he is necessary for the rest of the players in order to get the worth of 6. After the increase of the worth of the coalition formed by players 1 and 3 the situation remains the same since the new worth of the modified coalition is still lower than or equal to 6. That is, player 3 is still powerless and can be excluded (and should be excluded if we look for core allocations) when the rest of the players distribute this worth of 6. This observation motivates the introduction of the following restricted versions of monotonicity. Assume player l is a member of the coalition S and that this coalition increases its worth. If the new worth of the coalition is higher than the worth for which player l can be excluded we can expect that player l can receive a higher payoff in the new situation without violating core restrictions. Players that are in this situation can be equally benefited from the new game. We introduce the following two properties. Let / be a single-valued solution defined on C0. Property I (Restricted strong coalitional monotonicity): Let v, w 2 C0, such that v(T) < w(T), and for all S – T v(S) = w(S) and let P = {i 2 T; maxS # Nn{i}v(S) 6 v(T)}. Then / satisfies Property I if for all i, j 2 P,/i(w) /i(v) = /j(w) /j(v) P 0. Property II (Restricted strong monotonicity): Let v, w 2 C0 and let Q be a set of coalitions of N, such that v(T) < w(T), for all T 2 Q and v(S) = w(S) for all S R Q. Let T1 2 arg minT2Qv(T) and let
II
if
for
3.2. Selecting monotonic core allocations Let (N, v) be a game with veto players and let player 1 be a veto player. Define for each player i a value di as follows:
di ¼ maxS # Nnfig v ðSÞ: Then d1 = 0. Let dn+1 = v(N) and rename players according to the nondecreasing order of those values. That is, player 2 is the player with the lowest value besides player 1 and so on. We also use the notation di instead of di(N, v) for simplicity. The solution SR associates to each game with veto players, (N, v), the following payoff vector:
SRl ðN; v Þ ¼
all
i,
j 2 P,
/
Property I and Property II are equivalent on the class of veto balanced games4. Note that Property I (with efficiency) implies strong N-monotonicity. The properties can also be seen as fairness properties. All equal players that have been equally and positively affected by the changes should be equally treated. In our definitions we identify the set of players that should be equally treated. In the class of all TU games the Shapley value satisfies these properties and also the unrestricted versions of the properties (Strong coalitional monotonicity). In the class of veto balanced games the Shapley value is not a core solution. Hereafter we will use the term monotonic core solution5 to refer to any core solution satisfying these two properties. A monotonic 4 Let (N, v) be a veto balanced game. Let (N, w) be a veto balanced game where w(T) > v(T) for all T in Q. Then any game (N, q) where q(T) > v(T) for some T in Q is also a veto balanced game. In general, this is not true in the class of convex games and therefore the two properties are not equivalent in this class. 5 We use this term instead of restricted strongly monotonic core concept for simplicity.
Xn diþ1 di for all l 2 f1; . . . ; ng: i¼l i
Note that since d1 = 0 the solution is efficient. In case no player is veto the solution is not efficient. The following example illustrates how the solution behaves. Example 3. Let N = {1, 2, 3, 4} a set of players and consider the following 4-person veto balanced game (N, v) where
8 8 if S 2 ff1; 2; 3g; f1; 2; 4gg > > > < 6 if S ¼ f1; 3; 4g v ðSÞ ¼ > 12 if S ¼ N > > : 0 otherwise: Computing the vector of d-values we get:
ðd1 ; d2 ; d3 ; d4 ; d5 Þ ¼ ð0; 6; 8; 8; 12Þ: Applying the formula,
SR1 ¼ SR2 ¼ SR3 ¼ SR4 ¼
P ¼ fi 2 \T2Q T; maxS # Nnfig v ðSÞ 6 v ðT 1 Þg: Then / satisfies Property i(w) /i(v) = /j(w) /j(v) P 0.
core allocation for a game is any allocation selected by a monotonic core solution.
d2 d1 1
þ
d3 d2 2 d3 d2 2
þ þ
d4 d3 3 d4 d3 3 d4 d3 3
þ þ þ
d5 d4 4 d5 d4 4 d5 d4 4 d5 d4 4
¼8 ¼2 ¼1 ¼ 1:
The definition suggests that the solution follows a serial rule principle. Each player i has a right, a veto power, over the amount v(N) di and the amount is divided equally among the players with that right, veto power, over it. By a serial rule principle we refer to the ideas underlying on the definition of solution concepts such as the minimal overlapping rule for bankruptcy problems (O’Neill, 1982), the Shapley value for airport problems (Littlechild and Owen, 1973) or the serial cost rule for cost sharing problems (Moulin and Shenker, 1992 and Moulin and Shenker, 1994). Roughly speaking, a serial rule-like solution concept for TU games behaves as follows: the amount v(N) is divided in several parts and each part is divided equally among the group of players that have a demand over this part. Hence, the solution has two components: a vector of demands (claims or rights) is defined for a given TU game and a serial rule principle is applied over this vector. The question about which is the demand of each player does not need to have a unique answer and, therefore, different solutions can be defined under this principle of serial rule 6. 6 For example, the AVS can be seen as a serial rule where di = v(N) for all non veto players. It is not difficult to prove that if a serial rule satisfies core selection then di P maxS # Nn{i}v(S). Therefore the serial rule introduced in this paper gives to all non veto players the minimal demand compatible with core selection. In this case this minimal demand is the best situation for non veto players.
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The main result of the paper is that the introduced solution concept is the only monotonic core solution on the class of veto balanced games. First of all we prove the following lemma. Lemma 4. Let (N, v) be a veto balanced TU game. Let r be a P l1 monotonic core solution on CVB. Then r1 ri ¼ il¼2 dl d ¼ l1 SR1 SRi for all i 2 {2, . . . , n}. Proof. We first prove that r1 r2 = d2. Consider the game (N, w) where w(S) = min{v(S), d2}. Clearly (N, w) is a veto balanced game and its core contains only the allocation (d2, 0, . . . , 0) and therefore r(N, w) = (d2, 0, . . . , 0). From game (N, w) to game (N, v) all coalitions that have increased their worth contain players 1 and 2. By applying Property II to the pair {(N, v), (N, w)} we know that r1(v) r1(w) = r2(v) r2(w) P 0. Therefore, r1(w) r2(w) = r1(v) r2(v) = d2. Assume that the lemma is true for any k 2 {2, . . . , i 1}. We P l1 prove that r1 ri ¼ il¼2 dl d : Consider the game (N, q) where l1 q(S) = min{v(S), di}. Clearly (N, q) is a veto balanced game and if x is a core allocation then xl = 0 for l 2 {i, . . . , n} and by efficiency Pi1 Therefore rl(N, q) = 0 for l 2 {i, . . . , n} and k¼1 xk ¼ di . Pi1 k¼1 rk ðN; qÞ ¼ di : From game (N, q) to game (N, v) all coalitions that increase their worth contain the set of players {1, 2, . . . , i}. By applying Property II to the pair {(N, v), (N, q)} we know that r1(v) r1(q) = rk(v) rk(q) P 0 for all k 2 {2, . . . , i}. Therefore r1(q) rk(q) = r1(v) rk(v) for all k 2 {2, . . . , i}. By P l1 assumption r1 ðv Þ rk ðv Þ ¼ kl¼2 dl d for all k 2 {2, . . . , i 1}. l1 Pk dl dl1 Therefore r1 ðqÞ rk ðqÞ ¼ l¼2 l1 for all k 2 {2, . . . , i 1} and P we have seen that i1 k¼1 rk ðN; qÞ ¼ di : These two facts imply that Pi dl dl1 r1 ðqÞ ¼ SR1 ðqÞ ¼ l¼2 l1 . This is a consequence of solving r1(q) in the following linear system of equalities: i1 X
rk ðN; qÞ ¼ di and r1 ðqÞ rk ðqÞ ¼
k¼1
k X dl dl1 for all k l1 l¼2
211
and the serial rule provides non negative payoffs to all players. Therefore we focus on coalitions with veto players that have a P positive worth. Note that k1 l¼1 SRl ðN; v Þ P dk : Consider the smallest P k 2 NnS. Therefore, {1, 2, . . . , k 1} # S. Then v ðSÞ 6 dk 6 k1 l¼1 P SRl 6 l2S SRl . We now prove that SR satisfies Property I and therefore Property II. Let (N, v), (N, w) 2 CVB, such that v(T) < w(T), and for all S – T v(S) = w(S) and let P = {i 2 T;maxS # Nn{i}v(S) 6 v(T)}. Then for any l 2 P it holds that dl(N, v) = dl(N, w). Note that if l 2 P and dk(N, w) 6 dl(N, w) then dk(N, v) = dk(N, w). Therefore we get
SR1 ðN; v Þ SRl ðN; v Þ ¼ SR1 ðN; wÞ SRl ðN; wÞ: This is equivalent to
SR1 ðN; wÞ SR1 ðN; v Þ ¼ SRl ðN; wÞ SRl ðN; v Þ: That means that all players in P are equally affected when increasing the worth of coalition T. h If a veto balanced game is not positive then it is not necessarily true that its serial rule selects a core allocation. Consider a 3-person veto balanced game as follows:
8 > < 4 if S 2 ff1g; f1; 2g; f1; 3gg v ðSÞ ¼ > 3 if S ¼ N : 0 otherwise Clearly, SR(v) = (1, 1, 1) R C(v). The requirement of positivity may be replaced by monotonicity of the game in order to avoid these problems. We axiomatize the solution with a different set of axioms where to be core solution is not needed as an axiom but it is implied by the axiomatization. We introduce a new axiom. Let / be a single-valued solution defined on C0. / satisfies reasonability if v(i) 6 /i(v) 6 v(N) di for all i 2 N and for all (N, v) 2 C0.
2 f2; . . . ; i 1g: Adding both sides over all k = 2, 3, . . . , i 1 yields i1 X k i1 Xi1 X dl dl1 X ðr1 ðqÞ rk ðqÞÞ ¼ ði 2Þr1 ðqÞ r ðqÞ ¼ k¼2 k l 1 k¼2 l¼2 k¼2
¼ ði 2Þr1 ðqÞ ðdi r1 ðqÞÞ: Due to a straightforward combinatorial argument, it follows
ði 1Þr1 ðqÞ di ¼ ¼
i1 X k i1 X i1 X dl dl1 X dl dl1 ¼ l 1 l1 k¼2 l¼2 l¼2 k¼l i1 X il ðdl dl1 Þ: l 1 l¼2
Theorem 6. SR is the only solution on CVB that satisfies efficiency, reasonability and Property II. The proof is almost identical to the proof of Theorem 5 and therefore omitted. Lemma 4 can be rewritten replacing to be core concept by to be a solution satisfying reasonability and the proof is identical: we use jointly efficiency and reasonability to identify the outcome of the solution in the games (N, w) and (N, q). Note that any Core-Selector / satisfies reasonability since the upper bound /l(N, v) 6 v(N) dl arises from the core constraints applied to any coalition S # Nn{l}. Therefore the serial rule satisfies reasonability. Note that the Shapley value satisfies efficiency and Property II but not reasonability while the nucleolus satisfies efficiency and reasonability but not Property II.
So i1 i X d 1 X il dl dl1 r1 ðqÞ ¼ i þ ðdl dl1 Þ ¼ i 1 i 1 l¼2 l 1 l1 l¼2
by rearranging terms. h Theorem 5. SR is the only monotonic core solution on CVB. Proof. It is immediate that SR is the only efficient solution concept that satisfies the conditions of the previous lemma. We now prove that SR is a core solution. Let (N, v) be a veto balanced TU game. We need to prove that for any coalition S it holds that SR(S) P v(S). The worth of coalitions without the veto player is 0
4. Per capita nucleolus and strong N-monotonic core solutions The per capita nucleolus is a core solution that in the class of all balanced games satisfies strong N-monotonicity. This is why some authors consider this core solution a good candidate when trying to select monotonic core allocations. The following example shows that the per capita nucleolus violates the monotonicity principle and therefore, even if it does satisfy strong N-monotonicity, it can hardly be seen as a monotonic core solution (at least in the class of all balanced games and in the subclass of veto balanced games). Example 7. Let N = {1, 2, . . . , 6} a set of players and consider the following 6-person veto balanced games:
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8 8 > > < 6 if 1 2 S and jSj ¼ 5 < 8 if S 2 N n f6g; v ðSÞ ¼ > 6 and wðSÞ ¼ if S ¼ N N n f5g;N > : : v ðSÞ otherwise: 0 otherwise A core solution must choose the allocation (6, 0, 0, 0, 0, 0) in the first game. The per capita nucleolus of the second game is (5.75, 0.75, 0.75, 0.75, 0, 0). Therefore, the per capita nucleolus is not monotonic since player 1 receives a lower payoff in the second game. Therefore the per capita nucleolus does not satisfy monotonicity (see SubSection 2.2). In the class of veto balanced games there are core solutions (other than the serial rule introduced in the previous subsection) that satisfy monotonicity and strong N-monotonicity such as the following modified version of the all for veto players solution. Let (N, v) be a veto balanced game with T as the set of veto players and let rn be a solution for veto balanced games defined as follows:
rni ¼
8 < dn þ dnþ1 dn jTj
:
for all i 2 T
jNj
dnþ1 dn jNj
for all i 2 N n T:
Similarly, core solutions satisfying monotonicity can be defined all of them violating Property I. Let (N, v) be a veto balanced game with T as the set of veto players and let rk be a solution for veto balanced games defined as follows:
rkl ¼
8 > > > < > > > :
dk jTj
þ
n P i¼k
n P i¼maxfl;kg
The solution is continuous, satisfies the necessary conditions of the Theorem but is not monotonic. The following example illustrates that the solution is not monotonic. Example 10. Let N = {1, 2, 3, 4} a set of players and consider the following 4-person veto balanced games:
8 8 > > < 6 if 1 2 S and jSj ¼ 3 < 12 if 1 2 S and jSj ¼ 3 v ðSÞ ¼ > 24 and wðSÞ ¼ 24 if S ¼ N if S ¼ N > : : 0 otherwise 0 otherwise: Computing the solution for the two games we obtain:
rðN; v Þ ¼ ð19:5; 1:5; 1:5; 1:5Þ and rðN; wÞ ¼ ð16; 2; 2; 2Þ: Clearly, player 1 has received a lower payoff when the coalitions that have increased their worth contain him. 5. Games without veto players We have shown that in the class of veto balanced games there are monotonic core solutions and we have axiomatized the serial rule in this class. The serial rule is well defined for any TU game but in general is not efficient. An efficient modified serial rule can be defined as follows. Let (N,v) be a TU game. Define for each player i a value di as follows:
di ¼ maxS # Nnfig v ðSÞ:
diþ1 di i
for all l 2 T
diþ1 di i
for all l 2 N n T:
We conclude this section identifying some necessary conditions that any core solution that satisfies monotonicity (see SubSection 2.2) must fulfill. Theorem 8. Let (N, v) be a veto balanced TU game. Let r be a core P solution on CVB that satisfies monotonicity. Then lk¼1 rk P dlþ1 for all l 2 {1, . . . , n}. Proof. Assume on the contrary that the theorem is not true and P that for some l, l 2 {1, . . . , n} holds lk¼1 rk < dlþ1 : Since r is a core P solution this l is different than n. Let lk¼1 rk < dlþ1 and consider the game (N, w) where w(S) = min{v(S), dl+1}. Since r is a core soluP tion i2N ri ðwÞ ¼ dlþ1: By monotonicity of the solution, for all k 2 {1, . . . , l} it holds that rk(v) P rk(w) and therefore P P i2f1;...;lg ri ðv Þ P i2f1;...;lg ri ðwÞ ¼ dlþ1: This result contradicts the initial assumption and the proof is completed. h Similarly with the results of the previous subsection we also point out the following result that has almost an identical proof.
Let dn+1 = v(N) and rename players according to the nondecreasing order of those values. The solution SR associates to each TU game, (N, v), the following payoff vector:
SRl ðN; v Þ ¼
n X diþ1 di d1 þ for all l 2 f1; . . . ; ng: i n i¼l
In general, this solution is not a core solution, not even in the class of convex games. Therefore it is not immediately apparent that serial rule like concepts may be candidates for monotonic core solutions in domains other than the domain of veto balanced games. A different aspect of the problem is the existence of core solutions satisfying weak monotonicity properties. In particular we can consider the following property: Property III (Restricted weak coalitional monotonicity): Let v, w 2 C0, such that v(T) < w(T), and for all S – T v(S) = w(S) and let
P ¼ fi 2 T; maxS # Nnfig v ðSÞ 6 v ðTÞg: Then / satisfies Property III if for all i 2 P, /i(w) /i(v) P 0. The following result solves this question. Theorem 11. In the class of balanced games there is no core concept that satisfies restricted weak coalitional monotonicity.
Theorem 9. Let (N, v) be a veto balanced TU game. Let r be a solution concept on CVB that satisfies monotonicity, efficiency and reasonabilP ity. Then lk¼1 rk ðv Þ P dlþ1 for all l 2 {1, . . . , n}.
Proof. Let N = {1, 2, 3, 4} be a set of players and consider the following 4-person balanced games:
There are core solutions defined on CVB that satisfy the conditions of the theorem and do not satisfy monotonicity. Let (N, v) be a veto balanced game such that v(N) > 0. Let T be the set of veto players. Let r be a solution for veto balanced games defined as follows:
v ðSÞ ¼ >
rl ðN; v Þ ¼
8 > <
dn dnþ1 dn dnþ1 jNnTj d
for all l R T
v ðNÞd n ðdnþ1 dn Þ > nþ1 : for all l 2 T: jTj
8 > > > < 12 if S 2 24 > > : 0
f1; 3g; f1; 4g; f2; 3g; f2; 4g
f1; 2; 3g; f1; 2; 4g; f1; 3; 4g; f2; 3; 4g if S ¼ N otherwise
and
wi ðSÞ ¼
v ðSÞ
if S–N n fig
24
if S ¼ N n fig
where i 2 N. Let /a core solution satisfying restricted weak coalitional monotonicity. For the games w1 and w2 we need to consider the only
J. Arin, V. Feltkamp / European Journal of Operational Research 216 (2012) 208–213
core allocation of the two games, z1 = (0, 0, 12, 12) and for games w3 and w4 we need to consider the allocation z3 = (12, 12, 0, 0). By applying Property III we need to conclude that /1(v) = 0 and /2(v) = 0 (considering games w1 and w2). By applying Property III we also need to conclude that /3(v) = 0 and /4(v) = 0 (considering games w3 and w4). Therefore core solutions are not compatible with Property III. h In the class of convex games the Shapley value is a core solution and therefore a monotonic core solution. But, in general, the Shapley value is not a core solution. This motivates the following question: Is there a core solution defined for the class of all balanced games that is monotonic in the class of convex games and in the class of veto balanced games? This is an open question but we can assert that if such a solution does exist it will not coincide in general with the Shapley value of convex games. Let N = {1, 2, 3, 4} be a set of players and consider the following 4-person balanced games:
v ðSÞ ¼
4 if S 2 ff1; 2; 3g; f1; 2; 4g; f1; 3; 4g; Ng 0
otherwise
and
wðSÞ ¼
v ðSÞ
if S–N
8
if S ¼ N
The game w is convex and therefore its Shapley value, 3; 53 ; 53 ; 53 is a core allocation. The only core allocation of the game (N, v) is (4, 0, 0, 0). Therefore a monotonic core solution should not select an allocation where player 1 receives a payoff lower than 4. Both games are veto balanced games and it is immediately apparent that SR1(N, w) = 5 > 4. 6. Concluding remarks The monotonicity property used in the characterization of the serial rule may be interpreted in other terms. The property seeks to maintain a general principle that can be called equal treatment of equally affected agents (ETEA). If the worth of a coalition changes we can expect the members of this coalition to suffer the same consequences if there is no other change. In our paper
213
we try to find compatibility between this principle and the principle of core stability. This compatibility requires that no all members of the coalition should be treated equally. This principle of ETEA has inspired several relevant papers, so our analysis runs along broadly similar lines to those papers even though they may deal with different models and formalize the principle defining axioms of fairness (Brink, 2001) or equal responsibility (Maniquet (2003)). Therefore it is possible that the serial rule introduced in this paper may also satisfy these other axioms whenever they can be adapted to our model. References Arin, J., Feltkamp, V., 2005. Monotonicity properties of the nucleolus on the domain of veto balanced games. TOP 13 (2), 331–342. Brink, R. van den, 1997. An axiomatization of the disjunctive permission value for games with a permission structure. International Journal of Game Theory 26, 27–43. Brink, R. van den, 2001. An axiomatization of the Shapley value using a fairness property. International Journal of Game Theory 20, 183–190. Grotte, J., 1970. Computation of and Observations on the Nucleolus, the Normalised Nucleolus and the Central Games. Ph. D. thesis, Cornell University, Ithaca. Littlechild, S.C., Owen, G., 1973. A simple expression for the Shapley value of a spedial case. Management Science 3, 370–372. Maniquet, F., 2003. A characterization of the Shapley value in queuing problems. Journal of Economic Theory 109, 90–103. Meggido, N., 1974. On the monotonicity of the bargaining set, the kernel and the nucleolus of a game. SIAM Journal of Applied Mathematics 27, 355–358. Moulin, H., Shenker, S., 1992. Serial cost sharing. Econometrica 60, 1009–1037. Moulin, H., Shenker, S., 1994. Average cost pricing versus serial cost sharings; an axiomatic comparison. Journal of Economic Theory 64, 178–201. Muto, S., Nakayama, M., Potters, J., Tijs, S., 1988. On big boss games. International Journal of Game Theory 39, 303–321. Muto, S., Potters, J., Tijs, S., 1989. Information market games. International Journal of Game Theory 18, 209–226. O’Neill, B., 1982. A problem of rights arbitration from the Talmud. Mathematical Social Sciences 2, 345–371. Peleg, B., Sudholter, P., 2007. Introduction to the Theory of Cooperative Games. Kluwer Academic, Boston. Schmeidler, D., 1969. The nucleolus of a characteristic function game. SIAM Journal of Applied Mathematics 17, 1163–1170. Shapley, L.S., 1953. A value for n-person games. In: Kuhn, H.W., Tucker, A.W. (Eds.), Contributions to the Theory of Games, vol. II. Shapley, L.S., Shubik, M., 1967. Ownership and the production function. The Quarterly Journal of Economics 81, 88–111. Young, H.P., 1985. Monotonic solutions of cooperative games. International Journal of Game Theory 14, 65–72.
Contents lists available at ScienceDirect
European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor
Decision Support
Coalitional games: Monotonicity and core q J. Arin a,⇑,1, V. Feltkamp b a b
Dpto. Ftos. A. Económico I, University of the Basque Country, L. Agirre 83, 48015 Bilbao, Spain Maastricht School of Management, P.O. Box 1203, 6201 BE Maastricht, The Netherlands
a r t i c l e
i n f o
Article history: Received 3 January 2011 Accepted 13 July 2011 Available online 23 July 2011 Keywords: Monotonicity Core TU games Per capita nucleolus
a b s t r a c t We characterize a monotonic core solution defined on the class of veto balanced games. We also discuss what restricted versions of monotonicity are possible when selecting core allocations. We introduce a family of monotonic core solutions for veto balanced games and we show that, in general, the per capita nucleolus is not monotonic. Ó 2011 Elsevier B.V. All rights reserved.
1. Introduction Young (1985) formulates an impossibility result for the problem of finding core solutions satisfying monotonicity on the domain of balanced TU games. A positive result holds on the domain of convex games where the Shapley value (Shapley, 1953) is a core solution satisfying several monotonicity requirements. On the domain of veto balanced games, Arin and Feltkamp (2005) introduce a monotonic core solution that does not entirely accomplish the goals we would expect from a fair solution concept. The solution is very extreme and divides the worth of the grand coalition equally among the veto players (we call this the all for veto players solution). We also argue that the per capita nucleolus is not monotonic on the class of veto balanced games even if it does satisfy strong N-monotonicity. Therefore, the existence of monotonic core solutions in the class of veto balanced games is still an open question. This paper deals with this problem. We discuss the possibilities of combining monotonicity and core selection. We introduce restricted monotonicity requirements and given these restricted versions of monotonicity properties, we characterize a core solution on the class of veto balanced games. We call this core solution
serial rule and we also offer an alternative characterization where the requirement of being a core solution is not needed. The core solution characterized proves to be a member of a family of core solutions that also includes the all for veto players solution. The class of veto balanced games2 have been used to model social situations in which a group of agents has the power of veto. This and the class of convex games may be the most significant classes of TU games. Various economic problems have been studied using this class of games, and in this way, interesting subclasses of games have been generated. Some examples are Information market games (Muto et al., 1988), Clan games (Muto et al., 1989) and Landowner economies (Shapley and Shubik, 1967). The class is also useful for modeling hierarchical situations where a top player is necessary for a project to be developed (see, for example, Brink, 1997). A different approach opened up by the result of Young is to define new properties of monotonicity (weaker than the one formulated by Young) and to deal with the class of all TU balanced games. Following this approach we also discuss whether the serial rule may be a candidate for a monotonic core solution in the class of all balanced games. 2. Preliminaries
q We thank R. Velez-Cardona and two anonymous referees for their helpful comments. ⇑ Corresponding author. E-mail address: [email protected] (J. Arin). 1 This research is supported by the Spanish Ministerio de Ciencia e Inno-vación under projects SEJ2006-05455 and ECO2009-11213, co-funded by ERDF, and by Basque Government funding for Grupo Consolidado GIC07/146-IT-377-07. This author also thanks the Department of Economics of the University of Rochester for its hospitality and the ‘‘Salvador de Madariaga’’ Program of the Spanish Ministerio de Educación y Ciencia for the financial support provided.
0377-2217/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2011.07.025
2.1. TU Games A cooperative n-person game in characteristic function form is a pair (N, v), where N is a finite set of n elements and v : 2N ! R is a real-valued function on the family 2Nof all subsets of N with 2 This paper does not set out to analyze the solution concept characterized here in different settings. We consider that for some interesting subclasses of veto balanced games further research may be of interest.
J. Arin, V. Feltkamp / European Journal of Operational Research 216 (2012) 208–213
v(;) = 0. Elements of N are called players and the real valued function v the characteristic function of the game. Any subset S of N is called a coalition. For notational convenience we sometimes denote the singleton {i} by i. The number of players in S is denoted by jSj. Given S N we denote by NnS the set of players of N that are not in S. We only consider games where the worth of all coalitions is nonnegative. A distribution of v(N) among the players is a real-valued vector x 2 RN where xi is the payoff assigned by x to player i. A distribution satisfying xi P v(i) for all i 2 N is called an imputation and P the set of imputations is denoted by I(v). We denote i2S xi by x(S). The core of a game is the set of imputations that cannot be blocked by any coalition, i.e.
Cðv Þ ¼ fx 2 Iðv Þ : xðSÞ P v ðSÞ for all S Ng: It has been shown that a game with a non-empty core is balanced3 and therefore games with non-empty core are called balanced games. Player i is a veto player if v(S) = 0 for all S where player i is not present. A balanced game with at least one veto player is called a veto balanced game. We denote by CB the class of balanced games and by CVB the class of veto balanced games. A solution / on a class of games C0 is a correspondence that associates with every game (N, v) in C0 a set /(N, v) in RN such that x(N) 6 v(N) for all x 2 /(N, v). This solution is efficient if this inequality holds with equality. The solution is single-valued if the set contains a unique element for every game in the class. For notational convenience we write a single-valued solution as /ðv Þ 2 Rn : Also for notational convenience and if there is no confusion we write /i instead of /i(v). We say that a solution satisfies core selection if it selects core allocations whenever the core is non-empty. Schmeidler (1969) introduced the nucleolus of a game v, denoted by g(v). For any game v with a non-empty imputation set, the nucleolus is a single-valued solution, is contained in the kernel and lies in the core provided that the core is non-empty. The per capita nucleolus (Grotte, 1970) is related with the nucleolus and selects core allocations whenever the game is balanced. In this paper, we study solution concepts that select precisely one core allocation for each balanced game. We call such concepts core solutions. Let / be a single-valued solution concept defined on a class of games C0. We say that / satisfies the equal treatment property (ETP) if for each (N, v) 2 C0 interchangeable players i, j are treated equally, i.e., /i(N, v) = /j(N, v). Players i and j are interchangeable if v(S [ i) = v(S [ j) for all S # Nn{i, j}.
2.2. On monotonicity properties We present some monotonicity properties for single-valued solution concepts. Let / be a single-valued solution on a class of games C0. We say that solution / satisfies: Monotonicity: if for all v, w 2 C0, such that for all T containing player i, v(T) 6 w(T), and for all S # Nn{i}v(S) = w(S), then, /i(w) P /i(v). Coalitional monotonicity: if for all v, w 2 C0, if for all S – T, v(S) = w(S) and v(T) < w(T), then for all i 2 T, /i(v) 6 /i(w). Strong coalitional monotonicity: if for all v, w 2 C0, if for all S – T, v(S) = w(S) and v(T) < w(T), then for all i, j 2 T, /i(w) /i(v) = /j(w) /j(v) P 0. N-monotonicity (Meggido, 1974): if for all v, w 2 C0, if for all S – N, v(S) = w(S) and v(N) < w(N), then for all i 2 N, /i(v) 6 /i(w). 3
See Peleg and Sudholter (2007).
209
Strong N-monotonicity: if for all v, w 2 C0, if for all S – N, and v(N) < w(N), then for all i, j 2 N, /i(w) /i(v) = /j(w) /j(v) P 0. The strong versions of the properties capture the idea of equal treatment of players equally affected (by the monotonic change). Monotonicity and coalitional monotonicity are equivalent whenever each coalitional change generates games belonging to the domain in which the solution has been defined. Coalitional monotonicity implies N-monotonicity. It is also immediately apparent that strong coalitional monotonicity implies strong Nmonotonicity. In the class of veto balanced games there exist core solutions satisfying monotonicity. We call the following core solution the all for veto players solution (denoted by AVS). Let (N, v) be a veto balanced game with T as the set of veto players and let AVS be the solution for veto balanced games defined as follows:
v(S) = w(S)
( AVSi ðN; v Þ ¼
v ðNÞ jTj
for all i 2 T
0
for all i 2 N n T:
This solution does not satisfy the strong versions of the monotonicity properties. Young (1985) proves that in the class of balanced games there is no solution satisfying coalitional monotonicity and core selection. The next section discusses the impossibility of finding a core solution (on the class of veto balanced games) that satisfies the strong monotonicity requirements. But still is possible to find more core solutions satisfying different properties of monotonicity. It is also interesting to note that the per capita nucleolus satisfies strong N-monotonicity on the class of all TU games. The per capita nucleolus selects a core allocation whenever the game is balanced. 3. On monotonic core selections 3.1. Examples and properties We use two examples to illustrate what kind of restricted monotonicity properties are allowed for core solutions on the domain of veto balanced games. Example 1. Let N = {1, 2, 3} a set of players and consider the following 3-person veto balanced games:
8 8 > > < 6 if S ¼ f1; 2g < 6 if S 2 ff1; 2g; f1; 3gg and wðSÞ ¼ 6 v ðSÞ ¼ > 6 if S ¼ N if S ¼ N > : : 0 otherwise: 0 otherwise A core solution satisfying ETP should choose (6, 0, 0) and (3, 3, 0) in games (N, v) and (N, w). Player 3 (unlike player 1) does not receive any benefit from the fact that he is a member of the only coalition changing its worth by increasing it. Therefore, core solutions do not satisfy strong coalitional monotonicity. However there are core solutions that satisfy strong N-monotonicity. Example 2. Let N = {1, 2, 3, 4} a set of players and consider the following 4-person veto balanced games:
8 8 6 if 1 2 S and jSj ¼ 3 > > > > < 6 if1 2 S and jSj ¼ 3 < or S ¼ f1; 2g v ðSÞ ¼ > 6 and wðSÞ ¼ if S ¼ N > 6 if S ¼ N : > > 0 otherwise : 0 otherwise: A core selection should choose (6, 0, 0, 0) in the games (N, v) and (N, w).
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Players 1 and 2 do not receive any benefit from the fact that they are the members of the only coalition changing its worth by increasing it. Therefore, core selections impose some restrictions on the monotonicity properties we can ask for. The fact that, given a game, only one coalition increases its worth does not imply necessarily that all members of the coalition should benefit from this increase. The aim of a monotonic core solution, given these restrictions, should be to be as strongly monotonic as possible. This notion is formalized below. The first problem is to detect whether or not increasing the worth of a coalition is strictly profitable for some (maybe all) members of the coalition or not. Example 1 shows that player 3 does not receive any benefit from the fact that the only coalition that has increased its worth contains him. This is so because player 3 is not necessary for players 1 and 2 in order to obtain a worth of six. Player 3 has no power to claim that he is necessary for the rest of the players in order to get the worth of 6. After the increase of the worth of the coalition formed by players 1 and 3 the situation remains the same since the new worth of the modified coalition is still lower than or equal to 6. That is, player 3 is still powerless and can be excluded (and should be excluded if we look for core allocations) when the rest of the players distribute this worth of 6. This observation motivates the introduction of the following restricted versions of monotonicity. Assume player l is a member of the coalition S and that this coalition increases its worth. If the new worth of the coalition is higher than the worth for which player l can be excluded we can expect that player l can receive a higher payoff in the new situation without violating core restrictions. Players that are in this situation can be equally benefited from the new game. We introduce the following two properties. Let / be a single-valued solution defined on C0. Property I (Restricted strong coalitional monotonicity): Let v, w 2 C0, such that v(T) < w(T), and for all S – T v(S) = w(S) and let P = {i 2 T; maxS # Nn{i}v(S) 6 v(T)}. Then / satisfies Property I if for all i, j 2 P,/i(w) /i(v) = /j(w) /j(v) P 0. Property II (Restricted strong monotonicity): Let v, w 2 C0 and let Q be a set of coalitions of N, such that v(T) < w(T), for all T 2 Q and v(S) = w(S) for all S R Q. Let T1 2 arg minT2Qv(T) and let
II
if
for
3.2. Selecting monotonic core allocations Let (N, v) be a game with veto players and let player 1 be a veto player. Define for each player i a value di as follows:
di ¼ maxS # Nnfig v ðSÞ: Then d1 = 0. Let dn+1 = v(N) and rename players according to the nondecreasing order of those values. That is, player 2 is the player with the lowest value besides player 1 and so on. We also use the notation di instead of di(N, v) for simplicity. The solution SR associates to each game with veto players, (N, v), the following payoff vector:
SRl ðN; v Þ ¼
all
i,
j 2 P,
/
Property I and Property II are equivalent on the class of veto balanced games4. Note that Property I (with efficiency) implies strong N-monotonicity. The properties can also be seen as fairness properties. All equal players that have been equally and positively affected by the changes should be equally treated. In our definitions we identify the set of players that should be equally treated. In the class of all TU games the Shapley value satisfies these properties and also the unrestricted versions of the properties (Strong coalitional monotonicity). In the class of veto balanced games the Shapley value is not a core solution. Hereafter we will use the term monotonic core solution5 to refer to any core solution satisfying these two properties. A monotonic 4 Let (N, v) be a veto balanced game. Let (N, w) be a veto balanced game where w(T) > v(T) for all T in Q. Then any game (N, q) where q(T) > v(T) for some T in Q is also a veto balanced game. In general, this is not true in the class of convex games and therefore the two properties are not equivalent in this class. 5 We use this term instead of restricted strongly monotonic core concept for simplicity.
Xn diþ1 di for all l 2 f1; . . . ; ng: i¼l i
Note that since d1 = 0 the solution is efficient. In case no player is veto the solution is not efficient. The following example illustrates how the solution behaves. Example 3. Let N = {1, 2, 3, 4} a set of players and consider the following 4-person veto balanced game (N, v) where
8 8 if S 2 ff1; 2; 3g; f1; 2; 4gg > > > < 6 if S ¼ f1; 3; 4g v ðSÞ ¼ > 12 if S ¼ N > > : 0 otherwise: Computing the vector of d-values we get:
ðd1 ; d2 ; d3 ; d4 ; d5 Þ ¼ ð0; 6; 8; 8; 12Þ: Applying the formula,
SR1 ¼ SR2 ¼ SR3 ¼ SR4 ¼
P ¼ fi 2 \T2Q T; maxS # Nnfig v ðSÞ 6 v ðT 1 Þg: Then / satisfies Property i(w) /i(v) = /j(w) /j(v) P 0.
core allocation for a game is any allocation selected by a monotonic core solution.
d2 d1 1
þ
d3 d2 2 d3 d2 2
þ þ
d4 d3 3 d4 d3 3 d4 d3 3
þ þ þ
d5 d4 4 d5 d4 4 d5 d4 4 d5 d4 4
¼8 ¼2 ¼1 ¼ 1:
The definition suggests that the solution follows a serial rule principle. Each player i has a right, a veto power, over the amount v(N) di and the amount is divided equally among the players with that right, veto power, over it. By a serial rule principle we refer to the ideas underlying on the definition of solution concepts such as the minimal overlapping rule for bankruptcy problems (O’Neill, 1982), the Shapley value for airport problems (Littlechild and Owen, 1973) or the serial cost rule for cost sharing problems (Moulin and Shenker, 1992 and Moulin and Shenker, 1994). Roughly speaking, a serial rule-like solution concept for TU games behaves as follows: the amount v(N) is divided in several parts and each part is divided equally among the group of players that have a demand over this part. Hence, the solution has two components: a vector of demands (claims or rights) is defined for a given TU game and a serial rule principle is applied over this vector. The question about which is the demand of each player does not need to have a unique answer and, therefore, different solutions can be defined under this principle of serial rule 6. 6 For example, the AVS can be seen as a serial rule where di = v(N) for all non veto players. It is not difficult to prove that if a serial rule satisfies core selection then di P maxS # Nn{i}v(S). Therefore the serial rule introduced in this paper gives to all non veto players the minimal demand compatible with core selection. In this case this minimal demand is the best situation for non veto players.
J. Arin, V. Feltkamp / European Journal of Operational Research 216 (2012) 208–213
The main result of the paper is that the introduced solution concept is the only monotonic core solution on the class of veto balanced games. First of all we prove the following lemma. Lemma 4. Let (N, v) be a veto balanced TU game. Let r be a P l1 monotonic core solution on CVB. Then r1 ri ¼ il¼2 dl d ¼ l1 SR1 SRi for all i 2 {2, . . . , n}. Proof. We first prove that r1 r2 = d2. Consider the game (N, w) where w(S) = min{v(S), d2}. Clearly (N, w) is a veto balanced game and its core contains only the allocation (d2, 0, . . . , 0) and therefore r(N, w) = (d2, 0, . . . , 0). From game (N, w) to game (N, v) all coalitions that have increased their worth contain players 1 and 2. By applying Property II to the pair {(N, v), (N, w)} we know that r1(v) r1(w) = r2(v) r2(w) P 0. Therefore, r1(w) r2(w) = r1(v) r2(v) = d2. Assume that the lemma is true for any k 2 {2, . . . , i 1}. We P l1 prove that r1 ri ¼ il¼2 dl d : Consider the game (N, q) where l1 q(S) = min{v(S), di}. Clearly (N, q) is a veto balanced game and if x is a core allocation then xl = 0 for l 2 {i, . . . , n} and by efficiency Pi1 Therefore rl(N, q) = 0 for l 2 {i, . . . , n} and k¼1 xk ¼ di . Pi1 k¼1 rk ðN; qÞ ¼ di : From game (N, q) to game (N, v) all coalitions that increase their worth contain the set of players {1, 2, . . . , i}. By applying Property II to the pair {(N, v), (N, q)} we know that r1(v) r1(q) = rk(v) rk(q) P 0 for all k 2 {2, . . . , i}. Therefore r1(q) rk(q) = r1(v) rk(v) for all k 2 {2, . . . , i}. By P l1 assumption r1 ðv Þ rk ðv Þ ¼ kl¼2 dl d for all k 2 {2, . . . , i 1}. l1 Pk dl dl1 Therefore r1 ðqÞ rk ðqÞ ¼ l¼2 l1 for all k 2 {2, . . . , i 1} and P we have seen that i1 k¼1 rk ðN; qÞ ¼ di : These two facts imply that Pi dl dl1 r1 ðqÞ ¼ SR1 ðqÞ ¼ l¼2 l1 . This is a consequence of solving r1(q) in the following linear system of equalities: i1 X
rk ðN; qÞ ¼ di and r1 ðqÞ rk ðqÞ ¼
k¼1
k X dl dl1 for all k l1 l¼2
211
and the serial rule provides non negative payoffs to all players. Therefore we focus on coalitions with veto players that have a P positive worth. Note that k1 l¼1 SRl ðN; v Þ P dk : Consider the smallest P k 2 NnS. Therefore, {1, 2, . . . , k 1} # S. Then v ðSÞ 6 dk 6 k1 l¼1 P SRl 6 l2S SRl . We now prove that SR satisfies Property I and therefore Property II. Let (N, v), (N, w) 2 CVB, such that v(T) < w(T), and for all S – T v(S) = w(S) and let P = {i 2 T;maxS # Nn{i}v(S) 6 v(T)}. Then for any l 2 P it holds that dl(N, v) = dl(N, w). Note that if l 2 P and dk(N, w) 6 dl(N, w) then dk(N, v) = dk(N, w). Therefore we get
SR1 ðN; v Þ SRl ðN; v Þ ¼ SR1 ðN; wÞ SRl ðN; wÞ: This is equivalent to
SR1 ðN; wÞ SR1 ðN; v Þ ¼ SRl ðN; wÞ SRl ðN; v Þ: That means that all players in P are equally affected when increasing the worth of coalition T. h If a veto balanced game is not positive then it is not necessarily true that its serial rule selects a core allocation. Consider a 3-person veto balanced game as follows:
8 > < 4 if S 2 ff1g; f1; 2g; f1; 3gg v ðSÞ ¼ > 3 if S ¼ N : 0 otherwise Clearly, SR(v) = (1, 1, 1) R C(v). The requirement of positivity may be replaced by monotonicity of the game in order to avoid these problems. We axiomatize the solution with a different set of axioms where to be core solution is not needed as an axiom but it is implied by the axiomatization. We introduce a new axiom. Let / be a single-valued solution defined on C0. / satisfies reasonability if v(i) 6 /i(v) 6 v(N) di for all i 2 N and for all (N, v) 2 C0.
2 f2; . . . ; i 1g: Adding both sides over all k = 2, 3, . . . , i 1 yields i1 X k i1 Xi1 X dl dl1 X ðr1 ðqÞ rk ðqÞÞ ¼ ði 2Þr1 ðqÞ r ðqÞ ¼ k¼2 k l 1 k¼2 l¼2 k¼2
¼ ði 2Þr1 ðqÞ ðdi r1 ðqÞÞ: Due to a straightforward combinatorial argument, it follows
ði 1Þr1 ðqÞ di ¼ ¼
i1 X k i1 X i1 X dl dl1 X dl dl1 ¼ l 1 l1 k¼2 l¼2 l¼2 k¼l i1 X il ðdl dl1 Þ: l 1 l¼2
Theorem 6. SR is the only solution on CVB that satisfies efficiency, reasonability and Property II. The proof is almost identical to the proof of Theorem 5 and therefore omitted. Lemma 4 can be rewritten replacing to be core concept by to be a solution satisfying reasonability and the proof is identical: we use jointly efficiency and reasonability to identify the outcome of the solution in the games (N, w) and (N, q). Note that any Core-Selector / satisfies reasonability since the upper bound /l(N, v) 6 v(N) dl arises from the core constraints applied to any coalition S # Nn{l}. Therefore the serial rule satisfies reasonability. Note that the Shapley value satisfies efficiency and Property II but not reasonability while the nucleolus satisfies efficiency and reasonability but not Property II.
So i1 i X d 1 X il dl dl1 r1 ðqÞ ¼ i þ ðdl dl1 Þ ¼ i 1 i 1 l¼2 l 1 l1 l¼2
by rearranging terms. h Theorem 5. SR is the only monotonic core solution on CVB. Proof. It is immediate that SR is the only efficient solution concept that satisfies the conditions of the previous lemma. We now prove that SR is a core solution. Let (N, v) be a veto balanced TU game. We need to prove that for any coalition S it holds that SR(S) P v(S). The worth of coalitions without the veto player is 0
4. Per capita nucleolus and strong N-monotonic core solutions The per capita nucleolus is a core solution that in the class of all balanced games satisfies strong N-monotonicity. This is why some authors consider this core solution a good candidate when trying to select monotonic core allocations. The following example shows that the per capita nucleolus violates the monotonicity principle and therefore, even if it does satisfy strong N-monotonicity, it can hardly be seen as a monotonic core solution (at least in the class of all balanced games and in the subclass of veto balanced games). Example 7. Let N = {1, 2, . . . , 6} a set of players and consider the following 6-person veto balanced games:
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8 8 > > < 6 if 1 2 S and jSj ¼ 5 < 8 if S 2 N n f6g; v ðSÞ ¼ > 6 and wðSÞ ¼ if S ¼ N N n f5g;N > : : v ðSÞ otherwise: 0 otherwise A core solution must choose the allocation (6, 0, 0, 0, 0, 0) in the first game. The per capita nucleolus of the second game is (5.75, 0.75, 0.75, 0.75, 0, 0). Therefore, the per capita nucleolus is not monotonic since player 1 receives a lower payoff in the second game. Therefore the per capita nucleolus does not satisfy monotonicity (see SubSection 2.2). In the class of veto balanced games there are core solutions (other than the serial rule introduced in the previous subsection) that satisfy monotonicity and strong N-monotonicity such as the following modified version of the all for veto players solution. Let (N, v) be a veto balanced game with T as the set of veto players and let rn be a solution for veto balanced games defined as follows:
rni ¼
8 < dn þ dnþ1 dn jTj
:
for all i 2 T
jNj
dnþ1 dn jNj
for all i 2 N n T:
Similarly, core solutions satisfying monotonicity can be defined all of them violating Property I. Let (N, v) be a veto balanced game with T as the set of veto players and let rk be a solution for veto balanced games defined as follows:
rkl ¼
8 > > > < > > > :
dk jTj
þ
n P i¼k
n P i¼maxfl;kg
The solution is continuous, satisfies the necessary conditions of the Theorem but is not monotonic. The following example illustrates that the solution is not monotonic. Example 10. Let N = {1, 2, 3, 4} a set of players and consider the following 4-person veto balanced games:
8 8 > > < 6 if 1 2 S and jSj ¼ 3 < 12 if 1 2 S and jSj ¼ 3 v ðSÞ ¼ > 24 and wðSÞ ¼ 24 if S ¼ N if S ¼ N > : : 0 otherwise 0 otherwise: Computing the solution for the two games we obtain:
rðN; v Þ ¼ ð19:5; 1:5; 1:5; 1:5Þ and rðN; wÞ ¼ ð16; 2; 2; 2Þ: Clearly, player 1 has received a lower payoff when the coalitions that have increased their worth contain him. 5. Games without veto players We have shown that in the class of veto balanced games there are monotonic core solutions and we have axiomatized the serial rule in this class. The serial rule is well defined for any TU game but in general is not efficient. An efficient modified serial rule can be defined as follows. Let (N,v) be a TU game. Define for each player i a value di as follows:
di ¼ maxS # Nnfig v ðSÞ:
diþ1 di i
for all l 2 T
diþ1 di i
for all l 2 N n T:
We conclude this section identifying some necessary conditions that any core solution that satisfies monotonicity (see SubSection 2.2) must fulfill. Theorem 8. Let (N, v) be a veto balanced TU game. Let r be a core P solution on CVB that satisfies monotonicity. Then lk¼1 rk P dlþ1 for all l 2 {1, . . . , n}. Proof. Assume on the contrary that the theorem is not true and P that for some l, l 2 {1, . . . , n} holds lk¼1 rk < dlþ1 : Since r is a core P solution this l is different than n. Let lk¼1 rk < dlþ1 and consider the game (N, w) where w(S) = min{v(S), dl+1}. Since r is a core soluP tion i2N ri ðwÞ ¼ dlþ1: By monotonicity of the solution, for all k 2 {1, . . . , l} it holds that rk(v) P rk(w) and therefore P P i2f1;...;lg ri ðv Þ P i2f1;...;lg ri ðwÞ ¼ dlþ1: This result contradicts the initial assumption and the proof is completed. h Similarly with the results of the previous subsection we also point out the following result that has almost an identical proof.
Let dn+1 = v(N) and rename players according to the nondecreasing order of those values. The solution SR associates to each TU game, (N, v), the following payoff vector:
SRl ðN; v Þ ¼
n X diþ1 di d1 þ for all l 2 f1; . . . ; ng: i n i¼l
In general, this solution is not a core solution, not even in the class of convex games. Therefore it is not immediately apparent that serial rule like concepts may be candidates for monotonic core solutions in domains other than the domain of veto balanced games. A different aspect of the problem is the existence of core solutions satisfying weak monotonicity properties. In particular we can consider the following property: Property III (Restricted weak coalitional monotonicity): Let v, w 2 C0, such that v(T) < w(T), and for all S – T v(S) = w(S) and let
P ¼ fi 2 T; maxS # Nnfig v ðSÞ 6 v ðTÞg: Then / satisfies Property III if for all i 2 P, /i(w) /i(v) P 0. The following result solves this question. Theorem 11. In the class of balanced games there is no core concept that satisfies restricted weak coalitional monotonicity.
Theorem 9. Let (N, v) be a veto balanced TU game. Let r be a solution concept on CVB that satisfies monotonicity, efficiency and reasonabilP ity. Then lk¼1 rk ðv Þ P dlþ1 for all l 2 {1, . . . , n}.
Proof. Let N = {1, 2, 3, 4} be a set of players and consider the following 4-person balanced games:
There are core solutions defined on CVB that satisfy the conditions of the theorem and do not satisfy monotonicity. Let (N, v) be a veto balanced game such that v(N) > 0. Let T be the set of veto players. Let r be a solution for veto balanced games defined as follows:
v ðSÞ ¼ >
rl ðN; v Þ ¼
8 > <
dn dnþ1 dn dnþ1 jNnTj d
for all l R T
v ðNÞd n ðdnþ1 dn Þ > nþ1 : for all l 2 T: jTj
8 > > > < 12 if S 2 24 > > : 0
f1; 3g; f1; 4g; f2; 3g; f2; 4g
f1; 2; 3g; f1; 2; 4g; f1; 3; 4g; f2; 3; 4g if S ¼ N otherwise
and
wi ðSÞ ¼
v ðSÞ
if S–N n fig
24
if S ¼ N n fig
where i 2 N. Let /a core solution satisfying restricted weak coalitional monotonicity. For the games w1 and w2 we need to consider the only
J. Arin, V. Feltkamp / European Journal of Operational Research 216 (2012) 208–213
core allocation of the two games, z1 = (0, 0, 12, 12) and for games w3 and w4 we need to consider the allocation z3 = (12, 12, 0, 0). By applying Property III we need to conclude that /1(v) = 0 and /2(v) = 0 (considering games w1 and w2). By applying Property III we also need to conclude that /3(v) = 0 and /4(v) = 0 (considering games w3 and w4). Therefore core solutions are not compatible with Property III. h In the class of convex games the Shapley value is a core solution and therefore a monotonic core solution. But, in general, the Shapley value is not a core solution. This motivates the following question: Is there a core solution defined for the class of all balanced games that is monotonic in the class of convex games and in the class of veto balanced games? This is an open question but we can assert that if such a solution does exist it will not coincide in general with the Shapley value of convex games. Let N = {1, 2, 3, 4} be a set of players and consider the following 4-person balanced games:
v ðSÞ ¼
4 if S 2 ff1; 2; 3g; f1; 2; 4g; f1; 3; 4g; Ng 0
otherwise
and
wðSÞ ¼
v ðSÞ
if S–N
8
if S ¼ N
The game w is convex and therefore its Shapley value, 3; 53 ; 53 ; 53 is a core allocation. The only core allocation of the game (N, v) is (4, 0, 0, 0). Therefore a monotonic core solution should not select an allocation where player 1 receives a payoff lower than 4. Both games are veto balanced games and it is immediately apparent that SR1(N, w) = 5 > 4. 6. Concluding remarks The monotonicity property used in the characterization of the serial rule may be interpreted in other terms. The property seeks to maintain a general principle that can be called equal treatment of equally affected agents (ETEA). If the worth of a coalition changes we can expect the members of this coalition to suffer the same consequences if there is no other change. In our paper
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we try to find compatibility between this principle and the principle of core stability. This compatibility requires that no all members of the coalition should be treated equally. This principle of ETEA has inspired several relevant papers, so our analysis runs along broadly similar lines to those papers even though they may deal with different models and formalize the principle defining axioms of fairness (Brink, 2001) or equal responsibility (Maniquet (2003)). Therefore it is possible that the serial rule introduced in this paper may also satisfy these other axioms whenever they can be adapted to our model. References Arin, J., Feltkamp, V., 2005. Monotonicity properties of the nucleolus on the domain of veto balanced games. TOP 13 (2), 331–342. Brink, R. van den, 1997. An axiomatization of the disjunctive permission value for games with a permission structure. International Journal of Game Theory 26, 27–43. Brink, R. van den, 2001. An axiomatization of the Shapley value using a fairness property. International Journal of Game Theory 20, 183–190. Grotte, J., 1970. Computation of and Observations on the Nucleolus, the Normalised Nucleolus and the Central Games. Ph. D. thesis, Cornell University, Ithaca. Littlechild, S.C., Owen, G., 1973. A simple expression for the Shapley value of a spedial case. Management Science 3, 370–372. Maniquet, F., 2003. A characterization of the Shapley value in queuing problems. Journal of Economic Theory 109, 90–103. Meggido, N., 1974. On the monotonicity of the bargaining set, the kernel and the nucleolus of a game. SIAM Journal of Applied Mathematics 27, 355–358. Moulin, H., Shenker, S., 1992. Serial cost sharing. Econometrica 60, 1009–1037. Moulin, H., Shenker, S., 1994. Average cost pricing versus serial cost sharings; an axiomatic comparison. Journal of Economic Theory 64, 178–201. Muto, S., Nakayama, M., Potters, J., Tijs, S., 1988. On big boss games. International Journal of Game Theory 39, 303–321. Muto, S., Potters, J., Tijs, S., 1989. Information market games. International Journal of Game Theory 18, 209–226. O’Neill, B., 1982. A problem of rights arbitration from the Talmud. Mathematical Social Sciences 2, 345–371. Peleg, B., Sudholter, P., 2007. Introduction to the Theory of Cooperative Games. Kluwer Academic, Boston. Schmeidler, D., 1969. The nucleolus of a characteristic function game. SIAM Journal of Applied Mathematics 17, 1163–1170. Shapley, L.S., 1953. A value for n-person games. In: Kuhn, H.W., Tucker, A.W. (Eds.), Contributions to the Theory of Games, vol. II. Shapley, L.S., Shubik, M., 1967. Ownership and the production function. The Quarterly Journal of Economics 81, 88–111. Young, H.P., 1985. Monotonic solutions of cooperative games. International Journal of Game Theory 14, 65–72.