The Egalitarian Solution for Convex Games: Some Characterizations February 26, 1999 Flip Klijna,1 , Marco Slikkera, Stef Tijsa, and Jose Zarzuelob

a

Department of Econometrics and CentER, Tilburg University, Tilburg, The Netherlands b Department of Applied Mathematics, University of Pas Vasco, Bilbao, Spain

Abstract: The egalitarian solution for TU-games as introduced by Dutta and Ray [3]

is studied. Five characterizations of the restriction of this solution to the class of convex games are given. They all involve a stability property due to the concept of the equal division core from Selten [8] and all but the third characterization involve a property restricting maximum payos. The rst two characterizations use in addition eciency and the reduced game properties of Hart and Mas-Colell [6] and Davis and Maschler [5], respectively. The fourth and fth characterization only need in addition weak variants of the reduced game properties mentioned above. The third characterization involves besides the stability condition, eciency and a new consistency property. Keywords: Egalitarian solution; Convex games; Characterizations JEL classi cation: C71 The authors thank Herbert Hamers and two anonymous referees for useful suggestions and comments. Corresponding author. Department of Econometrics and CentER, Tilburg University, P.O.Box 90153, 5000 LE Tilburg, The Netherlands. Tel. (31) 13 466 2733; Fax. (31) 13 466 3280; e-mail: [email protected]  1

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1 Introduction Dutta and Ray [3] introduced the egalitarian solution as a solution concept for TU-games. This solution uni es the two con icting concepts of individualistic utility maximization and the social goal of equality. Under certain conditions it is non-empty, and then its outcome is unique, namely it is the Lorenz maximal element of the set of payos satisfying core-like participation constraints. If the egalitarian solution exists, then every other feasible allocation for the grand coalition is either blocked, or is Lorenz dominated by some allocation which, in turn, is not blocked. We refer to Dutta and Ray [3] for the details. For convex games Dutta and Ray [3] describe an algorithm to locate the unique egalitarian solution, and they show, in addition, that it is in the core. Nevertheless, for balanced games, which have a non-empty core, the egalitarian solution might not even exist. Dutta and Ray [4] consider a parallel concept, the S-constrained egalitarian solution, which is not a singleton in general. The construction of the S-constrained egalitarian solution is identical to the egalitarian solution mentioned above, except that in the concept of blocking it is required that every member of the blocking coalition is strictly better o rather than at least one member, as is the case for the original egalitarian solution. They show that in contrast to the original egalitarian solution, S-constrained egalitarian solutions exist under very mild conditions on the game. The two solutions are not completely unrelated, since, for example, for convex games either the egalitarian solution is the unique S-constrained egalitarian allocation or every S-constrained egalitarian allocation Lorenz-dominates the egalitarian solution. Dutta [2] characterizes the egalitarian solution over the class of convex games. The properties used are the reduced game properties due to Hart and Mas-Colell [6] and Davis and Maschler [5]. The egalitarian solution is the only solution concept satisfying either of the two reduced game properties and constrained egalitarianism, which is a prescriptive property on two person games. Arin and I~narra [1] introduce another solution concept that embodies a `bilaterally egalitarian' notion. Their solution concept is called the egalitarian set. They characterize this multi-valued solution over the class of all TU-games by extended constrained egalitarianism, the Davis-Maschler reduced game property, and the converse Davis-Maschler reduced game property. Moreover, they prove that it is non-empty for the class of balanced games. They show in addition that there is nevertheless no solution on the class of balanced games satisfying non-emptiness, extended constrained egalitarianism, and the Hart and Mas-Colell reduced game property. Arin and I~narra [1] show that in general the egalitarian solution of Dutta and Ray [3] does not satisfy the Davis-Maschler reduced game property, nor the Hart and Mas-Colell reduced game property. Finally, Arin and I~narra [1] show that the egalitarian solution belongs to the egalitarian set, and that for convex games it even coincides with the latter. Given the results above that for the class of convex games the egalitarian solution exists and coincides with the egalitarian set, we would like to reconsider the egalitarian solution on the class of convex games. 2

In this paper we provide ve characterizations of the egalitarian solution, without directly making use of Dutta's [2] prescriptive property constrained egalitarianism on twoperson games. All ve characterizations involve a stability property due to the concept of the equal division core from Selten [8] and all but the third characterization involve a property restricting maximum payos. The rst two characterizations use in addition eciency and a reduced game property, more speci cally, the reduced game property of Hart and Mas-Colell [6] and the reduced game property of Davis and Maschler [5], respectively. The fourth and fth characterization only need in addition weak variants of the reduced game properties mentioned above. Here, weak means that we only look at the reduced game where the players receiving most have been sent away. The third characterization uses besides the stability property due to the concept of the equal division core from Selten [8] eciency and another consistency property. The stability property due to the concept of the equal division core from Selten [8], called equal division stability, states that for any convex game and for any coalition there is some player in this coalition that gets at least the average of the value of the coalition in the game. Clearly, any core-allocation satis es this property. The intuitive reasoning behind this property is spelled out in Selten [8]. The principle of equal division is a strong distributive norm which in uences the behavior of the players. The attention of the players is attracted by coalitions with high equal shares. This is con rmed by the great number of cases of experimental games in Selten [8] in which the outcome is such that there is no coalition that can divide its value equally among its members giving all of them more than in the original outcome. The second property, concerning the boundedness of the payos, states that the payos of the players receiving most is bounded by imposing the condition that the sum of payos of these players does not exceed the value of the players in the game. This might be desirable from a social point of view. The work is organized as follows. Section 2 deals with notation and de nitions regarding TU-games and recalls the egalitarian solution for convex games. In section 3 we provide several characterizations of this solution concept.

2 Preliminaries A cooperative game with transferable utilities (TU-game) is a pair (N; v), where N = f1; : : : ; ng is the player set and v the characteristic function, which assigns to every subset S of N a value v(S ), with v(;) = 0. A game (N; v) is called convex if 2

v (S [ T ) + v (S \ T )  v (S ) + v (T )

for all S; T  N:

The core of a game (N; v) is de ned by C (N; v ) := fx 2 IRN : x(N ) = v (N ); and x(S )  v (S ) for all S  N g: 2 S  N denotes that S is a subset of N and S  N denotes that S is a strict subset of N .

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by

Throughout this paper we will denote the average worth of coalition S in game (N; v) a(S; v ) :=

v (S ) jS j :

We will recall the algorithm of Dutta and Ray [3] to locate the egalitarian solution for convex games. In every step of the algorithm a cooperative game is considered. The set of players in this game is the set of players that have not received a payo yet. The largest coalition with the highest average worth is selected and the players in this coalition receive this average worth. Let (N; v) be a convex TU-game. De ne N := N and v := v. Step 1: Let S be the largest coalition with the highest average worth in the game (N ; v ). De ne Ei (N; v ) := a(S ; v ) for all i 2 S : Step k : Suppose that S ; : : : ; Sk, have been de ned recursively and S [


[ Sk, 6= N . De ne a new game with player set Nk := Nk, nSk, = N n(S [


[ Sk, ). For all subcoalitions S  Nk , de ne vk (S ) := vk, (Sk, [ S ) , vk, (Sk, ). Convexity of (Nk, ; vk, ) implies convexity of (Nk ; vk ). De ne Sk to be the largest coalition with the highest average worth in this game. De ne Ei (N; v ) := a(Sk ; vk ) for all i 2 Sk : 1

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It can be checked that in every step convexity ensures the existence of a largest coalition with highest average worth. In at most n steps the algorithm ends, and the constructed allocation E (N; v) is called the egalitarian solution of the game (N; v). Dutta and Ray [3] show that E (N; v) is an element of the core of (N; v). Furthermore, they note that for each convex game (N; v) it holds that Ei (N; v ) > Ej (N; v ); for all i 2 Sk ; j 2 Sk : (1) +1

3 Characterizations of the egalitarian solution In this section we provide several characterizations of the egalitarian solution for convex games. The rst two characterizations are based on characterizations of Dutta [2] in the sense that we replace a prescriptive property on two-person games by some other properties. The third characterization uses a new consistency property, whereas the last two characterizations strengthen the rst two characterizations by weakening the respective consistency properties and not demanding eciency a priori. Our rst characterization of the egalitarian solution for convex TU-games involves the properties eciency, equal division stability, bounded maximum payo property, and HM consistency. We describe these properties below. Let C be the set of convex TUgames. A solution on C is a map assigning to each convex game (N; v) 2 C an element 4

(N; v) 2 IRN . Let (N; v) be a convex game. Given the solution , de ne S m(N; v; ) (or S m for short, if no confusion is possible) to be the set of players with the highest payo. Formally, S m = S m (N; v; ) := argmax j (N; v ): j 2N

A solution on C satis es  eciency (EFF) if for all games (N; v) 2 C : X i (N; v ) = v (N ): i2N

 equal division stability (EDS) if for all games (N; v) 2 C and all S  N , S 6= ; there exists i 2 S with i (N; v )  a(S; v ):  bounded maximum payo property (BMPP) if for all games (N; v) 2 C : X m i (N; v )  v (S ): i2S m

 HM consistency (HMC) if for all games (N; v) 2 C , all S  N , and all i 2 N nS : ,S i (N; v ) = i(N nS; v ); where v,S is the reduced subgame de ned by X v ,S (T ) := v (S [ T ) , i (S [ T; v ) i2S

for all subcoalitions T  N nS .  constrained egalitarianism (CE) if for all games (fi; j g; v) 2 C : (

v (fig); v fi;j g ) if v (fj g)  v fi;j g ; (i fi; j g; v) = max( v (fi; j g) , v (fig) otherwise. (

)

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(

)

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(EDS) plays a role in the concept of equal division core from Selten [8]. (BMPP) states that the payos of the players receiving most is bounded, which might be desirable from a social point of view. (HMC) is the well-known consistency property of Hart and Mas-Colell [6]. Dutta [2] characterizes the egalitarian solution with (HMC) and (CE). Theorem 3.1 shows that we can replace the prescriptive property (CE) by the properties (EFF), (EDS), and (BMPP). But rst we prove two lemmas.

Lemma 3.1 If a solution satis es (EDS) and (BMPP) then for all (N; v) 2 C and all i 2 Sm m i(N; v ) = a(S ; v ):

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Proof. Let be a solution that satis es (EDS) and (BMPP). Let (N; v) be a convex game. Suppose there is a player i 2 S m such that i(N; v) < a(S m; v). Since all players in S m receive the same payo we have a contradiction with (EDS). Hence, i(N; v)  a(S m; v) for all i 2 S m . By (BMPP), it then immediately follows that i(N; v) = a(S m; v) for all i 2 S m. 2 Lemma 3.2 If a solution satis es (EFF), (EDS), and (BMPP) then it also satis es (CE).

Proof. Let be a solution that satis es (EFF), (EDS), and (BMPP). Let (f1; 2g; v) be a convex game. Suppose without loss of generality that 1

(f1; 2g; v)  (f1; 2g; v):

(2)

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Suppose the inequality in (2) is an equality. Then (f1; 2g; v) = (f1; 2g; v) = by lemma 3.1. So, together with (EDS) it follows that v f ; g = i(f1; 2g; v)  v (fig). One easily veri es that satis es the condition in (CE). Now suppose the inequality in (2) is strict. Then, by lemma 3.1, (f1; 2g; v) = v(f1g). So, with (EFF), (f1; 2g; v) = v(f1; 2g) , v(f1g). Hence, 1

v(f1;2g) 2

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( 12 ) 2

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v (f1g) =

1

(f1; 2g; v) > (f1; 2g; v) = v(f1; 2g) , v(f1g): 2

So, v(f1g) > v f ; g . Furthermore, by convexity, ( 12 ) 2

v (f1; 2g)

2 + v(f2g): Hence, v(f2g) < v f ; g . Then again it is readily veri ed that satis es the condition in (CE). 2 v (f1; 2g)  v (f1g) + v (f2g) >

( 12 ) 2

We provide our rst characterization in the following theorem.

Theorem 3.1 A solution satis es (EFF), (EDS), (BMPP), and (HMC) if and only if = E.

Proof. First we show that E satis es the properties.

Since E assigns to every convex game a core element, it satis es (EFF) and (EDS). It follows from (1) that every player in S receives the maximum payo and that all other players receive less than this maximum. Since these players divide v(S ) it follows that E satis es (BMPP). From Dutta [2] it follows that E satis es the reduced game property of Hart and Mas-Colell [6]. Suppose a solution satis es the four properties in the theorem. Then, by lemma 3.2 also satis es (CE). Then, by Dutta's [2] characterization with (CE) and (HMC) it 1

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immediately follows that = E . 2 The previous characterization still holds true if we replace the consistency property of Hart and Mas-Colell [6] with the consistency property of Davis and Maschler [5]. A solution on C satis es  DM consistency (DMC) if for all games (N; v) 2 C , all S  N , and all i 2 N nS : i (N; v ) = i (N nS; v,S ); where v,S is the reduced subgame de ned by 8 > if T = ;; <0 P if T = N nS ; v,S (T ) := > v (N ) , i2S i (N; v ) P : maxQS fv (T [ Q) , i2Q i (N; v )g if ;  T  N nS .

Theorem 3.2 A solution satis es (EFF), (EDS), (BMPP), and (DMC) if and only if = E.

Proof. This immediately follows from lemma 3.2 and Dutta's [2] characterization with (DMC). 2

Another characterization of the egalitarian solution is obtained when we use the next, third consistency property. This property only puts a condition on the reduced game where the players receiving most are sent away. A solution on C satis es  max-consistency (MC) if for all games (N; v) 2 C and all i 2 N nS m : m ,S m ); i (N; v ) = i (N nS ; v where v,Sm is the reduced subgame de ned by m v ,S (T ) := v (S m [ T ) , v (S m) for all subcoalitions T  N nS m. Before we prove the characterization we prove another lemma. Lemma 3.3 If a solution satis es (EFF) and (MC) then for all (N; v) 2 C and all i 2 Sm

m i(N; v ) = a(S ; v ):

Proof. Let be a solution that satis es (EFF) and (MC). Let (N; v) bePa convex game. m m , S By eciency of (N; v) and (N nS ; v ) it follows using (MC) that i2Sm i(N; v) = v (S m ). By de nition of S m it follows that for all i 2 S m it holds that i (N; v ) = a(S m ; v ).

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We can now prove the third characterization. 7

Theorem 3.3 A solution satis es (EFF), (EDS), and (MC) if and only if = E . Proof. Since from the algorithm of the egalitarian solution it immediately follows that

E satis es (MC), we prove the "only if"-part. Suppose that a solution satis es the properties. We prove that = E . The proof will be by induction on the number of players. Clearly, for convex games (N; v) with jN j = 1 we have that (N; v) = v(f1g) = E (N; v ) by lemma 3.3. Suppose that for some p  2 we have (N; v ) = E (N; v ) for all convex games (N; v) with jN j  p , 1. We prove that (N; v) = E (N; v) also holds for all convex games (N; v) with jN j = p. Let (N; v) be a convex game with jN j = p. Let S be the largest coalition that maximizes the average worth function a(
; v). First we will show that a(S ; v) = a(S m; v). Since satis es (EDS) there exist i 2 S with i(N; v)  a(S ; v). Then for all j 2 S m we have a(S m; v ) = j (N; v )  i (N; v )  a(S ; v ); where the equality follows from lemma 3.3. The rst inequality follows by de nition of S m . Since the de nition of S implies a(S ; v )  a(S m; v ) we conclude 1

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a(S m; v ) = a(S1; v ):

(3)

Again by de nition of S this implies S m  S . We will show that S m = S . Suppose S m  S . With T = S nS m 6= ; (MC) gives 1

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m

v ,S (S1 nS m ) = v (S1) , v (S m );

because S m  S . But then v (S ) v (S ) , v (S m ) + v (S m) v ,S m (S nS m ) + v (S m ) jS j = jS nS mj + jS mj = jS nS mj + jS mj : From this and (3) it follows that 1

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v (S1) v ,S m (S1 nS m ) a(S1; v ) = jS1j = jS1nS mj :

(4)

Now, using the convexity of the reduced game (N nS m ; v,Sm ) it follows that i

(N nS m; v,Sm ) =

v ,S m (S1 nS m ) m i (N; v ) < a(S1 ; v ) = jS1nS mj for all i 2 S1nS ;

(5)

where the rst equality follows from (MC), the strict inequality from (3) and the de nition of S m , and the second equality from (4). Inequality (5) contradicts with (EDS) of S nS m m in the reduced game (N nS m ; v,S ). Hence, the assumption S m  S is false. Since S m  S , this completes the proof of S m = S . 1

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It remains to prove that indeed from S m = S it follows that (N; v) = E (N; v). Note rst that lemma 3.3, the de nition of S m, and S m = S yield v (S m) = E (N; v) for all i 2 S m : (6) (N; v) = 1

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i jS m j Then, if S m = N we are done. If S m = 6 N it holds that m ,S m ) = E (N nS m ; v ,S m ) = E (N; v ) for all i 2 N nS m ; (7) i (N; v ) = i (N nS ; v i i i

where the rst equality follows from (MC), the second equality from the induction hypothesis, and the third equality from S m = S . The theorem now follows from (6) and (7). 2 1

It follows from the following examples that the properties in theorem 3.3 are logically independent.  The solution that equally divides the worth of the grand coalition to the players satis es (EFF) and (MC), but does not satisfy (EDS).  The solution (N; v) = 2E (N; v) satis es (EDS) and (MC), but not (EFF).  The Shapley value satis es (EFF) and (EDS), but not (MC). Property (EFF) in theorem 3.3 can be replaced by (BMPP). The proof then follows the lines of the proof of theorem 3.3 replacing lemma 3.3 by lemma 3.1. A fourth and fth characterization are obtained by (EDS), (BMPP), together with either a weaker variant of the reduced game property of Hart and Mas-Colell [6] or a weaker variant of the reduced game property of Davis and Maschler [5]. Formally, a solution on C satis es  HM max-consistency (HMMC) if for all games (N; v) 2 C , and all i 2 N nS m: m ,S m ); i (N; v ) = i (N nS ; v where v,Sm is the reduced subgame de ned by X m m v ,S (T ) := v (S m [ T ) , i (S [ T; v ) i2S m

for all subcoalitions T  N nS m. DM max-consistency (DMMC) is de ned similar to (HMMC) but with the reduced game de ned in property (DMC). So, (HMMC) and (DMMC) are weak variants of (HMC) and (DMC), respectively. This restriction is quite natural since the players receiving most are the players that might have the greatest incentive to walk away from the grand coalition. Thus, our fourth and fth characterization of the egalitarian solution are as follows. 9

Theorem 3.4 A solution satis es (EDS), (BMPP), and (HMMC) if and only if =

satis es (EDS), (BMPP), and (DMMC) if and only if = E . The proofs of the characterizations are closely related to the proof of theorem 3.3 and can be found in Klijn et al. [7]. The following examples show that the properties in the two characterizations of theorem 3.4 are logically independent.  The solution that equally divides the worth of the grand coalition to the players satis es (BMPP), (HMMC), and (BMMC), but does not satisfy (EDS).  The Shapley value satis es (EDS) and (HMMC) (see Hart and Mas-Colell [6]), but not (BMPP).  Sobolev [9] showed that the prenucleolus satis es (DMMC). Furthermore, the prenucleolus belongs to the core for convex games and hence it satis es (EDS). Finally, the prenucleolus does not satisfy (BMPP).  We will de ne the solution . Let S1 be the coalition that is determined in the rst step of the algorithm to determine E . If S1 6= N , de ne: ( ) , v(S1) if T = N nS1; v,S1 (T ) := vv ((N T) if T  N nS1: Note that v(N ) , v(S1)  v(N nS1). Hence, (N nS1; v,S1 ) is convex. Now, let ( if i 2 S1; i (N; v ) i (N; v ) := E Ei (N nS1 ; v,S1 ) if i 62 S1 : Obviously, satis es (EFF). We will show that satis es (EDS). Let S  N , S 6= ;. If S \ S1 6= ; then with i 2 S \ S1 it holds that i(N; v) = Ei(N; v)  a(S; v). If S  N nS1 it follows that there exists i 2 S with i (N; v ) = Ei (N nS1 ; v,S1 )  a(S; v,S1 )  a(S; v ), where the last inequality follows by de nition of v. So, also satis es (EDS). Finally we will show that 6= E implying that satis es neither (HMMC) nor (DMMC). Consider the 3-person cooperative game (f1; 2; 3g; v) de ned by 8 10 if S 2 ff1g; f1; 3gg; > > < 16 if S = f1; 2g; v (S ) = > 20 if S = N ; > : 0 otherwise. E . A solution

Then it can be checked that S = f1g, E(N; v) = (10; 6; 4), and (N; v) = (10; 5; 5). Hence, satis es neither (HMMC) nor (DMMC). The main conclusion following from these examples is the following: considering (EFF), (EDS), (BMPP), and (HMMC)/(DMMC), then obviously, theorem 3.4 implies that E can be characterized by these four properties. Furthermore, theorem 3.4 implies that E can be characterized omitting (EFF), but the examples show that we cannot omit any of the other properties. Finally, note that theorem 3.4 strengthens theorems 3.1 and 3.2 by omitting eciency and requiring only a weak consistency property. 1

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References [1] J. Arin and E. I~narra, Consistency and egalitarianism: the egalitarian set, Discussion Paper 163, University of Pas Vasco, Bilbao, Spain, 1997. [2] B. Dutta, The egalitarian solution and reduced game properties in convex games, International Journal of Game Theory 19 (1990) 153-169. [3] B. Dutta and D. Ray, A concept of egalitarianism under participation constraints, Econometrica 57 (1989) 615-635. [4] B. Dutta and D. Ray, Constrained egalitarian allocations, Games and Economic Behavior 3 (1991) 403-422. [5] M. Davis and M. Maschler, The kernel of a cooperative game, Naval Research Logistics Quarterly 12 (1965) 223-259. [6] S. Hart and A. Mas-Colell, Potential, value and consistency, Econometrica 57 (1989) 589-614. [7] F. Klijn, M. Slikker, S. Tijs, and J. Zarzuelo, Characterizations of the egalitarian solution for convex games, CentER Discussion Paper 9833, Tilburg University, The Netherlands, 1998. [8] R. Selten, Equal share analysis of characteristic function experiments, in: H. Sauermann, ed., Beitrage zur experimentellen Wirtschaftsforschung, vol.3 (J.C.B. Mohr, Tubingen, 1972) pp. 130-165. [9] A. Sobolev, The characterization of optimality principles in cooperative games by functional equations, in: N. Vorobjev, ed., Mathematischeskie metody v socialnix naukakh, Proceedings of the seminar, vol. 6 (Vilnius: Institute of Physics and Mathematics, Academy of Sciences of the Lithuanian SSR, 1975) pp. 94-151 [In Russian, English summary].

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