Takumi Kongo Characterizations of the Per-capita Shapley Value

CAES Working Paper SeriesCharacterizations of the Per-capita Shapley ValueTakumi KongoFaculty of EconomicsFukuoka UniversityWP-2012-006Center for Advanced Economic StudyFukuoka University(CAES)8-19-1 Nanakuma, Jonan-ku, Fukuoka,JAPAN 814-0180

Characterizations of the Per-capita Shapley ValueTakumi Kongo ∗AbstractThree axiomatic characterizations of the per-capita Shapley value (Driessenand Radzik 2002; International Transactions in Operational Research) forTU games are presented. The first one is a characterization via a modificationof Myerson’s (1980; International Journal of Game Theory) balancedcontributions property, the second via a modification of the null playerproperty, and the third via a modification of Hart and Mas-Colell’s (1989;Econometrica) consistency. Comparison between the per-capita and usualShapley value for land corn production economies and claims problems isstudied.Keywords: balanced contributions property; per-capita Shapley value; characterization;consistency;JEL classification: C711 IntroductionWe study problems of allocation of surplus generated by multiple agents’ cooperation.Our concern is such problems represented by coalitional game with transferableutilities (henceforth, TU games), and especially, values for TU games. Avalue for TU games represents each player’s allocation of the surplus obtainedby participating in the games. Many studies have investigated properties ofdifferent values for TU games.Myerson’s (1980) balanced contributions property is a property related tofairness, that equalizes claims between any pair of players in TU games. Playerj’s claim against another player i is defined as the difference of i’s value for theoriginal game and i’s value for the game from which j is deleted. Hence, inclaims, we compare two values in games with different numbers of players. Inconjunction with an important property of efficiency, the number of players ingames affects to the value of games. Consider an efficient value and comparevalues for players in a game with n players, and those in the game obtained justby adding a new player. Unless the newly added player obtains ones marginalcontributions to n players’ cooperation in the value, there must be a player whoobtains different values in the two games under the efficient value.In order to avoid such influence of the number of players in games on values,we consider the normalized value regarding the number of players in gamesand measure the claim by the normalized values. In conjunction with efficiency,∗ Faculty of Economics, Fukuoka University. 8-19-1 Nanakuma, Jonan-ku, Fukuoka 814-0180, Japan E-mail: kongo@adm.fukuoka-u.ac.jp Tel: +81-92-871-66311

the new “normalized” balanced contributions property characterizes the “percapita”Shapley value introduced in Driessen and Radzik (2002). The valueis a modification of the Shapley value (Shapley 1953). By replacing the nullplayer property with a modified one, we succeed in characterizing it as Shapley’soriginal characterization of his value. The modified property is called theP-null player property and in this property, we focus on null players in “normalized”game. The characterization of the per-capita Shapley value in termsof a consistency property is also discussed. Our consistency property, is a kindof “normalized” version of Hart and Mas-Colell’s (1989) consistency.The paper is structured as follows. Notation and definitions are providedin Section 2. Some characterizations of per-capita Shapley value are studied inSection 3 and especially, a characterization by a consistency property is presentedin Section 4. The differences between the per-capita and usual Shapleyvalues in land corn production economies and claims problems are studied inSection 5.2 PreliminariesLet N ⊆ N be a set of players. Assume that N is a finite set. The number ofelements in a set is represented by | · |. We also use the corresponding lowercase letter to represent the number of elements in the set, i.e., for S, T, · · · ⊆ N,|S| = s, |T | = t and so on. A nonempty subset S ⊆ N is a coalition. If thereexists i ∈ N such that S = {i}, we write it just as i for simplicity.Let v : 2 N → R be a function with v(∅) = 0.A TU game is a pair (N, v). Let Γ denote the set of all games. A value φ isa function that assigns an n-dimensional vector to any game in Γ. A value φ isefficient if ∑ i∈N φ i(N, v) = v(N) for any game (N, v) ∈ Γ.The per-capita Shapley value ψ of the game (N, v) is defined as follows: Forany i ∈ N,or equivalently,ψ i (N, v) = v(i) +ψ i (N, v) = v(i) +∑S∋i;S≠i∑S⊆N\i;S≠∅(s − 1)!(n − s)!(n − 1)!s!(n − s − 1)!(n − 1)!( v(S)−s( v(S ∪ i)s + 1)v(S \ i),s − 1− v(S) ).sDriessen and Radzik (2002) showed that the above value is represented byusing the weighted pseudo-potential function P : Γ → R. For any (N, v) ∈ Γand any i ∈ N,ψ i (N, v) = P (N, v) −n P (N \ i, v),n − 1whereP (N, v) =∑S⊆N;S≠∅(s − 1)!(n − s)! v(S)(n − 1)! s ,for any N ≠ ∅ and P (∅, v) = 0.We consider another representation of the per-capita Shapley value by usingthe orders on the player set. Let π be an order on N, and let π(i) denote a2

position of player i in π, that is, player π −1 (1) appears first in π, π −1 (2) appearssecond, and so on. Let a π (N, v) ∈ R n where{v(i) if π(i) = 1a π i (N, v) = v({j∈N:π(j)≤π(i)})|{j∈N:π(j)≤π(i)}|− v({j∈N:π(j)

For the population normalized balanced contributions property, we considerthree cases. The case |N| = 1 is obvious. In the case |N| = 2, by definition,ψ i ({i, j}, v) = 1 2v({i, j}) + v(i) − v(j). By efficiency,12 ψ i({i, j}, v) − ψ i (i, v) = 1 2 ψ i({i, j}, v) − v(i)= 1 4 v({i, j}) − 1 (v(i) + v(j))2= 1 2 ψ j({i, j}, v) − v(j) = 1 2 ψ j({i, j}, v) − ψ j (j, v).In the case when |N| ≥ 3, by the weighted pseudo-potential function P ,1n ψ i(N, v) − 1n − 1 ψ i(N \ j, v)= 1 111P (N, v) − P (N \ i, v) − P (N \ j, v) + P (N \ {i, j}, v)n n − 1 n − 1 n − 2= 1 n ψ j(N, v) − 1n − 1 ψ j(N \ i, v)For the uniqueness of the value, let φ denote a value possessing the twoproperties. If |N| = 1, it is obvious by efficiency. Let n ≥ 2. Suppose that φcoincides with ψ if the number of players is less than n. We consider n playerscase. Given i ∈ N and for any j ∈ N \ i, by the population normalized balancedcontributions property and the supposition,1n φ i(N, v) − 1 n φ j(N, v) = 1n − 1 φ i(N\j, v) − 1n − 1 φ j(N\i, v)Hence we obtain,n − 1nφ i(N, v) − 1 n∑j∈N\i= 1n − 1 ψ i(N\j, v) − 1n − 1 ψ j(N\i, v)= 1 n ψ i(N, v) − 1 n ψ j(N, v).φ j (N, v) = n − 1nψ i(N, v) − 1 n∑j∈N\iψ j (N, v).Since both φ and ψ is efficient, the above equation equals to φ i (N, v) − v(N)n=ψ i (N, v)− v(N)nand thus φ i (N, v) = ψ i (N, v). This is applicable to any k ∈ N \i,and thus, φ = ψ.For the independence of the properties, the Shapley value satisfies efficiencybut not the population normalized balanced contributions property. Letφ 1 (N, v) = (0, 0, . . . , 0) ∈ R n for any (N, v) ∈ Γ. It is obvious that φ 1 satisfiespopulation normalized balanced contributions property but not efficiency.These two properties enable us to represent the per-capita Shapley value inthe following recursive manner.Theorem 2. For any (N, v) ∈ Γ and any i ∈ N,ψ i (N, v) = v(N)n−v(N \ i)n − 1+ ∑j∈N\iψ i (N \ j, v).n − 14

Proof. It is obvious if |N| = 1. Otherwise, by the population normalized balancedcontributions property,n − 1nψ i(N, v) = 1 nBy efficiency,and1n∑j∈N\i∑j∈N\i1n − 1ψ j (N, v) + 1n − 1∑j∈N\iψ j (N, v) = 1 n (v(N) − ψ i(N, v)),∑j∈N\iψ j (N \ i, v) = 1 v(N \ i).n − 1(ψ i (N \ j, v) − ψ j (N \ i, v)) .Substituting the above two equalities to eq.(2), the equality in Theorem 2 isobtained.Next, we give another axiomatic characterization of the per-capita Shapleyvalue. A P-null player in (N, v) is a player i ∈ N, satisfying v(i) = 0 andv(S)s−v(S \ i)s − 1 = 0,for any coalition S ⊆ N with S ∋ i and S ≠ {i}. The notion of P-null players isintroduced by Kamijo and Kongo (2012) 1 and a P-null player in a game (N, v) isa null player in the game (N, ¯v), such that for any coalition S ⊆ N, ¯v(S) = v(S)sand ¯v(∅) = 0. The following is the property related to the P-null players.P-null player property: For any (N, v) ∈ Γ and its any P-null player i ∈ N,φ i (N, v) = 0.In a game (N, v) ∈ Γ, two players i, j ∈ N are symmetric if for any S ⊆N \ {i, j}, v(S ∪ i) = v(S ∪ j). A value φ satisfies symmetry if i, j ∈ N aresymmetric in (N, v) ∈ Γ, then φ i (N, v) = φ j (N, v). A value φ satisfies additivityif for any two games (N, v), (N, v ′ ) ∈ Γ, φ i (N, v + v ′ ) = φ i (N, v) + φ i (N, v ′ ),where (v + v ′ )(S) = v(S) + v ′ (S) for any S ⊆ N.We use following four lemmas to obtain the result.Lemma 1. The per-capita Shapley value satisfies efficiency, the P-null playerproperty, symmetry, and additivity.Proof. Efficiency has already shown (see Theorem 1). The P-null player propertyand additivity is obvious by definition. As for the symmetry, by theweighted pseudo-potential function representation,1n (ψ i(N, v) − ψ j (N, v))= 1 1P (N, v) −n n − 1 P (N \ i, v) − 1 n= 1∑(P (N\j, v)−P (N\i, v)) =n − 1Therefore, it possesses symmetry.T ⊆N\{i,j}1P (N, v) + P (N \ j, v)n − 1(t!(n − t)! v(T ∪ i)−(n − 1)! t + 11 In Kamijo and Kongo (2012), P-null players are called proportional players.v(T ∪ j)t + 1(2))5

Lemma 2. Given a game (N, v) and a coalition S ⊆ N, let p S : 2 N → Rsatisfying,{tp S (T ) =sif T ⊇ S0 otherwise,for any T ⊆ N. Then, {p S |S ⊆ N, S ≠ ∅} is a basis for the linear space Γ.∑Proof. We show that {p S |S ⊆ N, S ≠ ∅} is linear independent, that is, ifS⊆N,S≠∅ d Sp S = 0 then for any coalition S ⊆ N, it holds that d S = 0.We prove it by a contradiction. Suppose there exists at least one coalition Ssuch that d S ≠ 0. Take one of a minimal coalition S ∗ that satisfies d S ∗ ≠ 0,that is, for any S ⊊ S ∗ , d S = 0. By the definition of p S , p S (S ∗ ) = 0 if S ∗ doesnot include S. Thus,∑d S p S (S ∗ ) =∑d S p S (S ∗ ) + d S ∗p S ∗(S ∗ ) = d S ∗.S⊆N,S≠∅S⊊S ∗ ,S≠∅By the definition of S ∗ , d S ∗ ≠ 0 and it is a contradiction.Lemma 3. For any (N, v) ∈ Γ, v is represented as follows:v =∑c v Sp S ,where c v S = ∑ T ⊆S;T ≠∅ s t (−1)s−t v(T ).Proof. For any coalition R ⊆ N,∑c v Sp S (R) = ∑ c v rSs = ∑ S⊆N;S≠∅ R⊇S R⊇S= ∑ ∑R⊇SS⊆N;S≠∅T ⊆S;T ≠∅∑T ⊆S;T ≠∅st (−1)s−t v(T ) r srt (−1)s−t v(T )= v(R) + ∑ ∑r−t( ) r − t t + k(−1) k rv(T )k tt + kT ⊊Rk=0= v(R) + ∑ − 1)T ⊊R(1 r−t r v(T ) = v(R)tLemma 4. Let φ be a value that possesses efficiency, the P-null player property,and the symmetry. For any (N, v) ∈ Γ, any coalition S ⊆ N, and any i ∈ N,{nφ i (N, p S ) =sif i ∈ S20 if i ∈ N \ SProof. Efficiency implies that ∑ i∈N φ i(N, p S ) = p S (N) = n s. We show that anyplayer in N \ S is a P-null player in (N, p S ). Let i ∈ N \ S. For any T ⊆ N \ i,if T ⊉ S, then T ∪ i ⊉ S and p S(T ∪i)t+1− p S(T )t= 0t+1 − 0 t= 0. Otherwise,T ⊇ S and p S(T ∪i)t+1− p S(T )t= 1 s − 1 s= 0. Thus, by the P-null player property,φ i (N, p S ) = 0 for any i ∈ N \ S.6

Next, we prove that all players in S are symmetric in (N, p S ). Let i, j ∈ S.For any T ⊊ S, p S (T ∪ i) = p S (T ∪ j) = 0 Thus, by symmetry, φ i (N, p S ) =φ j (N, p S ) for any i, j ∈ S.By these three properties, φ i (N, p S ) = n s 2 for any i ∈ S.Theorem 3. The per-capita Shapley value ψ is uniquely characterized by efficiency,the P-null player property, symmetry, and additivity.Proof. The four properties for the per-capita Shapley value has already shownin Lemma 1. Let φ be a value that satisfies the four properties. By Lemmas 3and 4 and additivity, φ i (N, v) = ∑ S⊆N;S≠∅ cv S n s. By Lemma 2, c v 2 S is uniquelydetermined. Thus φ is uniquely determined.The independence of the properties are as follows. φ 1 defined before satisfiesall the properties except efficiency. The Shapley value satisfies all the propertiesexcept the P-null player property. Let φ 2 be a value such that every P-nullplayers obtains 0, and a fixed non-P-null player always obtains v(N) (if thereexists). φ 2 satisfies all the property except symmetry. Let φ 3 (N, v) = n·χ(N, ¯v),where χ is prenucleolus, and (N, ¯v) of (N, v) is defined before. φ 3 satisfies allproperties except additivity.Comparing the above result and the results in Shapley (1953), Nowak andRadzik (1994), Brink and Funaki (2008), and Brink (2007) the differencesbetween the per-capita Shapley, Shapley, solidarity, delta-discounted Shapleyvalue, and the equal division are the differences between the P-null player, nullplayer, A-null player, delta-reducing player, and nullifying player properties. Itmeans that the difference of these values lies in which players obtain nothing.4 ConsistencyA consistency property of the per-capita Shapley value is discussed. The consistencyproperty we use here is a “population normalized” version of the consistencyproperty introduced by Hart and Mas-Colell (1989).Let φ be a value and (N, v) be a game. For any coalition S ⊆ N, we definethe reduced game (S, v φ S) as follows: For any T ⊆ S with T ≠ ∅⎛⎞v φ S (T ) = t⎝v(T ∪ (N \ S)) −∑φ j (T ∪ (N \ S), v) ⎠ . (3)t + n − sj∈N\Sand v φ S(∅) = 0.Comparing Hart and Mas-Colell’s (1989) reduced game, we consider normalizationof the coalitional worth with respect to t, n, and s; the number of playersin the coalition, and that in the original and the reduced games. Rearrangingeq.(3), we obtain the following:v φ S (T )t= v(T ∪ (N \ S)) − ∑ j∈N\S φ j(T ∪ (N \ S), v).t + n − sThus, we clearly focus on per-capita values.Given a game (N, v) ∈ Γ, a value φ satisfies population-normalized consistencyif for any (N, v) ∈ Γ, any coalition S ⊆ N, and any i ∈ S,φ i (N, v)n= φ i(S, v φ S ) .s7

Lemma 5. The per-capita Shapley value satisfies population normalized consistency.Proof. Since ψ is efficient, for any (N, v) ∈ Γ, any coalition S ⊆ N and anynonempty coalition T ⊆ S,v ψ S (T ) = v(T ∪ (N \ S)) − ∑ j∈N\S φ ∑j(T ∪ (N \ S), v)j∈T=ψ j(T ∪ (N \ S), v).tt + n − st + n − sThen,=ψ i (S, v ψ S ) =∑R⊆S\i;R≠∅+ ∑R⊆S\i;R≠∅∑R⊆S\i;R≠∅r!(s − r − 1)!(s − 1)!=∑R⊆S\i;R≠∅r!(s − r − 1)!(s − 1)!(r!(s − r − 1)!(s − 1)!(∑j∈R∪i ψ j(R ∪ i ∪ (N \ S), v)−r + 1 + n − sr!(s − r − 1)!(s − 1)!∑(ψj (R ∪ i ∪ (N \ S), v)r + 1 + n − sj∈R)v ψ S(R ∪ i)− vψ S (R) + v ψ Sr + 1 r(i)∑ψ i (R ∪ i ∪ (N \ S), v)r + 1 + n − s− ψ j(R ∪ (N \ S), v)r + n − sj∈R ψ j(R ∪ (N \ S), v)r + n − s)+v ψ S (i). (4))+v ψ S (i)R,By the population-normalized balanced contributions property, for any j ∈ψ j (R ∪ i ∪ (N \ S), v)r + 1 + n − s− ψ j(R ∪ (N \ S), v)r + n − s= ψ i(R ∪ i ∪ (N \ S), v)r + 1 + n − sThus, the r.h.s. of eq.(4) equals to the following:∑R⊆S\i;R≠∅r!(s − r − 1)!(s − 1)!When R = S \ i, it holds that− ψ i((R ∪ i ∪ (N \ S)) \ j, v).r + n − s( ∑(1 + r)ψi (R ∪ i ∪ (N \ S), v)−r + 1 + n − sr!(s − r − 1)! (1 + r)ψ i (R ∪ i ∪ (N \ S), v)= s (s − 1)! r + 1 + n − s n ψ i(N, v).Hence, the above equality is equal to the following:sn ψ i(N, v) +−∑R⊊S\i;R≠∅∑T ⊆S\i;T ≠∅r!(s − r − 1)! (1 + r)ψ i (R ∪ i ∪ (N \ S), v)(s − 1)! r + 1 + n − st!(s − t − 1)!(s − 1)!∑j∈Tψ i ((T ∪ i ∪ (N \ S)) \ j, v)t + n − sj∈R ψ i((R ∪ i ∪ (N \ S)) \ j, v)r + n − s+ v ψ S(i). (5))+v ψ S (i).8

For any coalition R ⊊ S \ i, the number of coalition T ⊆ S \ i with T \ j = R iss − r − 1, since j ∈ S \ R and j ≠ i. In this case t = r + 1, thus, for any coalitionR ⊊ S \ i,∑∑j∈T T ⊆S\i;T \j=Rt!(s − t − 1)! ψ i ((T ∪ i ∪ (N \ S)) \ j, v)(s − 1)!t + n − s= (s − r − 1)(r + 1)!(s − r − 2)!(s − 1)!ψ i (R ∪ i ∪ (N \ S), v).r + 1 + n − sHence, the second term and a part of the third term of eq.(5) are canceled.Thus, the r.h.s. of eq.(5) equals to the following:sn ψ i(N, v)−∑T ⊆S\i;t=1t!(s − t − 1)!(s − 1)!∑j∈TNow, the second term of eq.(6) is equal toand by definition and efficiency,⎛v ψ S (i) = 1⎝v(i ∪ (N \ S)) −1 + n − s− ψ i(i ∪ (N \ S), v),1 + n − sψ i ((T ∪ i ∪ (N \ S)) \ j, v)+v ψ S(i). (6)t + n − s∑j∈N\S=⎞ψ j ((T ∪ i ∪ (N \ S)), v) ⎠11 + n − s ψ i((T ∪ i ∪ (N \ S)), v).Therefore, the second and the third term of eq.(6) are canceled and we obtainψ i (S, v ψ S ) = s n ψ i(N, v) ⇐⇒ ψ i(S, v ψ S )sConsider the following property.= ψ i(N, v).n1-standardness for two-person games: For any (N, v) ∈ Γ with |N| = 2and any i ∈ N,φ i (N, v) = 1 v(N) + v(i) − v(j).2The above property equals to well-known α-standardness for two-persongames with respect to α = 1.Lemma 6. If a value φ satisfies population-normalized consistency and 1-standardness for two-person games, it is efficient.Proof. First, we consider the case when |N| = 1. Let ({i}, v) and v(i) = a. Considerthe two person game ({i, j}, ¯v) with ¯v(i) = a, ¯v(j) = 0, and ¯v({i, j}) = 2a.By 1-standardness for two-person games, φ i ({i, j}, ¯v) = 2a and φ j ({i, j}, ¯v) = 0.By the definition of the ¯v φ {i} ,¯v φ {i} (i) = 1 2 (¯v({i, j}) − φ j({i, j}, ¯v)) = a = v(i).9

By population-normalized consistency,v(i) = a = φ i({i, j}, ¯v)2= φ i ({i}, ¯v φ {i} ) = φ i({i}, v).Next, we consider the case when players are two or more. By the 1-standardnessfor two-person game, it is clear that the value is efficient when n = 2. Let n ≥ 3and assume that Lemma 6 holds for any game with less than n players. Weprove the fact by an induction regarding the number of players. Let φ satisfiesthe two properties. For a game (N, v) with n players, let i ∈ N. Bypopulation-normalized consistency, for any j ∈ N \ i,φ j (N, v) =nn − 1 φ j(N \ i, v φ N\i ).By assumption, φ is efficient for games with less than n players. Thus,∑φ j (N, v) = ∑j∈Nj∈N\iBy definition, we have the desired result.nn − 1 φ j(N\i, v φ N\i )+φ i(N, v) =nn − 1 vφ N\i (N\i)+φ i(N, v).Theorem 4. The per-capita Shapley value is uniquely characterized by populationnormalizedconsistency and 1-standardness for two-person games.Proof. Population-normalized consistency of the per-capita Shapley value is byLemma 5. 1-standardness for two-person games of it is straightforward. Wediscuss only the uniqueness.If |N| = 1, Lemma 6 and efficiency imply the uniqueness. In the case when|N| = 2, it is shown by 1-standardness for two person game. Let |N| = n > 2and assume that the two values coincides with each other if the number ofplayers is less than n. We show the fact by an induction regarding the numberof players. Take any {i, j} ⊊ N. By population-normalized consistency,andNow, by definition,φ i (N, v)nψ j (N, v)n= φ i({i, j}, v φ {i,j} ),2= ψ j({i, j}, v ψ {i,j} ).2⎛⎞v φ {i,j} (i) = 1 ⎝v(N \ j) −∑φ k (N \ j, v) ⎠n − 1k∈N\{i,j}⎛⎞= 1 ⎝v(N \ j) −∑ψ k (N \ j, v) ⎠ = v ψn − 1{i,j} (i).k∈N\{i,j}Similarly, v φ {i,j} (j) = vψ {i,j} (j). 10

Together with 1-standardness for two-person games,φ i (N, v)n− φ j(N, v)= φ i({i, j}, v φ {i,j} )− φ j({i, j}, v φ {i,j} )n22= v φ {i,j} (i) − vφ {i,j} (j) = vψ {i,j} (i) − vψ {i,j} (j)= ψ i({i, j}, v ψ {i,j} )2− ψ j({i, j}, v ψ {i,j} )2= ψ i(N, v)nFix i ∈ N. Summing up the above equality for all j ∈ N \ i,n − 1nφ i(N, v) − ∑ 1n φ j(N, v) = n − 1nψ i(N, v) − ∑j∈N\ij∈N\i− ψ j(N, v).n1n ψ j(N, v).By efficiency, φ i (N, v) = ψ i (N, v). Therefore uniqueness is shown.For the independence of properties, φ 1 defined before satisfies populationnormalizedconsistency but not 1-standardness for two-person games. Any valuethat is equal to the per-capita Shapley value for any game with less than threeplayers and is not equal to it for any game with more than or equal to threeplayers satisfies the latter property but not the former one.In the above, we characterized the per-capita Shapley value by means of“normalized” consistency. However, by the following reduced game, we cancharacterize it by the genuine consistency.Let φ be a value. For any coalition S ⊆ N, we define another reduced game(S, ṽ φ S) as follows: For any T ⊆ S with T ≠ ∅⎛⎞ṽ φ S (T ) = nt⎝v(T ∪ (N \ S)) −∑φ j (T ∪ (N \ S), v) ⎠s(t + n − s)j∈N\Sand ṽ φ S(∅) = 0. Comparing the reduced game we define in the earlier, it holdsthat ṽ φ S (T ) = n s vφ S(T ). By additivity of the per-capita Shapley value, we obtainψ i (S, ṽ ψ S ) = n s ψ i(S, v ψ S ) = ψ i(N, v), and it is a genuine consistency.5 The per-capita Shapley value for land cornproduction economiesWe compare the per-capita Shapley value with the Shapley value in two specificclasses of games: land corn production economies and claims problems.Land corn production economies (see Shapley and Shubik 1967) is productioneconomies with two types of agents. One is a landlord who owns the land thatis suitable for the production of corn but who does not want to work at his/herland. The other is a worker who does not own his/her land and who want towork at the land of landlord to produce corn. There is only one landlord andthere are m workers who are not differ in ability of the production. In orderto produce corn, both the land of landlord and workers are needed, and theamount of the production of corn depend only upon how many workers work.Thus, the amount of the production of corn is represented by a non-decreasingfunction f regarding the number of workers and f(0) = 0. A TU game with11

m + 1 players is obtained from a land corn production economy. In the game,if a coalition does not contain the landlord then it produces nothing and if itcontains the landlord and s ∈ {0, 1, . . . , m} workers it produces f(s).The Shapley value allocates to the landlordand to each worker1m + 1m∑f(k),k=1f(m)m − 1m(m + 1)m∑f(k).k=1Further, the Shapley value is a core allocation when f(k)kk = 1, . . . , m − 1.The per-capita Shapley value allocates to the landlordand to each workerf(m)mm∑k=1− 1 mf(k)k ,n∑k=1f(k)k .≤ f(m)mholds for anySimilarly, we can prove that the per-capita Shapley value is a core allocationwhen the same condition holds. Moreover, since f(k)k> f(k)m+1for any k =1, . . . , m, the per-capita Shapley value allocates the landlord more (workersless) than the Shapley value. Especially, if the production function exhibitsconstant marginal returns, that is, there exist a non-negative real number a andf(k) = ka for any k = 1, . . . , m, the Shapley value allocates to the landlordthe half of the production, f(m)2, and to each worker the equal division of thehalf of the production, f(m)2m, while the per-capita Shapley value allocates to thelandlord all of the production, f(m), and to each worker nothing. Note that inthis case, all workers are P-null players and thus they obtain nothing.Claims problems (O’Neill 1982) are division problems for some amount ofan infinitely divisible good among agents each of which possess a different right(claim) to be divided, but the amount of the good is not more (usually strictlyless) than the sum of all agents’ claims. Formally, let N be a set of agents andlet E ∈ N be an amount of a good to be divided. Each agent i ∈ N has itsclaim c i ∈ N and ∑ i∈N c i ≤ E.From claims problem, a TU game is obtained. In such a situation, playersS ⊆ N surely obtain the amount 0 or the rest of the amount of the goods whenplayers other than S obtain their claims. That is,v c (S) = max{0, E − ∑c i }.i∈N\SThe Shapley value defined on (N, v c ) is one of a well-investigated division rulein claims problems. For example, let N = {1, 2, 3}, (c 1 , c 2 , c 3 ) = (100, 200, 300),and E = 100, then the Shapley value vector is ( 1003 , 1003 , 1003). In case E = 300and 500, then the Shapley value vectors are (50, 100, 150) and ( 4006 , 11006 , 15006 ),12

espectively. While, the per-capita Shapley value vectors for each of the casesE = 100, 300, 500 are ( 1003 , 1003 , 1001753), (25, 100, 75) and, (−3 , 5003 , 11753), respectively.Therefore, as the amount of goods to be divided increases, the per-capitaShapley value allocates more to an agent whose claim is greater than the usualShapley value does.Acknowledgements: The author thank Youngsub Chun and Yukihiko Funakifor their helpful comments and discussions. Remaining errors are my ownresponsibility.ReferencesBrink R, van den (2007) Null or nullifying players: The difference betweenthe Shapley value and equal division solutions. Journal of Economic Theory136:767–775Brink R van den, Funaki Y (2008) Axiomatization and implementation of discountedShapley values. Tinbergen Institute Discussion Paper 2010-065/1Tinbergen Institute and VU University, AmsterdamDriessen T, Radzik T (2002) A weighted pseudo-potential approach to valuesfor TU-games. International Transactions in Operational Research 9:303–320Hart S, Mas-Colell A (1989) Potential, value and consistency. Econometrica57:589–614Kamijo Y, Kongo T (2012) Whose deletion does not affect your payoff? Thedifference between the Shapley value, the egalitarian value, the solidarityvalue, and the Banzhaf value. European Journal of Operational Research216:638–646Myerson RB (1980) Conference structures and fair allocation rules. InternationalJournal of Game Theory 9:169–182Nowak AS, Radzik T (1994) A solidarity value for n-person transferable utilitygames. International Journal of Game Theory 23:43–48O’Neill B (1982) A problem of rights arbitration from the Talmud. MathematicalSocial Sciences 2:345–371Shapley LS (1953) A value for n-person games. In: H. Kuhn, A. Tucker (Eds.),Contributions to the Theory of Games II. Princeton University Press, Princeton,pages 307–317.Shapley LS, Shubik M (1967) Ownership and the production function. QuarterlyJournal of Economics 81:88-11113