- Email: [email protected]
On axiomatizations of the Shapley value for assignment games
Journal of Mathematical Economics 60 (2015) 110–114
Contents lists available at ScienceDirect
Journal of Mathematical Economics journal homepage: www.elsevier.com/locate/jmateco
On axiomatizations of the Shapley value for assignment games✩ René van den Brink a,∗ , Miklós Pintér b,c,d a
VU University, Department of Econometrics and Tinbergen Institute, De Boelelaan 1105, 1081 HV Amsterdam, The Netherlands
b
Corvinus University of Budapest, Department of Mathematics, 1093, Budapest, Fovám tér 13-15, Hungary
c
MTA-BCE ‘‘Lendület’’ Strategic Interactions Research Group, 1093, Budapest, Fovám tér 13-15, Hungary
d
Department of Business and Economics, University of Pecs, Hungary
article
info
Article history: Received 14 March 2013 Received in revised form 21 November 2013 Accepted 26 June 2015 Available online 7 July 2015 Keywords: Game theory Assignment game Shapley value Graph game Submarket efficiency Valuation fairness
abstract We consider the problem of axiomatizing the Shapley value on the class of assignment games. It turns out that several axiomatizations of the Shapley value on the class of all TU-games do not characterize this solution on the class of assignment games. However, when considering an assignment game as a (communication) graph game where the game is simply the assignment game and the graph is a corresponding bipartite graph where buyers (sellers) are connected with sellers (buyers) only, we show that Myerson’s component efficiency and fairness axioms do characterize the Shapley value on the class of assignment games. Moreover, these two axioms have a natural interpretation for assignment games. Component efficiency yields submarket efficiency stating that the sum of the payoffs of all players in a submarket equals the worth of that submarket, where a submarket is a set of buyers and sellers such that all buyers in this set have zero valuation for the goods offered by the sellers outside the set, and all buyers outside the set have zero valuations for the goods offered by sellers inside the set. Fairness of the graph game solution boils down to valuation fairness stating that only changing the valuation of one particular buyer for the good offered by a particular seller changes the payoffs of this buyer and seller by the same amount. © 2015 Elsevier B.V. All rights reserved.
1. Introduction The history of assignment games goes back to the XIX century to Böhm-Bawerk’s (1923) horse market model. Later Shapley and Shubik (1972) introduced the formal, modern concept of assignment games. One of the most popular solution concepts for TU-games is the Shapley value (Shapley, 1953). Numerous axiomatizations of the Shapley value are known in the literature, for example (i) Shapley’s original axiomatization (Shapley, 1953) by efficiency, the null player property (originally stated together as the carrier axiom), symmetry and additivity (also discussed by Dubey, 1975 and Peleg and Sudhölter, 2003), (ii) Young’s (1985) axiomatization replacing additivity and the null player property by strong monotonicity (also discussed by Moulin (1988) and Pintér (2012)), (iii) Chun’s (1991) replacing strong monotonicity by coalitional strategic
✩ We thank an anonymous referee for her/his useful comments. Miklós Pintér gratefully acknowledges the Financial support by the Hungarian Scientific Research Fund (OTKA) grant number K 101224, SPOR grant number 4.2.2.B-15/1/konv and the János Bolyai Research Scholarship of the Hungarian Academy of Sciences. ∗ Corresponding author. E-mail addresses: [email protected] (R. van den Brink), [email protected] (M. Pintér).
http://dx.doi.org/10.1016/j.jmateco.2015.06.016 0304-4068/© 2015 Elsevier B.V. All rights reserved.
equivalence, (iv) van den Brink’s (2001) replacing (in Shapley’s original axiomatization) additivity and symmetry by fairness, and (v) Hart and Mas-Colell’s (1989) approaches using the potential function and a related reduced game consistency. It turns out that none of these characterizations are valid on the class of assignment games in the sense that they do not characterize a unique solution. In this paper, we show that when considering an assignment game as a (communication) graph game where the game is simply the assignment game and the graph is a corresponding bipartite graph where buyers (sellers) are connected with sellers (buyers) only, Myerson’s (1977) component efficiency and fairness axioms do characterize the Shapley value on the class of assignment games. Moreover, the axioms have a natural interpretation for these games. An assignment game is fully described by the assignment situation being a set of buyers, a set of sellers, and for every buyer a valuation of the good offered by each seller. Instead of defining an assignment game as a graph game, we will directly work on the class of these assignment situations. For such assignment situations, component efficiency of a graph game solution boils down to submarket efficiency stating that the sum of the payoffs of all players in a submarket equals the worth of that submarket, where a submarket in an assignment situation is a set of buyers and sellers such that all buyers in this set have zero valuation for the goods
R. van den Brink, M. Pintér / Journal of Mathematical Economics 60 (2015) 110–114
111
offered by the sellers outside the set, and all buyers outside the set have zero valuations for the goods offered by sellers inside the set. Fairness of the graph game solution boils down to valuation fairness stating that only changing the valuation of one particular buyer for the good offered by a particular seller changes the payoffs of this buyer and seller by the same amount. We show that these two axioms do characterize the Shapley solution for assignment situations being the solution that is obtained by applying the Shapley value to the corresponding assignment game. So, we obtain a positive result by viewing an assignment game as a graph game. Besides introducing and axiomatizing his solution, Myerson (1977) also shows that it is stable for superadditive graph games in the sense that two players never get worse off when building a link between them. The Shapley solution for assignment situations is valuation monotonic in the sense that the payoffs of a buyer i and a seller j do not decrease if only the valuation of buyer i for the good offered by seller j increases.
reservation value zero for every good. The nonnegative valuation (reservation value) of buyer i ∈ B for the good offered by seller j ∈ S is denoted by ai,j ≥ 0. So, buyer i and seller j can make a deal and earn worth ai,j . Buyers cannot trade among each other (since they do not own a good), and also sellers cannot earn a worth among themselves since their valuation is zero. Let A be the |B| × |S | non-negative matrix with ai,j its (i, j) element. We refer to this matrix A as an assignment situation or valuation matrix on (B, S ). We denote the collection of all assignment situations on (B, S ) by AB,S . Furthermore for all T ⊆ N, a matching on T is a set of sets M ⊆ {{i, j} ⊆ T | i ∈ B ∩ T , j ∈ S ∩ T } such that for every g ∈ T , |{{h, k} ∈ M | g ∈ {h, k}}| ≤ 1. So, buyers can only be matched with sellers, sellers can only be matched with buyers, and every buyer (seller) can be matched with at most one seller (buyer). Let M (T ) be the set of all matchings of T . Taking the sets of buyers B and sellers S fixed, the assignment game (see Shapley and Shubik, 1972) for valuation matrix A is the game vA on N = B ∪ S, given by3
2. Preliminaries
vA (T ) = max
M ∈M (T )
2.1. TU-games
ai,j
for all T ⊆ B ∪ S .
{i,j}∈M
The elements of Let N be a non-empty, finite set, let |N | be its cardinality, and let P (N ) denote the power set of N. A transferable utility (TU) game with player set N is a pair (N , v) with characteristic function v : P (N ) → R such that v(∅) = 0. Since we take the player set N to be fixed, we represent a TU-game (N , v) simply by its characteristic function v . The class of all characteristic functions on player set N is denoted by GN . Game v is superadditive if v(S ∪ T ) ≥ v(S ) + v(T ) for all S , T ⊆ N with S ∩ T = ∅. A (single-valued) solution on C ⊆ GN , is a function φ : C → RN . In this paper we focus on the Shapley value (Shapley, 1953) being the solution φ Sh : GN → RN , for every v ∈ GN , given by
φiSh (v) =
|T |!(|N \ T | − 1)! mTi (v) for all i ∈ N , |N |! T ⊆N \{i}
where, for any v ∈ GN , i ∈ N and T ⊆ N , mTi (v) = v(T ∪ {i}) − v(T ) is player i’s marginal contribution to coalition T in game v . We refer to φiSh (v) as the Shapley value of player i in game v ∈ GN . Several axiomatizations of the Shapley value can be found in the literature, such as the axiomatization1 by Pareto optimality (also known as efficiency), the null player property, the equal treatment property and additivity (Shapley, 1953), Pareto optimality, the equal treatment property and strong monotonicity (Young, 1985), Pareto optimality, the equal treatment property and marginality (also by Young, 1985), Pareto optimality, the equal treatment property and coalitional strategic equivalence (Chun, 1991) and Pareto optimality, the null player property and fairness (van den Brink, 2001).2 2.2. Assignment games Let B, S ⊆ N be two non-empty sets such that B ∩ S = ∅ and B ∪ S = N. The interpretation is the following. The sets B and S are the sets of buyers and sellers, respectively. Every buyer wants one good, and every seller owns one good. These goods are not exactly the same, so a buyer can have different valuations for the goods owned by different sellers. We assume that the sellers have
arg max
M ∈M (T ) {i,j}∈M
ai,j
are called the maximal matchings of coalition T . For any set of buyers and sellers T , the worth of this coalition is the maximum aggregated worth of the deals the involved players can achieve contingent on every player trading with at most one other player from the other type. Remark 2.1. Since in the definition of assignment games, B and S are non-empty, in this paper every assignment game has at least two players. 2.3. Graph games Myerson (1977) introduced a model in which it is assumed that the players in a game v are part of a communication structure that is represented by an undirected graph (N , L), with the player set N as the set of nodes and L ⊆ {{i, j} | i, j ∈ N , i ̸= j} being a collection of edges or links, that is, subsets of N such that each element of L contains precisely two elements. Since in this paper the nodes in a graph represent the players in a game, we use the same notation for the set of nodes as the set of players, and refer to the nodes in a graph just as players. If there is no confusion about the player set N, we denote a graph on N just by its set of links L and refer to this as the graph. We denote the class of all possible sets of links on N by LN . A sequence of k different nodes (i1 , . . . , ik ) is a path between players i1 and ik in L ∈ LN if {ih , ih+1 } ∈ L for h = 1, . . . , k − 1. A coalition S ⊆ N is connected in graph L if every pair of players in S is connected by a path that only contains players from S, that is, for every i, j ∈ S , i ̸= j, there is a path (i1 , . . . , ik ) such that i1 = i, ik = j and {i1 , . . . , ik } ⊆ S. Coalition T ⊆ S is a component of S in graph L if it is a maximally connected subset of S, that is, T is connected in L(S ) and for every h ∈ S \ T the coalition T ∪ {h} is not connected in L(S ), where L(S ) = {{i, j} ∈ L | {i, j} ⊆ S }. We denote the set of components of S ⊆ N in L by CL (S ). A pair (v, L) ∈ GN × LN is referred to as a graph game on N. Following Myerson (1977), in the graph game (v, L) players can cooperate if and only if they are able to communicate with each
1 We refer the reader to the mentioned literature for the definition of the axioms. 2 For games on variable player sets the Shapley value is characterized by, e.g. Pareto optimality, covariance, the equal treatment property and consistency in Hart and Mas-Colell (1989) who also expressed it by a potential.
3 We use the convention that the empty sum is 0.
112
R. van den Brink, M. Pintér / Journal of Mathematical Economics 60 (2015) 110–114
other, that is, a coalition S can realize its worth v(S ) if and only if S is connected in L. Whenever this is not the case, players in S can only realize the sum of the worths of the components of S in L. This yields the (Myerson) restricted game v L ∈ GN given by
v L (S ) =
v(T ),
S ⊆ N.
(1)
T ∈CL (S )
The Myerson value is the solution µ: GN × LN → RN that is obtained by taking the Shapley value of the restricted game v L , that is,
µ(v, L) = φ Sh (v L ),
for all v ∈ GN and L ∈ LN .
Besides introducing this value, Myerson (1977) gives a characterization showing that it is the unique solution for graph games satisfying component efficiency and fairness. Solution φ on C ⊆ GN × LN satisfies N N • component efficiency, if for every graph game (v, L) ∈ G × L and component S ∈ CL (N ), it holds that i∈S φi (v, L) = v(S ); • graph game fairness,4 if for every graph game (v, L) ∈ GN × LN and pair of players i, j ∈ N, it holds that φi (v, L) − φi (v, L \ {{i, j}}) = φj (v, L) − φj (v, L \ {{i, j}}).
Component efficiency states that the sum of the payoffs of all players in a component equals the worth of that component. Graph game fairness states that deleting the link between two players changes their payoffs by the same amount. Moreover, Myerson (1977) shows that his solution is stable in the sense that for superadditive games adding a link never hurts the two players incident with that link. 3. An axiomatization of the Shapley value for assignment situations Several solutions that allocate the surplus that can be earned in an assignment situation have been studied and axiomatized in the literature. The most famous set-valued solution, the core (Gillies, 1959), has been axiomatized in Sasaki (1995) and Toda (2003, 2005). The two most famous one-point solutions are the nucleolus (Schmeidler, 1967) and the Shapley value (Shapley, 1953). The nucleolus for assignment games has been considered in, e.g. Llerena and Núñez (2011) and Solymosi and Raghavan (1994), while an axiomatization of this solution for assignment games can be found in Llerena et al. (2015). The Shapley value has been considered in Hoffmann and Sudhölter (2007) for exact assignment games.5 Although the Shapley value of an assignment game does not necessarily belong to its core,6 as is the case for the nucleolus, an advantage of the Shapley value is that it rewards all players that have some influence on the game.7 For this see the following example: Example 3.1. Consider the three player assignment situation with B = {1, 2}, S = {3}, a1,3 = 4 and a2,3 = 5. Any core allocation divides the full surplus (of a2,3 = 5) between players 2 and 3, so player 1 has payoff zero. Moreover, player 3 earns at least 4. However, when player 1 leaves the ‘market’ then the core allows any distribution of the surplus between players 2 and 3, so player 3 can earn any value between 0 and 5.
4 Myerson (1977) just refers to this as fairness, but we call it graph game fairness to distinguish it from valuation fairness that is defined in the next section. 5 A limit approach to the Shapley value for assignment games has been considered by Ligett et al. (2009). 6 Hoffmann and Sudhölter (2007) show that the Shapley value for exact assignment games always belongs to the core. 7 Shapley and Shubik (1972) already mention as one of the weaknesses of the core (=competitive) solution that it excludes third-party payments although these parties might have an important role on the ‘‘leverage’’ of other players.
For example, the nucleolus of the three player game is (0, 12 , 92 ) while the nucleolus for the two player assignment game (without player 1) is ( 52 , 52 ). So, the presence of player 1 clearly has an effect on the allocation of the surplus. In some sense, player 2 might be willing to pay player 1 to leave the game. Alternatively, player 3 might be willing to pay player 1 to stay in the game. Here the Shapley solution yields ( 46 , 76 , 19 ) for the three player 6 game, and also ( 52 , 52 ) for the two player game (when player 1 leaves the game). Therefore, the Shapley solution expresses the intuition about the relevance of player 1 in the situation, while no core allocation does so.
Following the above mentioned literature, the purpose of the underlying paper is to study axiomatizations of the Shapley value restricted to the class of assignment games. It is straightforward to see that on the class of two-player assignment games, the Shapley value is the unique solution satisfying Pareto optimality8 and the equal treatment property9 since in such games there is exactly one buyer and one seller and thus the assignment game is a multiple of the unanimity game on N. Henceforth we assume that |B ∪ S | > 2, that is, the class of assignment games under consideration is not the ‘‘trivial’’ one. Let GB,S be the class of assignment games with buyer and seller sets B, S such that |B ∪ S | > 2. It turns out that all axiomatizations of the Shapley value mentioned in the preliminaries are not valid on classes of assignment games with at least three players in the sense that they do not characterize a unique solution. An example of an alternative solution that satisfies all axioms mentioned in the last paragraph of Section 2.1 is the following. Let OR(N ) be the set of all (linear) orderings on set N, let ORB = {τ ∈ OR(B ∪ S ) | τ (i) ≤ |B| ⇒ i ∈ B}, be the orders where the buyers come first, and let ORS = {τ ∈ OR(B ∪ S ) | τ (i) ≤ |S | ⇒ i ∈ S }, be the orders where the sellers come first. Next, for all v ∈ GB,S and i ∈ B ∪ S, let 1
φiB (v) =
|ORB |
(v({j ∈ B ∪ S | τ (j) ≤ τ (i)})
τ ∈ORB
− v({j ∈ B ∪ S | τ (j) < τ (i)})) , be the average marginal contribution of buyer or seller i over all orders where the buyers come first, and
φ (v) = S i
1
|ORS |
(v({j ∈ B ∪ S | τ (j) ≤ τ (i)})
τ ∈ORS
− v({j ∈ B ∪ S | τ (j) < τ (i)})) be the average marginal contribution of buyer or seller i over all orders where the sellers come first. Then the solution given by
φ B,S (v) =
φ B (v) + φ S (v) 2
for all v ∈ GB,S ,
8 Solution φ on C ⊆ GN is Pareto optimal (or efficient), if i∈N φi (v) = v(N ) for all v ∈ C . 9 Players i, j ∈ N are symmetric in game v ∈ GN , if mT (v) = mT (v) for all i j
T ⊆ N \ {i, j}. Solution φ on C ⊆ GN satisfies the equal treatment property, if φi (v) = φj (v) for all v ∈ C and symmetric players i, j in v .
R. van den Brink, M. Pintér / Journal of Mathematical Economics 60 (2015) 110–114
being the average of the ‘buyer-first’ and ‘seller-first’ solutions, satisfies all axioms mentioned in the last paragraph of Section 2.1 on the class GB,S .10 Thus, these axioms do not yield uniqueness of a solution for assignment games and the axiomatizations of the Shapley value mentioned in Section 2.1 are not valid for assignment games.11 Given an assignment situation with buyer–seller partition (B, S ), consider the undirected graph on N in which the links reflect all matching possibilities. So, the graph on N is the complete bipartite graph LB,S = {{i, j} ⊆ N | i ∈ B, j ∈ S }. Since every coalition that contains at least one seller and at least one buyer is connected in LB,S , and all coalitions that contain only buyers or only sellers have worth zero in the assignment game vA , for every graph restricted assignment game (vA , LB,S ), A ∈ AB,S , it holds that the B,S
Myerson restricted game (vA )L is equal to vA . A general bipartite graph on (B, S ) is a graph L ⊆ LB,S with {i, j} ∈ L only if i ∈ B and j ∈ S or vice versa. We denote the class of all bipartite graphs on (B, S ) by LB,S . Typically, a matching is a bipartite graph that is not complete. Note that if for an assignment situation A ∈ AB,S it holds that vA = (vA )L for some bipartite ′ graph L ∈ LB,S , then also vA = (vA )L for every bipartite graph Lm A L′ ⊃ L. The minimal bipartite graph Lm A such that vA = (vA ) is Lm = {{i, j} ⊂ N | i ∈ B, j ∈ S and ai,j > 0}. Since A vA ({i, j}) = ai,j > 0 = (vA )L ({i, j}) for every A ∈ AB,S with ai,j > 0 m m and L ⊂ Lm A , L ̸= LA and l = {i, j} ∈ LA \ L, we have the following proposition. Proposition 3.2. Consider assignment situation A ∈ AB,S . Then (i) vA m m L = (vA )L for all L ⊇ Lm A , and (ii) vA ̸= (vA ) for all L ⊂ LA , L ̸= LA . Next, we show that component efficiency and graph game fairness characterize the Shapley value for assignment games when we consider this class as graph games on bipartite graphs GB,S × LB,S . From its definition every assignment situation A yields a unique assignment game vA . On the other hand, consider assignment game v ∈ GB,S . Then v = vA where A ∈ AB,S is determined by ai,j = v({i, j}) for all i ∈ B, j ∈ S.12 Because of this one-toone correspondence between assignment games and assignment situations in the sense that every assignment situation yields a unique assignment game, but also for every assignment game the two player coalitions containing a buyer and a seller uniquely determine the assignment situation, we state this axiomatization directly for assignment situations instead of the corresponding class of graph games. We refer to the solution that assigns to every assignment situation A ∈ AB,S the Shapley value of the corresponding assignment game vA as the Shapley value for assignment situations, and denote it by f Sh (A) = φ Sh (vA ). A submarket in assignment situation A ∈ AB,S is a set of buyers and sellers such that all buyers in the set have zero valuation for the goods offered by the sellers outside the set, and all buyers outside the set have zero valuation for the goods offered by sellers inside the set. Definition 3.3. Let A ∈ AB,S be an assignment situation. Then B′ ∪ S ′ , B′ ⊆ B, S ′ ⊆ S , B′ ∪ S ′ ̸= ∅, is a submarket of A if ai,j = 0 for all (i, j) ∈ (B′ × (S \ S ′ )) ∪ ((B \ B′ ) × S ′ ).
10 The proof can be obtained from the authors on request. 11 Although it is not surprising that axioms that do characterize a unique solution on the class of all TU-games, do not yield uniqueness on subclasses of games, this is not straightforward. For example, the axiomatization of Young (1985) also yields uniqueness on the class of strictly monotone games. However, the axiomatization of Shapley (1953) does not. Therefore, when we argue for or against a solution by an axiomatization on a subclass of games, we must be very cautious. 12 Note that for v ∈ GB,S it holds that v(T ) = max v({i, j}) for all T ⊆ B ∪ S with |T | > 2.
M ∈ M (T )
{i,j}∈M
113
Applied to assignment situations, component efficiency of a graph game solution implies that the sum of the payoffs of all players in a submarket equals the worth of that submarket. Definition 3.4. Solution f on AB,S satisfies submarket efficiency, if B ,S for submarkets (B′ , S ′ ) of A, it holds that all A ∈ A and ′ for all ′ i∈B′ ∪S ′ fi (A) = vA (B ∪ S ). It turns out that applying any graph game solution satisfying component efficiency yields a solution for assignment situations that satisfies submarket efficiency. Proposition 3.5. Consider player set N and B, S ⊂ N such that {B, S } is a partition of N. Let ϕ : GN × LN → RN be a graph game solution, and let f : AB,S → RN+ be a solution for assignment situations on N given by f (A) = ϕ(vA , Lm A ). If ϕ satisfies component efficiency, then f satisfies submarket efficiency. Proof. Note that (B′ , S ′ ) being a submarket in the assignment situation A ∈ AB,S implies that C = B′ ∪ S ′ is a component in Lm A . Obviously, f satisfying submarket efficiency for assignment situations follows from ϕ satisfying component efficiency for graph games. Graph game fairness of a solution applied to assignment situations implies that decreasing the valuation of one particular buyer for the good offered by a particular seller to zero, changes the payoffs of this buyer and seller by the same amount. Definition 3.6. Solution f on AB,S satisfies valuation fairness, if for every buyer i ∈ B, every seller j ∈ S and every pair of assignment situations A, A ∈ AB,S such that ai,j = 0 and ag ,h = ag ,h for all (g , h) ∈ ((B \ {i}) × S ) ∪ (B × (S \ {j})), it holds that fi (A) − fi (A) = fj (A) − fj (A). Proposition 3.7. Consider player set N and B, S ⊂ N such that {B, S } is a partition of N. Let ϕ : GN × LN → RN be a graph game solution, and let f : AB,S → RN+ be a solution for assignment situations on N given by f (A) = ϕ(vA , Lm A ). If ϕ satisfies graph game fairness, then f satisfies valuation fairness. Proof. If ag ,h = ag ,h for all (g , h) ∈ ((B \ {i}) × S ) ∪ (B × (S \ {j})), then Lm = Lm A \ {{i, j}}. But then f satisfying graph game fairness A
m m implies that fi (A) − fi (A) = ϕi (vA , Lm A ) − ϕi (vA , L ) = ϕj (vA , LA ) − A
ϕj (vA , Lm ) = fj (A) − fj (A), where the second equality follows from A graph game fairness of ϕ , and thus f satisfies valuation fairness. Next, a similar characterization as the Myerson value for graph games holds for the Shapley value for assignment situations.13 , 14 Theorem 3.8. The Shapley value f Sh is the unique solution for assignment situations that satisfies submarket efficiency and valuation fairness. Proof. The Shapley value satisfying the two axioms follows from Propositions 3.5 and 3.7 and the Myerson value satisfying component efficiency and graph game fairness. We prove uniqueness by induction on the number of non-zero valuations k(A) = |K (A)|, where K (A) = {(i, j) ∈ B × S | ai,j > 0}.
13 Since there is a one-to-one correspondence between assignment games and assignment situations on given sets of buyers and sellers, it follows that submarket efficiency and valuation fairness also give an axiomatization of the Shapley value on the class of assignment games. 14 Although not explicitly written, the characterization of the Myerson value by component efficiency and graph game fairness holds more general in the sense that component efficiency and graph game fairness characterize the Myerson value on any restricted class of graph games GN × C such that C ⊆ LN is comprehensive, that is, for any L ∈ C and L′ ⊆ L it holds that L′ ∈ C . For example, the class LB,S of all bipartite graphs between the sets of buyers B and sellers S satisfies this property.
114
R. van den Brink, M. Pintér / Journal of Mathematical Economics 60 (2015) 110–114
Suppose that f : AB,S → RN satisfies submarket efficiency and valuation fairness. If k(A) = 0, then all singleton player sets form a submarket, and thus f (A) is determined (uniquely) by submarket efficiency. Proceeding by induction, assume that f (A′ ) is (uniquely) determined whenever 0 ≤ k(A′ ) < k(A). Take any submarket C = B′ ∪ S ′ , B′ ⊆ B, S ′ ⊆ S. If |C | = 1, then submarket efficiency determines fi (A) for i ∈ C . Otherwise, if |C | > 1, then C is connected in Lm A . Then for all (i, j) ∈ (C ∩ B) × (C ∩ S ) such that ai,j > 0, let Ai,j ∈ AB,S denote the matrix where for all (k, l) ∈ B × S: i,j ak,l
=
0, ak,l
if (k, l) = (i, j) otherwise.
From valuation fairness it follows that for all (i, j) ∈ (C ∩ B) ×
(C ∩ S ) such that ai,j > 0,
fi (A) − fi (Ai,j ) = fj (A) − fj (Ai,j ).
(2)
Since C is a component, there is a tree T ⊆ (C ) connecting all nodes in C . Since the values fi (Ai,j ) and fj (Ai,j ) are determined by the induction hypothesis, taking the equations (2) for the buyer–seller pairs (i, j) such that (i, j) ∈ T , yields |C | − 1 linear independent equations in the |C | unknown payoffs fi (A), i ∈ C . Together with the equation i∈C fi (A) = vA (C ) which follows from submarket efficiency, the |C | unknown payoffs fi (A), i ∈ C , are uniquely determined. Since this can be done for all submarkets, the payoffs f (A) are uniquely determined. Lm A
The Shapley value satisfies an even stronger valuation fairness property which states that changing the valuation of one particular buyer for the good offered by a particular seller in any way, changes the payoffs of this buyer and seller by the same amount. This follows from fairness15 of the Shapley value and the fact that this buyer and seller are symmetric players in the difference game (vA − vAi,j ). However, for the characterization the weaker version looking at valuation zero is sufficient. The Shapley value for assignment situations also satisfies a monotonicity property similar to the stability property for graph games.16 This property states that the payoffs of buyer i and seller j do not decrease if we only increase the valuation of buyer i for the good offered by seller j from zero to a positive valuation. Definition 3.9. Solution f on AB,S satisfies valuation monotonicity, if for all assignment situations A, A ∈ AB,S such that for some i ∈ B, j ∈ S it holds that ai,j = 0, and ag ,h = ag ,h for all (g , h) ∈ ((B \ {i}) × S ) ∪ (B × (S \ {j})), we have fi (A) ≥ fi (A) and fj (A) ≥ fj (A). It turns out that applying any graph game solution satisfying stability on the class of assignment games yields a solution for assignment situations that satisfies valuation monotonicity. Proposition 3.10. Consider player set N and B, S ⊂ N such that {B, S } is a partition of N. Let ϕ : GN × LN → RN be a graph game
15 Solution φ on C ⊆ GN satisfies fairness, if φ (v +w)−φ (v) = φ (v +w)−φ (v) i i j j for all v, w ∈ C and i, j ∈ N such that i, j are symmetric in w and v + w ∈ C . 16 Graph game solution ϕ : GN × LN → RN satisfies stability, if ϕ (v, L) ≥ ϕ (L \ i
{{i, j}}) and ϕj (v, L) ≥ ϕj (L \ {{i, j}}) whenever game v is superadditive.
i
solution, and let f : AB,S → RN+ be a solution for assignment situations on N given by f (A) = ϕ(vA , Lm A ). If ϕ satisfies stability, then f satisfies valuation monotonicity. This follows immediately from stability of ϕ and the fact that m Lm A = L \ {{i, j}}. A
Corollary 3.11. The Shapley value for assignment situations satisfies valuation monotonicity. As with valuation fairness, the Shapley value for assignment situations also satisfies a stronger valuation monotonicity property stating that the payoffs of buyer i and seller j do not decrease if we only increase the valuation of buyer i for the good offered by seller j, i.e. for all assignment situations A, A ∈ AB,S such that for some i ∈ B, j ∈ S it holds that ai,j ≥ ai,j , and ag ,h = ag ,h for all (g , h) ∈ ((B \ {i}) × S ) ∪ (B × (S \ {j})), we have fiSh (A) ≥ fiSh (A) and fjSh (A) ≥ fjSh (A). References Böhm-Bawerk, E. von, 1923. Positive Theory of Capital (translated by W. Smart), G.E. Steckert, New York, (original publication 1891). Brink, R. van den, 2001. An axiomatization of the Shapley value using a fairness property. Internat. J. Game Theory 30, 309–319. Chun, Y., 1991. On the symmetric and weighted shapley values. Internat. J. Game Theory 20, 183–190. Dubey, P., 1975. On the uniqueness of the Shapley value. Internat. J. Game Theory 4, 131–139. Gillies, D.B., 1959. Solutions to General Non-zero-sum Games. Princeton University Press. Hart, S., Mas-Colell, A., 1989. Potential, value, and consistency. Econometrica 57, 589–614. Hoffmann, M., Sudhölter, P., 2007. The Shapley value of exact assignment games. Internat. J. Game Theory 35, 557–568. Ligett, T.M., Lippman, S.A., Rumelt, R.P., 2009. The asymptotic Shapley value for a simple market game. Econom. Theory 40, 333–338. Llerena, F., Núñez, M., 2011. A geometric characterization of the nucleolus of the assignment game. Econ. Bull. 31, 3275–3285. Llerena, F., Nunez, M., Rafels, C., 2015. An axiomatization of the nucleolus of assignment markets. Internat. J. Game Theory 44, 1–15. Moulin, H., 1988. Axioms of Cooperative Decision Making. Cambridge University Press. Myerson, R.B., 1977. Graphs and cooperation in games. Math. Oper. Res. 2, 225–229. Peleg, B., Sudhölter, P., 2003. Introduction to the Theory of Cooperative Games. Kluwer Academic Publishers, Boston, Dordrecht, London. Pintér, M., 2012. Young’s axiomatization of the Shapley value — A new proof. arXiv:0805.2797v3. Sasaki, H., 1995. Consistency and monotonicity in assignment problems. Internat. J. Game Theory 24, 373–397. Schmeidler, D., 1967. The nucleolus of a characteristic function game. SIAM J. Appl. Math. 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 Volume II. In: Annals of Mathematical Studies, vol. 28. pp. 307–317. Shapley, L.S., Shubik, M., 1972. The assignment game I: the core. Internat. J. Game Theory 1, 111–130. Solymosi, T., Raghavan, T.E.S., 1994. An algorithm for finding the nucleolus of assignment games. Internat. J. Game Theory 23, 119–143. Toda, M., 2003. Consistency and its converse in assignment games. Int. J. Math. Game Theory Algebra 13, 1–14. Toda, M., 2005. Axiomatization of the core of assignment games. Games Econom. Behav. 53, 248–261. Young, H.P., 1985. Monotonic solutions of cooperative games. Internat. J. Game Theory 14, 65–72.
Contents lists available at ScienceDirect
Journal of Mathematical Economics journal homepage: www.elsevier.com/locate/jmateco
On axiomatizations of the Shapley value for assignment games✩ René van den Brink a,∗ , Miklós Pintér b,c,d a
VU University, Department of Econometrics and Tinbergen Institute, De Boelelaan 1105, 1081 HV Amsterdam, The Netherlands
b
Corvinus University of Budapest, Department of Mathematics, 1093, Budapest, Fovám tér 13-15, Hungary
c
MTA-BCE ‘‘Lendület’’ Strategic Interactions Research Group, 1093, Budapest, Fovám tér 13-15, Hungary
d
Department of Business and Economics, University of Pecs, Hungary
article
info
Article history: Received 14 March 2013 Received in revised form 21 November 2013 Accepted 26 June 2015 Available online 7 July 2015 Keywords: Game theory Assignment game Shapley value Graph game Submarket efficiency Valuation fairness
abstract We consider the problem of axiomatizing the Shapley value on the class of assignment games. It turns out that several axiomatizations of the Shapley value on the class of all TU-games do not characterize this solution on the class of assignment games. However, when considering an assignment game as a (communication) graph game where the game is simply the assignment game and the graph is a corresponding bipartite graph where buyers (sellers) are connected with sellers (buyers) only, we show that Myerson’s component efficiency and fairness axioms do characterize the Shapley value on the class of assignment games. Moreover, these two axioms have a natural interpretation for assignment games. Component efficiency yields submarket efficiency stating that the sum of the payoffs of all players in a submarket equals the worth of that submarket, where a submarket is a set of buyers and sellers such that all buyers in this set have zero valuation for the goods offered by the sellers outside the set, and all buyers outside the set have zero valuations for the goods offered by sellers inside the set. Fairness of the graph game solution boils down to valuation fairness stating that only changing the valuation of one particular buyer for the good offered by a particular seller changes the payoffs of this buyer and seller by the same amount. © 2015 Elsevier B.V. All rights reserved.
1. Introduction The history of assignment games goes back to the XIX century to Böhm-Bawerk’s (1923) horse market model. Later Shapley and Shubik (1972) introduced the formal, modern concept of assignment games. One of the most popular solution concepts for TU-games is the Shapley value (Shapley, 1953). Numerous axiomatizations of the Shapley value are known in the literature, for example (i) Shapley’s original axiomatization (Shapley, 1953) by efficiency, the null player property (originally stated together as the carrier axiom), symmetry and additivity (also discussed by Dubey, 1975 and Peleg and Sudhölter, 2003), (ii) Young’s (1985) axiomatization replacing additivity and the null player property by strong monotonicity (also discussed by Moulin (1988) and Pintér (2012)), (iii) Chun’s (1991) replacing strong monotonicity by coalitional strategic
✩ We thank an anonymous referee for her/his useful comments. Miklós Pintér gratefully acknowledges the Financial support by the Hungarian Scientific Research Fund (OTKA) grant number K 101224, SPOR grant number 4.2.2.B-15/1/konv and the János Bolyai Research Scholarship of the Hungarian Academy of Sciences. ∗ Corresponding author. E-mail addresses: [email protected] (R. van den Brink), [email protected] (M. Pintér).
http://dx.doi.org/10.1016/j.jmateco.2015.06.016 0304-4068/© 2015 Elsevier B.V. All rights reserved.
equivalence, (iv) van den Brink’s (2001) replacing (in Shapley’s original axiomatization) additivity and symmetry by fairness, and (v) Hart and Mas-Colell’s (1989) approaches using the potential function and a related reduced game consistency. It turns out that none of these characterizations are valid on the class of assignment games in the sense that they do not characterize a unique solution. In this paper, we show that when considering an assignment game as a (communication) graph game where the game is simply the assignment game and the graph is a corresponding bipartite graph where buyers (sellers) are connected with sellers (buyers) only, Myerson’s (1977) component efficiency and fairness axioms do characterize the Shapley value on the class of assignment games. Moreover, the axioms have a natural interpretation for these games. An assignment game is fully described by the assignment situation being a set of buyers, a set of sellers, and for every buyer a valuation of the good offered by each seller. Instead of defining an assignment game as a graph game, we will directly work on the class of these assignment situations. For such assignment situations, component efficiency of a graph game solution boils down to submarket efficiency stating that the sum of the payoffs of all players in a submarket equals the worth of that submarket, where a submarket in an assignment situation is a set of buyers and sellers such that all buyers in this set have zero valuation for the goods
R. van den Brink, M. Pintér / Journal of Mathematical Economics 60 (2015) 110–114
111
offered by the sellers outside the set, and all buyers outside the set have zero valuations for the goods offered by sellers inside the set. Fairness of the graph game solution boils down to valuation fairness stating that only changing the valuation of one particular buyer for the good offered by a particular seller changes the payoffs of this buyer and seller by the same amount. We show that these two axioms do characterize the Shapley solution for assignment situations being the solution that is obtained by applying the Shapley value to the corresponding assignment game. So, we obtain a positive result by viewing an assignment game as a graph game. Besides introducing and axiomatizing his solution, Myerson (1977) also shows that it is stable for superadditive graph games in the sense that two players never get worse off when building a link between them. The Shapley solution for assignment situations is valuation monotonic in the sense that the payoffs of a buyer i and a seller j do not decrease if only the valuation of buyer i for the good offered by seller j increases.
reservation value zero for every good. The nonnegative valuation (reservation value) of buyer i ∈ B for the good offered by seller j ∈ S is denoted by ai,j ≥ 0. So, buyer i and seller j can make a deal and earn worth ai,j . Buyers cannot trade among each other (since they do not own a good), and also sellers cannot earn a worth among themselves since their valuation is zero. Let A be the |B| × |S | non-negative matrix with ai,j its (i, j) element. We refer to this matrix A as an assignment situation or valuation matrix on (B, S ). We denote the collection of all assignment situations on (B, S ) by AB,S . Furthermore for all T ⊆ N, a matching on T is a set of sets M ⊆ {{i, j} ⊆ T | i ∈ B ∩ T , j ∈ S ∩ T } such that for every g ∈ T , |{{h, k} ∈ M | g ∈ {h, k}}| ≤ 1. So, buyers can only be matched with sellers, sellers can only be matched with buyers, and every buyer (seller) can be matched with at most one seller (buyer). Let M (T ) be the set of all matchings of T . Taking the sets of buyers B and sellers S fixed, the assignment game (see Shapley and Shubik, 1972) for valuation matrix A is the game vA on N = B ∪ S, given by3
2. Preliminaries
vA (T ) = max
M ∈M (T )
2.1. TU-games
ai,j
for all T ⊆ B ∪ S .
{i,j}∈M
The elements of Let N be a non-empty, finite set, let |N | be its cardinality, and let P (N ) denote the power set of N. A transferable utility (TU) game with player set N is a pair (N , v) with characteristic function v : P (N ) → R such that v(∅) = 0. Since we take the player set N to be fixed, we represent a TU-game (N , v) simply by its characteristic function v . The class of all characteristic functions on player set N is denoted by GN . Game v is superadditive if v(S ∪ T ) ≥ v(S ) + v(T ) for all S , T ⊆ N with S ∩ T = ∅. A (single-valued) solution on C ⊆ GN , is a function φ : C → RN . In this paper we focus on the Shapley value (Shapley, 1953) being the solution φ Sh : GN → RN , for every v ∈ GN , given by
φiSh (v) =
|T |!(|N \ T | − 1)! mTi (v) for all i ∈ N , |N |! T ⊆N \{i}
where, for any v ∈ GN , i ∈ N and T ⊆ N , mTi (v) = v(T ∪ {i}) − v(T ) is player i’s marginal contribution to coalition T in game v . We refer to φiSh (v) as the Shapley value of player i in game v ∈ GN . Several axiomatizations of the Shapley value can be found in the literature, such as the axiomatization1 by Pareto optimality (also known as efficiency), the null player property, the equal treatment property and additivity (Shapley, 1953), Pareto optimality, the equal treatment property and strong monotonicity (Young, 1985), Pareto optimality, the equal treatment property and marginality (also by Young, 1985), Pareto optimality, the equal treatment property and coalitional strategic equivalence (Chun, 1991) and Pareto optimality, the null player property and fairness (van den Brink, 2001).2 2.2. Assignment games Let B, S ⊆ N be two non-empty sets such that B ∩ S = ∅ and B ∪ S = N. The interpretation is the following. The sets B and S are the sets of buyers and sellers, respectively. Every buyer wants one good, and every seller owns one good. These goods are not exactly the same, so a buyer can have different valuations for the goods owned by different sellers. We assume that the sellers have
arg max
M ∈M (T ) {i,j}∈M
ai,j
are called the maximal matchings of coalition T . For any set of buyers and sellers T , the worth of this coalition is the maximum aggregated worth of the deals the involved players can achieve contingent on every player trading with at most one other player from the other type. Remark 2.1. Since in the definition of assignment games, B and S are non-empty, in this paper every assignment game has at least two players. 2.3. Graph games Myerson (1977) introduced a model in which it is assumed that the players in a game v are part of a communication structure that is represented by an undirected graph (N , L), with the player set N as the set of nodes and L ⊆ {{i, j} | i, j ∈ N , i ̸= j} being a collection of edges or links, that is, subsets of N such that each element of L contains precisely two elements. Since in this paper the nodes in a graph represent the players in a game, we use the same notation for the set of nodes as the set of players, and refer to the nodes in a graph just as players. If there is no confusion about the player set N, we denote a graph on N just by its set of links L and refer to this as the graph. We denote the class of all possible sets of links on N by LN . A sequence of k different nodes (i1 , . . . , ik ) is a path between players i1 and ik in L ∈ LN if {ih , ih+1 } ∈ L for h = 1, . . . , k − 1. A coalition S ⊆ N is connected in graph L if every pair of players in S is connected by a path that only contains players from S, that is, for every i, j ∈ S , i ̸= j, there is a path (i1 , . . . , ik ) such that i1 = i, ik = j and {i1 , . . . , ik } ⊆ S. Coalition T ⊆ S is a component of S in graph L if it is a maximally connected subset of S, that is, T is connected in L(S ) and for every h ∈ S \ T the coalition T ∪ {h} is not connected in L(S ), where L(S ) = {{i, j} ∈ L | {i, j} ⊆ S }. We denote the set of components of S ⊆ N in L by CL (S ). A pair (v, L) ∈ GN × LN is referred to as a graph game on N. Following Myerson (1977), in the graph game (v, L) players can cooperate if and only if they are able to communicate with each
1 We refer the reader to the mentioned literature for the definition of the axioms. 2 For games on variable player sets the Shapley value is characterized by, e.g. Pareto optimality, covariance, the equal treatment property and consistency in Hart and Mas-Colell (1989) who also expressed it by a potential.
3 We use the convention that the empty sum is 0.
112
R. van den Brink, M. Pintér / Journal of Mathematical Economics 60 (2015) 110–114
other, that is, a coalition S can realize its worth v(S ) if and only if S is connected in L. Whenever this is not the case, players in S can only realize the sum of the worths of the components of S in L. This yields the (Myerson) restricted game v L ∈ GN given by
v L (S ) =
v(T ),
S ⊆ N.
(1)
T ∈CL (S )
The Myerson value is the solution µ: GN × LN → RN that is obtained by taking the Shapley value of the restricted game v L , that is,
µ(v, L) = φ Sh (v L ),
for all v ∈ GN and L ∈ LN .
Besides introducing this value, Myerson (1977) gives a characterization showing that it is the unique solution for graph games satisfying component efficiency and fairness. Solution φ on C ⊆ GN × LN satisfies N N • component efficiency, if for every graph game (v, L) ∈ G × L and component S ∈ CL (N ), it holds that i∈S φi (v, L) = v(S ); • graph game fairness,4 if for every graph game (v, L) ∈ GN × LN and pair of players i, j ∈ N, it holds that φi (v, L) − φi (v, L \ {{i, j}}) = φj (v, L) − φj (v, L \ {{i, j}}).
Component efficiency states that the sum of the payoffs of all players in a component equals the worth of that component. Graph game fairness states that deleting the link between two players changes their payoffs by the same amount. Moreover, Myerson (1977) shows that his solution is stable in the sense that for superadditive games adding a link never hurts the two players incident with that link. 3. An axiomatization of the Shapley value for assignment situations Several solutions that allocate the surplus that can be earned in an assignment situation have been studied and axiomatized in the literature. The most famous set-valued solution, the core (Gillies, 1959), has been axiomatized in Sasaki (1995) and Toda (2003, 2005). The two most famous one-point solutions are the nucleolus (Schmeidler, 1967) and the Shapley value (Shapley, 1953). The nucleolus for assignment games has been considered in, e.g. Llerena and Núñez (2011) and Solymosi and Raghavan (1994), while an axiomatization of this solution for assignment games can be found in Llerena et al. (2015). The Shapley value has been considered in Hoffmann and Sudhölter (2007) for exact assignment games.5 Although the Shapley value of an assignment game does not necessarily belong to its core,6 as is the case for the nucleolus, an advantage of the Shapley value is that it rewards all players that have some influence on the game.7 For this see the following example: Example 3.1. Consider the three player assignment situation with B = {1, 2}, S = {3}, a1,3 = 4 and a2,3 = 5. Any core allocation divides the full surplus (of a2,3 = 5) between players 2 and 3, so player 1 has payoff zero. Moreover, player 3 earns at least 4. However, when player 1 leaves the ‘market’ then the core allows any distribution of the surplus between players 2 and 3, so player 3 can earn any value between 0 and 5.
4 Myerson (1977) just refers to this as fairness, but we call it graph game fairness to distinguish it from valuation fairness that is defined in the next section. 5 A limit approach to the Shapley value for assignment games has been considered by Ligett et al. (2009). 6 Hoffmann and Sudhölter (2007) show that the Shapley value for exact assignment games always belongs to the core. 7 Shapley and Shubik (1972) already mention as one of the weaknesses of the core (=competitive) solution that it excludes third-party payments although these parties might have an important role on the ‘‘leverage’’ of other players.
For example, the nucleolus of the three player game is (0, 12 , 92 ) while the nucleolus for the two player assignment game (without player 1) is ( 52 , 52 ). So, the presence of player 1 clearly has an effect on the allocation of the surplus. In some sense, player 2 might be willing to pay player 1 to leave the game. Alternatively, player 3 might be willing to pay player 1 to stay in the game. Here the Shapley solution yields ( 46 , 76 , 19 ) for the three player 6 game, and also ( 52 , 52 ) for the two player game (when player 1 leaves the game). Therefore, the Shapley solution expresses the intuition about the relevance of player 1 in the situation, while no core allocation does so.
Following the above mentioned literature, the purpose of the underlying paper is to study axiomatizations of the Shapley value restricted to the class of assignment games. It is straightforward to see that on the class of two-player assignment games, the Shapley value is the unique solution satisfying Pareto optimality8 and the equal treatment property9 since in such games there is exactly one buyer and one seller and thus the assignment game is a multiple of the unanimity game on N. Henceforth we assume that |B ∪ S | > 2, that is, the class of assignment games under consideration is not the ‘‘trivial’’ one. Let GB,S be the class of assignment games with buyer and seller sets B, S such that |B ∪ S | > 2. It turns out that all axiomatizations of the Shapley value mentioned in the preliminaries are not valid on classes of assignment games with at least three players in the sense that they do not characterize a unique solution. An example of an alternative solution that satisfies all axioms mentioned in the last paragraph of Section 2.1 is the following. Let OR(N ) be the set of all (linear) orderings on set N, let ORB = {τ ∈ OR(B ∪ S ) | τ (i) ≤ |B| ⇒ i ∈ B}, be the orders where the buyers come first, and let ORS = {τ ∈ OR(B ∪ S ) | τ (i) ≤ |S | ⇒ i ∈ S }, be the orders where the sellers come first. Next, for all v ∈ GB,S and i ∈ B ∪ S, let 1
φiB (v) =
|ORB |
(v({j ∈ B ∪ S | τ (j) ≤ τ (i)})
τ ∈ORB
− v({j ∈ B ∪ S | τ (j) < τ (i)})) , be the average marginal contribution of buyer or seller i over all orders where the buyers come first, and
φ (v) = S i
1
|ORS |
(v({j ∈ B ∪ S | τ (j) ≤ τ (i)})
τ ∈ORS
− v({j ∈ B ∪ S | τ (j) < τ (i)})) be the average marginal contribution of buyer or seller i over all orders where the sellers come first. Then the solution given by
φ B,S (v) =
φ B (v) + φ S (v) 2
for all v ∈ GB,S ,
8 Solution φ on C ⊆ GN is Pareto optimal (or efficient), if i∈N φi (v) = v(N ) for all v ∈ C . 9 Players i, j ∈ N are symmetric in game v ∈ GN , if mT (v) = mT (v) for all i j
T ⊆ N \ {i, j}. Solution φ on C ⊆ GN satisfies the equal treatment property, if φi (v) = φj (v) for all v ∈ C and symmetric players i, j in v .
R. van den Brink, M. Pintér / Journal of Mathematical Economics 60 (2015) 110–114
being the average of the ‘buyer-first’ and ‘seller-first’ solutions, satisfies all axioms mentioned in the last paragraph of Section 2.1 on the class GB,S .10 Thus, these axioms do not yield uniqueness of a solution for assignment games and the axiomatizations of the Shapley value mentioned in Section 2.1 are not valid for assignment games.11 Given an assignment situation with buyer–seller partition (B, S ), consider the undirected graph on N in which the links reflect all matching possibilities. So, the graph on N is the complete bipartite graph LB,S = {{i, j} ⊆ N | i ∈ B, j ∈ S }. Since every coalition that contains at least one seller and at least one buyer is connected in LB,S , and all coalitions that contain only buyers or only sellers have worth zero in the assignment game vA , for every graph restricted assignment game (vA , LB,S ), A ∈ AB,S , it holds that the B,S
Myerson restricted game (vA )L is equal to vA . A general bipartite graph on (B, S ) is a graph L ⊆ LB,S with {i, j} ∈ L only if i ∈ B and j ∈ S or vice versa. We denote the class of all bipartite graphs on (B, S ) by LB,S . Typically, a matching is a bipartite graph that is not complete. Note that if for an assignment situation A ∈ AB,S it holds that vA = (vA )L for some bipartite ′ graph L ∈ LB,S , then also vA = (vA )L for every bipartite graph Lm A L′ ⊃ L. The minimal bipartite graph Lm A such that vA = (vA ) is Lm = {{i, j} ⊂ N | i ∈ B, j ∈ S and ai,j > 0}. Since A vA ({i, j}) = ai,j > 0 = (vA )L ({i, j}) for every A ∈ AB,S with ai,j > 0 m m and L ⊂ Lm A , L ̸= LA and l = {i, j} ∈ LA \ L, we have the following proposition. Proposition 3.2. Consider assignment situation A ∈ AB,S . Then (i) vA m m L = (vA )L for all L ⊇ Lm A , and (ii) vA ̸= (vA ) for all L ⊂ LA , L ̸= LA . Next, we show that component efficiency and graph game fairness characterize the Shapley value for assignment games when we consider this class as graph games on bipartite graphs GB,S × LB,S . From its definition every assignment situation A yields a unique assignment game vA . On the other hand, consider assignment game v ∈ GB,S . Then v = vA where A ∈ AB,S is determined by ai,j = v({i, j}) for all i ∈ B, j ∈ S.12 Because of this one-toone correspondence between assignment games and assignment situations in the sense that every assignment situation yields a unique assignment game, but also for every assignment game the two player coalitions containing a buyer and a seller uniquely determine the assignment situation, we state this axiomatization directly for assignment situations instead of the corresponding class of graph games. We refer to the solution that assigns to every assignment situation A ∈ AB,S the Shapley value of the corresponding assignment game vA as the Shapley value for assignment situations, and denote it by f Sh (A) = φ Sh (vA ). A submarket in assignment situation A ∈ AB,S is a set of buyers and sellers such that all buyers in the set have zero valuation for the goods offered by the sellers outside the set, and all buyers outside the set have zero valuation for the goods offered by sellers inside the set. Definition 3.3. Let A ∈ AB,S be an assignment situation. Then B′ ∪ S ′ , B′ ⊆ B, S ′ ⊆ S , B′ ∪ S ′ ̸= ∅, is a submarket of A if ai,j = 0 for all (i, j) ∈ (B′ × (S \ S ′ )) ∪ ((B \ B′ ) × S ′ ).
10 The proof can be obtained from the authors on request. 11 Although it is not surprising that axioms that do characterize a unique solution on the class of all TU-games, do not yield uniqueness on subclasses of games, this is not straightforward. For example, the axiomatization of Young (1985) also yields uniqueness on the class of strictly monotone games. However, the axiomatization of Shapley (1953) does not. Therefore, when we argue for or against a solution by an axiomatization on a subclass of games, we must be very cautious. 12 Note that for v ∈ GB,S it holds that v(T ) = max v({i, j}) for all T ⊆ B ∪ S with |T | > 2.
M ∈ M (T )
{i,j}∈M
113
Applied to assignment situations, component efficiency of a graph game solution implies that the sum of the payoffs of all players in a submarket equals the worth of that submarket. Definition 3.4. Solution f on AB,S satisfies submarket efficiency, if B ,S for submarkets (B′ , S ′ ) of A, it holds that all A ∈ A and ′ for all ′ i∈B′ ∪S ′ fi (A) = vA (B ∪ S ). It turns out that applying any graph game solution satisfying component efficiency yields a solution for assignment situations that satisfies submarket efficiency. Proposition 3.5. Consider player set N and B, S ⊂ N such that {B, S } is a partition of N. Let ϕ : GN × LN → RN be a graph game solution, and let f : AB,S → RN+ be a solution for assignment situations on N given by f (A) = ϕ(vA , Lm A ). If ϕ satisfies component efficiency, then f satisfies submarket efficiency. Proof. Note that (B′ , S ′ ) being a submarket in the assignment situation A ∈ AB,S implies that C = B′ ∪ S ′ is a component in Lm A . Obviously, f satisfying submarket efficiency for assignment situations follows from ϕ satisfying component efficiency for graph games. Graph game fairness of a solution applied to assignment situations implies that decreasing the valuation of one particular buyer for the good offered by a particular seller to zero, changes the payoffs of this buyer and seller by the same amount. Definition 3.6. Solution f on AB,S satisfies valuation fairness, if for every buyer i ∈ B, every seller j ∈ S and every pair of assignment situations A, A ∈ AB,S such that ai,j = 0 and ag ,h = ag ,h for all (g , h) ∈ ((B \ {i}) × S ) ∪ (B × (S \ {j})), it holds that fi (A) − fi (A) = fj (A) − fj (A). Proposition 3.7. Consider player set N and B, S ⊂ N such that {B, S } is a partition of N. Let ϕ : GN × LN → RN be a graph game solution, and let f : AB,S → RN+ be a solution for assignment situations on N given by f (A) = ϕ(vA , Lm A ). If ϕ satisfies graph game fairness, then f satisfies valuation fairness. Proof. If ag ,h = ag ,h for all (g , h) ∈ ((B \ {i}) × S ) ∪ (B × (S \ {j})), then Lm = Lm A \ {{i, j}}. But then f satisfying graph game fairness A
m m implies that fi (A) − fi (A) = ϕi (vA , Lm A ) − ϕi (vA , L ) = ϕj (vA , LA ) − A
ϕj (vA , Lm ) = fj (A) − fj (A), where the second equality follows from A graph game fairness of ϕ , and thus f satisfies valuation fairness. Next, a similar characterization as the Myerson value for graph games holds for the Shapley value for assignment situations.13 , 14 Theorem 3.8. The Shapley value f Sh is the unique solution for assignment situations that satisfies submarket efficiency and valuation fairness. Proof. The Shapley value satisfying the two axioms follows from Propositions 3.5 and 3.7 and the Myerson value satisfying component efficiency and graph game fairness. We prove uniqueness by induction on the number of non-zero valuations k(A) = |K (A)|, where K (A) = {(i, j) ∈ B × S | ai,j > 0}.
13 Since there is a one-to-one correspondence between assignment games and assignment situations on given sets of buyers and sellers, it follows that submarket efficiency and valuation fairness also give an axiomatization of the Shapley value on the class of assignment games. 14 Although not explicitly written, the characterization of the Myerson value by component efficiency and graph game fairness holds more general in the sense that component efficiency and graph game fairness characterize the Myerson value on any restricted class of graph games GN × C such that C ⊆ LN is comprehensive, that is, for any L ∈ C and L′ ⊆ L it holds that L′ ∈ C . For example, the class LB,S of all bipartite graphs between the sets of buyers B and sellers S satisfies this property.
114
R. van den Brink, M. Pintér / Journal of Mathematical Economics 60 (2015) 110–114
Suppose that f : AB,S → RN satisfies submarket efficiency and valuation fairness. If k(A) = 0, then all singleton player sets form a submarket, and thus f (A) is determined (uniquely) by submarket efficiency. Proceeding by induction, assume that f (A′ ) is (uniquely) determined whenever 0 ≤ k(A′ ) < k(A). Take any submarket C = B′ ∪ S ′ , B′ ⊆ B, S ′ ⊆ S. If |C | = 1, then submarket efficiency determines fi (A) for i ∈ C . Otherwise, if |C | > 1, then C is connected in Lm A . Then for all (i, j) ∈ (C ∩ B) × (C ∩ S ) such that ai,j > 0, let Ai,j ∈ AB,S denote the matrix where for all (k, l) ∈ B × S: i,j ak,l
=
0, ak,l
if (k, l) = (i, j) otherwise.
From valuation fairness it follows that for all (i, j) ∈ (C ∩ B) ×
(C ∩ S ) such that ai,j > 0,
fi (A) − fi (Ai,j ) = fj (A) − fj (Ai,j ).
(2)
Since C is a component, there is a tree T ⊆ (C ) connecting all nodes in C . Since the values fi (Ai,j ) and fj (Ai,j ) are determined by the induction hypothesis, taking the equations (2) for the buyer–seller pairs (i, j) such that (i, j) ∈ T , yields |C | − 1 linear independent equations in the |C | unknown payoffs fi (A), i ∈ C . Together with the equation i∈C fi (A) = vA (C ) which follows from submarket efficiency, the |C | unknown payoffs fi (A), i ∈ C , are uniquely determined. Since this can be done for all submarkets, the payoffs f (A) are uniquely determined. Lm A
The Shapley value satisfies an even stronger valuation fairness property which states that changing the valuation of one particular buyer for the good offered by a particular seller in any way, changes the payoffs of this buyer and seller by the same amount. This follows from fairness15 of the Shapley value and the fact that this buyer and seller are symmetric players in the difference game (vA − vAi,j ). However, for the characterization the weaker version looking at valuation zero is sufficient. The Shapley value for assignment situations also satisfies a monotonicity property similar to the stability property for graph games.16 This property states that the payoffs of buyer i and seller j do not decrease if we only increase the valuation of buyer i for the good offered by seller j from zero to a positive valuation. Definition 3.9. Solution f on AB,S satisfies valuation monotonicity, if for all assignment situations A, A ∈ AB,S such that for some i ∈ B, j ∈ S it holds that ai,j = 0, and ag ,h = ag ,h for all (g , h) ∈ ((B \ {i}) × S ) ∪ (B × (S \ {j})), we have fi (A) ≥ fi (A) and fj (A) ≥ fj (A). It turns out that applying any graph game solution satisfying stability on the class of assignment games yields a solution for assignment situations that satisfies valuation monotonicity. Proposition 3.10. Consider player set N and B, S ⊂ N such that {B, S } is a partition of N. Let ϕ : GN × LN → RN be a graph game
15 Solution φ on C ⊆ GN satisfies fairness, if φ (v +w)−φ (v) = φ (v +w)−φ (v) i i j j for all v, w ∈ C and i, j ∈ N such that i, j are symmetric in w and v + w ∈ C . 16 Graph game solution ϕ : GN × LN → RN satisfies stability, if ϕ (v, L) ≥ ϕ (L \ i
{{i, j}}) and ϕj (v, L) ≥ ϕj (L \ {{i, j}}) whenever game v is superadditive.
i
solution, and let f : AB,S → RN+ be a solution for assignment situations on N given by f (A) = ϕ(vA , Lm A ). If ϕ satisfies stability, then f satisfies valuation monotonicity. This follows immediately from stability of ϕ and the fact that m Lm A = L \ {{i, j}}. A
Corollary 3.11. The Shapley value for assignment situations satisfies valuation monotonicity. As with valuation fairness, the Shapley value for assignment situations also satisfies a stronger valuation monotonicity property stating that the payoffs of buyer i and seller j do not decrease if we only increase the valuation of buyer i for the good offered by seller j, i.e. for all assignment situations A, A ∈ AB,S such that for some i ∈ B, j ∈ S it holds that ai,j ≥ ai,j , and ag ,h = ag ,h for all (g , h) ∈ ((B \ {i}) × S ) ∪ (B × (S \ {j})), we have fiSh (A) ≥ fiSh (A) and fjSh (A) ≥ fjSh (A). References Böhm-Bawerk, E. von, 1923. Positive Theory of Capital (translated by W. Smart), G.E. Steckert, New York, (original publication 1891). Brink, R. van den, 2001. An axiomatization of the Shapley value using a fairness property. Internat. J. Game Theory 30, 309–319. Chun, Y., 1991. On the symmetric and weighted shapley values. Internat. J. Game Theory 20, 183–190. Dubey, P., 1975. On the uniqueness of the Shapley value. Internat. J. Game Theory 4, 131–139. Gillies, D.B., 1959. Solutions to General Non-zero-sum Games. Princeton University Press. Hart, S., Mas-Colell, A., 1989. Potential, value, and consistency. Econometrica 57, 589–614. Hoffmann, M., Sudhölter, P., 2007. The Shapley value of exact assignment games. Internat. J. Game Theory 35, 557–568. Ligett, T.M., Lippman, S.A., Rumelt, R.P., 2009. The asymptotic Shapley value for a simple market game. Econom. Theory 40, 333–338. Llerena, F., Núñez, M., 2011. A geometric characterization of the nucleolus of the assignment game. Econ. Bull. 31, 3275–3285. Llerena, F., Nunez, M., Rafels, C., 2015. An axiomatization of the nucleolus of assignment markets. Internat. J. Game Theory 44, 1–15. Moulin, H., 1988. Axioms of Cooperative Decision Making. Cambridge University Press. Myerson, R.B., 1977. Graphs and cooperation in games. Math. Oper. Res. 2, 225–229. Peleg, B., Sudhölter, P., 2003. Introduction to the Theory of Cooperative Games. Kluwer Academic Publishers, Boston, Dordrecht, London. Pintér, M., 2012. Young’s axiomatization of the Shapley value — A new proof. arXiv:0805.2797v3. Sasaki, H., 1995. Consistency and monotonicity in assignment problems. Internat. J. Game Theory 24, 373–397. Schmeidler, D., 1967. The nucleolus of a characteristic function game. SIAM J. Appl. Math. 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 Volume II. In: Annals of Mathematical Studies, vol. 28. pp. 307–317. Shapley, L.S., Shubik, M., 1972. The assignment game I: the core. Internat. J. Game Theory 1, 111–130. Solymosi, T., Raghavan, T.E.S., 1994. An algorithm for finding the nucleolus of assignment games. Internat. J. Game Theory 23, 119–143. Toda, M., 2003. Consistency and its converse in assignment games. Int. J. Math. Game Theory Algebra 13, 1–14. Toda, M., 2005. Axiomatization of the core of assignment games. Games Econom. Behav. 53, 248–261. Young, H.P., 1985. Monotonic solutions of cooperative games. Internat. J. Game Theory 14, 65–72.