The core allocations presented so far may provide a solution for a specic allo-cation problem (assuming the core is non-empty), yet it is not always easy to choose a solution, should there be more than one. In certain cases the nucleolus is a good choice, but more often than not, it is dicult to determine. We would like to present another allocation rule, Shapley's famous solution concept, the Shapley value (Shapley, 1953).
In this section we dene the Shapley value and examine its properties. Shapley studied the value gained by a player due to joining a coalition. In other words, what metric denes the value of the player's role in the game. We introduce the notions solution and value, and present some axioms that uniquely dene the Shapley value on the sub-class in question.
Let GN denote the set of TU games with the set of players N. Let X∗(N, v) denote the set of feasible payo vectors, i.e.X∗(N, v) =
x∈RN| x(N)≤v(N) . Using the above notations, consider the following denition.
Denition 3.18 A solution on the set of games GN is a set-valued mapping σ, that assigns to each game v ∈ GN a subset σ(v) of X∗(N, v). A single-valued solution (henceforth, value) is dened as a function ψ : GN → RN that maps to all v ∈ GN the vector ψ(v) = (ψi(v))i∈N ∈RN, in other words, it gives the value of the players in any game v ∈ GN.
In the following we will be focusing primarily on single-valued solutions, i.e.
values.
Denition 3.19 In game (N, v) player i's individual marginal contribution to coalition S is v0i(S) =v(S∪ {i})−v(S). Player i's marginal contribution vector is vi0 = (vi0(S))S⊆N.
Denition 3.20 We say that value ψ on class of games A ⊆ GN is / satises
ecient (Pareto-optimal), if P
i∈Nψi(v) = v(N),
individually acceptable, if ψi(v)≥v({i}) for all i∈N,
equal treatment property, if ∀i, j ∈ N, ∀S ⊆ N \ {i, j} and v(S ∪ {i}) = v(S∪ {j}) implies ψi(v) =ψj(v),
null-player property, if for all i∈N such that vi0 = 0 implies that ψi(v) = 0, where vi0(S) = v(S∪ {i})−v(S) is the individual contribution of i to S,
dummy, if ψi(v) =v({i}), where i∈N is a dummy player in v, i.e. v(S∪ {i})−v(S) = v({i}) for all S ⊆N \ {i},
additive, if ψ(v+w) = ψ(v) +ψ(w) for all v, w∈ A, where (v+w)(S) = v(S) +w(S) for all S ⊆N,
homogeneous, if ψ(αv) =αψ(v) for all α ∈R, where (αv)(S) =αv(S) for all S ⊆N,
covariant, if ψ(αv+β) = αψ(v) +b for all α > 0 and b ∈ RN, where β is the additive game generated by vector b,
given that the above conditions hold for all v, w∈A for all properties.
Eciency implies that we get an allocation as result. Individual acceptability represents that each player is worth at least as much as a coalition consisting of only the single player. The equal treatment property describes that the payo of a player depends solely on the role played in the game, in the sense that players with identical roles receive identical payos. The null-player property ensures that a player whose contribution is0to all coalitions does not receive payo from
the grand coalition. The dummy property expresses that a player that neither increases nor decreases the value of any coalition by a value other than the player's own, shall be assigned the same value as this constant contribution. The dummy property also implies the null-player property.
Covariance ensures that a potential change in scale is properly reected in the valuation as well. The additive and homogeneous properties, on the other hand, are not straightforward to satisfy. In case both are satised, the linearity property holds, which is much stronger than covariance.
Let us now consider the denition of the solution of Shapley (1953).
Denition 3.21 In a game (N, v) the Shapley value of player i∈N is φi(v) = X
S⊆N\{i}
|S|!(|N\S| −1)!
|N|! vi0(S), and the Shapley value of the game is
φ(v) = (φi(v))i∈N ∈RN.
Claim 3.22 The Shapley value is / satises all the above dened properties, namely: ecient, null-player property, equal treatment property, dummy, additive, homogeneous, and covariance.
It is important to note that the Shapley value is not necessarily a core alloca-tion. Let us consider the previously discussed horse market game (Section 3.17).
Example 3.23 The horse market game has been dened as seen in Table 3.2.
S A B C AB AC BC ABC
v(S) 0 0 0 80 100 0 100
Table 3.2: The horse market game and its Shapley value
The Shapley value for the individual players can be calculated as follows φA= 0!2!
3! (0−0) + 1!1!
3! (80−0) + 1!1!
3! (100−0) + 0!2!
3! (100−0) = 631 3,
1 1 1 1 1
φC = 1 The core of the game was the set containing the following payos:
{(xA, xB = 0, xC = 100−xA)| 80≤xA≤100}.
However, since the core contains only payos where the value of player B is 0, clearly, in this case the Shapley value is not a core allocation.
The following claim describes one of the possible characterizations of the Shap-ley value.
Claim 3.24 (Shapley, 1953) A value ψ on a class of games GN is ecient, dummy, additive and satises the equal treatment property, if and only if ψ =φ, i.e. it is the Shapley value.
Further axiomatization possibilities are discussed in Section 5.2, for details see the papers of Pintér (2007, 2009) (both in Hungarian), and Pintér (2015).
With |N| = n let π : N → {1, . . . , n} an ordering of the players, and let ΠN be the set of all possible orderings of the players. Let us assume that a coalition is formed by its members joining successively according to the ordering π ∈ ΠN. Then π(i) denotes the position of player i in ordering π, and Piπ = {j ∈N | π(j)≤π(i)}denotes the players precedingi. The marginal contribution of player i in the game v where the ordering π is given is xπi(v) =v(Piπ ∪ {i})− v(Piπ). The payo vector xπ(v) = (xπi(v))i∈N ∈RN dened by these components is called the marginal contribution vector by ordering π.
It can be shown that in a generalized form (see for example Solymosi, 2007), the Shapley value can be given as:
φi(v) = 1
|N|!
X
π∈ΠN
xπi(v).
Consequently, the Shapley value is the mean of the marginal contribution vectors.
Remark 3.25 In an arbitrary convex game the core is never empty (Shapley, 1971), and the Shapley value always provides a core allocation. Namely, based on the Shapley-Ichiishi theorem (Shapley, 1971; Ichiishi, 1981), we know, that the marginal contribution vectors are extremal points of the core if and only if the game is convex, while the Shapley value is the mean of the marginal contribution vectors.