Shapley value as a measurer of shareholders decision power
Available online at www.worldscientificnews.com
WSN 90 (2017) 231-242
EISSN 2392-2192
Shapley value as a measurer of shareholders
decision power
Dariusz Karaś
Department of Management, The Cardinal Wyszynski University in Warsaw, Poland
E-mail address:
ABSTRACT
Making decisions in joint-stock company on stockholders meeting is an example of cooperative
game. Cooperative game theory focuses on the coalition players may form. Voting at the general
meeting of shareholders is a special kind of cooperative game. We assume each coalition may attain
some payoffs, and then we try to predict which coalitions will form. To determine the solution and
measure the ability of shareholders to create victorious coalitions we can use Shapley value. Among
the shareholders it assigns a unique distribution of a total surplus generated by the coalition of all
players.
Keywords: Game theory, Shapley value, joint-stock company, decision power
1. INTRODUCTION
Von Neumann and Morgenstern introduced the cooperative game in the form of a cooperative characterization of the coalition [Copeland, 1945]. Consider the interaction between
a potential seller and two potential buyers of some object that the seller who is the current
owner values at ten euro, the first buyer values at twenty euro, and the second buyer values at
thirty euro. If the players can freely transfer money among themselves, and if they are risk
neutral, this situation can be modeled as the game with players N = {1,2,3} and v given by
v(1) = 10, v(2) = v(3) = v(23) = 0, v(12) = 20, v(13) = v(123) = 30. This situation shows that
only coalitions containing the seller (player 1) and at least one buyer can create transaction
( Received 21 November 2017; Accepted 04 December 2017; Date of Publication 05 December 2017 )
World Scientific News 90 (2017) 231-242
that gives profit. A coalition that contains player 1 is worth the maximum that the object in
question is worth to any member of the coalition. The tools of cooperative game theory
applied to this model reflect some of the important features of such an interaction. For
example, the core of the game corresponds to the set of outcomes at which the seller sells to
the buyer with the higher reservation price, at some price between twenty and thirty euro, and
no other transfers are made. Von Neumann and Morgenstern proposed solution for such a
game, which today is called a stable set or a von Neumann-Morgenstern solution [Roth,
1988].
Shapley proposed to summarize the complex possibilities facing each player in a game
in characteristic function form by a single number representing the value of playing the game
[Shapley, 1953]. Thus the value of a game with a set N = {1, . . . , n} of players would be a
vector of n numbers representing the value of playing the game in each of its n positions.
Formally defined, we say that for every set of players N = {1,2, ..., n}, we assign a certain
value of v(K) to the union, which expresses the payoff that members of the coalition can work
together. This game has a certain property, which we call superadditivity. It means that each
of the two players can get more acting together than they would get together, but acting
separately. Let us assume that K and L are disjoint coalitions of the set N. The characteristic
function v will have the property of superadditivity if it fulfilled the following condition:
( K L) v( K ) v( L), if K L .
(1)
and we have:
v( K L) v( K ) v( L),
(2)
then it is not worthwhile to create coalitions.
Superadditivity is justified when coalitions can always work without interfering with
one another. The value of two coalitions will be no less than the sum of their individual
values. Superadditivity implies that the grand coalition has the highest payoff.
Consider a situation in which agents need to get connected to the public good in order to
enjoy its benefit. The example from [Shoham, Leyton-Brown, 2008] is the problem of
multicast cost sharing and based on characteristic function with the property of
superadditivity. A group of customers must be connected to a critical service provided by
some central facility, such as a power plant or an emergency switchboard. In order to be
served, a customer must either be directly connected to the facility or be connected to some
other connected customer. Let us model the customers and the facility as nodes on a graph,
and the possible connections as edges with associated costs. This situation can be modeled as
a coalitional game (N, v). N is the set of customers, and v(S) is the cost of connecting all
customers in S directly to the facility minus the cost of the minimum spanning tree that spans
both the customers in S and the facility.
Another example from [Shoham, Leyton-Brown, 2008] concerns sharing the cost of a
public good, along the lines of the road-building referendum. A number of cities need airport
capacity. If a new regional airport is built the cities will have to share its cost, which will
depend on the largest aircraft that the runway can accommodate. Otherwise each city will
have to build its own airport. This situation can be modeled as a coalitional game (N,v), where
N is the set of cities, and v(S) is the sum of the costs of building runways for each city in S
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World Scientific News 90 (2017) 231-242
minus the cost of the largest runway required by any city in S. This airport game is also an
example of superadditive game.
In the n-personal cooperative game, players have to divide the total utility of v(N). This
value can be divided in any way, but it is obvious that no rational player will agree to a
division in which he obtains less than he would have obtained by acting alone. The division
called also the imputation in the n-personal co-operative game is the vector x = (x1, ..., xn)
that meets the conditions:
xi v(N ),
(3)
xi v({i}) for every i N .
(4)
iN
Condition (4) means optimality in the sense of Pareto.
The essence of solving a given n-personal game is to indicate such a division or to
define a set of divisions that will satisfy all the players. Considering the concepts of npersonal game solutions, Shapley in [Shapley, 1953] formulated three axioms that reflect the
idea of a fair division and proved that in every n-personal game with the property of a
superadditivity characteristic function, only one payoff system v(S) can be clearly identified
and there is exactly one imputation defined for all axiomatic games:
i[v] v(S ).
(5)
S
For any permutation we have:
i[v] i[v].
(6)
For any game we have:
i[u v] i[u] i[v].
(7)
The Shapley value is calculated as follows:
(t 1)!(n t )!
[v( S ) v( S {i})] .
iT
n!
i[v] T N
(8)
The summation runs after all coalitions to which system S belongs. When we have a
simple game, the value of Shapley is simplified because the equation gives always value 0 or
1:
(t 1)!(n t )!
.
iT
n!
i[v] T N
(9)
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