Bicooperative Games: Applications in Management and a Simple Solution∗ Surajit Borkotokey†
Rajnish Kumar‡
Sudipta Sarangi§
Abstract In the management of organizations both public and private, many decisions are made by committees. The most common decision-making procedure in committees is by voting. In this paper we consider a generalization of voting games called bicooperative games where the strength of coalitions is taken into account. We show the existence and uniqueness of the potential function for such games. An example, where networks among players are used to illustrate the strength of coalitions, is also developed in the paper.
Keywords: Bicooperative games, Potential function, Shapley value JEL classification: C71, D61, D71 ∗ Surajit Borkotokey acknowledges the Indo-US Science and Technology Forum for providing him a fellowship to visit
Louisiana State University, during 2011-12 and also expresses his gratitude for the hospitality provided by the Department of Economics, Louisiana State University during his stay there. † Department
of Mathematics, Dibrugarh University, India-786004, email:
[email protected] of Computing and Information Sciences, FIU, Miami, USA-33199, email:
[email protected] § School of Management, KIIT University, Bhubaneswar, India and Department of Economics, Louisiana State ‡ School
University, USA-70803, email:
[email protected]
1
1
Introduction:
In the management of organizations, both public and private, many decisions are made by committees. The most common decision making procedure in committees is by voting. This situation has been well studied in terms of a voting game (equivalently a simple game) introduced by Shapley and Shubik [15, 14] and later by a voting game with abstention introduced by Felsenthal and Machover [5] under the cooperative game theory framework. A cooperative game is fully characterized by the player set and a real valued function defined over its subsets (coalitions) and called the characteristic function. In a voting game (with or without abstention), a decision is accepted when the coalition in its favor is a part of the winning coalition. The worth (assigned by the characteristic function) of such a coalition is taken as 1 and that of any other coalition is taken as 0. However, such a characteristic function being binary, does not distinguish between the strengths of the coalitions in terms of their support or oppositions to a decision. In this paper, we study a generalized form of voting games with abstention called Bicooperative games. In this games the worth of a coalition given by the characteristic function ranges over the set of real numbers (instead of {0, 1}) and thus represents the strength of a coalition over its opponents. We show that the potential function can be used to find the Shapley value for such a game. To explain the importance this notion plays in the management of organizations, consider for example, the decision to introduce a new Pensioners’ benefit scheme in a company’s board meeting. Suppose the scheme has different policies for employees joining the company before and after a specific date.1 The stakeholders are the beneficiaries of the scheme. It is natural to imagine that a subset of beneficiaries would be in favor of the scheme while another subset might oppose it depending on their expectations from the policy. A third group of stakeholders being unaffected by the move may remain indifferent to the issue. This situation is represented by a voting game with abstention. However, if we consider the strengths or power of a supporting coalition when a particular group of people oppose, which may be the case when voters have different levels of influences over other voters, or occupy key positions in some sense to attract the undecided voters then it can be modeled with the notion of a bicooperative game more effectively.2 Bicooperative games were introduced by Bilbao et al. [1] where players have three options: support a decision, oppose it, or remain indifferent. Various aspects of bicooperative games have been 1A
similar Pension rule has been formulated and is in action for the employees of Govt. of India where the rules
differ for employees joining before and after the year 2006. 2 In Section 4, we illustrate this idea through an example which assumes that a voter’s influence over her peers to accept a decision can be a function of her degree of networking (number of direct connections she has). We then illustrate our solution for this example.
2
studied in the literature in [12], [2], [17], [4], [11], and [7]. In the game theoretic literature, though credit for the introduction of several levels of cooperations goes to the work on multi-choice games [9]. The major domain of application of games with several levels of participations has been the study of cost allocation problems [16] which have significant applications in many areas of Management and Economics. The introduction of potential functions by Hart and Mas-Colell ([13], [8]) for cooperative and non-cooperative games has been useful in many technical aspects. The potential function for cooperative games ([8]) has been useful in providing a characterization of the Shapley value, a key solution concept in the theory of cooperative games. A remarkable aspect of the potential function approach in cooperative games is that the potential function turns out to be unique. The purpose of our paper is to show the existence and uniqueness of potential functions for a special class of bicooperative games. We use the value formulation proposed in ([3]). This class is important in voting like situations where number of players in support and against of an issue is crucial. A significant step in accomplishing this task is to identify upper and lower restricted bicooperative games using the chain structure of these games. We apply our findings to solve a decision making problem where the underlying network structure between agents in a coalition matters. We obtain the Shapley value of this game directly and then using the potential function approach. It is seen that all the advantages of using the potential function to compute the Shapley value under the cooperative environment shown in [13] and [8] extend to bicooperative games. The rest of the paper is organized as follows. Section 2 provides the necessary mathematical preliminaries, includes a brief description of bicooperative games and the related properties. Section 3 develops our notion of a potential function and presents the main result of the paper. Section 4 illustrates the findings of the paper by means of an example and is followed by concluding remarks in Section 5.
2
Model Formulation
Let N = {1, 2, 3, ..., n} be the set of players. We will use upper case letters to denote the subsets of N and the corresponding lower case letters for their cardinality. We consider pairs (A, B), with A, B ⊆ N and A ∩ B = ∅. Thus, (A, B, N \(AU B)) yields a partition of the set N of all players in three groups. Players in A are in favor of an issue called the defenders; players in B do not agree to the issue and will take action against any change. We call the members of B, the detractors. Finally, the members of N \(A ∪ B) are not affected by the profits derived from the proposal, and therefore neither object nor support the proposal: the absentees. Thus, in our model we consider
3
the set of all ordered pairs of disjoint coalitions 3N = {(A, B) : A, B ⊆ N, A ∩ B = ∅}, and define a function b : 3N → R. For each (A, B) ∈ 3N , the worth b(A, B) represents the maximal gain (whenever b(A, B) > 0) or minimal loss (whenever b(A, B) < 0) that is obtained where A and B are as defined above, and the players in N \(A ∪ B) are not taking part because they are indifferent. Thus, b(∅, N ) indicates the cost (or expense) that is obtained when all players decide to follow with the initial situation and b(N, ∅) is the maximal gain that is obtained if all players want a change in the situation. As a consequence, the total profit of the bicooperative game (N, b) is given by b(N, ∅) − b(∅, N ). This setup of bicooperative games follows the formulation proposed in Bilbao ([1]). Formally, Definition 1 A bicooperative game is a pair (N, b) where N is a finite set, and b a function b : 3N → R with b(∅, ∅) = 0. The space of all bicooperative games is denoted by BG. Definition 2 A value Ψ is a function Ψ : BG → Rn assigning each bicooperative game (N, b) an n−dimensional vector where n represents the number of players in N . A number of values as possible solution concepts for a bicooperative game have been proposed so far (see for example, [11], [12], [3]). In the following, we define the Shapley value proposed by Bilbao et al.([3]) and restrict attention to it. Definition 3 The Shapley value for the bicooperative game (N, b) ∈ BG is a vector ΨSh (N, b) ∈ RN defined for each i ∈ N , by ΨSh i (N, b) =
X
[pa,b (b(A ∪ {i}, B) − b(A, B)) + pa,b (b(A, B) − b(A, B ∪ {i}))]
(A,B)∈3N \i
where, for all (A, B) ∈ 3N \i , pa,b =
(n + a − b)!(n + b − a − 1)! n−a−b 2 , (2n)!
pa,b =
(n + b − a)!(n + a − b − 1)! n−a−b 2 . (2n)!
and
In order to characterize the Shapley value for a bicooperative game, Bilbao et al.([3]) introduced a set of axioms similar to their counterparts in cooperative games. Prior to stating the characterization we need to introduce the following concepts and definitions. It has been shown by Grabisch and Labreuche ([6]) that the set (3N , v) is a partially ordered set (poset) where the symbol “v” implies the following: (A, B) v (C, D) ⇔ A ⊆ C, B ⊇ D. 4
Definition 4 A chain of 3N is an induced subposet of 3N in which any two elements are comparable. Definition 5 The identity game δ(A,B) : 3N → R is given by, 1 δ(A,B) (A, B) = 0
if (A, B) = (A, B)
(1)
otherwise
Definition 6 A player i ∈ N is a dummy for (N, b) ∈ BG if, for every (A, B) ∈ 3N \i , it holds, b(A ∪ {i}, B) − b(A, B)) = b({i}, ∅)
(2)
b(A, B) − b(A, B ∪ {i}) = −b(∅, {i}).
(3)
We now state the characterization of the Shapley value for bicooperative games. Theorem 1 (Theorem 7 in [3], page 112) Let Ψ be a value for members in BG. The value Ψ is the Shapley value if and only if Ψ satisfies the following axioms: 1. (Efficiency):
Pn
i=1
φi (b) = b(N, ∅) − b(∅, N ) for every (N, b) ∈ BG.
2. (Dummy) : If player i ∈ N is dummy for(N, b) ∈ BG, then φi (b) = b({i}, ∅) − b(∅, {i}). 3. (Anonymity): For all (N, b) ∈ BG and for any permutation π over N , it holds that φπi (πb) = φi (b) for all i ∈ N , where πb(πA, πB) = b(A, B) and πA = {πi : i ∈ A}. 4. (Linearity) : For α, β ∈ R and (N, b), (N, b0 ) ∈ BG such that (N, αb + βb0 ) ∈ BG, we have, Ψ(αb + βb0 ) = αΨ(b) + βΨ(b0 ). 5. (Structural Axiom): Given (N, b) ∈ BG, for every (A, B) ∈ 3N \{i} , j ∈ A and k ∈ B, it holds, Ψj (δ(A,B) ) c([(∅, N ), (A \ {j}, B)]) =− c([(∅, N ), (A, B ∪ {i})]) Ψi (δ(A,B∪{i}) )
(4)
Ψk (δ(A,B) ) c([(A, B \ {k}), (N, ∅)]) =− c([(A ∪ {i}, B), (N, ∅)]) Ψi (δ(A∪{i},B)
(5)
where the symbol c([(A, B), (A0 , B 0 ]) represents the length of the maximal chain from (A, B) to (A0 , B 0 ) for all (A, B), (A0 , B 0 ) ∈ 3N .
5
Except the Structural Axiom, all the other four axioms are standard in cooperative game theory and therefore need no particular explanation as such. Bilbao et al. [3] justify the Structural Axiom by arguing that the number of maximal chains in the sublattice [(∅, N ), (A \ j, B)] is not the same as the number of maximal chains in [(∅, N ), (A, B \ i)]. Hence, beginning with the coalition (∅, N ), the probability of formation of the coalition (A, B) with the incorporation of one player j to (A \ j, B) must be distinct from the probability of formation (A, B) with the desertion of one player i in (A, B ∪ i). Therefore the probability of formation of (N, ∅) beginning from (A, B \ k) when one player k leaves the coalition B must be distinct from the probability of formation of (N, ∅) when one player i forms the coalition (A ∪ i, B). Consequently, the values that a player must obtain in the identity games must be proportional to the number of maximal chains in the corresponding sublattices. It must be also considered that a value assigns a non-negative real number to one player i in the identity game δ(A,B) if this player belongs to A and a non-positive real number if the player i belongs to B. This idea leads to the Structural Axiom.
3
Result
Let BG0 be the class of bicooperative games (N, b) such that, b(S, T ) = f (|S|, |T |) ∀S, T ∈ 3N
(6)
for some bilinear function f : Rn → R. As mentioned in the Introduction, games in BG 0 are extensions of voting games with abstention and are widely applied to decision making in management problems. For (N, b) ∈ BG0 , let us define the upper and lower restricted bicooperative games b(N,∅) and b(∅,N ) in BG0 as, b(A, B) b(N,∅) (A, B) = 0 and
b(A, B) b(∅,N ) (A, B) = 0
if B = ∅
(7)
otherwise if A = ∅
(8)
otherwise
for all (A, B) ∈ 3N . Thus the game b(N,∅) (similarly b(∅,N ) ) assigns non-zero values only along all chains from (∅, ∅) to (N, ∅) (from (∅, N ) to (∅, ∅)). Given a bicooperative game (N, b), and a subset of players W ⊂ N , we write (W, b) for the subgame obtained by restricting b to 3W . Given a function P : BG0 → R, which associates a real number to every bicooperative game (N, b), the superior marginal contribution ∆+ i P (N, b) of a player i ∈ N is defined to be: ∆+ i P (N, b) = P (N, b(N,∅) ) − P (N \ {i}, b(N \{i},∅) ), ∀ i ∈ N 6
where the sub-game (N \ {i}, b(N \{i},∅) ) is obtained from (N, b(N,∅) ) by removing player i from the defender coalition to join the absentees. Similarly the inferior marginal contribution ∆− i P (N, b) of a player i ∈ N is defined as: ∆− i P (N, b) = P (N \ {i}, b(∅,N ) ) − P (N, b(∅,N \{i}) ), ∀ i ∈ N where the sub-game (N \ {i}, b(∅,N \{i}) ) is obtained from (N, b(∅,N ) ) by adding absentee i to the detractor coalition. Definition 7 A function P : BG0 → R with P (∅, b) = 0 is called a potential function if it satisfies the following conditions: n X i=1 n X
∆+ i P (N, b) = b(N, ∅)
(9)
∆− i P (N, b) = b(∅, N )
(10)
i=1
for all games (N, b) ∈ BG0 . The potential function for a bicooperative game is defined recursively by using the superior and inferior marginal contributions on the upper and lower restrictions of the game. As we will see below, the recursive nature of the definition uniquely determines the superior and inferior marginal contributions. Hence the potential function is unique on this restricted domain and uniquely determines the Shapley value as its marginal contribution vector. Theorem 2 For every bicooperative game (N, b), there exists a unique potential function P such − that the resulting payoff vector (∆+ i P (N, b) + ∆i P (N, b))i∈N coincides with the Shapley value of
the game. Moreover, the potential of any game (N, b) is uniquely determined by equations (9) and (10) applied to the game and its subgames only along the maximal chains from (∅, ∅) to (N, ∅) and from (∅, N ) to (∅, ∅). Proof. Equations (9) and (10) can be re-written as: " # X 1 b(N, ∅) + P (N \ {i}, b(N \{i},∅) ) P (N, b(N,∅) ) = |N | i∈N " # X 1 P (N, b(∅,N ) ) = −b(∅, N ) + P (N \ {i}, b(∅,N \{i}) ) |N |
(11)
(12)
i∈N
Starting with P (∅, b) = 0, equations (11) and (12) determine P (N, b(N,∅) ) and P (N, b(∅,N ) ). This proves the uniqueness of the potential function P which is uniquely determined by equation (9)
7
− and (10)(or equivalently equation (11) and (12)). To prove that ∆+ i P (N, b) + ∆i P (N, b) is the − Shapley value, we show that ∆+ i P (N, b) + ∆i P (N, b) satisfies all the Shapley axioms and since
Shapley value is the unique value satisfying all these axioms, equality follows at once.
Efficiency: It follows from equations (9) and (10).
Dummy Axiom: Let player i be dummy. We use induction on the player set N . We assume that the following hold for all games with less than |N | players, P (N, b(N,∅) ) − P (N \ {i}, b(N \{i},∅) ) = b({i}, ∅)
(13)
P (N \ {i}, b(∅,N \{i}) ) − P (N, b(∅,N ) ) = −b(∅, {i})
(14)
In particular, P (N \ {j}, b(N \{j},∅) ) − P (N \ {i, j}, b(N \{i,j},∅) ) = b({i}, ∅),
(15)
P (N \ {i, j}, b(∅,N \{i,j}) ) − P (N \ {j}, b(∅,N \{j}) ) = −b(∅, {i}), ∀j 6= i
(16)
We shall show that they hold for all games with exactly |N | players. In doing so, we will only consider equation (13) since a similar argument can be made for equation (14) because of symmetry. Using equation (11) for the players’ set N \ {i}, we get, X 1 b(N \ {i}, ∅) + P (N \ {i}, b(N \{i},∅) ) = |N | − 1
P (N \ {i, j}, b(N \{i,j},∅) )
(17)
j∈N \{i}
Subtracting (17) from (11) and using induction hypothesis, we get the required result.
Anonymity Axiom: Let π be a permutation on N , such that πb(πA, πB) = b(A, B), πA = {πi|i ∈ A}. We show that, + ∆+ πi P (πN, πb) = ∆i P (N, b)
(18)
− ∆− πi P (πN, πb) = ∆i P (N, b)
(19)
We show that equation (18) holds and equation (19) follows directly from symmetry. Let us apply induction on the set of players and assume that equation (18) holds for every set with less than |N | − 1 players. In particular, this would imply that, + ∆+ πi P (πN \ {πj}, πb) = ∆i P (N \ j, b) ∀j ∈ N : j 6= i ∈ N.
8
Now, using equation (11) we have, X 1 πb(πN, ∅) + P (πN \ {πj}, πb) P (πN, πb) = |N | πj∈πN X 1 P (πN \ {πi}, πb) = πb(πN \ {πi}, ∅) + P (πN \ {πj, πi}, πb) |N | − 1
(20)
(21)
πj∈πN \{πi}
The result follows by subtracting equation (21) from equation (20) and on simplifications after using the induction hypothesis.
Linearity: Inductive arguments on equations (10) and (11) lead to linearity.
Structural Axiom: We will show equation (4) only as equation (5) follows from symmetry. c([(∅,N ),(A\{j},B)]) c([(∅,N ),(A,B∪{i})]) Ψj (δ(A,B) ) Ψi (δ(A,B∪{i}) ) = −2 or
Since
= 2, for every(A, B) ∈ 3N \{i} and j ∈ A, it remains to show that, equivalently, Ψj (δ(A,B) ) = −2 × Ψi (δ(A,B∪{i}) ). Now, by hypothesis,
Ψj (δ(A,B) ) = P (N, (δ(A,B) )(N,∅) ) − P (N \ {j}, (δ(A,B) )(N \{j},∅) ) and Ψi (δ(A,B∪{i}) ) = P (N, (δ(A,B∪{i}) )(N,∅) ) − P (N \ {i}, (δ(A,B∪{i}) )(N \{i},∅) ) for all (A, B) ∈ 3N \{i} and j ∈ A. We also observe that the result holds trivially for |N | = 2. Let it be true for all player sets with |N | − 1 players. In particular for every k ∈ N \ {i, j}, P (N \ {k}, (δ(A,B) )(N \{k},∅) ) − P (N \ {k, j}, (δ(A,B) )(N \{k,j},∅) ) = −2 × P (N \ {k}, (δ(A,B∪{i}) )(N \{k},∅) ) − P (N \ {i, k}, (δ(A,B∪{i}) )(N \{k,i},∅) ) Equation (10) applied to δ(A,B) , for N and N \ {j} yields that, |N | × P (N, (δ(A,B) )(N,∅) ) − (|N | − 1) × P (N \ {j}, (δ(A,B) )(N \{j},∅) ) X = δ(A,B) (N, ∅) + P (N \ {k}, (δ(A,B) )(N \{k},∅) ) k∈N
−δ(A,B) (N \ {j}, ∅) −
X
P (N \ {j, k}, (δ(A,B) )(N \{j,k},∅) )
k∈N \{j}
= δ(A,B) (N, ∅) − δ(A,B) (N \ {j}, ∅) + P (N \ {j}, (δ(A,B) )(N \{j},∅) ) i X h + P (N \ {k}, (δ(A,B) )(N \{k},∅) ) − P (N \ {k, j}, (δ(A,B) )(N \{k,j}) , ∅) k∈N \{j}
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Using the induction hypothesis and its repeated application up to N \ {i} (i.e. using equation (9) repeatedly), we have, |N | × P (N, (δ(A,B) )(N,∅) ) − P (N \ {j}, (δ(A,B) )(N,∅) ) = (δ(A,B) )(N, ∅) − (δ(A,B) )(N \ {j}, ∅) i X h + (−2) × P (N \ {k}, (δ(A,B∪i) )(N \{k},∅) ) − P (N \ {i, k}, (δ(A,B∪{i}) )(N \{i,k},∅) ) k∈N \{i}
(22) Since, A ⊂ N (A 6= N ), B 6= ∅ as i 6∈ A and j ∈ B, we have from the definition of the identity game δ(A,B) , δ(A,B) (N, ∅) = δ(A,B∪{i}) (N, ∅) = 0
(23)
δ(A,B) (N \ {j}, ∅) = δ(A,B∪{i}) (N \ {i}, ∅) = 0
(24)
Similarly, equation (10) applied to δ(A,B∪{i}) for N and N \ {i} we get, |N |{P (N, (δ(A,B∪{i}) )(N,∅) ) − P (N \ {i}, (δ(A,B∪{i}) )(N \{i},∅) )} = δ(A,B∪{i}) (N, ∅) − δ(A,B∪{i}) (N \ {i}, ∅) X + [P (N \ {k}, (δ(A,B∪{i}) )(N \{k},∅) ) − P (N \ {k}, (δ(A,B∪{i}) )(N \{k},∅) )] k∈N \{i}
(25) Combining equation (22), (23), (24) and (25), we get the desired result. This completes the proof.
We will now specify the connection between our paper and the OR aspects of the literature. Note that, the computational complexity of the Shapley value for bicooperative games can be substantially reduced if we adopt the potential function approach. In addition to this, in the bicooperative setup, incorporation of all possible maximal chains between (∅, N ) and (N, ∅) for calculating the Shapley value is difficult especially when N is sufficiently large. However, the potential approach requires only a few of those maximal chains where players leave the defenders’ coalition (or joins the detractors’ coalition) only to join (leave) the absentees. As a result, the application of the potential approach is more appealing for addressing real life decision making problems.
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4
An Example
In this section, we model a decision problem along the lines described at the beginning of the paper. The Shapley value of a player in our model measures her individual power or strength to affect the decision. A natural choice for computing such a measure is to evaluate how she influences her peers which can depend on the way she is related to them. Typically the theory of coalitions does not consider such interactions among players inside a coalition. A player may have different relationships with different individuals within a coalition and this can be described by a network. By a network, we mean a finite set of elements called nodes corresponding to players and a finite set of pairs of nodes called links which corresponds to the bilateral relationships between players. Let N be the player set and g be a network through which the players from N are connected among themselves. The network g is thus a list of unordered pairs of players {i, j}, where {i, j} ∈ g indicates that i and j are linked in the network g. For simplicity, we write ij to represent the link {i, j}. The degree of a player in a network is the number of direct links she has in the network. Thus each player k ∈ N is endowed with a positive real degree dk , in the network g that signifies her power of communication with her peers.3 We assume that this power linearly measures her influence in building support for an issue. Define a bi-cooperative game bg : Q(N ) → R given by, bg (A, B) = d(A) − d(B) where for every K ⊆ N, d(K) =
P
i∈K
di . Call this the Net-degree bicooperative game or NBG
in short. The value bg (A, B) can be interpreted as the strength of the coalition A over B in the network g whenever players in A support a proposal, players in B oppose and players in N \ A ∪ B abstain from taking any decision. We compute the Shapley value for NBG directly and also illustrate the use of Potential functions for doing so. Prior to this, let us deduce the the following. From the definition of NBG, we have for every (A, B) ∈ 3N \i , bg (A ∪ i, B) − b(A, B) = di bg (A, B) − b(A, B ∪ i) = di 3 Here
we use the degree of a node in a network as the measure of player’s influence over her peers just as an
illustration of the applicable domains of our model. However, we can take any other measures instead of degree and obtain a corresponding expression (see [10] for instance).
11
Therefore, (n + a − b)!(n + b − a − 1)! (n + b − a)!(n + a − b − 1)! + 2n! 2n! (A,B)∈3N \i X (n + a − b − 1)!(n + b − a − 1)! =di × 2n−a−b (2n − 1)! (A,B)∈3N \i X ((n − 1) + (a − b))!((n − 1) − (a − b)! (26) =di × 2n−a−b (2n − 1)! N \i X
Ψi(b) =di ×
2n−a−b
(A,B)∈3
Similarly, the upper and lower restricted bicooperative games (bg )(N,∅) and (bg )(∅,N ) of the NBG take the form of, (bg )(N,∅) (A, B) = and
P
i∈A
di
0
P − i∈B di (bg )(∅,N ) (A, B) = 0
if B = ∅
(27)
otherwise if A = ∅
(28)
otherwise
for all (A, B) ∈ 3N . We take a particular example with N = {1, 2, 3} and g = {12, 13} and obtain the Shapley value directly and by using the Potential function approach respectively. Using equation 26, we get, ΨSh (N, bg ) = (4, 2, 2) Next, compute the same using potential approach in the following three steps: Step 1: From equation 11, we have, P (N, (bg )(N,∅) ) =P ({1, 2, 3}, (bg )({1,2,3},∅) ) o 1n bg ({1, 2, 3}, ∅) + P ({2, 3}, (bg )({2,3},∅) ) + P ({1, 2}, (bg )({1,2},∅) ) + P ({1, 3}, (bg )({1,3},∅) ) = 3 Step 2: Using the recurrence relation on n, we have, o 1n P ({2, 3}, (bg )({2,3},∅) ) = bg ({2, 3}, ∅) + P ({2}, (bg )(2,∅) ) + P ({3}, (bg )(3,∅) ) 2 o 1n P ({1, 2}, (bg )({1,2},∅) ) = bg ({1, 2}, ∅) + P ({1}, (bg )(1,∅) ) + P ({2}, (bg )(2,∅) ) 2 o 1n P ({1, 3}, (bg )({1,3},∅) ) = bg ({1, 3}, ∅) + P ({1}, (bg )(1,∅) ) + P ({3}, (bg )(3,∅) ) 2 Step 3: and finally, P ({1}, (bg )( 1, ∅)) = bg (1, ∅) = 2 P ({2}, (bg )( 2, ∅)) = bg (2, ∅) = 1 P ({3}, (bg )( 3, ∅)) = bg (3, ∅) = 1 Using these values in the expressions of Step 2 and Step 1, we get, ∆+ P (N, bg ) + ∆− P (N, bg ) = (4, 2, 2) = ΨSh (N, bg ) 12
5
Conclusion
This paper shows that bicooperative games have many applications in the world of management since they generalize voting games to include for the possibility of abstaining. We show that a existence of a unique potential function for such games and establish its relationship to the Shapley value. To illustrate this idea we develop an example where an individual’s influence depends on her network on individuals in favor and against the issue to be considered for voting. This approach could form the starting point for exploring how network structure and coalition structure affect each other in determining the outcome of a voting game.
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