Chapter 4 Analysis and Discussion
4.4 Solution concepts
In this chapter, we discuss the commonly used solution concepts for multi-player cooperative games. When the necessary conditions for stability of a multi-player coalition are satisfied, the coalition would be stable if the allocation is fair to each player in the system. To find fairly allocating cost savings which all members in the coalition
adopt major solution concepts in the theory of cooperative games to assure fairness in allocation of extra cost savings defined as the difference between cost savings of coalitions and the sum of cost savings of individual members.
4.4.1 Essential game
Because our final goal is to discuss the relationship between the O2O information sharing model and cost saving effects, we must seek for a game in coalitional form realizing maximum efficiency, Pareto optimality, achieved when specific criterion is maximized and no allocation of resources could yield a higher value according to that criterion. In the theory of cooperative game, to create more efficiency; moreover, to make the grand coalition stable and the members have more incentive to form a grand coalition, we have to apply the concept of essential game, where ∑𝑃𝑃∈𝑇𝑇v𝑃𝑃 >
v(𝑇𝑇) 𝐼𝐼𝑜𝑜𝑟𝑟 𝑖𝑖𝑜𝑜𝑚𝑚𝑐𝑐𝑖𝑖𝑡𝑡𝑖𝑖𝑜𝑜𝑀𝑀 𝑇𝑇, to find an allocation scheme. In our model, we can find that, under the stable condition:
� v𝑃𝑃
𝑃𝑃∈𝑇𝑇
= v(P) + v(L) + v(M) + v(R) < v(PLMR).
Therefore, the allocation scheme also qualifies as an essential game.
4.4.2 Shapley value with Banzhaf power index
In the cooperative game theory, Shapley value is a solution concept assigning a unique distribution of a total surplus generated by the coalition among all players. That is to say, The Shapley value distribute the total gains and provides unique imputations in assumption that all members collaborate fairly by an arbitrator. The unique Shapley values ∅ = (∅1,…, ∅𝑛𝑛) are determined by ∅𝑃𝑃 = ∑ (|𝑇𝑇 − 1|)! (𝑀𝑀 − |𝑇𝑇|)𝑃𝑃∈𝑇𝑇 ! [v(𝑇𝑇) −
v(𝑇𝑇 − 𝑖𝑖)]/𝑀𝑀! , where T denotes an information sharing coalition, |𝑇𝑇| is the size of T, n is the total number of players and the sum extends over all coalitions T not containing player i, The formula can be interpreted as follows: imagine the coalition being formed one player at a time, with each player demanding their contribution [v(𝑇𝑇) − v(𝑇𝑇 − 𝑖𝑖)]as a fair compensation, and then for each player take the average of this contribution over the possible permutations in which the coalition can be formed.
However, for the situations associated with practical applications, the amounts of feasible input coalitions can often be reduced. In some cases, subtraction of a member from a coalition may also result an infeasible coalition. In our paper, since the online channel is indispensable in the practice of O2O business models, removing platform that dominates information sharing of coalitions in supply chain would then be infeasible. The coalition among other players, exclusive of platform, though has the power to make decisions, betray the definition of O2O, coordination of virtual and physical channel. Therefore, we are going to block these infeasible coalitions in O2O business model from our coalition sets.
To truly combine the real world with theorem, we adopt the concept of Shapley value, yet with Banzhaf power index. Our paper is not the first to disallow certain coalitions in values or power indices. Aumann and Dr`eze (1975) assume that property rights may make it impossible to form every coalition. Though the application of such restrictions to power indices are more recent, to obtain an index a further normalization is required. The Banzhaf measure (Penrose 1946; Banzhaf 1965), originally designed for changing an outcome of a vote where voting rights are not necessarily equally divided among the voters, is the probability that a party is critical for a coalition, that its desertion can turn winning coalitions into losing ones. That is, in real world, some
strategic behaviors could influence the formation of some coalitions; therefore, through the concept we can block some infeasible coalitions in the business models.
In our paper, we are going to adopt the concept of the Banzhaf measure as power index to weight winning coalitions, defined by enough quota to win. To properly distribute allocation of cost savings among members in different feasible coalitions, we follow the procedures below:
Step 1: According to the definition of O2O business models, platform would always play the role of one of critical players. We try to block some infeasible coalitions where the platform does not involve in. Then there are seven feasible coalitions:
{(PL)MR}, {(PM)LR}, {(PR)ML}, {(PML)R}, {(PLR)M}, {(PMR)L}, {(PMLR)}.
Step 2: After blocking some infeasible coalitions, we need to determine quota, the minimum number to become winning coalitions. We assume min{v(PM), v(PL), v(PR)}
the minimum allocation of cost savings as quota, which stands for entry barrier of O2O model. That is, with the involvement of platform in a supply chain, we can at least gain these cost savings. If the characteristic function is larger than min{v(PM), v(PL), v(PR)}, we regard it as winning solution. Then we can find that all the feasible coalitions are winning coalitions:
v(PL) ≥ min {v(PM),v(PL),v(PR)}, v(PM) ≥ min {v(PM),v(PL),v(PR)},
v(PR) ≥min {v(PM),v(PL),v(PR)},
v(PML) =�𝑃𝑃𝑃𝑃1−𝑃𝑃𝑝𝑝5� + (𝑀𝑀𝑃𝑃1−𝑀𝑀𝑃𝑃5) + (𝐿𝐿𝑃𝑃1−𝐿𝐿𝑃𝑃5) ≥ min {v(PM),v(PL),v(PR)}, v(PLR)= �𝑃𝑃𝑝𝑝1−𝑃𝑃𝑝𝑝7� + (𝐿𝐿𝑃𝑃1−𝐿𝐿𝑃𝑃7) + (𝑅𝑅𝑃𝑃1−𝑅𝑅𝑃𝑃7) ≥ min {v(PM),v(PL),v(PR)},
v(PMR) =(𝑃𝑃𝑃𝑃1−𝑃𝑃𝑃𝑃6) + (𝑀𝑀𝑃𝑃1−𝑀𝑀𝑃𝑃6) + (𝑅𝑅𝑃𝑃1−𝑅𝑅𝑃𝑃6) ≥ min {v(PM),v(PL),v(PR)}, v(PLMR)= �𝑃𝑃𝑝𝑝1−𝑃𝑃𝑝𝑝8� + (𝐿𝐿𝑃𝑃1−𝐿𝐿𝑃𝑃8) + (𝑅𝑅𝑃𝑃1−𝑅𝑅𝑃𝑃8) + (𝑀𝑀𝑃𝑃1−𝑀𝑀𝑃𝑃8) ≥ min
{v(PM),v(PL),v(PR)}.
Table 2. The feasible, infeasible coalitions and winning coalitions of O2O model Infeasible coalitions Feasible coalitions Winning coalitions
{(ML)PR} {(PM)LR} {(PM)LR}
{(MR)PL} {(PL)MR} {(PL)MR}
{(LR)PM} {(PR)ML} {(PR)ML}
{(MLR)P} {(PML)R} {(PML)R}
{P, M, L, R} {(PMR)L} {(PMR)L}
{(PLR)M} {(PLR)M}
{(PMLR)} {(PMLR)}
Since some of infeasible coalitions in O2O business model would be considered in the calculation of marginal contribution of the platform in the next step, we also list these coalitions in the table.
Step 3: We now can start to identify the critical players in whole winning coalitions. In each of the winning coalitions, there would be critical members, which provide the required allocation of cost savings for the coalition, and unnecessary members. Now we can find out critical players (underlined) below. The set winning coalitions with critical players underlined is
{{(PL)MR}, {(PM)LR}, {(PR)LM}, {(PML)R}, {(PLR)M}, {(PMR)L}, {(PLMR)}}
Obviously, the coalition is able to provide the required production, even when one of these unnecessary members goes out of the winning coalition. However, when one necessary member leaves, the winning coalition becomes insufficient. Since the Banzhaf index is derived by simply counting, we can find that there are 10 total swing players, the coalitions in which participate would win, or would lose, and the power is divided as:
P = 7/10, L = 1/10, M = 1/10, R = 1/10.
The player P is necessary for whole seven winning coalitions, L is necessary for one winning coalitions, M also for one winning coalitions, R for one winning coalitions.
Therefore, P is necessary in 0.7 of the total cases (10 = 7+1+1+1, so 7/10= 0.7), L in 0.1, M in 0.1, and R in 0.1. Obviously, platform dominates the weight of distribution of cost allocation. As the main source of cost allocation, P is definitely the critical player of the game, or it would be meaningless to construct a platform as well as adopt O2O model. The importance of platform also corresponds to that, in our O2O model, platform is the coordinator of information flow, and the cost of P would definitely decrease by a wider margin than other players’ cost due to its larger base of fixed cost.
After calculating the Banzahf power index, next, we will compute one of the most important part of Shapley value, marginal contributions of individual players (MC) to coalitional scheme. The following table displays the marginal contributions of players:
Table 3. The marginal cost allocations of different players
Player MC
P
[v(P) − v(∅)] + [v(PL) − v(L)] + [v(PM) − v(M)] + [v(PR) − v(R)]
+ [v(PML) − v(ML)] + [v(PLR) − v(LR)]
+ [v(PMR) − v(MR)] + [v(PMLR) − v(MLR)]
L
[v(L) − v(∅)] + [v(PL) − v(P)] + [v(PML) − v(PM)] + [v(PLR)
− v(PR)] + [v(PMLR) − v(PMR)]
M
[v(M) − v(∅)] + [v(PM) − v(P)] + [v(PML) − v(PL)] + [v(PMR)
− v(PR)] + [v(PMLR) − v(PLR)]
R
[v(R) − v(∅)] + [v(PR) − v(P)] + [v(PLR) − v(PL)] + [v(PMR)
− v(PM)] + [v(PMLR) − v(PML)]
Now, we can use the results from Banzhaf measure to calculate the allocated cost saving to the supply chain member i: P, L, M, R completely.
∅𝑝𝑝 = 7/10{[v(P) − v(∅)] + [v(PL) − v(L)] + [v(PM) − v(M)] + [v(PR) − v(R)] + [v(PML) − v(ML)] + [v(PLR) − v(LR)] + [v(PMR) − v(MR)] + [v(PMLR) −
v(MLR)]}
∅𝑙𝑙 = 1/10{[v(L) − v(∅)] + [v(PL) − v(P)] + [v(PML) − v(PM)]
+ [v(PLR) − v(PR)] + [v(PMLR) − v(PMR)] }
∅𝑚𝑚= 1/10{[v(M) − v(∅)] + [v(PM) − v(P)] + [v(PML) − v(PL)]
+ [v(PMR) − v(PR)] + [v(PMLR) − v(PLR)] }
∅𝑟𝑟 = 1/10{[v(R) − v(∅)] + [v(PR) − v(P)] + [v(PLR) − v(PL)]
+ [v(PMR) − v(PM)] + [v(PMLR) − v(PML)]}
4.4.3 Core
From the Shapley value with Banzahf power index above, we now can continually discuss commonly used solution concepts in cooperative game theory to analyze and find fair unique allocation scheme for our cooperative game. We use the concept of core to assure the stability of coalitions. Before that, we first analyze the imputations, defined as an acceptable distributions of the payoff of the grand coalition. the imputations distributions must satisfy two properties: efficiency and are individually rational. To make the grand coalition stable, we define x𝑃𝑃 as the allocated cost savings to the supply chain member i = P, L, M, & R. To meet the condition of imputations, the allocation of cost savings(𝑚𝑚𝑝𝑝, 𝑚𝑚𝑙𝑙, 𝑚𝑚𝑚𝑚, 𝑚𝑚𝑟𝑟) must be (1) individual rational:
𝑚𝑚𝑝𝑝 > v(P), 𝑚𝑚𝑀𝑀 > v(M), x𝐿𝐿 > v(L), x𝑅𝑅 > v(R)
Obviously, in our paper, we satisfy the condition due to that the characteristics value of P, L, M, & R all equals zero, and is smaller than (𝑚𝑚𝑝𝑝, 𝑚𝑚𝑙𝑙, 𝑚𝑚𝑚𝑚, 𝑚𝑚𝑟𝑟).
The other property (2) collective rationality, i. e. , 𝑚𝑚𝑝𝑝+ 𝑚𝑚𝑀𝑀+ 𝑚𝑚𝐿𝐿+ 𝑚𝑚𝑅𝑅 = v(PLMR), is not satisfied in our model with the unique allocation scheme suggested by Shapley value with Banzhaf power index method. However, we use linear programming (LP) method to get the constrained solution, which makes the grand coalition stable, and the result will be presented in the next section.
After we get the imputations, then we can apply the concept of core to assure the stability of allocation scheme. In game theory, the core is the set of imputations under which no coalition has a value greater than the sum of member s' payoffs in grand coalition to block it. Therefore, no coalition has incentive to leave the grand coalition and receive a larger payoff. The core of multi-player cooperative game is defined as the set of imputations (xp; 𝑚𝑚𝐿𝐿; 𝑚𝑚𝑚𝑚 ; 𝑚𝑚𝑟𝑟) such that for all coalitions, we have
∑ 𝑚𝑚𝑃𝑃∈𝑇𝑇 𝑃𝑃 ≥ v(𝑇𝑇) (Shapley 1967). In our model, we can easily find that our allocation scheme is suggested by the core:
x𝑝𝑝+x𝑀𝑀 + x𝐿𝐿+ x𝑅𝑅 ≥ v(PLMR) x𝑝𝑝+x𝑀𝑀 + x𝐿𝐿 ≥ v(PLM) x𝑝𝑝+x𝑀𝑀+ x𝑅𝑅 ≥ v(PMR) x𝑝𝑝+ x𝐿𝐿 + x𝑅𝑅 ≥ v(PLR)
x𝑝𝑝+x𝑀𝑀 ≥ v(PM) x𝑝𝑝+ x𝐿𝐿 ≥ v(PL) x𝑝𝑝+ x𝑅𝑅 ≥ v(PR)
Even if the core exists, we face the problem of which allocation scheme would be best to be divided cost savings among whole stakeholders.
4.4.4 Nucleolus
Another interesting value function for multi-person cooperative games may be found in the nucleolus, a concept introduced by Schmeidler (1969). Instead of applying a general method of fairness to the set of all characteristic functions, we try to find an imputation x = (𝑚𝑚1,..., 𝑚𝑚𝑛𝑛) that minimizes worst inequity, the maximum dissatisfaction among members in the information sharing coalition. The nucleolus is defined as a measure of the inequity of an imputation x for a coalition T, excess, e(𝑚𝑚, 𝑇𝑇) = v(𝑇𝑇) −
∑ 𝑚𝑚𝑃𝑃∈𝑇𝑇 𝑃𝑃. Since we have discussed the core above: ∑ 𝑚𝑚𝑃𝑃∈𝑇𝑇 𝑃𝑃 ≥ v(𝑇𝑇), we immediately have that an imputation x is in the core if and only if all its excesses are negative or zero. Then we can find the nucleolus by looking first at the largest excess of those coalitions. Then we try to adjust x, to make the largest excess smaller. When the largest excess has been made as small as possible, we concentrate on the next largest excess, and adjust x to make it as small as possible, and so on. In our model, we use LP to solve the nucleolus solution.