_{1}

^{*}

The delivery of the natural gas obtained by drilling, fracking and sending the product to consumers is done usually in two phases: in the first phase, the gas is collected from all wells spread on a large area, and belonging to several companies, and is sent to a depot owned by the city; then, in the second phase, another company is taking the gas on a network of ducts belonging to the city, along the streets to the neighborhoods and the individual consumers. The first phase is managed by the gas producing companies on the ducts owned by each company, possibly also on some public ducts. In this paper, we discuss only this first phase, to show why the benefits of these companies depend on the cooperation of the producers, and further, how a fair allocation of the total gas obtained, to the drilling companies, is computed. Following the model of flow games, we generate a cooperative transferable utilities game, as shown in the first section, and in this game any efficient value gives an allocation of benefits to the owners of ducts in the total network. However, it may well happen that the chosen value is not coalitional rational, in the game, that is, it does not belong to the Core of the game. By using the results obtained in an earlier work of the author, sketched in the second section, we show in the last section how the same allocation may be associated to a new game, which has the corresponding value a coalitional rational value. An example of a three person flow game shows the game generation, as well as the procedure to be used for obtaining the new game in which the same value, a Shapley Value, will give a coalitional rational allocation.

The flow games have been introduced by E. Kalai and E. Zemel (1982), in [

Example 1. Consider the case of a city which has wells belonging to three companies, K_{1}, K_{2}, K_{3}, at the nodes A, B, C, of a graph, respectively. The companies are able to extract natural gas as follows: company K_{1}, 11 tones at A; company K_{2}, 13 tones at B; and company K_{3}, 9 tones at C. The set of nodes in the graph is

while the cooperation of all companies will give

The maximal amounts for coalitions of size two are obvious from the figure; the maximal amount provided by the grand coalition should be computed in the network by a max-flow-min-cut algorithm, for example the Ford- Fulkerson algorithm (see [

The capacities of the ducts represented by the arcs of the graph are shown in the following figure:

A graph with capacities associated to the above model.

The graph has the arcs for which there are positive capacities in the table, plus the entering and exiting auxiliary arcs, and one public arc discussed below. Now, note that the columns for S and T and the rows for G and T have been omitted, because they are empty, except a public duct, belonging to the city, [G, T], with a capacity of 25. The graph and the capacities are taken arbitrarily by the author.

As shown by the numbers that give the characteristic function of the game, the three companies should cooperate in order to get a maximum benefit. Now, the three-person cooperative TU game and any fair schedule of allocations, are obtained by using an efficient value of the game:

where we replaced the names of the companies by the indices and the brackets by parantheses, to keep the notation simple.

As the worth of singletons is zero, a simpler value like the Center of the Imputation Set, expressed by the formula:

gives

which is not fair. Therefore, we shall use the Shapley Value, given by the formula:

which has more properties due to its axiomatic definition (see [

Obviously, the Shapley Value does not belong to the Core, as

and all the other inequalities for coalitions of size two do not hold, that is the Shapley Value is not coalitional rational, even though it is efficient, as the sum of components makes 25. Of course, any pair of companies may break the grand coalition and make a two person coalition which may give more gain to each player, while the third player is left out of the deal, an unfair solution. This instability is due exactly to the fact that the solution is not coalitional rational, which could be a motivation for the present work.

To be able to give a fair allocation, we shall solve a connected problem that has been discussed in an earlier work of the author (see [

In [

It is well known that the set of TU games with the set of players N, forms a vector space of dimension^{7} for our game of Example 1. The results of Linear Algebra will be useful in this respect. A basis for the vector space was introduced in [

where the basic vectors are defined for

As mentioned above all these vectors, invented in [

where the coefficients are constants which may be determined from this vector equation written in scalar form. To solve the Inverse Problem we used the results:

and the linearity of the Shapley Value. By applying these results in the above expansion of

After eliminating the constants

(see [

As mentioned above, we shall be looking for a new game in the Inverse Set relative to the Shapley Value L, which is coalitional rational. We shall confine ourselves to find such a game in the subfamily of the Inverse Set defined by all

This subset of the Inverse Set will be called the almost null subfamily. Taking into account the expressions of the basic games shown above, the games in the family given by this vector formula, may be rewritten in scalar form as

Now, the Core conditions written together give an inequality determining the values of parameter

where the minimum is taken over the index i.

The results already shown provide a procedure for solving our above stated problem:

・ Compute the Shapley Value of the given game. Check whether, or not, the Shapley Value is in the Core. If yes, stop, the problem is solved; otherwise.

・ Compute the game in the Inverse Set given by (*), for the Shapley Value L.

・ Take a value for c_{N} in formula (*). to satisfy the inequality (**), and find out the desired game in the almost null subfamily.

Now, we intend to apply the results to the gas routing problem.

Return to the Example 1 given in the first section, in which we associated to the network the cooperative TU game obtained by finding for each coalition the maximum flow that could be delivered by using the ducts owned by each company in the coalition and the final public arc. The TU game is

and we can compute the Shapley Value:

Obviously, this is an efficient vector, but it is not belonging to the Core, because

are not satisfied. Then, start the procedure described above, by deriving the inequality (**):

We choose the largest value of the parameter that satisfies the inequality,

where the null worth of the singletons is omitted. We obtain

Further, we can compute again the Shapley Value and find out the same vector as before. This is not only efficient, but also coalitional rational, because the Core conditions hold. A similar situation and a similar strategy may be used for the Shapley Value in the general case of games with any number of players. Moreover, the procedure will work also for any other value which is efficient. A value which is not efficient is requiring a new definition of the coalitional rational value.

The procedure may be extended to this case by using the results following from the ideas developed in the joint paper [

Note that the present paper is an application of the results obtained in the previous paper [

IrinelDragan, (2015) On the Gas Routing via Game Theory. American Journal of Operations Research,05,288-292. doi: 10.4236/ajor.2015.54022