This paper focuses on games on augmenting systems with a coalition structure that can be seen as an extension of games with a coalition structure and games on augmenting systems. Considering the player payoffs, the quasi-Owen value is defined. To show the rationality of this payoff index, five representative axiomatic systems are established. The population monotonic allocation scheme (PMAS) and the core are introduced. Moreover, the relationships between the PMAS and quasi-Owen value as well as the core and quasi-Owen value are discussed. Finally, an illustrative example is given to show the concrete application of the new payoff indices.

In some cooperative situations, the players join in coalitions that form a partition or coalitional structure of the set of players to get more payoffs or to gain the competitive advantage. Aumann and Dreze (

Different from games with a coalitional structure (Aumann and Dreze,

In general, games with a coalition structure are formed by the players’ internal factor for obtaining more payoffs, while games under precedence constraints are due to the external factor as listed above. Considering these two aspects simultaneously, Meng and Zhang (

The rest of this paper is organized as follows: In Section

Let

Let

The set of coefficients

For the finite set

Let

Let

Let

A set system on

An augmenting system is a set system

If

If

Because the power set of

A compatible ordering of an augmenting system

Similar to Faigle and Kern (

In this section, we discuss cooperative games on augmenting systems with a coalition structure, which can be seen as an extension of games with a coalition structure (Owen,

Similar to the concept of augmenting system on

If

If

The number of maximal chains from

From Definition

A game on augmenting system with a coalition structure is a set function

Let

For

Similarly, we define the

Let

Because

Let

Because

Similar to Owen (

From

For any

Let

If there is only one coalition in Γ, then

Next, we apply the above listed axioms to show the existence and uniqueness of the quasi-Owen value. First, let us consider the following lemma:

From the expression of the quasi-Owen value, we have

Case (2): If

Case (3): If

Thus,

Existence. From Eq. (

From Definition

From Lemma

Further, according to Lemma

Uniqueness. From Lemma

Let

Similar to Bilbao and Ordonez (

From

Next, let us consider another axiomatization of the quasi-Owen value. Young (

From Theorem

If

If there is one

Assume that

Let

On the other hand, for any

Next, we will give another two axiomatic systems to characterize the quasi-Owen value from the perspective of the

Hart and Mas-Colell (

Let

Existence. For any

Let

Uniqueness. Note that Eq. (

When there is only one union in the coalition structure, we can conclude that there is the Hart-Mas-Colell potential function (Hart and Mas-Colell,

Different to

Existence. Obviously,

Thus,

Similarly, one can show that

To prove uniqueness, we just need to show that

Then,

From Eq. (

In this subsection, we focus on the axioms of the quasi-Owen value and give five axiomatic systems. These axiomatic systems can be divided into two categories in view of the axiom of linearity. The first two are based on

In this section, we introduce the core and the PMAS for games on augmenting systems with a coalition structure. Further, the relationship between the quasi-Owen value and the core is discussed, and the conditions for the quasi-Owen value to be a PMAS are given.

In a similar way to the core of games with a coalition structure (Pulido and Sánchez-Soriano,

Let

Let

From reduced games, we further offer the following concepts of the coalitional reduced game property (

Let

Let

Next, we show that the core satisfies

If for any

From the assumption, we have

According to Lemma

If

The following corollary is immediate from Lemma

To build the axiomatic system of the core

From the above analysis, one can check that the core of games on atomic augmenting systems with a coalition structure satisfies:

The proof of Theorem

If there is only one coalition in Γ, then

Similar to the Owen value for games with a coalition structure, we can prove that the quasi-Owen value for games on augmenting systems with a coalition structure belongs to the core. Based on the work of Pulido and Sánchez-Soriano (

Let

Following the work of Pulido and Sánchez-Soriano (

Let

Notice that this game is not quasi coalitional strong-convex as

Let

According to Definition

For any

If

If

By recursive relation, we get

The above theorem shows that when games on augmenting systems with a coalition structure are convex, there is no player who can make his own payoff larger than the quasi-Owen value without reducing other players’ payoff. Hence, there are no incentive to deviate from this allocation scheme.

Inspired by Sprumont (

Let

Next, we study the conditions under which the quasi-Owen value is a PMAS.

From Eq. (

By condition (ii), we have

By condition (i), we obtain

Hence,

If there is only one coalition in Γ, we get the conditions for the Shapley value for games on augmenting systems to be a PMAS. If all subsets of

In this section, we provide an application of games on augmenting system with a coalition structure in the food supply chain. Set up a supply chain consisting of food raw material supplier, food packaging supplier, food processing manufacturer, wholesaler and retailer. For the convenience of expression, the above members are set as 1, 2, 3, 4 and 5, respectively. The model of the food supply chain is shown in Fig.

The model of the food supply chain.

In this food supply chain, to gain more profits with lower cost, companies 1, 2 and 3 decide to cooperate and form the production union

The coalition values (million dollars/week).

∅ | 0 | 9 | |

3 | 13 | ||

4 | 17 | ||

8 | 20 |

From Eq. (

This example shows that

From the relationships among augmenting system, antimatroid and convex geometry (Bilbao and Ordonez,

The above relationships about different types of games show that the quasi-Owen value can be seen as a payoff index for them under the corresponding special conditions. Further, all listed axiomatic systems still hold for the quasi-Owen value in the setting of the above mentioned cooperative games, where axiomatic systems are defined under the associated conditions. This paper only studies a special kind of games under precedence constraints with a coalition structure, and it will be interesting to take into account other types of games under precedence constraints. Moreover, similar to the offered numerical example, we can apply the quasi-Owen value into other practical cooperative cases.