Guardiola et al. (2009) introduced Production-Inventory games (henceforth PI-games) as a new class of totally balanced combinatorial optimization games. That paper proposed the so-calledOwen point
core-allocation that allows all players to operate at minimum cost at the price of not compensating essential users by the cost reduction that they induce over the remaining players (fans). This allocation has proven to be rather appealing and in another paper, Guardiola et al. (2008) analyze its properties and propose three axiomatic characterizations for the Owen point. These papers also contribute to a better knowledge of the core of PI-games. Nevertheless, it was missing a deeper analysis of its complexity. Specifically speaking the two following aspects were not considered: testing core membership and the extreme points structure of the core of these games. Complexity issues in cooperative game theory raise important questions only partially answered for particular classes of games. The core of any convex game is the convex hull of its marginal vectors (Shapley 1971), and the same property holds true for those games satisfying the Co-Ma property which include, among others, assignment and information games, see Hamers et al. (2002) and Kuipers (1993) respectively. It is also well-known that the core of assignment games coincide with the allocations induced by dual solutions and it is a complete lattice with only two extreme points, see Sotomayor (2003). Also, for transportation games, which constitute an extension of the assignment games, some results about the relationship between the core and the allocations induced by dual solutions are provided by Sánchez-Soriano et al. (2001). Moreover, Perea et al. (2012) study cooperation situations in linear production problems. In particular, that paper proposes a new solution concept called EOwen set as an improvement of the Owen set that contains at least one allocation that assigns a strictly positive payoff to players necessary for optimal production plans.
For minimum cost spanning tree games, flow games, linear production games, cooperative facility location games or min-coloring games among others, testing whether a given allocation is in the core is an NP-complete problem (see Faigle et al. (1997), Fang et al. (2002), Goemans and Skutella (2004) and Deng et al. (1999), respectively). On the other hand, there are some classes of games for which testing core membership is polynomially solvable as for instance for routing games, see Derks and Kuipers (1997), connectivity games, -arborescence games, max matching games, min vertex cover games, min edge cover games or max independent set games, see e.g., Deng et al. (1999). However, for many other classes of cooperative games answering that question is still open, as it is the case of PI-games.
In this paper we investigate the structure of the core of PI-games by determining its algorithmic complexity. Our contribution is to prove that testing core membership is an NP-complete problem and moreover that the number of extreme points of the core of PI-games is exponential on the number of players. Specifically, we characterize an exponential size subset of them. In addition, we look for alternative cost allocations improving the fairness properties of the Owen point in that they recognize the role of the essential players on reducing the costs of the remaining players.
To present our results the rest of the paper is organized as follows. We start by introducing some preliminary concepts in section 2. In section 3 we prove that testing core membership of PI-games in an NP-complete problem, and we analyze the core structure of PI-games. We define what we call the extreme functions, which help us to prove that the core of a PI-game, in general, has an exponential number of extreme points. In section 4 we introduce a new core-allocation for PI-games, the Omega point, and provide an axiomatic characterization. Finally, in section 5 we define the set of Quid Pro Quo allocations (henceforth, QPQ allocations). Every QPQ allocation is a convex combination of the Owen and the Omega point. We focus then on the equally weighted QPQ allocation, the Solomonic allocation, and we provide some necessary conditions for the coincidence of the latter with the Shapley value and the Nucleolus.
A cost game with transferable utility (henceforth TU cost game) is a pair , where
is the finite set of players, and the characteristic function, is defined over the set of nonempty coalitions of . By agreement, it always satisfies For all , we denote by the cardinal of the set .
A distribution of the costs of the grand coalition, usually called cost-sharing vector, is a vector . For every coalition we denote by the cost-sharing of coalition (where The core of a TU cost game consists of those cost-sharing vectors which allocate the cost of the grand coalition in such a way that no coalition has incentives to leave because is smaller than the original cost of , . Formally, the core of is given by In the following, core-allocations will be cost-sharing vectors belonging to the core. A cost game is balanced if and only if has a nonempty core (see Bondareva 1963 or Shapley 1967). Shapley and Shubik (1969) describe totally balanced games as those games whose subgames are also balanced; i.e., the core of every subgame is nonempty. A cost game is concave if for all and all such that with then
The Shapley value (Shapley, 1953) is a linear function on the class of all TU games and for a cost game it is defined as where for all
The Nucleolus (Schmeidler, 1969) is the allocation that lexicographically minimizes the vector of excesses. It is well-known that the Nucleolus is a core-allocation provided that the core is nonempty.
Let be a bounded convex polyhedron in . We say that is an extreme point if and imply . From now on, we denote, respectively, by and by the set of extreme points and the boundary of the set of . Moreover, for the sake of readability, we use to refer to the -th element of the canonical basis of and stands for the optimal value of the mathematical programming problem .
It is well-known that if and only if satisfies as equalities at least linearly independent constraints of those defining . Since the core is a bounded convex polyhedron, it has a finite number of extreme points. Moreover, the core is a convex set. Therefore, characterizing the extreme core-allocations is important to know its intrinsic structure.
From now on, and for the sake of readability, we follow the same notation as Guardiola et al. (2009) to describe Production-Inventory situations (henceforth: PI-situations) and PI-games. Consider first a situation with several agents facing each one a Production-Inventory problem. Then, they decide to cooperate to reduce costs. Here the cooperation is considered as sharing technologies in production, inventory carrying and backlogged demand. We mean that if a group of agents agree on cooperation then at each period they will produce and pay inventory carrying and backlogged demand at the cheapest costs among the members of the coalition. This situation is called a PI-situation.
Formally, let be an infinite set, the universe of players. A PI-situation is a 3-tuple where is a finite set of players and an integer matrix of demands with , , is the demand of the player during period and is the planning horizon. In addition, is a cost matrix, so that , and where , is the unit inventory carrying costs of the player in period , , is the unit backlogging carrying costs of the player in period , and , the unit production costs of the player in period , for . The decision variables of the model, which are required to be integer quantities, are the production during period (), the inventory at hand at the end of period (), and the backlogged demand at the end of period (). We denote by the set of PI-situations defined over , being and an integer matrix.
Now given a PI-situation , we can associate the corresponding TU cost game with the following characteristic function : and for any , where is given by
Every TU cost game defined as above is called a Production-Inventory game. Guardiola et al. (2009) points out that the problem has integer optimal solutions provided that the demands are integer. We know that the dual problem of , for any coalition , is the following mathematical programming problem,
Moreover, Guardiola et al. (2009) also proves that an optimal solution of problem is , for all with
It is important to note that those optimal solutions satisfy a monotonicity property with respect to coalitions : for all and all . Moreover, the characteristic function of PI-games can be rewritten as follows: for any .
PI-games are not concave in general as shown by Example 4.4 in Guardiola et al. (2009). The allocation is called the Owen point, and it is denoted by . At times, for the sake of simplicity, we use to refer to the Owen point. That same paper also proves that the Owen point is a core-allocation which can be reached through a PMAS (Sprumont, 1990); hence every PI-game is a totally balanced game. In some situations we will use instead of , in order to denote that the game comes from the situation .
We say that a player is essential if there exists with such that . An essential player is the one for which there exists at least one period in which he is needed by the rest of players in order to produce a certain demand at a minimum cost. The set of essential players is denoted by . Those players not being essential are called inessential. We can easily check that for each inessential player , . Guardiola et al. (2009) showed that the core of PI-games shrinks to a single point, the Owen point, just only when all players are inessential for the PI-situation.
Finally, to conclude this section devoted to preliminaries, we recall the class of PS-games introduced by Kar et al. (2009). A PS-game is a TU cost game satisfying that for all player there exists a real constant such that for all where The above mentioned paper proves that, for this class of games, the Shapley value and the Nucleolus coincide; i.e. .
3 Extreme points of the core of PI-games
Guardiola et al. (2009) demostrated that the core of PI-games without essential players () shrinks to a singleton, the Owen point. However, for those PI-games with essential players (
), the core is large. We focus here on those PI-games with large cores and study the structure of its core by analyzing its extreme points. First of all, we remark that testing core membership for PI-games cannot be done in polynomial time. One can adapt the reduction proposed in Fang et al. (2002) to prove that checking if an imputation belongs to the core of a PI-game is an NP-complete decision problem. In spite of that, it is important to know the structure of the core and still very little is known about the extreme points complexity of PI-games. This is the goal of this section.
We begin this analysis by defining the essential player fan set.
Let be a PI-situation with being an integer matrix (), and let be an essential player. We define the fan set of as follows:
The fan set of player consists of all players who need him to operate at a lower cost. It is always a non-empty set. Indeed, since taking there exists such that and . In that case, there must be, at least, a player such that and .
In addition, you may notice that there is a pairwise relationship among essential players and their fans, in the sense that the latter are interested in taking on a portion of the costs of the former. This relationship allows us to introduce the concept of essential-fan pair.
Let . The essential-fan pair set, denoted by , is:
We are now interested in determining the cost that can be transferred within every essential-fan pair with a cost allocation; i.e., the maximum portion of the essential player cost that his fan could assume while maintaining cooperation.
Given a essential-fan pair and a allocation the transferred cost induced by regarding is:
can be interpreted as the maximum portion of cost of player that can be awarded by player while maintaining the cooperation of the group. It is worth nothing that if then .
Next result states that there are always a positive transferred cost within every essential-fan pair with the Owen point.
Let and be the corresponding PI-game. Then for all .
Proof. As , we can take and therefore . Let such that and let . By definition, there exists such that and . Then and moreover . Thus, . Hence, .
We introduce now a function that transforms any cost allocation into a new cost allocation in which a fan player charges with the maximum cost of his essential player. That is, for each , the function transforms any allocation into a new allocation in which the fan player assumes as much cost as possible from his essential player It is called the extreme function.
Definition 3.2 (extreme function)
Let and be the corresponding PI-game. For any , the extreme function is defined by:
Let us denote by the -fold cartesian product of the set . We consider now the composition of extreme functions. For each we define the extreme composite function, , as the composition of extreme functions for all the pairs in , that is,
Notice that if then
The following table shows a PI-situation with three periods and three players:
We can easily check that for all . Hence, the characteristic function of the corresponding PI-game is given in the following table:
In this example, and are the optimal solution for and respectively. Then, the Owen point is . Moreover, , and .
The transferred cost within every essential-fan pair in with the Owen point are,
Therefore, the extreme functions are
In this case, both the extreme functions, , , and the Owen point, , are extreme points of the core.
The previous example shows that the Owen point is an extreme point of the core, and that the extreme functions transform it into other extreme points of the core. We wonder then if this fact occurs in general for any PI-game. First, we find a very interesting property that relates the extreme functions to the core boundary.
Let and be the corresponding PI-game. For all
Proof. Let and take Then . Applying the extreme function at , we have:
To prove that we distinguish four possibilities:
Hence, since for any coalition . Let us proof now that belongs to the frontier of the core.
If then there exists such that . Since belongs to the core and satisfies as equality one of the constraints defining the core, we can conclude that .
If then for all Take with to have
We can check that if is such that then , therefore . Hence, is not an interior point.
It follows straightforward from the above proposition, that for all .
The main Theorem of this Section provides a partial answer to our previous question about the transformation of the Owen point into extreme points of the core of PI-games. It states that for PI-situations with a single essential player, all the different compositions of extreme functions over the Owen point generate extreme points of the core.
Let and be the corresponding PI-game. If , then for all .
Proof. Let then the pair . is an extreme point if for any such that
By definition, we know that
Let us suppose that for any then However this is not possible, therefore for all . Now, apply (1) to get that . Moreover, since is the maximum possible increment for (see Lemma 3.1). Then by (1) we have that and hence .
Now, we consider , and apply the corresponding extreme function for this pair. We have that
We distinguish two possibilities:
attains its minimum in a coalition that contains player . In this case , thus and by the argument above is an extreme point of .
attains its minimum in a coalition that does not contain player . This case implies that . Take and assume that
Using the same argument as above we conclude that for all Consider now the -th coordinate. Suppose that . The coalition does not contain neither nor , which implies Since this is a contradiction, it means that (Notice that the same argument applies to and thus ). Therefore, by (2) we get that .
Next, consider the -th coordinate. Assume that , and let be the coalition where attains its minimum, then . Again using the same argument as in the -th coordinate we conclude that
Finally, we get the same conclusion for the -th coordinate since must be efficient. In conclusion Hence, . Notice that is different from since we have assumed that .
This construction can be repeated a finite number of times for each . Specifically, for any the transformation .
Let with , and be the corresponding PI-game. The Owen point is always an extreme point.
Proof. Take for all , therefore and
At this point we know that PI-games with a single essential player have, at least, extreme points. Next example shows that the core of a PI-game, in general, cannot be explicitly described in polynomial time.
Now we consider a PI-situation with n periods and n players:
The corresponding PI-game is given by , for all . Moreover, it is easy to see that and . Then, we can rewrite the characteristic function as follows:
In this example, the Owen point is . For all ,
For all ,
Hence, we have as many extreme points as possible ways to place and in positions; i.e. in this example the core has extreme points.
Therefore, we can conclude that the cardinality of the extreme points is exponential in the number of players. Hence, we cannot explicitly describe the core of a PI-game in polynomial time.
We propose below an alternative core allocation to the Owen point that recognizes the role played by essential players on reducing the cost of their fans.
4 Omega point
Guardiola et al. (2009) proposed the Owen point as a natural core allocation for PI-games that arises when focusing on shadow prices of each period that each player must pay to meet their demand in that period. It makes it possible for all players in the joint venture to operate at minimum cost. If there is no essential player, the Owen point is the unique core allocation. However, for those PI-situations with at least one essential player, the Owen point reveals the altruistic character of them because of it does not take into account the role that these essential players play in reducing the cost of their fans. As the core of the PI-games with essential players is large, we are looking for a core allocation that motivates the essential players to continue in the join venture obtaining a reduction in their demand costs in each period.
Let be a PI-situation with being an integer matrix (), and . Remember that for all , there is a period such that and there also exists at least one player such that . We denote by and the sets of essential players and fans for every period . We note in passing that .
First, we consider the marginal contribution of the shadow prices of a player to the grand coalition , that is, We then define the cost reduction that a player can produce in another player in a period as follows:
The reader may notice that only if and otherwise That is to say that only essential players can reduce their fan costs in a given period. Alternatively, the amount of the cost can be interpreted as the maximum cost increase that a fan is able to assume, in a certain period to incentivize the essential player
Next we define a new cost allocation rule, the Omega point, that considers the maximum cost increase mentioned above.
Definition 4.1 (Omega point)
Let and be the corresponding PI-game. The Omega point is defined as for all player where for each period ,
The Omega point means that, in each of the periods with a single essential player, i.e. without competition, this essential player gets a cost reduction from his fans. The amount represents the cost reduction or increase, depending on the sign, for player in the period . Notice that only if is an essential player, otherwise In addition, for all
The reader may also note that where with