Daniel Granot

Minimum cost spanning tree games

We consider the problem of cost allocation among users of a minimum cost spanning tree network. It is formulated as a cooperative game in characteristic function form, referred to as a minimum cost spanning tree (m.c.s.t.) game. We show that the core of a m.c.s.t. game is never empty. In fact, a point in the core can be read directly from any minimum cost spanning tree graph associated with the problem. For m.c.s.t. games with efficient coalition structures we define and construct m.c.s.t. games on the components of the structure. We show that the core and the nucleolus of the original game are the cartesian products of the cores and the nucleoli, respectively, of the induced games on the components of the efficient coalition structure.

A generalized linear production model: A unifying model

We introduce a generalized linear production model whose attractive feature being that the resources held by any subset of producersS is not restricted to be the vector sum of the resources held by the members ofS. We provide sufficient conditions for the non-emptiness of the core of the associated generalized linear production game, and show that if the core of the game is not empty then a solution in it can be produced from a dual optimal solution to the associated linear programming problem. Our generalized linear production model is a proper generalization of the linear production model introduced by Owen, and it can be used to analyze cooperative games which cannot be studied in the ordinary linear production model framework. We use the generalized model to show that the cooperative game induced by a network optimization problem in which players are the nodes of the network has a non-empty core. We further employ our model to prove the non-emptiness of the core of two other classes of cooperative games, which were not previously studied in the literature, and we also use our generalized model to provide an alternative proof for the non-emptiness of the core of the class of minimum cost spanning tree games. Thus, it appears that the generalized linear production model is a unifying model which can be used to explain the non-emptiness of the core of cooperative games generated by various, seemingly different, optimization models.

A Three-Stage Model for a Decentralized Distribution System of Retailers

We present and study a three-stage model of a decentralized distribution system consisting ofn retailers, each of whom faces a stochastic demand for an identical product. In the first stage, before the demand is realized, each retailer independently orders her initial inventory. In the second stage, after the demand is realized, each retailer decides how much of her residual supply/demand she wants to share with the other retailers. In the third stage, residual inventories are transshipped to meet residual demands, and an additional profit is allocated. Our model is an extension of the two-stage model of Anupindi et al. (ABZ) (2001), which implicitly assumes that all residuals enter the transshipment stage. We show, however, that allocation rules in the third stage based on dual solutions, which were used in the ABZ model, may induce the retailers to hold back some of their residual supply/demand. In general, we study the effect of implementing various allocations rules in the third stage on the values of the residual supply/demand the retailers are willing to share with others in the second stage, and the trade-off involved in achieving an optimal solution for the corresponding centralized system.

Formation of Alliances in Internet-Based Supply Exchanges

In different industries, such as automobiles, chemicals, or retailing, competitors are joining forces in establishing electronic marketplaces to reduce inefficiencies in the purchasing process and cut costs by combining their buying power. Joining such an alliance leads to reduced costs, including those of possible rivals, because members share the development and operating costs. A company that joins an alliance agrees to share its suppliers with others, which may lead to more intense competition among the increased number of suppliers, and it may further benefit an alliance member at the expense of companies left outside the alliance. Natural questions that could arise, then, are when would a firm prefer to take part in an electronic marketplace joint venture; when would it prefer that other firms, possibly rivals, join the venture; and what are the financial consequences of either joining an alliance or remaining independent? In an attempt to gain a better understanding of the issues, we have developed a model of three retailers whose products may have a certain degree of substitutability. We provide some conditions, in terms of product substitutability and compatibility of retailers, that would lead to the formation of a three-member alliance, or a two-member alliance, or no alliance at all. We also study the effect of alliance structure and compatibility of retailers on the profit of a company.

Price and Order Postponement in a Decentralized Newsvendor Model with Multiplicative and Price-Dependent Demand

We analyze the effect of price and order postponement in a decentralized newsvendor model with multiplicative and price-dependent demand, wherein the manufacturer sets the wholesale price, and possibly offers a buyback rate, and the retailer determines the order quantity and retail price. Such postponement strategies can be used by the retailer by delaying his operational decisions (order quantity and retail price) until after demand uncertainty is observed. We show how the equilibrium values of the contract parameters and profits are affected by (i) vertical competition, (ii) type of contract (wholesale price-only or buyback), (iii) demand distribution, (iv) form of the expected demand function, and (v) the timing of the retailers operational decisions. Although in most cases postponement is quite beneficial for the channel members, we show that for some model parameters, due to vertical competition, the expected value of perfect information about demand for price postponement and order postponement may be negative for the channel and even, surprisingly, for both members. We also show that when a buyback option is offered, neither order postponement nor price postponement has an effect on the equilibrium wholesale price, profit allocation ratio between the manufacturer and the retailer, and channel efficiency, and that the equilibrium wholesale price, expected retail price, profit allocation ratio between the manufacturer and the retailer, and channel efficiency in the model with buyback options under either order or price postponement further coincide with their counterparts in the corresponding deterministic model.

On Some Network Flow Games

We analyze three subclasses of cooperative games arising from network optimization problems in which the resources, such as arcs or nodes in the network, are controlled by individuals who have conflicting objectives. The first subclass of cooperative games is induced by network optimization problems over directed augmented trees. We show that for this subclass of games the kernel coincides with the nucleolus, and that the nucleolus can be characterized as the unique revenue allocation vector in which every pair of arc owners who are adjacent in the tree are located symmetrically with respect to their bargaining range. We further give a linear characterization of the core of this subclass of games, which is then used to provide a more explicit representation of the nucleolus and to construct a strongly polynomial algorithm for generating it. The second subclass of cooperative games is induced by maximum flow problems in simple undirected networks. We provide a useful parametric representation of the core of this subclass of games, which is used to characterize the nucleolus and the intersection of the core and the kernel. Explicitly, we show that the intersection of the core and the kernel consists of all revenue allocations in the core which assign equal payoffs to any pair of unseparated arc owners. We further demonstrate that among the core vectors, the nucleolus is the unique revenue allocation vector in which the smallest allocations are maximized in a lexicographical sense. The third cooperative game that we study is the assignment game, introduced by Shapley and Shubik 1972. This game is induced by the assignment problem which can be cast as a network optimization problem. We investigate the relationship between the kernel and the core of the assignment game, and provide a necessary and sufficient condition for the core to be contained in the kernel. We further show that, in general, the intersection of the kernel and the core is not a convex set. We also exhibit that under certain conditions the nucleolus has a simple characterization as the unique vector in the core in which the smallest revenue allocations are maximized in a lexicographical sense. Finally, we consider the horse market example of Bohm-Bawerk 1923, for which it is shown that the core is contained in the kernel and that the nucleolus is the midpoint of the core.

The kernel/nucleolus of a standard tree game

In this paper we characterize the nucleolus (which coincides with the kernel) of a tree enterprise. We also provide a new algorithm to compute it, which sheds light on its structure. We show that in particular cases, including a chain enterprise one can compute the nucleolus in O(n) operations, wheren is the number of vertices in the tree.

The Relationship Between Convex Games and Minimum Cost Spanning Tree Games: A Case for Permutationally Convex Games

Notwithstanding the apparent differences between convex games and minimum cost spanning tree (m.c.s.t.) games, we show that there is a close relationship between these two types of games. This close relationship is realized with the introduction of the group of permutationally convex (p.c.) games. It is shown that a p.c. game has a nonempty core and that both convex games and m.c.s.t. games are permutationally convex.

On sequential commitment in the price-dependent newsvendor model

Abstract We investigate the effect of sequential commitment in the decentralized newsvendor model with price-dependent demand. Sequential commitment allows the self-profit maximizing parties to commit to contract parameters (e.g., wholesale price, retail price, buyback price and order quantity) sequentially and alternately, and we investigate its effect on the equilibrium profits of the channel and its members. Sequential commitment introduces more flexibility to contracting in the supply chain and our analysis can provide some insight to channel members who follow a bargaining process to determine the values of contract parameters. We show that the introduction of sequential commitment to the price-dependent (PD) newsvendor model with buybacks can significantly improve the manufacturer’s and the channel expected profits, but it can also decrease the retailer’s expected profit. Finally, we demonstrate that with sequential commitment, under some conditions, the choice of the first mover is endogenized and we identify the unique sequence of commitments by channel members that would arise in equilibrium.

Computational complexity of a cost allocation approach to a fixed cost spanning forest problem

We present a computational analysis of a game theoretic approach to a cost allocation problem arising from a graph optimization problem, referred to as the fixed cost spanning forest FCSF problem. The customers in the FCSF problem, represented by nodes in a graph G, are in need of service that can be produced at some facilities yet to be constructed. The cost allocation problem is concerned with the fair distribution of the cost of providing the service among customers. We formulate this cost allocation problem as a cooperative game, referred to as the FCSF game. In general, the core of a FCSF game may be empty. However, for the case when G is a tree, it is shown that the core is not empty. Moreover, we prove that in this case core points can be generated in strongly polynomial time. We further provide a nonredundant characterization of the core of the FCSF game defined over a tree in the special case when all nodes are communities. This is shown to lead, in some instances, to a strongly polynomial algorithm for computing the nucleolus.