A discrete cost sharing model with technological cooperation

This article proposes a setting that allows for technological cooperation in the cost sharing model. Dealing with discrete demands, we study two properties: additivity and dummy. We show that these properties are insufficient to guarantee a unit-flow representation similar to that of Wang (Econ Lett 64:187–192, 1999). To obtain a characterization of unit flows, we strengthen the dummy axiom and introduce a property that requires the cost share of every agent to be non-decreasing in the incremental costs generated by their demand. Finally, a fairness requirement as to the compensation of technological cooperation is examined.     

Two Approaches to the Problem of Sharing Delay Costs in Joint Projects

This paper concentrates on cost sharing situations which arise when delayed joint projects involve joint delay costs. The problem here is to determine fair shares for each of the agents who contribute to the delay of the project such that the total delay cost is cleared. We focus on the evaluation of the responsibility of each agent in delaying the project based on the activity graph representation of the project and then on solving the important problem of the delay cost sharing among the agents involved. Two approaches, both rooted in cooperative game theory methods are presented as possible solutions. In the first approach delay cost sharing rules are introduced which are based on the delay of the project and on the individual delays of the agents who perform activities. This approach is inspired by the bankruptcy and taxation literature and leads to five rules: the (truncated) proportional rule, the (truncated) constrained equal reduction rule and the constrained equal contribution rule. By introducing two coalitional games related to delay cost sharing problems, which we call the pessimistic delay game and the optimistic delay game, also game theoretical solutions as the Shapley value, the nucleolus and the -value generate delay cost sharing rules. In the second approach the delays of the relevant paths in the activity graph together with the delay of the project play a role. A two-stage solution is proposed. The first stage can be seen as a game between paths, where the delay cost of the project has to be allocated to the paths. Here serial cost sharing methods play a role. In the second stage the allocated costs of each path are divided proportionally to the individual delays among the activities in the path.     

Strategy-proofness and public good provision using referenda based on unequal cost sharing

The paper studies strategy-proof cost sharing rules for public good provision based on referenda with different threshold quotas. By appropriately relaxing the assumptions of individual rationality and anonymity we provide a complete characterization of the family of quota rules with (possibly) unequal pricing. We prove that these quota rules are the only cost sharing rules satisfying four conditions: strategy-proofness, non-bossiness, weak continuity and weak anonymity. In addition, the specification of the degree to which individual rationality may be violated results in the selection of a specific “quota” for the referendum. While all these rules are “almost” always efficient when providing the public good and they are also almost everywhere coalitionally strategy-proof, only one family of rules from this class satisfies these two properties everywhere. The rules satisfying these two properties are Moulin’s Conservative Equal Costs Rule and unequal cost sharing variants of Moulin’s rule.     

Two main utility-sharing methods in joint businesses: A discussion

We compare two methods of sharing costs or profits among agents in cooperative games: the well–known proportional rule and the Shapley value. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The $$$$-seril cost shring rule

A new family of cost sharing rules for cost sharing problems is proposed. This family generalizes the family of \(\alpha \)-serial cost sharing rules (Albizuri in Math Soc Sci 60:24–29, 2010) which contains the serial cost sharing rule (Moulin and Shenker in Econometrica 60:1009–1037, 1992) among others. Every rule of the family is characterized by means of two properties.     

Truth-telling and Nash equilibria in minimum cost spanning tree models

In this paper we consider the minimum cost spanning tree model. We assume that a central planner aims at implementing a minimum cost spanning tree not knowing the true link costs. The central planner sets up a game where agents announce link costs, a tree is chosen and costs are allocated according to the rules of the game. We characterize ways of allocating costs such that true announcements constitute Nash equilibria both in case of full and incomplete information. In particular, we find that the Shapley rule based on the irreducible cost matrix is consistent with truthful announcements while a series of other well-known rules (such as the Bird-rule, Serial Equal Split, and the Proportional rule) are not.     

Cost allocation in asymmetric trees

Agents are connected each other through a tree. Each link of the tree has an associated cost and the total cost of the tree must be divided among the agents. In this paper we assume that agents are asymmetric (think on countries that use aqueducts to bring water from the rainy regions to the dry regions, for example). We suppose that each agent is entitled with a production and demand of a good that can be sent through the tree. This heterogeneity implies that the links are not equally important for all the agents. In this work we propose, and characterize axiomatically, two rules for sharing the cost of the tree when asymmetries apply.     

Agency selling or reselling: E-tailer information sharing with supplier offline entry

Nowadays, some suppliers are looking for offline expansion in addition to their preexisting online channels relying on e-tailers. This study focuses on the e-tailer’s demand information sharing strategy with the supplier who may build upon brick-and-mortar stores. Both prevailing agreements between the supplier and the e-tailer are investigated: agency selling and reselling. The equilibrium results are quite different under these two agreements. Specifically, when the supplier’s offline entry cost is very small or large, the e-tailer shares information under agency selling while keeps information private under reselling. When the entry cost is intermediate, channel substitution rate is large and information uncertainty is small, the e-tailer withholds the demand information under agency selling while shares information under reselling to deter the supplier from entering an offline channel. Furthermore, two extensions about consumer behavior in multichannel selection are discussed: showrooming and webrooming. With showrooming or webrooming, the e-tailer’s information sharing decisions qualitatively hold, while with showrooming the drive factor behind may change; that is, withholding information under agency selling and sharing information under reselling may also serve as measures to encourage supplier offline entry when the effect of showrooming is strong.     

Capital allocation for portfolios with non-linear risk aggregation

Existing risk capital allocation methods, such as the Euler rule, work under the explicit assumption that portfolios are formed as linear combinations of random loss/profit variables, with the firm being able to choose the portfolio weights. This assumption is unrealistic in an insurance context, where arbitrary scaling of risks is generally not possible. Here, we model risks as being partially generated by Lévy processes, capturing the non-linear aggregation of risk. The model leads to non-homogeneous fuzzy games, for which the Euler rule is not applicable. For such games, we seek capital allocations that are in the core, that is, do not provide incentives for splitting portfolios. We show that the Euler rule of an auxiliary linearised fuzzy game (non-uniquely) satisfies the core property and, thus, provides a plausible and easily implemented capital allocation. In contrast, the Aumann–Shapley allocation does not generally belong to the core. For the non-homogeneous fuzzy games studied, Tasche’s (1999) criterion of suitability for performance measurement is adapted and it is shown that the proposed allocation method gives appropriate signals for improving the portfolio underwriting profit.     

Simple random order methods to share costs

A simple random order method (SROM) is an extension of Webers random order values (ROVs), which allows the convex weights on orderings of agents to depend on the set of agents with strictly positive demands. Thus, a SROM permits different coalitions of agents adopting exogenously different ROVs to take into account the differences in, for example, bargaining abilities, rights or status of the agents in a cost sharing problem. Within the family of additive methods satisfying the dummy axiom, we characterize SROMs by Measurement Invariance in the discrete cost sharing model where demands are indivisible, and Ordinality in the continuous model where demands are divisible, respectively.I am indebted to Yves Sprumont for initiating and providing the starting point of this paper, and for his great help in this work. I thank Hervé Moulin, Ahmet Alkan, Anirban Kar, an associate editor, and a referee whose comments greatly improved the paper. I gratefully acknowledge the support from Sabanci University Research Fund.     

Bankruptcy and the per capita nucleolus: The claim-and-right rules family

This article describes the per capita nucleolus for bankruptcy games as a bankruptcy rule. This rule, called the clights rule, is based on the well-known constrained equal awards principle and it takes into account a vector of clights, a new term which is a blend of claims and rights. These clights only depend on the vector of claims while the height of the estate determines whether the clights should be interpreted as modified claims or as rights. It is shown that both the clights rule and the Aumann–Maschler rule can be captured within the family of so-called claim-and-right rules.     

Cost allocation in inventory transportation systems

In this paper, we deal with the cost allocation problem arising in an inventory transportation system with a single item and multiple agents that place joint orders using an EOQ policy. In our problem, the fixed-order cost of each agent is the sum of a first component (common to all agents) plus a second component which depends on the distance from the agent to the supplier. We assume that agents are located on a line route, in the sense that if any subgroup of agents places a joint order, its fixed cost is the sum of the first component plus the second component of the agent in the group at maximal distance from the supplier. For these inventory transportation systems, we introduce and characterize a rule which allows us to allocate the costs generated by the joint order. This rule has the same flavor as the Shapley value, but requires less computational effort. We show that our rule has good properties from the point of view of stability.     

On weighted Shapley values

Nonsymmetric Shapley values for coalitional form games with transferable utility are studied. The nonsymmetries are modeled through nonsymmetric weight systems defined on the players of the games. It is shown axiomatically that two families of solutions of this type are possible. These families are strongly related to each other through the duality relationship on games. While the first family lends itself to applications of nonsymmetric revenue sharing problems the second family is suitable for applications of cost allocation problems. The intersection of these two families consists essentially of the symmetric Shapley value. These families are also characterized by a probabilistic arrival time to the game approach. It is also demonstrated that lack of symmetries may arise naturally when players in a game represent nonequal size constituencies.     

Profit maximizing coalitions with shared capacities in distribution networks

We study the distribution network structure of multiple firms in the context of demand sensitivity to market offers. The problem consists in determining the profitability of horizontal collaboration between firms in a collaborative distribution schema. It considers the case of a set of regional distribution centers (DCs) where each DC is initially dedicated solely to one firm’s distribution activities and studies when it is beneficial that the DC owners collaborate through sharing their storage-throughput capacity. Such strategic decisions are made in order to improve the distribution capabilities of firms in terms of response time and cost-efficiency compared to the stand-alone situation. The problem is modeled as a coalition formation game in a cooperative framework, and we propose a collaborative distribution game with profit maximization. Three sharing mechanisms are modeled and tested: egalitarian allocation, proportional allocation, and Shapley value. The collaboration decision conditions for a given firm are analytically derived according to the sharing method considered and used to enhance the solution approach. Our numerical results clearly highlight the impact of this innovative collaboration opportunity on the firms’ performance in terms of distribution cost savings and revenue increases. An observed behavior is that the formation of several sub-coalitions prevails over the formation of a grand coalition, and that different cost sharing methods can lead to different sub-coalitions. We also provide managerial insights on the appropriate size of a coalition in various business instances tested, and on the key drivers that foster horizontal collaborative behavior among firms.     

基于Shapley值的壳资源溢价影响因素的实证研究

本文选取了我国1997-2003年证券市场上符合买壳上市条件的63家目标企业作为样本,对影响买壳上市溢价因素进行了理论分析和实证检验。论文基于合作博弈理论Shapley值分担法建立了多元回归模型以解决多重共线性变量产生偏倚性和负的多重性问题。研究显示:上市公司"壳"资源的溢价比率与每股净资产、资产负债率、货币性资产占总资产比率、市净率、股权集中度、净资产收益率等变量之间存在着显著的相关性。     

Optimal allocation of stock levels and stochastic customer demands to a capacitated resource

This paper considers a new class of stochastic resource allocation problems that requires simultaneously determining the customers that a capacitated resource must serve and the stock levels of multiple items that may be used in meeting these customers’ demands. Our model considers a reward (revenue) for serving each assigned customer, a variable cost for allocating each item to the resource, and a shortage cost for each unit of unsatisfied customer demand in a single-period context. The model maximizes the expected profit resulting from the assignment of customers and items to the resource while obeying the resource capacity constraint. We provide an exact solution method for this mixed integer nonlinear optimization problem using a Generalized Benders Decomposition approach. This decomposition approach uses Lagrangian relaxation to solve a constrained multi-item newsvendor subproblem and uses CPLEX to solve a mixed-integer linear master problem. We generate Benders cuts for the master problem by obtaining a series of subgradients of the subproblem’s convex objective function. In addition, we present a family of heuristic solution approaches and compare our methods with several MINLP (Mixed-Integer Nonlinear Programming) commercial solvers in order to benchmark their efficiency and quality.     

Heterogeneous cost sharing, the directional serial rule

The directional serial rule is introduced as a natural serial extension, generalizing the Moulin–Shenker cost sharing rule to heterogeneous cost sharing models. It is the unique regular rule compatible with the radial serial principle. In particular, this shows the incompatibility of the serial principle with differentiability of a cost sharing rule as a function of the individual demands.I would like to thank the editor and the referee for their comments which have been most useful.     

SHARING THE BENEFITS OF COOPERATION IN HIGH SEAS FISHERIES: A CHARACTERISTIC FUNCTION GAME APPROACH

In the current paper we examine a game-theoretic setting in which three countries have established a regional organization for the conservation and management of straddling and highly migratory fish stocks. A characteristic function game approach is applied to describe the sharing of the surplus benefits from cooperation. We demonstrate that the nucleolus and the Shapley value give more of the benefits to the coalition with substantial bargaining power than does the Nash bargaining scheme. We also compare the results that are obtained by using the nucleolus and the Shapley value as solution concepts. The outcomes from these solution concepts depend on the relative efficiency of the most efficient coalition. Furthermore, the question of fair sharing of the benefits is considered in the context of straddling stocks.     

Planning the supply of rewards with cooperative promotion considerations in coalition loyalty programmes management

Coalition loyalty programmes (CLPs) are owned and operated as for-profit enterprises. We consider the ordering decisions of rewards that arise in this context, under a general setting in which not only is the demand for rewards uncertain, but also the CLP firm offers bonus points, a very common cooperative promotion mechanism used in loyalty programmes. The rewards are acquired either at a wholesale ‘discounted’ cost or at a wholesale ‘non-discounted’ cost by the CLP firm from its multiple commercial partners and supplied to customers seeking to redeem their accumulated ‘reward points’, subject to commercial partners’ capacities for offering rewards, the firm’s overall budget for purchasing rewards, and its control policy on points liability. We formulate the problem as a stochastic linear programme with recourse and solve it using a sampling-based heuristic solution procedure previously discussed in the literature. We report on the managerial applicability of our model in dealing with the redemption budget spending resulting from changes in demand variability, changes in the redemption budget, and the control of liability levels within a reasonable range.