A Game Theoretical Approach to Sharing Penalties and Rewards in Projects

This paper analyzes situations in which a project consisting of several activities is not realized according to plan. If the project is expedited, a reward arises. Analogously, a penalty arises if the project is delayed. This paper considers the case of arbitrary monotonic reward and penalty functions on the total expedition and delay, respectively. Attention is focused on how to divide the total reward (penalty) among the activities: the core of a corresponding cooperative project game determines a set of stable allocations of the total reward (penalty). In the definition of project games, surplus (cost) sharing mechanisms are used to take into account the specific characteristics of the reward (penalty) function at hand. It turns outs that project games are related to bankruptcy and taxation games. This relation allows us to establish the nonemptiness of the core of project games.Project planning;delay;expedition;cost sharing mechanism;surplus sharing mechanism;bankruptcy problems;taxation problems;cooperative game;core

New Characterizations for Largeness of the Core

In this paper, we provide three new characterizations of largeness of the core. The first characterization is based on minimal covers of the grand coalition and associated inequalities. The second characterization shows the relation between the bases that provide core elements of the game and the bases that provide core elements of the games that are obtained from the original one by increasing the value of the grand coalition. The third characterization is based on the idea that if a base of the grand coalition does not provide a core element of the game, it should not provide a core element of a game which differs from the original one only by an increase of the value of the grand coalition. Based on these new characterizations, we show the equivalence between largeness of the core and stability of the core for games with at most 5 players. © 2012 Elsevier Inc.

On the Core of Multiple Longest Traveling Salesman Games

In this paper we introduce multiple longest traveling salesman (MLTS) games. An MLTS game arises from a network in which a salesman has to visit each node (player) precisely once, except its home location, in an order that maximizes the total reward.First it is shown that the value of a coalition of an MLTS game is determined by taking the maximum of suitable combinations of one and two person coalitions.Secondly it is shown that MLTS games with ¯ve or less players have a nonempty core.However, a six player MLTS game may have an empty core.For the special instance where the reward between a pair of nodes is equal to 0 or 1, we provide relations between the structure of the core and the underlying network.game theory;traveling salesman problem;games;core

Proportionate Flow Shop Games

In a proportionate flow shop problem several jobs have to be processed through a fixed sequence of machines and the processing time of each job is equal on all machines.By identifying jobs with agents, whose costs linearly depend on the completion time of their jobs, and assuming an initial processing order on the jobs, we face an additional problem: how to allocate the cost savings obtained by ordering the jobs optimally?In this paper, PFS games are defined as cooperative games associated to proportionate flow shop problems.It is seen that PFS games have a nonempty core.Moreover, it is shown that PFS games are convex if the jobs are initially ordered in decreasing urgency.For this case an explicit expression for the Shapley value and a specific type of equal gain splitting rule which leads to core elements of the PFS game are proposed.Proportionate flow shop problems;core;convexity

Project Games

This paper studies situations in which a project consisting of several activities is not executed as planned.It is divided into three parts.The first part analyzes the case where the activities may be delayed; this possibly induces a delay on the project as a whole with additional costs.Associated delayed project games are defined and are shown to have a nonempty core.The second part considers the case where the activities may be expedited; this possibly induces an expedition of the project as a whole creating profits. Corresponding expedited project games are introduced and are shown to be convex. The third and last part studies situations where some activities may be delayed and some activities may be expedited.Related project games are defined and shown to have a nonempty core.project planning;delay;expedition;cooperative games;convexity

Competitive Environments and Protective Behaviour

The class of two-person competition games is introduced and analyzed.For any game in this class the set of Nash equilibria is convex, equilibrium strategies are exchangeable, and all Nash equilibria lead to the same payoff vector. Competition games are compared to other competitive environments such as unilaterally competitive games and rivalry games.Moreover, protective behaviour within competitive environments is analyzed.For matrix games it is known that protective strategies pro¯les exactly correspond to proper equilibria.It is shown that this result can be extended to the class of unilaterally competitive games.competitive environments;unilaterally competitive games;rivalry games;competition games;protective strategies

Job Scheduling, Cooperation and Control

This paper considers one machine job scheduling situations or sequencing problems, where clients can have more than a single job to be processed in order to get a final output.Moreover, a job can be of interest for different players. This means that one of the main assumptions in classic sequencing problems is dropped: the one to one correspondence between clients and jobs.It is shown that the corresponding cooperative games are balanced for specific types of cost criteria.scheduling;cooperation;game theory;cooperative games