Classical cooperative game theory is no longer a suitable tool for those situations where the values of coalitions are not known with certainty. We consider a dynamic context where at each point in time the coalitional values are unknown but bounded by a polyhedron. However, the average value of each coalition in the long run is known with certainty. We design “robust” allocation rules for this context, which are allocation rules that keep the coalition excess bounded while guaranteeing each player a certain average allocation (over time). We also present a joint replenishment application to motivate our model
For cooperative games with transferable utility, convexity has turned out to be an important and widely applicable concept. Convexity can be defined in a number of ways, each having its own specific attractions. Basically, these definitions fall into two categories, namely those based on a supermodular interpretation and those based on a marginalistic interpretation. For games with nontransferable utility, however, the literature mainly focuses on two kinds of convexity, ordinal and cardinal convexity, which both extend the supermodular interpretation. In this paper, we analyse three types of convexity for NTU games that generalise the marginalistic interpretation of convexity.
We consider networks of queues in which the independent operators of individual queues may cooperate to reduce the amount of waiting. More specifically, we focus on Jackson networks in which the total capacity of the servers can be redistributed over all queues in any desired way. If we associate a cost to waiting that is linear in the queue lengths, it is known how the operators should share the available service capacity to minimize the long run total cost.We answer the question whether or not (the operators of) the individual queues will indeed cooperate in this way, and if so, how they will share the cost in the new situation. One of the results is an explicit cost allocation that is beneficial for all operators. The approach used also works for other cost functions, such as the server utilization.
This paper introduces a new model concerning cooperative situations in which the payoffs are modeled by random variables. First, we study adequate preference relations of the agents. Next, we define corresponding cooperative games and we introduce and study various basic notions like an allocation, the core and marginal vectors. Furthermore, we introduce three types of convexity, namely coalitional-merge, individual-merge and marginal convexity. The relations between these definitions are studied and in particular, as opposed to the deterministic counterparts for TU games, we show that these three types of convexity are not equivalent. However, all types imply that the core of the game is nonempty and the first two types even imply that each subgame has a nonempty core. In particular, we show that the Shapley value, the average of the marginal vectors, belongs to the core of the convex game for certain types of preferences and for any type of convexity. Journal of Economic Literature Classification Number: C71.1991 Mathematics Subject Classification Number: 90D12.
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