Cost Allocation Game Research Articles

Sharing congestion management costs among system operators using the Shapley value

With energy generation becoming increasingly decentralized, the need for congestion management across grid voltage levels is also increasing. To enable fair sharing of congestion costs among grid operators, these costs must be allocated to congested grid elements. We propose using the Shapley value for this purpose. The Shapley value is a cooperative game theory concept that was developed to share a total surplus generated by a coalition of players between the players based on their marginal contributions to the coalition. We apply this concept to share the costs of congestion management between grid elements based on their contributions to overall congestion management costs. To reduce the computational complexity of the Shapley value, we introduce two novel simplification approaches and compare them to existing methods using a numerical example based on CIGRE benchmark grids. The first method exploits the fact that the characteristic function for the congestion costs is obtained from an optimal power flow computation (i.e., a constrained optimization problem). It utilizes knowledge about which constraints are non-binding in the optimization to derive the values of related coalitions without calculating them. The second method takes advantage of the fact that the congestion management cost-allocation game is monotone and derives the values of coalitions based on this property. Both methods are implemented and compared to sampling. Using the first method, we are able to reduce computational complexity to less than 20% of that of the original problem while maintaining exact results. Our second approach is not dependent on detailed knowledge of the underlying optimization problem and can reduce the computational time by almost half with exact results and much further when compromising precision. While the methods are presented through an application example, they can be applied to other games with similar properties.

A Demand Response Trade Model Considering Cost and Benefit Allocation Game and Hydrogen to Electricity Conversion

With the widespread utilization of hydrogen in industry and the development of hydrogen-to-electricity (H2E) technology, how to design an efficient market-based demand response trade mechanism draws lots of attention for incentivizing industrial users to promote their demand response quantities and willingness with rational use of hydrogen. This article proposes an auction model of buyers and sellers considering the cost and benefit allocation game among electric transaction manager (ETM), electric response agents (ERAs), and electric-hydrogen industrial users (EHIUs) in demand response trade. First, a participant-led distributed trading framework of electricity demand response (EDR) with ETM, ERAs, and EHIUs is proposed. Second, in order to inspire ERAs to facilitate EDR trade and enhance EHIUs’ willingness to participate in the EDR, a bargain game of the cost and benefit allocation among ETM, ERAs, and EHIUs is established. Then, a co-optimization model with electricity and hydrogen considering production constraints is formed for promoting the ability and economy of EDR of EHIUs. Finally, a modified IEEE-69 node system is employed to illustrate the validity of the proposed method.

Correction to: Fair ticket pricing in public transport as a constrained cost allocation game

The authors wish to correct the acknowledgement in page 68 from The work of Nam-Dũng Hoang is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED).

A cooperative game approach to cost allocation in a rapid-transit network

We consider the problem of allocating costs of a regional transit system to its users, who employ shortest path routes between all pairs of nodes in the system network. We provide an axiomatic set of conditions that a solution should satisfy and use cooperative game theory to model the cost allocation problem. We provide an allocation, called the equal cost share solution, which is efficient to compute and is the unique solution that satisfies the conditions. In addition, we show not only that the cost allocation game has a nonempty core, but further, that the game is concave, meaning that the Shapley value allocation, which coincides with the equal cost share solution, always lies in the core of the game. We provide an application of the equal cost share solution to the Washington, D.C. Metro transit network and compare it to the existing fare pricing structure. As compared to equal cost share pricing, the Metro overcharges for short downtown trips and undercharges for very long commutes. The equal cost share solution is easy to update in real time as the cost data and user distribution change, or when the transit network expands.

Stable solutions for multiple scenario cost allocation games with partial information

Multiple scenario cooperative games model situations where the worth of the coalitions is simultaneously valued in several scenarios or under different states of nature. For multiple scenario cost allocation games, we focus on the identification of those allocations which are stable in the sense that agents have no incentives to abandon the group. The stability of an allocation depends on how the quantities assigned to the coalitions are compared with the vector-valued worth of the coalition. Therefore, different extensions of the notion of core emerge, depending on how these comparisons are made. In this paper different notions of core for multiple scenario cost allocation games are studied, and the relationships between them and the existing core solutions are analyzed. We also address the inclusion in the model of partial information about the relative importance of the scenarios which often refers to the probabilities of occurrence. In order to identify allocations which are also stable in the presence of partial information, we extend the notions of core to this new setting, and provide results which permit the calculation of the corresponding sets of allocations. The potential applicability of our analysis is illustrated with the allocation of the total cost of the waste collection service between the different urban areas in the city of Seville.

Fair ticket pricing in public transport as a constrained cost allocation game

Ticket pricing in public transport usually takes a welfare maximization point of view. Such an approach, however, does not consider fairness in the sense that users of a shared infrastructure should pay for the costs that they generate. We propose an ansatz to determine fair ticket prices that combines concepts from cooperative game theory and linear and integer programming. The ticket pricing problem is considered to be a constrained cost allocation game, which is a generalization of cost allocation games that allows to deal with constraints on output prices and on the formation of coalitions. An application to pricing railway tickets for the intercity network of the Netherlands is presented. The results demonstrate that the fairness of prices can be improved substantially in this way.

The [formula omitted]-centrum Chinese Postman delivery problem and a related cost allocation game

We analyze the cost allocation cooperative game, denoted by (N,ck), induced by the k-centrum Chinese Postman (k-centrum CP) delivery problem in connected undirected and strongly connected directed graphs. For the undirected case, we show, for example, that for k=1,2, (N,ck) is submodular for all graphs having non-negative edge-weights, for all locations of the post-office. For k≥3, we prove that (N,ck) is submodular for all non-negative edge-weights and for all locations of the post office if and only if the graph is either the cyclic graph on three edges or it is a graph wherein each edge is contained in at most one cycle and these cycles, if any, have only two edges. For the directed graph case we show, for example, that the k-centrum CP game induced by a strongly connected graph G is submodular for all k≥2 if and only if every arc in G is contained in precisely one directed cycle.

Cost Allocation Under Competition: How to Reduce Externalities?

We consider a setting in which a not-for-profit rail infrastructure provider divides the costs of track maintenance and renewals and the costs of managing track access proportionally among above-rail operators based on their usage. We study a proposed access charge policy aimed at incentivizing the above-rail operators to use long train configurations in order to reduce congestion in the system. The proposed access charge policy sets a target length of a train configuration and gives a discount on an operator's access charge if the operator employs a train configuration of at least the target length. As a result, the access charge of an operator is no longer independent of the actions of other operators, and operators need to be strategic in their decision making. We discuss the different situations that may arise in the induced cost allocation game for two operators for a given discount factor and establish conditions under which a discount factor incentivizes both operators to employ train configurations of at least the target length. If there does not exist a discount factor that incentivizes both operators to employ long train configurations, then we demonstrate that setting a sub-optimal target and a discount factor that incentivizes both train operators to reach that target may be beneficial in terms of reducing congestion.

REDUCTION OF ULTRAMETRIC MINIMUM COST SPANNING TREE GAMES TO COST ALLOCATION GAMES ON ROOTED TREES

A minimum cost spanning tree game is called ultrametric if the cost function on the edges of the underlying network is an ultrametric. We show that every ultrametric minimum cost spanning tree game is reduced to a cost allocation game on a rooted tree. It follows that there exist O(n 2 ) time algorithms for computing the Shapley value, the nucleolus and the egalitarian allocation of the ultrametric minimum cost spanning tree games, where n is the number of players.

Allocation of gains from inventory centralization in newsvendor environments

A company considers centralizing a single item inventory ordering for a number of stores whose demand fluctuates randomly. First, there must be savings passed on to the stores from this centralization arrangement. Then, the savings must be divided among the participating stores in a way that no store (or a subset) will have an incentive to order separately. We model this problem as a cooperative game whose players are the stores. When holding and penalty shortage costs are identical for all subsets of stores, a game based on optimal expected costs (or the corresponding benefits) is subadditive (there are savings from centralization), and for normally distributed demands, whatever their correlations the core is never empty When the stores' holding and penalty costs differ, the corresponding game may have an empty core, and in fact, centralization may not be beneficial. We give conditions on the holding and penalty costs that ensure subadditivity. Given inventory centralization and a cost allocation game based on demand realizations, even in the case of identical holding and penalty costs the cost game in each period might have an empty core. We give sufficient conditions for such an allocation to be justifiable and subsidy-free (nonempty core) and examine properties of a number of ex ante—ex post cost allocation procedures.

Allocating the Cost of Congestion with the Nucleolus

A natural approach to solve resource sharing problems is to model them as cooperative games and use the results to allocate the costs of the shared resource. The nucleolus, a solution concept derived from cooperative game theory, requires an exponential number of computations since the solution must adhere to individual and coalitional rationality conditions. Littlechild (1974) provides a linear algorithm which produces the nucleolus of a cost allocation game when the cost of a coalition is the cost of the largest player in that coalition. In this paper, we show that for a large class of congestion cost allocation games, where each player has an independent impact on the shared resource, we can nonetheless bypass all the computational complexity and derive allocations in closed form.

Some Results in Cost Allocation Games

Abstract The paper is mainly devoted to the separable cost allocations of a cost allocation game. The first part of the paper (Section 2) deals with the conditions which are sufficient for the (n–1)-person coalitions to be effective at the separable cost allocations. The second part (Section 3) considers the results related to the dummy players and the last part (Section 4) concerns the results related to the core catcher in cost allocation game.

On the core of a traveling salesman cost allocation game

Let G = ( V, E) be a connected undirected graph with positive edge lengths. Let V = {0} ∪ N, where N = {1,…, n}. Each node in N is identified as a customer, and 0 is the home location of a traveling salesman or repairman who serves the customers in N. Each subset of customers S can hire the repairman to serve its members only. In that case the cost incurred by S, c( S), is the minimum length of a tour traversed by the repairman who starts at node 0, visits each node in S at least once and returns to 0. We consider the core of the cooperative cost allocation game ( N; c) defined by the cost function c( S), S ⊆ N. We show that the core can be empty even if G is series parallel by presenting the unique minimal counter example for such graphs. We then use a recent result of Fonlupt and Naddef, and prove that the core is nonempty for a class of graphs that properly contains the subclass of cycle tress, i.e. graphs which have no edge included in more than one simple cycle.

Conflict Analysis of Water Resources Allocation

This paper focusses the significance of the total load control scheme which was recently introduced by the Japanese Government, to apply to designated closed water bodies in Japan. By showing that part of the problem is the resolution of conflicts in the allocation of environmental loads to be discharged by interested parties, a game-theoretic model by use of cooperative game theory has been presented. Mathematical analyses have been performed to derive theoretical properties of the two-person game that has been formulated as a prototype model of the conflict.It has been shown that what originally appears to be a game of COD load allocation can be reduced to what is called a cost allocation game. This latter type of allocation game has been proved to be largely determined by an a priori assignment of total COD loads dischargeable to each player.Illustration has been made of the necessity of setting up on a higher level than this game, some fairness norms that may bind what follows as a cost allocation game. The paper concludes by suggesting possible further generalization of the proposed two-person game into an n-person game.