Aumann-Shapley Value Research Articles

Collaborative scheduling and benefit allocation for waste-to-energy, hydrogen storage, and power-to-gas under uncertainties with temporal relevance

Waste-to-energy (WTE) is a technology that converts inexhaustible waste into electricity, thus significantly alleviating energy crisis. Integrating hydrogen storage into WTE provides an effective way to store and utilize surplus electricity from WTE on-site (by producing hydrogen). Additionally, integrating power-to-gas (P2G) into WTE provides an effective way to reduce CO2 emission from WTE on-site (by producing natural gas). To this end, a novel waste-driven renewable energy system (WRES), which is consisted of WTE, hydrogen storage, and P2G, has been proposed. However, the uncertain waste supply and electrical load markedly interfere with WRES operation. Meanwhile, when WTE, hydrogen storage, and P2G belong to different agents, the collaborative benefit from WRES operation should be rationally allocated among agents for system sustainability. This paper endeavors to achieve WRES scheduling and collaborative benefit allocation for WTE, hydrogen storage, and P2G under uncertainties. Firstly, a temporal relevance based distributionally robust optimization model is proposed for WRES scheduling under uncertainties, in which the possible range of the joint distribution for uncertainties is depicted by data covariance matrices involved ambiguity set. Secondly, collaborative benefit is allocated according to WTE, hydrogen storage, and P2G contributions, in which Gauss-Legendre quadrature formula is integrated with Aumann-Shapley value method to reduce calculation complexity. Finally, simulation results show that 1) the proposed scheduling model guarantees the economic and stable operation of WRES under uncertainties. 2) after considering the temporal relevance, WRES operation benefit is 8.35 % higher, which indicates that the proposed scheduling model has superiority in decision conservatism by introducing temporal relevance to remove impractical distributions in ambiguity set. 3) the proposed allocation model rationally distributes the collaborative benefit according to contributions, and presents lower calculation complexity, e.g., the calculation time is 95.52 % lower than the Shapely value method. This paper provides policy insights to promote widespread application and sustainable operation of WRES.

Benefit allocation for combined heat and power dispatch considering mutual trust

Combined heat and power dispatch (CHPD) has attracted wide attention recently, which exploits the potential flexibility in the district heating system (DHS) and provides an economic-effective solution for better wind accommodation. However, DHS operators bear extra operation cost in CHPD, which is proved theoretically in this paper, so that they have insufficient incentive to coordinate with electrical power system. To cope with this issue, a cost allocation strategy based on the Aumann-Shapley value method is proposed, which can rationally allocate the benefit among multiple agents. And a new solution scheme with the Gauss-Legendre quadrature formula is developed to enhance computation efficiency. Also, considering the potential malicious behavior of dishonest agents in the process of allocation, the blockchain framework with Proof of Solution consensus mechanism is introduced to ensure mutual trust among multiple agents. Numerical simulations based on different cases demonstrate the effectiveness of the proposed method of benefit allocation among multiple agents. In addition, the validity of the proposed blockchain framework in preventing malicious behavior when CHPD is verified.

Responsibility Attribution in Parameterized Markovian Models

We consider the problem of responsibility attribution in the setting of parametric Markov chains. Given a family of Markov chains over a set of parameters, and a property, responsibility attribution asks how the difference in the value of the property should be attributed to the parameters when they change from one point in the parameter space to another. We formalize responsibility as path-based attribution schemes studied in cooperative game theory. An attribution scheme in a game determines how a value (a surplus or a cost) is distributed among a set of participants. Path-based attribution schemes include the well-studied Aumann-Shapley and the Shapley-Shubik schemes. In our context, an attribution scheme measures the responsibility of each parameter on the value function of the parametric Markov chain. We study the decision problem for path-based attribution schemes. Our main technical result is an algorithm for deciding if a path-based attribution scheme for a rational (ratios of polynomials) cost function is over a rational threshold. In particular, it is decidable if the Aumann-Shapley value for a player is at least a given rational number. As a consequence, we show that responsibility attribution is decidable for parametric Markov chains and for a general class of properties that include expectation and variance of discounted sum and long-run average rewards, as well as specifications in temporal logic.

Incremental Benefit Allocation for Joint Operation of Multi-Stakeholder Wind-PV-Hydro Complementary Generation System With Cascade Hydro-Power: An Aumann-Shapley Value Method

In order to ensure joint operation of wind-PV-hydro complementary generation system with cascade hydro-power owned by multiple stakeholders, an equitable and efficient allocating mechanism for incremental benefit is of great importance. Firstly, the incremental benefit from joint operation is quantified for wind-PV-hydro complementary generation system with cascade hydro-power owned by multiple stakeholders in this paper. Then, different allocation methods such as proportional method, marginal benefits method, last addition method, Shapley value method, and Aumann-Shapley value method are applied to allocate the incremental benefit for each stakeholder. Next, a 4-stakeholder wind-PV-hydro complementary generation test system is employed to investigate the advantages and disadvantages of the presented different allocation methods and a 9-stakeholder wind-PV-hydro complementary generation test system is used to verify that Aumann-Shapley value method can reduce computational burden with regard to the participation of large number of stakeholders compared with traditional cooperative game-based method such as Shapley value method.

A generalization of the Aumann–Shapley value for risk capital allocation problems

The paper proposes a new method to allocate risk capital to divisions or lines of business within a firm. Existing literature advocates an allocation rule that, in game-theoretic terms, is equivalent to using the Aumann–Shapley value as allocation mechanism. The Aumann–Shapley value, however, is only well-defined if a specific differentiability condition is satisfied. The rule that we propose is characterized as the limit of an average of path-based allocation rules with grid size converging to zero. The corresponding allocation rule is equal to the Aumann–Shapley value if it exists. If the Aumann–Shapley value does not exist, the allocation rule is equal to the weighted average of the Aumann–Shapley values of “nearby” capital allocation problems.

Shapley Value-Based Techno-Economic Framework for Harmonic and Loss Mitigation

This study presents a multi-objective policy for harmonic distortion and loss mitigation in micro-grid and active distribution grid. The implementation of this policy in networks including Distributed Generation (DG), is put forward based on the determination of the locational marginal price of each DG bus, considering their impacts on the loss and harmonic mitigation. In this process, each DG receives an incentive in the form of price increment, based on the Aumann-Shapley value game-theoretic method as an iterative algorithm, considering the effectiveness of reducing loss or harmonic distortion. Furthermore, as a decision-making tool, the operator can change the contribution of the incentives used to mitigate the loss or harmonic distortion and make an appropriate estimation for the next step conditions. The simulations are organized based on real data on the modified Taiwan Power Company grid (TPC). The modified TPC consists of renewable energy sources (RES) and fossil-fuel based DG units. Also, to achieve a realistic framework, generation of RES, loads, and market prices are considered as uncertain parameters. The results demonstrate the competence of the proposed method for the TPC network in terms of triple pricing methods comparison, total loss reduction, harmonic mitigation, and merchandising surplus controlling.

Loss and emission reduction allocation in distribution networks using MCRS method and Aumann–Shapley value method

In order to provide reasonable economic signals for distribution companies (DISCOs) and further compensate for distributed generators (DGs) integrated in distribution networks equitably, the contributions of DGs to loss and emission reduction in distribution networks should be allocated according to their own responsibilities. Generally, the loss and emission reduction of the network can be allocated based on traditional cooperative-game-based allocation methods such as nucleolus method and Shapley value method. However, traditional cooperative-game-based allocation methods will result in the combinational explosion problem with the integration of a large number of DGs. In order to tackle this problem, minimum costs-remaining savings (MCRS) method and Aumann-Shapley value method are employed for loss and emission reduction allocation. Simulation results of two cases show that compared with the allocation results of traditional cooperative-game-based allocation methods, the proposed MCRS method and Aumann-Shapley value method both have the characteristics of individual rationality, coalition rationality, and global rationality. Furthermore, neither MCRS method nor Aumann-Shapley value method has the problem of combinational explosion, which can reduce computational burden with regard to the integration of a large number of DGs.

A Fair Pool Sharing Mechanism for Illiquid Catastrophe Risk Markets

For assuming risk in a multi-party risk pool of several insurance companies, a pool sharing mechanism, i.e., a type of pricing mechanism, is needed to determine each participant’s share of the pool’s losses. Based on axiomatic capital allocation principles, a fair and unique pool sharing mechanism is derived that differs from the current pooling solutions. Insights from game theory show that the mechanism is optimal in the sense of the Aumann-Shapley value. For this new application of these two theories, only small adaptations are necessary which are related to the differentiation between expected and unexpected losses. It is discussed why multi-party risk pools with this type of a pool sharing mechanism are particularly suitable for illiquid catastrophe risk markets. Special attention is given to the market for terrorism risk. A case study for portfolios of concentrated group life exposures shows that this type of a multi-party risk pool is feasible and economically beneficial.

A Generalization of the Aumann-Shapley Value for Risk Capital Allocation Problems

This paper analyzes risk capital allocation problems. For risk capital allocation problems, the aim is to allocate the risk capital of a firm to its divisions. Risk capital allocation is of central importance in risk-based performance measurement. We consider a case in which the aggregate risk capital is determined via a coherent risk measure. The academic literature advocates an allocation rule that, in game-theoretic terms, is equivalent to using the Aumann-Shapley value as solution concept. This value is however not well-defined in case a differentiability condition is not satisfied. As an alternative, we introduce an allocation rule inspired by the Shapley value in a fuzzy setting. We take a grid on a fuzzy participation set, define paths on this grid and construct an allocation rule based on a path. Then, we define a rule as the limit of the average over these allocations, when the grid size converges to zero. We introduce this rule for a broad class of coherent risk measures. We show that if the Aumann-Shapley value is well-defined, the allocation rule coincides with it. If the Aumann-Shapley value is not defined, which is due to non-differentiability problems, the allocation rule specifies an explicit allocation. It corresponds with the Mertens value, which is originally characterized in an axiomatic way (Mertens, 1988), whereas we provide an asymptotic argument.

Comparison between Financial Theory and Cooperative Game Theory in Risk Capital Allocation

We compare two prominent approaches to capital allocation in insurance firms. The financial theory approach includes Merton and Perold (1993) and Myers and Read (2001). The cooperative game theory approach utilizes concepts such as the Shapley value and the Aumann-Shapley value. We argue that, when an entire division is added or when the effect of a decision is discrete, the Shapley value approach provides an improvement over the Merton and Perold approach in that it properly accounts for the order in which divisions are added, and resoles the unallocated capital problem. When the effect of a decision is continuous, we show that the Auman-Shapley value approach not only provides game theoretic support for, but also conceptually extends, the Myers and Read approach.

Comparison between Financial Theory and Cooperative Game Theory in Risk Capital Allocation

We compare two prominent approaches to capital allocation in insurance firms. The financial theory approach includes Merton and Perold (1993) and Myers and Read (2001). The cooperative game theory approach utilizes concepts such as the Shapley value and the Aumann-Shapley value. We argue that, when an entire division is added or when the effect of a decision is discrete, the Shapley value approach provides an improvement over the Merton and Perold approach in that it properly accounts for the order in which divisions are added, and resoles the unallocated capital problem. When the effect of a decision is continuous, we show that the Auman-Shapley value approach not only provides game theoretic support for, but also conceptually extends, the Myers and Read approach.

On the Aumann–Shapley value

Let L be a D-lattice, i.e. a lattice ordered effect algebra, and let BV be the Banach space of all real-valued functions of bounded variation on L (vanishing at 0) endowed with the variation norm. We prove the existence of a continuous Aumann–Shapley value φ on \(\mathfrak{bv}^\prime\)NA, the subspace of BV spanned by all functions of the form \(f\circ\mu\), where \(\mu: L \to [0,1]\) is a non-atomic σ-additive modular measure and \(f: [0,1] \to {\mathbb{R}}\) is of bounded variation and continuous at 0 and at 1.

To split or not to split: Capital allocation with convex risk measures

Convex risk measures were introduced by Deprez and Gerber [Deprez, O., Gerber, H.U., 1985. On convex principles of premium calculation. Insurance: Math. Econom. 4 (3), 179–189]. Here the problem of allocating risk capital to subportfolios is addressed, when convex risk measures are used. The Aumann–Shapley value is proposed as an appropriate allocation mechanism. Distortion-exponential measures are discussed extensively and explicit capital allocation formulas are obtained for the case that the risk measure belongs to this family. Finally the implications of capital allocation with a convex risk measure for the stability of portfolios are discussed. It is demonstrated that using a convex risk measure for capital allocation can produce an incentive for infinite fragmentation of portfolios.

To Split or Not to Split: Capital Allocation with Convex Risk Measures

Convex risk measures were introduced by Deprez and Gerber (1985). Here the problem of allocating risk capital to subportfolios is addressed, when aggregate capital is calculated by a convex risk measure. The Aumann-Shapley value is proposed as an appropriate allocation mechanism. Distortion-exponential measures are discussed extensively and explicit capital allocation formulas are obtained for the case that the risk measure belongs to this family. Finally the implications of capital allocation with a convex risk measure for the stability of portfolios are discussed.

Procedure for Bridge Construction Cost Allocation Based on Game Theory

Traditional procedures for bridge construction cost allocation divide costs between vehicle classes on the basis of gross vehicle weight (traffic load) and ignore the effect of traffic capacity (traffic lane) requirements. This paper presents a new cost allocation method for bridge construction that integrates both traffic capacity and traffic load requirements. The proposed method is based on three allocation criteria known as Aumann-Shapley value, Shapley value, and incremental allocation. The first two criteria are game theory calculations defined as average marginal values associated with all possible ordered arrangements of a set of players. Depending on the criteria, players are defined as vehicle classes, 18-kip single-axle load applications, or traffic lanes. The incremental allocation method considers the vehicle classes in a specific order, usually on the basis of vehicle weight, and determines allocation costs as marginal costs after sequentially including individual vehicle classes and assessing total costs for bridges accommodating all vehicle classes included. An example is provided to illustrate the proposed procedure. For comparison purposes, cost allocations by other available methods are also provided for the example. This paper also provides an insight into the relationship between traffic capacity and bridge construction cost allocation.

A Non-standard Analysis of Aumann-Shapley Random Order Values of Non-atomic Games

Using techniques from the non-standard analysis, a non-standard analogue of the Aumann-Shapley random order value of non-atomic games is provided. The paper introduces the notion of effectively ergodic family of automorphism groups. It is shown that for a wide class of games, the non-standard random order value with respect to an effectively ergodic family of automorphism groups coincides with the standard Aumann-Shapley value.

An Aumann-Shapley Value Based Congestion Cost Allocating Method In Bilateral Transaction Environment

In bilateral transaction market, the congestion relief cost should be allocated in an equitable manner among all the market participants contributing to the congestion. Unfortunately in the allocation of the congestion relief cost to the congested lines, which is the first step of cost allocation, the previously proposed shadow price method and the aggregated allocation method both have some shortcomings. To overcome these shortcomings, a cost allocation method based on the Aumann-Shapley value is proposed. By means of the Aumann-Shapley value, a solution concept in the cooperative game with infinitely many players, the proposed method takes all orders of relieving congestion on the lines into consideration and satisfies the consistency of the allocation, thus guarantees equitable allocation of the congestion relief cost to the congested lines. In addition, the calculating procedure is both simple and accurate due to the use of the Gauss Integration method. Test results show that this method can lead to better allocation than do the shadow price method and the aggregated allocation method.

Potential, Consistency, and Cost Allocation Prices

In a typical cost allocation problem, a decision maker must allocate the joint cost of producing a specified vector of goods or services using prices. In earlier work, Billera, Heath, and Raanan applied the theory of values of nonatomic games of Aumann and Shapley to develop a pricing mechanism called Aumann-Shapley pricing. The Aumann-Shapley value for nonatomic games has an axiomatic foundation that was adapted to the cost allocation context by Billera and Heath and Mirman and Tauman. In this paper, we develop the notions of “reduced cost function,” “consistency,” and “potential” for cost allocation problems and we provide a new axiomatic characterization of symmetric and weighted Aumann-Shapley pricing in terms of consistency.

Continuous integration congestion cost allocation based on sensitivity

Congestion cost allocation is a very important topic in congestion management. Allocation methods based on the Aumann-Shapley value use the discrete numerical integration method which needs to solve the incremented OPF solution many times and as such it is not suitable for practical application to large-scale systems. The optimal solution and its sensitivity change tendency during congestion removal using a DC optimal power flow (OPF) process is analysed. A simple continuous integration method based on the sensitivity is proposed for the congestion cost allocation. The proposed sensitivity analysis method needs a smaller computation time than the method based on using the quadratic method and inner point iteration. The proposed congestion cost allocation method uses a continuous integration method rather than discrete numerical integration. The method does not need to solve the incremented OPF solutions; which allows it use in large-scale systems. The method can also be used for AC OPF congestion management.

Risk capital allocation and cooperative pricing of insurance liabilities

The Aumann–Shapley [Values of Non-atomic Games, Princeton University Press, Princeton] value, originating in cooperative game theory, is used for the allocation of risk capital to portfolios of pooled liabilities, as proposed by Denault [Coherent allocation of risk capital, J. Risk 4 (1) (2001) 1]. We obtain an explicit formula for the Aumann–Shapley value, when the risk measure is given by a distortion premium principle [Axiomatic characterisation of insurance prices, Insur. Math. Econ. 21 (2) (1997) 173]. The capital allocated to each instrument or (sub)portfolio is given as its expected value under a change of probability measure. Motivated by Mirman and Tauman [Demand compatible equitable cost sharing prices, Math. Oper. Res. 7 (1) (1982) 40], we discuss the role of Aumann–Shapley prices in an equilibrium context and present a simple numerical example.