Aumann-Shapley Pricing Research Articles

Real-Time Bidding Strategy of Energy Storage in an Energy Market with Carbon Emission Allocation Based on Aumann-Shapley Prices

Real-Time Bidding Strategy of Energy Storage in an Energy Market with Carbon Emission Allocation Based on Aumann-Shapley Prices

Applied cost allocation: The DEA–Aumann–Shapley approach

This paper deals with empirical computation of Aumann–Shapley cost shares for joint production. We show that if one uses a mathematical programing approach with its non-parametric estimation of the cost function there may be observations in the data set for which we have multiple Aumann–Shapley prices. We suggest to overcome such problems by using lexicographic goal programing techniques. Moreover, cost allocation based on the cost function is unable to account for differences between efficient and actual cost. We suggest to employ the notion of rational inefficiency in order to supply a set of assumptions concerning firm behavior. These assumptions enable us to connect inefficient with efficient production and thereby provide consistent ways of allocating the costs arising from inefficiency.

Monotonicity and the Aumann–Shapley cost-sharing method in the discrete case

We give an axiomatization of the Aumann–Shapley cost-sharing method in the discrete case by means of monotonicity and no merging or splitting (Sprumont, 2005). Monotonicity has not yet been employed to characterize this method in such a case, by contrast with the case in which goods are perfectly divisible, for which Monderer and Neyman (1988) and Young (1985b) characterize the Aumann–Shapley price mechanism.

Allocating the CO 2 emissions of an oil refinery with Aumann–Shapley prices: A reply

In this reply, I oppose and further debate some of the points raised in Mr Tehrani's comment (2010). In addition, I show that, when dealing with short-run linear-programming models with not-adjusted-to-demand capacities, Aumann–Shapley prices can be considered as an attempt to recreate long-run marginal costs.

Allocating the CO 2 emissions of an oil refinery with Aumann–Shapley prices: Comment

The allocation of CO 2 emissions of petroleum refineries to their oil products is a necessary step in the retrospective Well-to-Tank (WTT) analysis. These allocated emissions are used to evaluate the environmental impacts of automotive fuels' production within the refinery. Oil refining is a complex joint production system and there exists no simple and unique answer to this allocation question. Recently, Pierru proposed adapting the Aumann–Shapley cost sharing method to deal with this issue. Our paper aims at describing the conceptual and technical difficulties of this adaptation to the WTT context. Moreover, we show that this approach, as proposed by Pierru, is not applicable to any real-type refinery model. Different suggestions are provided to improve its applicability (when it is possible) in real situations. A simple numerical example as well as a real-type refinery case study is provided for illustrations. Finally, we discuss an alternative allocation approach which we believe more adapted to the WTT context.

Highway Toll Pricing

For a tolled highway where consecutive sections allow vehicles enter and exit unrestrictedly, we propose a simple toll pricing method. We show that the method is the unique method that satisfies the properties of Transit-proofness (No Merging or Splitting) and Cost Recovery. The Transit-proofness property rules out strategic respones of road users against a toll pricing scheme while the Cost Recovery rules out cross subsidizations between road sections with regard to their costs. Furthermore, it is shown that the toll pricing method is the discrete version of Aumann-Shapley pricing rule. When there is unit traffic (vehicle) for each (feasible) pair of entrance and exit, we show that our toll pricing method is the Shapley value of an associated game to the problem. Moreover, if there is unit traffic entering at each entrance but they all exit at the last exit, the toll pricing method coincides with the well-known airport landing fee solution-the Sequential Equal Contribution rule by Littlechild and Owen (1973).

An Optimal Congestion and Cost-sharing Pricing Scheme for Multiclass Services

We study in this paper a social welfare optimal congestion-pricing scheme for multiclass queuing services which can be applied to telecommunication networks. Most of the literature has focused on the marginal price. Unfortunately, it does not share the total cost among the different classes. We investigate here an optimal Aumann–Shapley congestion-price which verifies this property. We extend the work on the Aumann–Shapley price for priority services, based on the results on the marginal price: instead of just determining the cost repartition among classes for given rates, we obtain the rates and charges that optimize the social welfare.

Allocating the CO 2 emissions of an oil refinery with Aumann–Shapley prices

Linear programming is widely used by multiproduct oil-refining firms, which minimize a refinery's variable cost under a set of constraints. In addition to operating costs, this variable cost can include the cost associated with the refinery's CO 2 emissions. We suggest a quite general approach combining use of Aumann–Shapley cost-sharing method and breakdown of the objective function of the linear program. This approach determines an appropriate rule for the allocation of the refinery's CO 2 emissions (or, in general, variable costs) among the various finished products, which can be used for purposes of Life Cycle Assessment. A numerical application to a simplified refining model is presented.

On the Discrete Version of the Aumann-Shapley Cost-Sharing Method

Each agent in a finite set requests an integer quantity of an idiosyncratic good; the resulting total cost must be shared among the participating agents. The Aumann–Shapley prices are given by the Shapley value of the game where each unit of each good is regarded as a distinct player. The Aumann–Shapley cost-sharing method charges to an agent the sum of the prices attached to the units she consumes. We show that this method is characterized by the two standard axioms of Additivity and Dummy, and the property of No Merging or Splitting: agents never find it profitable to split or to merge their consumptions. We offer a variant of this result using the No Reshuffling condition: the total cost share paid by a group of agents who consume perfectly substitutable goods depends only on their aggregate consumption. We extend this characterization to the case where agents are allowed to consume bundles of goods.

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.

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.

Marginal cost price rule for homogeneous cost functions

Mirman and Tauman (1982) show that axioms of cost sharing, additivity, rescaling invariance, monotonicity, and consistency uniquely determine a price rule on the class of continuously differentiable cost problems as the Aumann-Shapley price mechanism. Here we prove that standard versions of these axioms determine uniquely the marginal cost price rule on the class of homogeneous and convex cost functions, which are, in addition, continuously differentiable. This result persists even if the cost functons are not required to be convex.

The Aumann–Shapley price mechanism for allocating congestion costs

This paper shows how to apply the Aumann–Shapley price mechanism for charging customers for the congestion costs that they impose on others. The mechanism is demonstrated on a single server queue with Poisson arrivals under various queueing disciplines.

Prices in Mixed Cost Allocation Problems

We consider mixed cost allocation problems, i.e., joint cost problems that involve two types of heterogeneous outputs, divisible and indivisible. The Aumann–Shapley price mechanism is extended to this setting. We also present a set of properties which characterize this cost sharing rule. Journal of Economic Literature Classification numbers: D63, C79.

Aumann-Shapley transmission congestion pricing

This paper describes a methodology for congestion cost allocation based on marginal costs and game-theoretic principles (Aumann-Shapley pricing). The methodology provides fair, economically efficient prices and ensures that the revenue collected from the transmission users will be equal to the cost of relieving the congestion.

Three Methods to Share Joint Costs or Surplus

We study cost sharing methods with variable demands of heterogeneous goods, additive in the cost function and meeting the dummy axiom. We consider four axioms: scale invariance (SI); demand monotonicity (DM); upper bound for homogeneous goods (UBH) placing a natural cap on cost shares when goods are homogeneous; average cost pricing for homogeneous goods (ACPH). The random order values based on stand alone costs are characterized by SI and DM. Serial costsharing, by DM and UB; the Aumann–Shapley pricing method, by SI and ACPH. No other combination of the four axioms is compatible with additivity and dummy. Journal of Economic Literature Classification Numbers: C71, D62, D63

Weighted Aumann-Shapley pricing

Cost allocation problems arise in many contexts in economics and management science. In a typical problem that we have in mind, a decision maker must decide how to allocate the joint cost of production among several commodities using prices. Furthermore, these prices must satisfy certain reasonable postulates among which is the requirement that total revenue associated with these prices must cover total cost. In this paper, we investigate a generalization of Aumann-Shapley pricing, called Weighted Aumann-Shapley pricing, that allows for asymmetric pricing of commodities even when those commodities affect costs in a symmetric fashion. Weighted AS pricing is a natural extension of (symmetric) Aumann-Shapley pricing, and may be considered a non-atomic analogue of Owen's modified diagonal formula (with respect to the multilinear extension) for the weighted TU Shapley Value.

Probabilistic value pricing

In this paper we consider the class of probabilistic value pricing mechanisms for cost allocation problems, which are related to probabilistic values for finite games with transferable utility. We characterize probabilistic value pricing axiomatically, as well as several related pricing mechanisms, including semivalue pricing (symmetric pricing without cost sharing), quasivalue pricing (cost sharing pricing without symmetry), and weighted Shapley value pricing. We also describe a class of problems in which (symmetric) Shapley value pricing coincides with Aumann-Shapley pricing, and a class of problems for which every quasivalue pricing mechanism is supportable.

ON ADDITIVE METHODS TO SHARE JOINT COSTS

The Shapley value theory is extended to cost functions with multiple outputs (or to production functions with multiple inputs) where each output is demanded by a different agent and the level of demand varies. Beyond the Additivity and Dummy axioms (Shapley's original axioms) we insist that the cost-share of an agent should not decrease when she increases her demand (Demand Monotonicity). This property rules out the Aumann-Shapley pricing formula, as well as any method charging average cost for homogeneous goods. We characterize the class of cost sharing methods satisfying Additivity, Dummy, Demand Monotonicity and Cross Monotonicity. The last says that when outputs i and j are cost complements (resp-cost substitutes) the cost share of i is non decreasing (resp-non increasing) in the demand of j. Two prominent methods in the class are the Shapley-Shubik method (i.e. the Shapley value of the Stand Alone cost game) and serial cost sharing (which extends to multiple goods a formula due to Moulin and Shenker). They are characterized respectively by a lower bound and by an upper bound on individual cost shares.

An approach to the pricing of broadband telecommunications services

In this paper, we describe two cost allocation methodologies which may be used to determine prices for a regulated telecommunications supplier offering heterogeneous services on a broadband network. Both methodologies can be characterized by sets of plausible axioms that one could argue should be satisfied by any pricing rule. One of the approaches leads to the Aumann-Shapley pricing rule, which is well known in the literature. The other approach leads to a pricing rule based on the Shapley value of a related cooperative game. While these approaches are similar in motivation, they differ in the technical requirements which must be imposed on the underlying cost function, and we argue that Shapley value pricing is more appropriate in a telecommunications context. We are able to explicitly determine Shapley value prices for customers who differ from one another in terms of their arrival rates, service rates, costs of lost work, and the number of simultaneous channels which they require.