The Lorenz order is commonly used to compare rules for claims problems. In this paper, we incorporate the average of awards rule, the mean value of the set of awards vectors for a claims problem, to the ranking of the standard rules by proving some properties that are satisfied by this rule. We define a pair of coefficients, inspired by the Gini index, aimed at measuring, for any given claims problem, the discrepancy between the awards assigned by a rule and the proportional division. We generalize the proportionality deviation indices by introducing coefficients that measure the deviation between the awards selected by any two division rules. We show how these deviation indices are related to the Lorenz order.

Given a claims problem, the average-of-awards rule ($${\text {AA}}$$ AA ) selects the expected value of the uniform distribution over the set of awards vectors. The $${\text {AA}}$$ AA rule is the center of gravity of the core of the coalitional game associated with a claims problem, so it corresponds to the core-center. We show that this rule satisfies a good number of properties so as to be included in the inventory of division rules. We also provide several representations of the $${\text {AA}}$$ AA rule and a procedure to compute it in terms of the parameters that define the problem.

The set of awards vectors for a claims problem coincides with the core of the associated coalitional game. We analyze the structure of this set by defining for each group of claimants a, so called, utopia game, whose core comprises the most advantageous imputations available for the group. We show that, given a claims problem, the imputation set of the associated coalitional game can be partitioned by the cores of the utopia games. A rule selects for each claims problem a unique allocation from the set of awards vectors. The average-of-awards rule associates to each claims problem the geometric center of the corresponding set of awards vectors. Based on the decomposition of the imputation set, we obtain an interpretation of the average-of-awards rule as a point of fairness between stable and utopia imputations and provide a backward recurrence algorithm to compute it. To illustrate our analysis, we present an application to the distribution of CO$$_2$$ 2 emissions.

The airport problem is a classic cost allocation problem that has been widely studied. Several rules have been proposed to divide the total cost among the agents, attending to the characteristics of the problem or via game theory. The axiomatic approach provides a way to choose among rules. Our main goal is to provide some tools to evaluate how rules differentially treat larger airlines as compared to smaller airlines. We use the Lorenz and no-subsidy orderings to compare rules. We introduce some monotonicity and boundedness properties that imply a specific ranking with respect to the nucleolus and the Shapley value.

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