On membership and marginal values

In Kleinberg and Weiss (1986b), the authors used the representation theory of the symmetric groups to characterize the space of linear and symmetric values. We call such values ‘membership’ values, as a player’s payoff depends on the worths of the coalitions to which he belongs and not necessarily on his marginal contributions. This could mean that the player would get some share of $v(N)$ regardless of whether or not he makes a marginal contribution to the welfare of society. In this paper it is demonstrated that the set of (non‐marginal) membership values include those that embody numerous widely held notions of fairness, such as partial ‘benefit equalization’, individual rationality and ‘greater rewards follow from greater contributions’, where one’s contributions are not measured marginally. We also present a very simple and revealing way of interpreting all values, including those having a marginal interpretation. Finally, we obtain a mapping which effectively embeds the space of marginal values in the space of all membership values.