A note on compromise values.
Several solution concepts for cooperative games are based on the following idea. For each player \(i\), we can define his/her dream gain \(M_i\) in some reasonable way: e.g., as the largest possible gain \(v(S\cup\{i\})-v(S)\) that a coalition \(S\) can get by involving \(i\). Then, we can define the guaranteed payoff \(m_i\), e.g., as the amount that he/she can gain by starting a coalition \(S\) and promising every other player from \(S\) their dream gain. In the ideal world, every player would get their dream gain, but usually, \(\sum\limits_{i\in N} M_i>v(N)\), so we have to use a compromise payoff \(x_i=\alpha\cdot M_i+(1-\alpha)\cdot m_i\), where \(\alpha\) is determined from the condition that \(\sum\limits_{i\in N} x_i=v(N)\). The author provides a general description of all such solutions, in which the two main properties are \textit{covariance} \(M_i(a\cdot v+b)=a\cdot M_i(v)+b\) (i.e., independence on the choice of a unit and starting point when measuring utility), and \textit{proportionality} -- which, crudely speaking, mean that we select a linear combination of the vectors \(m_i\) and \(M_i\). It turns out that many known properties of compromise solutions are valid in this general framework; interestingly, the Shapley vector becomes another particular case of this general definition.
scientific article; zbMATH DE number 1972741