A new solution for n-person games using coalitional theory. I. The conditions

In coalitional theory, coalitions have the structure of a rooted tree and a re called superplayers; because of a behavioural principle, the preference rule, such a tree is binary. When pay-offs are associated with the coalitio ns, these trees are called c-trees. After defining the operations of revers ible moves and transitions on c-trees, it is proved that an equilibrium cla ss of c-trees exists. Bargaining arguments show that in superadditive games , players choose, in general, an equilibrium state containing all possible c-trees. This state requires that from each c-tree there must be a reversib le transition to another c-tree in the state and it must satisfy some other conditions. This does not specify the c-trees completely and it is necessa ry to decide how the gains at the root are allocated. An example is the 'sp lit the difference' (STD) allocation. It is proved that the coalitional sol ution, i.e. the pay-offs in all the c-trees, are then uniquely determined a nd we provide a means of calculating them. With the STD allocation, dummy p layers gain their characteristic values in the solution, and it is proved t hat the averages over all c-trees are the Shapley value. If the pay-offs in all c-trees for any given player are constant, the imputation lies in the core. The coalitional solution provides via each c-tree a solution, thus gi ving a set of possible solutions and a statistical approach, not only provi ding the expectation value but also the standard deviations of the pay-offs . It is also argued that the bicameral meet of two simple voting games is n ot a realistic model for a bicameral legislature. Examples are also given w here the STD coalitional solution violates the preference rule and where no t all c-trees are in the solution.