Assignment of Heterogeneous Agents in Trees under the Permission Value

We investigate assignment of heterogeneous agents in trees where the payoff is given by the permission value. We focus on optimal hierarchies,namely those, for which the payoff of the top agent is maximized. For additive games, such hierarchies are always cogent, namely, more productive agents occupy higher positions. The result can be extended to non-additive games with appropriate restrictions on the value function.Next, we consider auctions where agents bid for positions in a vertical hierarchy of depth 2. Under standard auctions, usually this results in a non-cogent hierarchy.