Abstract
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.
Highlights
The paper addresses the issue of allocation of positions in a fixed hierarchical firm where the payoffs are determined by a certain cooperative game theoretic solution concept—the permission value seminally introduced by van den Brink and Gilles (1996)
We study a vertical hierarchy of depth 3 where positions are auctioned
We have shown that in hierarchical situations where payoff is determined by the permission value in trees under additive games, cogent and optimal allocations coincide
Summary
The paper addresses the issue of allocation of positions in a fixed hierarchical firm where the payoffs are determined by a certain cooperative game theoretic solution concept—the permission value seminally introduced by van den Brink and Gilles (1996). As the depth of the hierarchy increases, the exploitation of the lowest level of workers is greater reducing their wages They first analyze the structure of the firm under linear and Cobb-Douglas production technology under fixed reservation wage and profit. In our paper the focus is not so much as what firm structure would emerge (which we assume is fixed exogenously), but how positions in the fixed hierarchy are allocated Such allocation issues have previously been studied using a non-cooperative game theoretic framework by Lazear and Rosen (1981). They use rank order tournaments to allocate positions in a vertical hierarchy of depth 2 to workers whose productivity is a function of his/her investment plus a random component. Some of the more esoteric proofs are relegated to the Appendix
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