Uncertain production environment with coalition structure

In this paper, we consider the problem of valuing inputs in a production environment introduced by Pongou and Tondji (2018). In this framework, each input has a finite set of actions it can supply, governed by a probability distribution. For each input profile, the production function assigns the expected output. This production environment can be, for instance, a firm where the inputs are workers and the set of actions is the number of hours a worker can supply. Another example is a financial market where the inputs are banks and the set of actions is the quantities of different types of securities bought or sold by each bank. Pongou and Tondji (2018) proposed and characterized two valuation procedures, the a priori Shapley value (when the output is unknown) and the Bayesian Shapley value (when the output is known). However, these procedures are not suitable for some input valuation problems because they do not account for a restricted interaction among input factors. We extend this production environment by assuming that not all interactions between inputs are possible. In this case, we propose that the set of input factors can be partitioned into different groups, which we call a coalition structure. We generalize the known valuation procedures, naming them, respectively, the a priori Owen value and the Bayesian Owen value. Finally, we propose two axiomatic characterizations. When coalition structure is not taken into consideration, our results provide new characterizations of the a priori Shapley value and the Bayesian Shapley value.