Weighted Shapley values of efficient portfolios

Shapley value theory, which originally emerged from cooperative game theory, was established for the purpose of measuring the exact contribution of agents playing the game. Subsequently, the Shapley value was used in finance to decompose the risk of optimal portfolios, attributing to the various assets their exact contribution to total risk and return. In the present paper, the Shapley value results of Shalit [Annals of Finance 17(1) (2021), 1–25] are extended by using weighted Shapley values to decompose the risk of optimal portfolios. The weighted concept, as axiomatized by Kalai and Samet [Journal of Game Theory 16(3) (1987), 205–222], provides a solution to cooperative games when the symmetry of players cannot be justified. The weighted Shapley value theory is applied to model efficient mean-variance portfolios and price their constituents. The computation is carried out for the 13 most traded US stocks in 2020 and the results are compared with the standard Shapley values.