Shapley Value and its Modified Solutions: Axiomatic and Non-cooperative Characterizations

The Shapley value is one of the most well-known solutions of cooperative games. Shapley(1953) originally characterized it by axioms. There are many studies which characterize it by other axioms. Young(1985) gave it using a strong monotonicity axiom. Hart and Mas-colell(1989), Sobolev(1973) gave it using different consistency axioms. Perez-Casterillo and Wettstein(2001) also gave an extensive form bargaining game which implements the Shapley value. Firstly we review these previous results.

Recently some modifications of the Shapley value were studied, the egalitarian Shapley value and the discounted Shapley value given by Joosten(1996). These are also characterized by modified consistency axioms and monotonicity axioms, and implemented by modified extensive form games. These were studied by van den Brink, Funaki and Ju(2013), van den Brink and Funaki(2015), Casajus and Huettner(2013). Secondly we introduce these results.

Most recently we also characterized consensus value(Ju et al. ,2007), which is a convex combination of the Shapley value and the CIS(center of the imputation set) value, by using another monotonicity axiom. Lastly we briefly present this result and an additional result.