Second (and higher)-order properties of solutions for cooperative games

• Applicant: Professor Dr. André Casajus
• Subject Area: Economic Theory
• Term: since 2025
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 554904895

A finite cooperative game with transferable utility (TU game) consists of a finite player set and a coalition function that assigns to any coalition (subset of the player set) a worth, which reflects its productive potential in case of cooperation. A (one-point) solution assigns to any TU game and any player a payoff. In order to motivate solutions, their properties are discussed in the literature, in particular, properties that establish a relation between payoffs and the players' individual productivity. Productivity is typically measured by the players' marginal contributions, that is, the difference of the worth of a coalition after a player has joined and its worth before she joined. For example, strong monotonicity (Young, 1985, IJGT) requires a player's payoff to weakly increase whenever her individual productivity weakly increases. The Shapley value (Shapley, 1953) probably is the most eminent solution for TU games. Young (1985, IJGT) characterizes it with three properties, efficiency, symmetry, and strong monotonicity. Efficiency: the worth generated by the grand coalition is distributed among the player. Symmetry: equally productive players obtain the same payoff. The most eminent generalizations of the Shapley value probably are the positively weighted Shapley values (Shapley, 1953) and the egalitarian Shapley values (Joosten, 1986). Casajus (2021, DAM) refers to one player's influence on a second player's productivity and payoff as her second-order productivity and second-order payoff with respect to the second player. Based on these notions, Casajus defines a second-order versions of symmetry and strong monotonicity. It turns out that the Shapley value is characterized by efficiency, second-order symmetry, and second-order strong monotonicity. That is, it is the unique efficient solution that reflects second-order productivities in terms of second-order payoffs. This project is devoted to a comprehensive study of properties of solutions for TU games that relate second-order productivities and second-order payoff. In particular, but not limited to this, second-order versions/strengthenings of the following characterizations of solutions will be investigated: (i) Shapley value: Casajus (2021, DAM), van den Brink (2001, IJGT)/Casajus (2011, T&D), Casajus and Yokote (2017, JET), Shan, Cui, and Yu (2024, EL) (ii) positively weighted Shapley values: Casajus (2018, JET) (iii) egalitarian Shapley values: van den Brink, Funaki, and Yu (2013, SCWE)/Casajus and Huettner (2014, JET)

• DFG Programme: Research Grants