The balanced contributions property for symmetric players

The equal collective gains value in cooperative games

“…This relaxation involves intra-personal utility comparisons but avoids inter-personal utility comparisons. The third one with the (van den Brink 2001)'s fairness, and the fourth one, applying the equal collective gains property only to symmetric players, following Yokote and Kongo (2017) for the Shapley value. We need some additional definitions.…”

“…Shapley shows that properties that characterize his rule are efficiency (players' rewards cover the total value of the game), the null player property (players who do not contribute to the game receive nothing), symmetry (players who contribute the same receive the same), and additivity (the value of the sum of games is the sum of their values). In addition to Shapley's original characterization, other axiomatizations are due to Myerson (1980), Mas-Colell (1989, 1996), Young (1985), van den Brink (2001), Casajus (2017), and Yokote and Kongo (2017) among others.…”

“…Moreover, since the ECG-value satisfies the stronger differential marginality, which is equivalent to the van den Brink's ( 2001) fairness, we also prove that the ECGvalue is the unique solution that satisfies efficiency, the AP-null player axiom and fairness. Finally, we can relax the principle of equal collective gains, applying it only to symmetric players, following Yokote and Kongo (2017) for the Shapley value. We add a new property called AP-marginality.…”

Necessary versus equal players in axiomatic studies

“…This is left for future work. Finally, Yokote and Kongo (2017) has weakened the axiom of balanced contributions (Myerson, 1980) by applying it only to pairs of equal players. They called it balanced contributions for symmetric players.…”

“…Not surprisingly, this recent literature has been very active with regard to the Shapley value. Casajus (2011) introduces a differential version of the axiom of marginality (Young, 1985), Yokote and Kongo (2017) and Yokote et al (2018) weaken the axiom of balanced contributions (Myerson, 1980), Casajus (2018Casajus ( , 2019 weakens equal treatment of equals by requiring only identical signs for the payoffs of equal players or mutually dependent players, among others. To sum up, there are perhaps two ways to weaken the axiom of equal treatment of equals: weakening the requirement of the axiom by imposing equal signs instead of equal payoffs and restricting the set of players to which this requirement applies by considering only certain equal players but not all.…”