Monotonicity of power in weighted voting games with restricted communication

Indices that evaluate the distribution of power in simple games are commonly required to be monotonic in voting weights when the game represents a voting body such as a shareholder meeting, parliament, etc. The standard notions of local or global monotonicity are bound to be violated, however, if cooperation is restricted to coalitions that are connected by a communication graph. This paper proposes new monotonicity concepts for power in games with communication structure and investigates the monotonicity properties of the Myerson value, the restricted Banzhaf value, the position value, and the average tree solution.

► We consider measures of power in simple games with restricted communication.
► Standard notions of monotonicity are bound to be violated.
► We define new monotonicity conditions for power in games with restricted communication.
► The Myerson value and restricted Banzhaf value satisfy all considered monotonicity properties.
► The average tree solution satisfies monotonicity only in weights; position value violates all.