  This paper studies power indices based on average representations of a
weighted game. If restricted to account for the lack of power of dummy voters,
average representations become coherent measures of voting power, with power
distributions being proportional to the distribution of weights in the average
representation. This makes these indices representation-compatible, a property
not fulfilled by classical power indices. Average representations can be
tailored to reveal the equivalence classes of voters defined by the Isbell
desirability relation, which leads to a pair of new power indices that ascribes
equal power to all members of an equivalence class.
