Cooperative bargaining foundations of the Shapley-Shubik index

In a recent paper, we provide an interpretation of the Shapley-Shubik index as a measure of the 'bargaining power' that a voting rule gives to each member of a committee negotiating in search of consensus. It is assumed that negotiation takes place under the condition that every winning coalition can impose any agreement. In this paper we further investigate the axiomatic foundations of this interpretation of the Shapley-Shubik index. To this end a wider framework admitting random voting rules is considered.